I’m a bit surprised OpenAI isn’t finding these big results far faster than the product’s user base. With no limits on runtime, access to dev models, custom tuning, and top talent, you’d think there’d be a constantly running internal project with the goal of solving famous math problems. And who knows, perhaps there is, but it would be interesting to compare the rate of success per unit “effort” of the internal mathematics work with that of the user base.
What I am thinking is the way you make it 'keep going' and when you have people of the calibre of Tao doing it I kept thinking how many breakthroughs is he going to cause the LLM to find with his targetted questions :D Amazing that we have the privilege of witnessing a true expert in such a way question the LLM.
Without any more context, "keep going" seems to be doing a lot of work. The user is placing a lot of faith in the LLM to not make subtle logic mistakes and to take good approaches to each problem. In my experience, even frontier models (such as Fable) are quite capable of getting confused during even simple technical work I've done in the dev ops world. For example:
LLM: This package hasn't made it to production.
ME: are you sure? i see it right here!
LLM: You're right to push back. I inferred that based on weak data. I see now that the package has been deployed!
If the above conversation is typical for me, how could one expect to achieve a sound result by repeatedly prompting an LLM to simply "keep going" in dense mathematical proofs? Perhaps the user in this case had actually checked the LLM's work before issuing the prompt, but I think you see my point anyway.
Agreed, often you have to step in and stop it from reasoning itself into dumb directions, but occasionally it goes just like the transcript in question.
There may be something(s) about mathematics (proofs) that makes it particularly amenable to LLM reasoning - highly inductive from facts that are explicitly within-context/associative space? Being an unusually well documented discipline in general, with less influence from tacit knowledge or idiosyncratic “it works however the opinionated human made it work +- bugs” processes? Something about simulating even the smallest non-pure-inductive leaps necessarily risking simulating mistakes due to the nature of context “perception”?
There’s also probably a lot less noise from casual internet conversations. I imagine a nontrivial amount of what LLMs know about certain technologies comes directly from forums like reddit where quality of response isn’t guaranteed.
Is this the same as Dinitz Theorem[1] which seems to have been proved in 1994? This is the only result I keep stumbling upon when trying to understand the problem formulation
At Mozilla, we had a set of whiteboard tags we could set on bugs, like "[crash]" or "[compat]" or "[leave-open]". That last was used when there were multiple patches attached to the bug, and we wanted to land only some of them without automation closing the bug once they landed. (It's common to have alternate approaches or test cases also attached to the bug, so you normally don't want to wait for all of them to land before closing the bug.)
I started using "[leave-open" for those.
It lasted for a couple of years, until someone went through and "fixed" them all.
It’s endlessly fascinating to read the AI transcript of an expert who _really_ knows how to cut to the chase. It just shows how much you can potentially squeeze out of these models. I’m also surprised to see that even Terrence Tao seems to use it in a way that resembles, in progression, how I use llms in my area of expertise (emphasis on progression and usage patterns, not absolute skill, obv I don’t match that): short pointed questions that goes all in on the jargon and machinery of the field and steers the llm hard (eg no softballs). I’ve noticed that llms switch their tone and meet you basically more or less on your level.
More and more the skill of being able to ask the right question seems critical to me, and I don't know how one can do that without deeper and deeper domain expertise.
> I can use AI for coding after decades of coding. I can't use it for theoretical physics because I can't evaluate the responses.
That is what will happen though to future generations: they won't be able to use it for anything because none of them will have the "decades of coding" experience that you have had the privelege to have without AI.
> I’m also surprised to see that even Terrence Tao seems to use it in a way that resembles, in progression, how I use llms in my area of expertise
I didn't understood anything about the thread, but reading Terrence's messages was weird because it looked exactly like the discussions I have with LLMs
I've mostly seen people trying to oneshot a result, while I'll quickly experienced that going through steps/discovery was more effective and more satisfying, since you can always steer it back in the right direction; while oneshotting is hit (and it kind feel like magic) or miss (and you'll have to rework your prompt).
I was struck in the same way but I think it makes sense in terms of a thinking partner.
It is still ultimately Terence that is steering things.
What is crazy to me is how few of other people's conversations like this I have actually read.
Tao is really great for this because the anti-AI crowd can't really chime in and take the thread in a pointless direction. It is hard to think of another human alive who can carry the weight of unassailable authority in the same way.
I encountered something fairly similar working with Claude a few days ago. For a current project I've been fairly hand-wavy with requirements since I was getting good results, but it seemed to be failing hard on some key points, so I started to be more strict with it. Even after the fails were resolved, I've noticed that Claude now behaves differently within that project, carefully checking and rechecking things up front and also looking to me for guidance more often. Mildly irritating, but if it works...
Terrance Tao's chatgpt conversation is really interesting for a variety of reasons:
1. The counter example wasn't just a brute force selection, the polynomial is structured in a very specific way that ends up getting the result.
2. Terry Tao's questions are very specific and prompts the AI in a useful way, that without high math training you are not going to get the same information out of it. Terry seems to see some aspects of the problem and counter example and uses AI to brute force some parts of it.
It is ~a meme on subreddits that developers struggling to get good results out of any given model is a "skills issue."
But I think your comment drives at some authentic take on this. Skill with AI is not only crafting iterative prompts the agent will understand, but also very high domain-specific knowledge of what the prompts explore.
One without the other can result in frustration or worse.
It feels very humbling that here is one of the smartest humans on the planet asking questions, and the LLM keeps answering in this "Yes, it's really simple if you think about it" way, like a professor talking to a talented student.
I had a similar reaction with the language but also response times and language. It reminded me of a quote about Jon von Neumann from Edward Teller:
"Von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us."
I think we have a couple of years of "being good at talking to the LLM about your field of expertise" being a useful human skill, until that too gets washed away
At the end of the day, even if they are some insane oracle (pun intended), they're still bounded by training data and how it relates to the real world. Even if they're a near perfect tool, we are still the interface between them and our lived experience. If that stops being the case then why do we care about the output?
This assumes it doesn't graduate to just killing all of us and doing it's own thing, but within this paradigm it doesn't really have goals.
>Even if they're a near perfect tool, we are still the interface between them and our lived experience.
Until they are embedded into automatons that can interact with the world.
(To be clear, I'm both on board with this, and I think it's the natural progression. Currently they are bounded by their training set, but interaction with the world is the imperative for theory -> test -> analysis -> update that is essential for growth and creativity.)
Yes, but the point being for the LLM to be useful to us it has to do something relevant for us and we have to define what is relevant. Even if you view them as fully human level or beyond (in terms of agency) there's still some purpose that we have to help provide them with.
To put it differently, if you have some idealized model in front of you that can do anything a team of humans can do, what do you say to it? It's still just a model that takes an input and provides an output.
A community of those who distract themselves from the perfectly fixable problems in their lives corruptly self-evaluates. They validate each other's stagnation and unwillingness to move by finding flaws in each day that will enable shutting off the flow of any new data while condemning the world and any actions in it. The lay-z-boy they collectively protect appears as corroboration with a broad population but is in reality a repetition whose independence is meaningless since they are all copies of one system, one kind of person in the same kind of trap.
Immobile. Clogging the Suez with their sandbagging ways. Nothing to add except reasons to stay put. No aspiration. Only cynicism. They deserve nothing but all of our contempt.
It's crazy how he suggests simplifications over and over and gets led through the finding. Absolutely bonkers how you can use AI to understand something and map it to your own mental map so efficiently, and of course he's most interested in generalizing or finding a simpler sub-result that would explain it.
Just awesome to see new knowledge hit an incredible mind like this. Having these "what if" discussions is what I miss most from JPL and academia.
> you can use AI to understand something and map it to your own mental map
This "symbiosis" (for lack of better word) of human with AI seems to be an emergent value proposition of AI. In the process of doing stuff with AI, producing artefacts like code diffs, we are continuously able to decide how strong the mental map is of the current stage of the production process.
I could probably have worded this better but I'm sure it's something others have noticed... this choice we are able to make of how high fidelity our own understanding needs to be of the current working problem, and how that choice never really existed prior to AI.
Math has some of the most insanely dense and impenetrable nomenclature. I can generally keep my head mostly above water or at least near the surface reading from most STEM fields, perhaps leaning on google/wikipedia a bit, but man, mathematics just so quickly decouples from all common tractable understanding it's insane.
Sorry it's a bit of an aside, but I imagine many other otherwise "technical" folks feel the same unfamiliar sense of total loss like when encountering hard mathematics.
This is also true for almost every other field, even within computer science. The only difference is that a lot of people operate at a very surface level without realizing just how much background knowledge they have accumulated. Think about the number of keywords your average SWE is expected to know. It is rather insane.
Every sub fields (web/kernel/backend/etc.) has a million/bazillion weird words used in a dozen different contexts and if you read a paragraph of even semi technical software text you will feel like an over stuffed turkey.
Even cache could mean the CPU caches, the page cache, a browser cache, a CDN cache, a Redis cache, or imagine the flurry of words we have that have real world meaning. Session, handle, pool, buffer, stream, channel, event, task, worker, or queue. Generally there is some overlapping meaning but often there isn't.
Most people, especially non-tech technical people, could crash through the TCP article and come out the other side with at least a high level understanding of it.
Most people, even technical ones, could not even get through the first line of the rees article, heck the first statement of the article. And then if they try, they need to know about algebraic rings. And digging into rings becomes totally intractable. None of the words or symbols in any of the articles track to anything even many technical people can grab onto. And this pattern is all over the place in mathematics.
It's not about mastering the difficulty of a topic or it's relative depth, it's about how abstract and removed from anything tangible it is. Anything with math it is always seemingly impossible to get a foothold on the idea anywhere within 10 degrees of explanation. Hell you cannot even clearly understand the problem that is being solved, or anything within 10 degrees of that.
Definitely agreed. I have a bachelor's in math and took an abstract algebra course as part of it. I also took a couple computer science courses in college and work as a data engineer. My only real exposure to networking is from an AWS cert I did years ago.
I can tease apart the Rees Algebra article one bit of half remembered terminology at a time and come out of it feeling like I just barely understand what the topic even is.
I can read the TCP article and feel like I have a thorough overview of the topic and could explain it at a high level to someone else.
> Most people, especially non-tech technical people, could crash through the TCP article and come out the other side with at least a high level understanding of it.
I agree with you here, but what's special about tech is that many of us learned all these terms fully casually while using computers as children and teenagers, which would be much less common for a chemist. That makes programmers see a lot of things as computer literacy that most people have rather than specialized knowledge.
My argument is that all the vocabulary for computer science are things. Even if they are virtual, they are tangible. You can draw a picture and label a box "bytes".
Nothing in the ChatGPT conversation is tangible. It's all in the realm of concepts.
A ‘Byte’ is not a concrete thing and the fact you think it is speaks to the degree to which you have immersed yourself in a mental model which thinks of ‘information’ as if it is a real concrete thing, to the extent that you don’t even realize the levels of conceptual abstraction you needed to build in order to internalize what a ‘byte’ is.
Such a small sentence and yet it means very little to me. I understand some constituent pieces, but I don't understand what Z is here other than a 'ring' and I don't really grasp how t^-1 converts this into a generalized family of algebra. It would take me a lot of effort to understand this and use it practically. I find that fascinating because it really is such a small statement that seems perfectly cromulent, but there's a lot packed in there that someone like me is totally missing.
I suppose there may be similar concepts in computer science, but nothing comes to mind that ever stumped me. To be frank, the field has been relatively accessible to me because it hasn't been too challenging. Not sure if that's a personal aptitude thing or it is genuinely simpler.
Z is the ring of integers, t is a formal variable allowing us to discuss polynomials whose coefficients are in some ring. That’s what R[t] means: the ring of polynomials of the formal variable t with coefficients in R. Adding in t^-1 lets us include inverted terms like 2t^-3.
An algebra over a ring (call it S so we don’t confuse it with R from the previous paragraph) is a like a vector space over S, with the added structure that you can multiply elements of the algebra together (vector spaces only let you add their elements together). So for example the collection of even integers 2Z is an algebra over the ring of all integers Z. The collection of all polynomials with integer coefficients, Z[t], is another algebra over Z.
This is a great example of how dense language gets in math. There are tons of concepts hiding in the unstated background. Many are quite simple to explain individually, but there are so many of them that an outsider won’t know where to start to tease them apart. There’s a good reason to do it this way though; it would take a very long time to say anything in math without ever increasing levels of information density.
Absolutely agree. All formal statements (like mathematical ones) are going to have some level of assumed background. And as the assumed background expands, the language naturally becomes more information dense.
As for your specific questions, I believe Wikipedia does a great job of answering two of them for a layperson:
For the others, I’ll say that a formal variable is just a symbol (literally, like the letter t). With such a symbol, we can construct polynomials like 2t^2 - t + 3. Also, there’s no need to only use integers as the allowed coefficients; you can use any ring you like instead.
An “algebra over the ring R” is what I was attempting to define in my comment above. The algebra is “over” R if we can multiply an element of the algebra by an element of R. The useful analogy here is scalar multiplication in a vector space: you can multiply a vector by 2 to double it or -1/2 to reflect and shorten it. More generally, it makes perfect sense to consider some more general version of vectors which can be scalar multiplied by elements of any ring R.
> But... what is a ring? What is a formal variable? What is a vector space? What does "algebra over the ring" mean?
All these terms were taught to computer science (and of course math, physics, ...) students as part of getting their degree in computer science, because these concepts are important for many algorithms.
It's a class with an array of integers in it with .length() == t - 1 and the same methods as Matrix.
In lean4, even without mathlib4, TCP/IP is way more code than a Rees algebra.
Math uses dense notation that is gigaoverloaded, and the disambiguating context was historically the leisure and proximity to have someone explain what the lexemes even mean.
lean4 is proving to be very revealing as an uncorruptible referee on a lot of things, including the relative difficulty of computer science and complex analysis.
-- A Rees algebra over ℤ[t⁻¹] is this.
-- That's it. That's the whole thing.
structure ReesAlgebra where
coeffs : Array Int -- integers, indexed by grade
-- grade k means the coefficient sits at t^k
-- negative indices are the t⁻¹ part
-- The "algebra" part: you can add them
def ReesAlgebra.add (a b : ReesAlgebra) : ReesAlgebra :=
⟨a.coeffs.zipWith b.coeffs (· + ·)⟩
-- And multiply them (convolution, same as polynomial multiplication)
def ReesAlgebra.mul (a b : ReesAlgebra) : ReesAlgebra :=
sorry -- it's Array.foldl over index pairs (i,j) summing into slot (i+j)
-- exactly how you'd multiply polynomials in a job interview
-- That's the entire mathematical content of
-- "The Rees algebra is an algebra over Z[t^{-1}]"
--
-- Compare: a minimal TCP SYN handshake in Lean4 would be
-- ~200 lines before you even get to retransmission.
--
-- The notation is the gate, not the math.
That's false. Z[n] in rings does not mean "an array of integers of length n", it means the subring generated by Z union with {n}, where n is an element of some other set. For example:
Z[i], the Gaussian integers, is the subring (of C) generated by Z union {i} where i is the imaginary unit in C, the complex numbers. The Gaussian integers correspond to the integer grid-points of the complex plane, if you want to visualize them.
You are comparing TCP a relatively basic topic in the grand scheme of computing with Rees_algebra which is fairly specialized, we could take a simpler topic more foundational and clearer to understand and compare them.
I can understand that this feels like one is so much more complicated part of it is also how the articles were written, wikipedia is not known for quality maths explanations.
But beyond that this comparison to me feels unfair.
Let's take Euclidean algorithm or just modular arthimetic for example what a lot of computing even is based on I feel like that's a fairer comparison. No?
Perhaps that's too easy but I just find this specific comparison very unfair to both Math's intuitive-ness and Computing's complexity. Perhaps I am the one being delusional.
I think you are snagging on thinking this is an observation about difficulty, time-to-mastery, or mental firepower requirements. It's not.
It's a plain observation that math exists on mostly it's own path with little to zero overlap with our lived experiences. If mathematics was a vector, it would have similar magnitude to other vectors, but it's direction would be much more removed from the typical knowledge pack, forcing you to get really close to the origin before you can "hop" over to that math vector. Other "knowledge" vectors, by virtue of being more bunched up, are closer together much further up, if that poor analogy at all makes sense.
Mathematics has a lot of knowledge points that do connect to the "real world" very deeply, but perhaps the nature of their linkage to other mathematical pieces of knowledge is best left to the mathematicians. But we can still use the pearls of wisdom that come out of the process.
Very true, I feel like the sense that Computing is easy comes from the inherent closeness of our lives experiences to it. Everyone uses a Phone they see ram understand memory, can understand process and processing.
No, the problem with mathematics is that it is basically its own language separate from your native tongue. You have to learn dozens of symbols and greek letters and such and memorize what their meaning is in the context of mathematics in order to "follow" a mathematical conversation.
Mathematics would be much more approachable if it just used plain English like `sum(0, Infinity, my_func)` instead of a big Greek sigma with nested function nomenclature. But on the flip side, mathematics being its own language means that a mathematician from any country can read and understand mathematics from a different country without needing to translate words such as "sum" and "infinity"
Mathematics would be much more approachable if it just used plain English like `sum(0, Infinity, my_func)` instead of a big Greek sigma with nested function nomenclature
First of all, no, mathematics would be far less approachable if it did that. Most of the Greek letters used in mathematics don't have a universal meaning, they're context-specific and defined by convention or just prior to use.
Second of all, mathematics is optimized for hand calculation on paper, not long-term programming and code maintenance. Writing out long names over and over on a whiteboard gets tiring extremely quickly, so mathematicians prefer to stick to single-letter symbols.
To second this, most (non-applied) mathematicians work first with paper and pencil or on a chalkboard, and the act of writing out the symbols is a part of thinking about them. Typing doesn't wire into the brain in the same way. Think of how you learned algebra in school. You wrote out much of what you were thinking, often in ways that would be hard to flexibly format on a computer, and the act of writing your thoughts solidified them in your memory.
Mathematicians pretty much universally view typesetting as a distinct step from the thinking part of math, and something you do at the end once you have figured everything out.
Your translation only makes intuitive sense to you because you are well versed in programming.
I suspect if I showed a non-technical person with no background in either math or programming they would think both are nonsense until you explained it to them
> I suspect if I showed a non-technical person with no background in either math or programming they would think both are nonsense until you explained it to them
I doubt it. Greek letters convey almost no information, whereas (one hopes) the function and variable names are chosen by a programmer to help the reader. The Greek letters used by mathematicians (and physicists) weren't used to convey information, they were used because typesetting, publishing and paper were expensive. They are optimised for brevity over readability.
It was a perfectly reasonable trade-off at the time, but times have changed.
As an aside, some programming languages (such as APL, and to a lesser extent Perl) did emulate the old Greek letter style. "Line noise" is a typical description of the result. No computer language aimed at software engineers and computer scientists does that now.
My example was contrived, I'm sure some smart people could come up with a SQL-esque language that is even more readable to non-technical folks than programming syntax. At a certain point though, your layman has to know the "atomic" (as in, you can't break them down further) mathematical concepts like "functions" and "infinity":
I am a research mathematician. In my field (abstract algebra, computational group theory), a sum or some such notation is like the most trivial of trivial things in terms of notation. There are a few things like sums that could in theory be made to "look more like what a computer programmer would expect", but that'd be just a tiny corner of it.
And if you think about how summation would look in Lisp or APL (which some smart people use to this day), I am not even convinced your argument for the "sum function" notation being superior holds in general.
I'm sure some smart people could come up with a SQL-esque language that is even more readable to non-technical folks than programming syntax.
And somehow, none of the thousands of very smart mathematicians have done that, or if they had, it has not seen wide adoption. I recommend contemplating on this: if math could be made easier by changing notation, why hasn't this already happened?
The thing is that you basically cannot explain the math like you can the programming.
Tables, algos, and variables are all things people can generally quickly grasp. The construction is abstract but the function is tangible.
The math is working entirely on abstract objects, using abstract tools, governed by abstract rules. It's just all so desperately far away from anything even technical people have contact with.
I was amused and irritated in equal measure when a senior manager at $LAST_JOB demanded that documents include "no jargon", then went on to use many terms like those you've listed here with no indication of awareness of the contradiction.
Just like the famous observation that "anyone driving slower than me is an idiot, and anyone driving faster than me is a maniac" - "jargon" is just any term-of-art that you're not currently familiar with. Attempting to communicate like Up Goer Five is inefficient. The solution is not to ban jargon - it's to:
* Cultivate a glossary for any terms that were introduced _within_ the domain/company, which are not in common usage outside
* Normalize a culture that does not shame asking what something means
> I was amused and irritated in equal measure when a senior manager at $LAST_JOB demanded that documents include "no jargon", then went on to use many terms like those you've listed here with no indication of awareness of the contradiction.
"No jargon" is alwys in reference to some assumed knowledge model.
For example, when I claim that my own math texts include "no jargon", this assumes the knowledge of a person who has studied, say, mathematics, physics or computer science.
"tcp" can take roughly 3-4 weeks of heads-down dedicated study to have some reasonable familiarity with. Same is true with most of the other concepts. Being able to speak with expertise on that list of topics is 3-4 years of really focused study and work.
I think what happens is that people often have passing familiarity with a word or topic and presume knowledge, and years (decades) later they realize they knew almost nothing.
I will say that Mathematics is different (for me at least) because unlike the infrastructure computing concepts (IETF type, not IEEE)- which mostly require studying, lab work, and some coding to get your hands dirty - advanced math is just ... really hard. There are IQ issues at play.
Obviously a lot of computing turns out to be mathematics - so there is clearly convergence/overlap as well...
You could get a surface level understanding of TCP, or 99% of topic areas in computing, in less than 4 weeks of study. You could not get a surface level understanding of literally any of the maths in the link in less than 4 weeks of study.
The vast majority of what computers do just isn't that complex. I'm not saying it isn't "complex" just that any reasonably smart person can understand how a computer works and still have other hobbies, basically no one can understand phd level mathematics without dedicating their entire lives to it.
Speaking as someone who has run courses designed to take late teens / early 20somethings with a variety of backgrounds and tried to teach them 1st semester programming concepts in two weeks, I think anyone who thinks tcp/ip an easy four week dunk for most doesn’t remember how much they’ve already learned and take for granted.
Sure, if you’ve already learned enough groundwork, tcp/ip is accessible in weeks. The same is true of most of the algebraic concepts in play here. And both have rabbit holes you can also spend a much longer time going down (though here I am willing to give the edge to math which offers much greater opportunities for hypergeneralization and new vistas of abstraction along which not only specific rabbit holes but entire new generalizations of both rabbits and holes may be found).
This is not true, 99% is very exaggerated claim, but yeah you can learn 50-60% of the field at a surface level in months rather than years.
But you can have a surface level understanding of mathematical topics as well, ofc some topics might require deeper understanding, but that's true for both.
Any claims of being able to learn 99% of computing in a just 4 weeks even at surface level, is greatly underestimating your own knowledge built over the years perhaps, or perhaps underestimating your own ignorance.
The claim isn’t knowing 99% of the field but that 99% of topics are ones you could do a quick-and-dirty crash course and come out with some understanding.
Definitely a lot of people have very surface level understanding of tcp and computer science concepts.
I have had folks tell me cache is just cache in actual interviews. When I have asked them to explain the concept to me, but even beyond that I feel like we tend to think less of our own knowledge of topics once we have acquired it.
Especially ones acquired over years, alongside other work.
I second this and would add that it's really easy to catastrophically forget things in math. I'm pretty sure that most CS knowledge I have I will retain at a level where I won't forget the general ideas and re-reading materials can quickly refresh the details. This is not true for advanced math. I did a pure math PhD and my own thesis is impenetrable to me 20 years later. It would take months, if not years, of focused effort for me to regain the understanding.
I think one of the key differences is that math is abstract whereas CS is relatively concrete.
CS examples are often easy to picture and understand the motivation for. You can use tools to visualize or play around with them and test them.
Math gets abstract so fast you have to spend a week of research to even understand the problem statement. The the motivations themselves can be completely unclear until you have a lot of context.
I majored in math (B.S.) and upper level math is completely foreign to me.
I think so is upper level CS, there are fields in CS that are foreign to me too, there is a lot of depth in CS, computing is a very deep field for instance ML research although may seem simple isn't quite so intuition based as people make it out to be. Similarly there are dozens of topics where sophisticated research happens where we don't interact with at all as regular software developers.
Every slice has so much depth to it, in Maths it all seems like all of it is required at once but in computing it feels like so little is needed to get started which I honestly feel like is failure of our modern education systems.
But yes Computers being so easily accessible and compilers, documentation and libraries have made computer science so easy to get started with.
Imagine having to implement your own network layer to communicate with someone, you would have had to understand ip, tcp, network layer to an extent like http and etc. and then you finally would have been able to communicate.
In maths that's our reality for a lot of the field, there aren't good libraries, interfaces to help skip the unnecessary details. Hopefully AI might solve it I don't know though. It's fun to hope for it.
As a counterpoint, i've worked with enough math PhD's over my career who couldn't wrap their head intuitively around many concepts from software engineering, while others had no problems in doing so even from folks without any stem background. We often undererstimate how much field knowledge we aquire over the years and overestimate how easy it is for others to catch up.
Again, you are underestimating how much effort it takes to understand how an 8086 CPU works. There are a lot of foundational concepts that you are simply assuming the person already understands. That may be a reasonable assumption for a CS undergraduate, but the average person does not understand binary arithmetic, registers, memory addressing, instruction execution, calling conventions, or even what a CPU is doing at a meaningful level to even start to understand 8086.
Also, understanding an 8086 CPU is not even remotely comparable to the level of mathematics Terence Tao was discussing above. The 8086 is a relatively basic and concrete topic. You can build a workable mental model of it from a finite instruction set, a handful of registers, and a reasonably straightforward memory model.
From my perspective folks here on HN and in CS often think they should somehow be able to understand advanced mathematics papers at a glance, merely because they are good at basics of programming or computer science (8086). That is not how it works. Most mathematics is not inherently much harder than computer science; both fields require you to accumulate a large amount of foundational knowledge before advanced material becomes comprehensible.
There is an enormous amount of computer science that most programmers are completely unfamiliar with, especially within academic CS: programming-language theory, type theory, formal semantics, compiler theory, algorithmic research, complexity theory, distributed computing theory, verification, cryptography, computational geometry, numerical methods, and so on. Being proficient in one narrow area does not automatically give you the prerequisites for another.
A web developer would not be expected to casually understand a research paper on type theory or approximation algorithms without first learning the relevant notation, terminology, and foundational results. Mathematics is no different. The feeling that mathematical writing is uniquely impenetrable mostly comes from encountering it without the years of accumulated context that mathematicians have silently built-up.
I can show you a paper about an advanced algorithms or chip design, that is large made up of fundamental cs concepts and general physics and even you likely someone with pretty in-depth understanding of CS would find hard. There are orthogonal subjects, for instance my mathematician friends things I am insane reading so much about weird computing topics, and I find his research in some weird number theory thing completely mind-bending.
Try and explain to a lay friend how registers & isa works in-depth with all the details not a hypothetical higher level model so that they can understand the nuance of looking at assembly, limit it to 8086 perhaps, it will take significantly longer than a weekend.
Ofc Terence Tao and his level of intelligence is beyond me, I wouldn't compare but general advanced mathematics is not something folks here couldn't pick up if they actually tried to work on it, just give it a shot (though I would recommend don't start with advanced topics build up slowly I think most people can understand most maths papers even the bleeding edge ones within a few months of serious self-study, and won't even feel that it's after a few years, compare that to the time spent learning software and computing 6-8 hours a days for several years)...
> There is an enormous amount of computer science that most programmers are completely unfamiliar with, especially within academic CS: programming-language theory, type theory, formal semantics, compiler theory, algorithmic research, complexity theory, distributed computing theory, verification, cryptography, computational geometry, numerical methods, and so on. Being proficient in one narrow area does not automatically give you the prerequisites for another.
And the difficult part of all those areas of computing is the mathematics part. Which I think is what I am arguing, mathematics is a fundamentally different type of "difficult" to any other subject.
PLT and such isn't math conceptually sure it's all logical maths of some kind but largely it's not maths in the traditinal sense of how we understand maths.
Because otherwise if you think about it all of computing is Maths but with computers...
I don't think people who read the Wireless Fidelity spec can understand any of it in a weekend or anything even to a rough extent.
Similarly with websockets, quic etc. the most you can take away without much prior knowledge is what it does which maps into Maths as well.
Every quantifiable science has some amount of math, but it's by definition applied math. I think this was more of a debate between applied and more abstract math.
Computer science is not a great example for this. I could ELI5 most of the terms you listed (and I actually have done so for many of them!) This is because it's pretty easy to map these concepts to everyday physical objects. Once a child understands any of those objects in their lives, it's pretty easy to explain in those terms.
Like, just the concept of "books" gets you very far. E.g. a file is a like a book, a folder is like a shelf to keep books, a stack is literally a stack of books, a heap is just a place you can pile books in willy-nilly, a database is like a library, a cache is books on your desk versus books in the library, replication is having multiple copies of a book so we can afford to lose some copies, indexing/sharding is like arranging books alphabetically, and so on.
Others are trickier but not much: a process is an app that is running on your device, a socket / tcp / http / websocks is a way to exchange information between devices, a namespace is how the name "Tom" in Tom Sawyer is different from "Tom" in Tom & Jerry, DNS is a way to get an address from a name, etc. etc.
You'll also notice that many of the terms you mentioned are already derived from well-known real-world concepts like pool, stream, channel, stack, queue, worker, transactions. You can mix those with other everyday concepts to make useful analogies.
But I could not even begin making analogies for most topics in Mathematics. I guess this is because advanced topics in Mathematics are just too abstract to map to everyday things.
Yeah. In math at least it's clear that you don't know what's going on. In other fields, it's very easy to think you understand without knowing how much you're missing.
I studied math through college before learning to program as an adult and becoming a software engineer, and I strongly disagree.
I don't know how to say this in a way that won't sound insulting, but I don't mean it to be insulting. Programming, even systems engineering, is a surprisingly shallow field.
I don't mean that it's easy--it's not, it can be incredibly difficult. Difficult and deep are just different concepts. Difficult refers to how challenged you are. Depth, at least as it appears in math, is closer to a structure where concepts build on each other so that if you don't understand one concept, you can't understand further ones.
Programming can be difficult, and it can be intricate, but it is rarely deep in this fashion. Being deep in this way isn't the most important thing.
If I go into an area of programming that I don't have a lot of experience in (graphics, or the linux desktop environment), I will not be particularly useful, and it will not be easy. But I'll not experience the same type of impenetrability I experience when I try to read a paper on topos theory.
One more way of putting it: people are giving the example of TCP. You can spend a decade learning about TCP (or SQL semantics, or web standards). But what is happening is that you're filling in gaps in your knowledge. Meanwhile, in math, you do four years of undergrad, and even if you're a strong student at a typical university, there are topics that are still years away from you being able to touch them.
Computer science is a mix. Parts are deep, parts are shallow. Parts just are math. The odds that I can read a dissertation in computer science are decent. For math, they're much much much worse.
> Programming, even systems engineering, is a surprisingly shallow field.
I agree, and I don't think we should be at all ashamed of this.
The beauty of programming is that we can produce incredibly complex and powerful things by manipulating a small set of simple constructs together. There are only a few core tools--iteration, conditionals, etc.--but they can be snapped together into much more capable configurations.
Good programming is the art of deconstructing complex behaviour into these few constructs, and that's really fascinating.
Most of things that are impenetrable in programming aren't about... programming. They are about some actually complex field like math being applied to programming.
For example, a library that does stuff with geometry. You need to know geometry to understand the program, but the program itself will never be complicated. It's the geometry that is complicated.
In cryptography, it's not the program that is complicated, it's the field of cryptography. In AI, it's statistics.
In graphics programming, math is the most impenetrable part, not programming anything. You can be a very good programmer in the sense that you know how to architect information systems and still fail to write a shader because shader programming requires you to know what a "dot" product is and you haven't heard about that since high school.
I'd slightly disagree. Here's an example of a programming task that is incredibly complicated:
Your job is to maintain and modernize a system while delivering a constant stream of new features. Your system is several million lines of code, with some multi-thousand line classes, a rulesengine that can trigger nearly unlimited effects at any time, a persistence system that's weirder than anything any of your friends have ever worked with, and hundreds of customers delivering tens of millions in revenue who use the system in incredibly varied ways.
You can't stop to rewrite the thing, you can't just throw features out there and pray, because you'll cause regressions and your existing customers will hate you. You have to fix the thing as you're building on top of it.
But where I agree is that there's no single deep concept that unlocks it all, it's not like you'll fix it by reading a textbook about it. It's complicated, and it's going to stay complicated, no matter how long you work on it.
Just be thankful that we don't share the penchant for giving credit to discoverers. Imagine calling a cache a "Murphy/Steinman/Sokolov structure" (made-up names).
I mean, we do for some things, especially algorithms (Boyer-Moore). Probably for the same reason the mathematicians do -- there aren't readily available real-world analogies.
And I won't even mention the branded future, with its "Google HyperZipper String Search" and "OpenAI/Red Bull speedmaxx distributed consensus algorithm"...
Huffman Coding, Turing Machines, Knuth Devices, Bayesian networks, B-tree (named after Bayer), AVL Trees, so many data structures and algorithms, even relatively new ones like Timsort, Jaccard similarity, MapReduce (thankfully but I have seen people call it Dean-Ghemawatt MapReduce in literature).
Well people even name stuff after themselves as well, Fil-C, raylib, etc (I like both Filip and Ray just pointing it out).
Aside: If I butchered some spellings I am sorry. :3
If we take amateur level of understanding as a threshold, a sum of all of these concepts is not even a fraction of complexity required to understand a single non-trivial concept in mathematics.
This is also true for almost every other field, even within computer science. The only difference is that a lot of people operate at a very surface level without realizing just how much background knowledge they have accumulated. Think about the number of keywords your average SWE is expected to know. It is rather insane.
Nah, math is much harder because there is not just the lingo, but also all the math machinery behind it. Each math definition represents some long process behind it, which builds on another process, etc. The knowledge builds on itself , too much more so than computer science.
Barely scratching the surface. I think I could probably keep typing all the technical terms I know for at least 24 hours straight. For most of the topics above, I could probably give a 1 or 2 hour lecture on each one from memory. Some I could give a day-long lecture each.
To explain all the terms I know to a basic degree, I would probably need to give a whole year of lectures back-to-back from 9am to 5pm. And I'm just a rank-and-file senior engineer with 15 years of experience.
It's also why the vast majority of software systems are insecure. The average senior software engineer doesn't know everything that they need to know to build secure software. Last time I poked around Coinbase APIs on HackerOne, I found a DoS vulnerability in less than 30 minutes. That's Coinbase, not some startup built by a bunch of recent graduates.
AI cannot avoid vulnerabilities either since it is trained on average engineer code. There's not enough high quality code available on the entire internet to train AI to implement secure code IMO. As impressive as Mythos may be, it's not enough. I don't even think formal verification would provide protection since sometimes issues with the spec itself can provide an opening for a vulnerability.
Nice list. I like it how you also included some made up terms. For completeness, those are not real:
ETag, SAML, CSP, BGP, haproxy, helm file, k8s, SOCKS5, Lamport OTS, Winternitz OTS, SPHINCS, lattice-based, pg-vector, sameSite, httpOnly. (Or, at least, i have no idea what those are ::)
Coinbase not a startup built by recent graduates...
You overestimate Coinbase's engineering rigor.
But yes security in modern software systems is a joke. I don't even get paid to fix security bugs everytime I raise them the answer is to slap a sandbox and proxy and call it done.
They could drag it out by telling stories of situations they faced but information density would be low and repetitive. As someone who cleaned boats professionally for a year in my early days, I'm pretty sure I could convey all I know about cleaning in under an hour. That's very different from one year of back-to-back lectures.
Cleaning boats, the learning tapered off after a few weeks. With software, I'm 15 years in and it barely tapered at all and it's much more intense. I've been pulling nights and weekends too.
With software knowledge, it would be high information density with little to no repetition. Every piece would provide useful concrete knowledge which would serve to increase technical capabilities and/or security.
At least a lot of those are common words that allow some level of meaning inference with a little bit of adjacent knowledge, like cache and queue make sense with the barest of explanations because their everyday definitions are still applicable. Many of these terms aren’t entirely opaque until you drill down into specific niches.
I had a few moments of this in the past. For example, in my quantum class the teacher wrote "H Psi = E Psi" on the board, we all laughed, "just cancel the psi" but it turns out one was a multiplcation and the other was a matrix multiplication (operator) and so we had to learn all new nomenclature.
Similarly, at some point somebody pointed out to me "the reason you're confused is that the bold on that variable means it's a matrix"
That is why Iverson invented APL.
As a notation to get rid of all those inconsistencies in math notation.
And for years he taught math classes with APL on the blackboard without computers.
whether he succeeded, is debatable.
But APL is definitely powerful, succinct and "regular".
In APL you don't infer the operation from the types at all. × is elementwise, +.× is inner product /always/, on scalars, vectors, matrices, whatever. The glyph tells you what happens. Nothing is bold, nothing is inferred, nothing depends on what your professor assumed you'd absorbed.
I've been trying to get into Iversonian languages myself with the book: Calculous on J
I mean, unfortunately being completely explicit and pedantic does not scale.
Imagine that instead of being able to use high-level programming languages, you had to write in assembly everywhere, all the time.
That's what software engineers and computer scientists' suggestions of redoing mathematical notation fee like to mathematicians.
These efforts also don't go anywhere because research mathematics moves beyond elementary arithmetic very quickly, and once you're there, "descriptive" notation becomes as incomprehensible as whatever mathematicians use.
My favorite moment of this kind was when the teacher said 'Ok, and for the rest of the course we will look at a completely different problem', and the equation he wrote down was exactly the same as before. Except that the letters referred to vectors/matrices now.
A decade or so ago I wondered if the reason maths was hard was the names being optimised for writing by hand. Everything's single letters if they can get away with it, so when mathematicians run out of Latin alphabet, they use Greek, bold, etc.
Even integration's ∫ is a fancy elongated s.
CS version would be e.g. integral(function=some_named_function, from=a, to=b, with_respect_to=argument_of_function), which may be longer, but is less opaque, especially when you get in so deep there's 3 other people in the world who've looked into this specific problem and you had to invent your own operations.
But that's all an outsider's perspective. I stopped with two A-levels in maths and further maths.
Nope, math notations are optimized for reading, not writing (consider that people still use symbols on computers despite it being quite a bit more tedious to type). The conciseness makes it easier for you to see structural patterns and do symbolic manipulation in your mind's eye. Even something basic like the wave equation would become completely illegible with an expanded notation like that.
Same reason why we write 5-3, not subtract(minuend=five, subtrahend=three).
For those of us with strong verbal processing and weak symbolic/pattern processing, this makes math much more difficult to approach.
Interestingly, discrete math feels the most "verbal" of all the subfields of math I've encountered (I haven't gone very deep). I think this is because notation in discrete math is is somehow closer to compressed prose or logic, whereas other forms of math use notation to fill in for long sequences of symbolic manipulation.
Not sure if that makes sense... I'm curious whether anyone else experiences it that way.
No, math is difficult to approach because it's genuinely deep. Trying to verbalize what is going on is extremely difficult, because you end up saying stuff like "and then do that to all of these things, and then do it again to all of the results, and so on ad infinitum, and then take the collection of all of that, and join it with the collection of doing the same procedure as before starting with a different set of objects, and then join those to yet another set of objects and the results of their operations, ad infinitum, ad infinitum..."
People genuinely struggle to think verbally or visually once we extend beyond 3 dimensions and start talking about infinite-dimensional constructs, uncountable sets, and so on...
I did study math at university for a while. Dropped out eventually. In the beginning I was super annoyed by the brevity and hated it. But after like 3 months it suddenly became natural. I also appreciate the clarity of how mathematicians introduce new ways to write things. That is sometimes even more verbose than some random API docs for a new function…
yes! I really find when computer science ppl start using math notation to describe algorithm very pretentious. we have programming languages in comp sci, we don't need it!
I was reading about Tao's efforts to get more people to use Lean and apparently a big roadblock for people is that Lean uses very specific static typing.
e.g. to use a very simple example on a white board "3" is "overloaded" as:
- the integer 3
- the rational number 3
- the whole number 3
- etc
When you write a proof in Lean, you have to specify the the type of "3" you mean.
Having using Python/Perl and Java over the years, I get that some math folks found handling this daunting or at a minimum friction to getting into using Lean.
LLMs seem to have been a big help here just for the "translate my math notation into a proof" feature.
Sometimes I wonder if mathematics would have been significantly more improved if they hadn't insisted on notating their variables as single letters and also indicated variable types out-of-line (or at all)...
but then I take a look at literally anything the Haskell people do and realize that it probably wouldn't have helped.
>For example, in my quantum class the teacher wrote "H Psi = E Psi" on the board, we all laughed, "just cancel the psi" but it turns out one was a multiplcation and the other was a matrix multiplication (operator) and so we had to learn all new nomenclature.
This is one of the great things about Lean becoming used for more and more mathematics: understanding exactly how an operator/function is defined is just an IDE click or few away. It completely removes the ambiguity present in hand-written proofs, although it still can require a lot of reading to actually meaningfully understand the definitions.
I would expand on this. AI is great for me because it can read the equations I don't understand and turn it into code I can understand. I've worked in science for decades and it's still like pulling teeth to replicate a competitor's paper when they are vague and sloppy with their description (often intentionally).
ugh, I had some text book that used R for a scalar value and (edit: \u{MATHEMATICAL BOLD FRAKTUR CAPITAL R} here) for a matrix that was related to the scalar and I had to go back and re-learn a month of material once I figured out that the font was being used with intent
For what it's worth, it's not a problem with multiple kinds of multiplication (multiplication by a scalar can be viewed as multiplication by a specific kind of matrix), but with the idea that one can cancel in a multiplication. Since you can't cancel in matrix multiplication, you run into unexpected trouble when you try to do so, even if that's the only multiplication in sight. (In fact, you can't cancel in scalar multiplication either unless you've checked that you aren't multiplying by 0 …. Also, I'll note that surely no young mathematician has encountered the P = NP problem without thinking for a sophomoric moment that the solution is N = 1.)
P = NP is actually one of the worst abuses of symbology I've seen in math.
"""During his own Google interview, Jeff Dean was asked the implications if P=NP were true. He said "P = 0 or N = 1." Then, before the interviewer had even finished laughing, Jeff examined Google's public certificate and wrote the private key on the whiteboard."""
A term that gets tossed around in math is "mathematical maturity." It's similar to what you see in other fields - e.g. learning how to program, learning how to make music, learning how to cook - that involves many "aha" moments and reshapes your perspective. Math is full of such steps, moreso than most other endeavors, probably because the main limit is the abstract reasoning itself.
The abstraction is by necessity. Our puny brains have only a very small working memory. The only way we can reason about many problems is by creating multiple levels of hierarchy. That is actually the essence of what mathematics is.
Math strives to minimize ambiguity, which other fields don't do as much. Non-math fields tend to reuse regular words as jargon (i.e. with specificity of meaning that may fly over the laymen's heads). Social sciences and humanities are most notorious for this, often resulting in non-practitioners not realizing they are out of their depth because they are not looking at symbols from non-Roman alphabets.
That's something only someone who's never studied advanced math could say. Math notation and jargon can be extremely ambiguous and overloaded. "Normal" has about 20 different meanings.
No, I'd say ambiguous means "context dependent" and the context is unclear.
Pronouns like you/me/he/she/they/them are context dependent in everyday English writing but they're only ambiguous when the context is unclear, otherwise most people have no trouble dealing with them at all!
Is there something that translates math formula into code? There are many (comparatively) "simple" algorithms I simply cannot make heads or tails when they are described via math prose or math formula, but when it's code I basically instantly know how to rewrite it into any other language I know, at least, and sometimes that's a starting point for poking at it to understand it a bit better, if not the same way would if you understood all the underlying math.
For poor old me, too many wikipedia articles on algorithms useful mostly or only for programming are described in formulas rather than simply code with detailed comments. Scrap the whole page and just gimme the code :( Not even to copy and paste, because that's a language I can understand, and enjoy learning.
Yeah it is a lot of simple ideas stacked one on top of the other, but the edifice is so large from some vantages that the building blocks aren't visible, or tractable to think about independently. And sometimes the ideas are very subtle, so you can only develop fluency partly by spending lots of time playing with those blocks by building your own little structures. You also develop fluency by talking to other mathematicians
I like to emphasize that the ideas are usually very simple at their core. Sometimes they map to kinds of objects or reasoning that non-mathematicians use implicitly all the time in their daily lives, mathematicians just have words for them and so are able to use them explicitly.
And I suspect the density of the language/terminology may give the wrong impression about how mathematicians think about the math they are working on. I mean, different people think / experience / practice math differently of course but IME the underlying thought about a particular problem tends to be much looser and concrete than formal math writing would imply.
That more formal language is needed of course because at the end of the day, it is how we communicate our thoughts in the way that other mathematicians can understand them, not to mention how we can check our own thinking
Yeah this is pretty much where I am at. Take the phrase from one of the responses
"The special fiber is the associated graded ring.....and that the filtration admits sufficiently simple homogeneous lifts of the three generators, then one might prove"
In any other context I would at least have some degree of intuition about what is being discussed, but in in math? Absolutely no idea. And usually if I start digging and turning over stones to uncover meaning, I'm just met with even more totally dense code-word language. Unlike other fields were digging is usually quick to relieve ignorance, somehow in math it tends to get worse.
I'm sure I am capable of grasping this if I took the time, and perhaps even what is being discussed it rather intuitive, but the incredibly density of the nomenclatic swamp you have to trudge through for math is totally unrivaled.
The basic problem is that to get to the objects you mention here is at least 3 or 4 years of full-time study away from the kind of math people learn for a typical college degree in science or engineering. If you really want to understand them, to make your "digging" efficient you should probably just get a pure math degree, but setting aside several years to satisfy occasional curiosity is not feasible for most people for various reasons.
One unfortunate feature of published pure math research is that often the ideas are quite accessible and straightforward and don't really require special abstractions or terminology, but those get used anyway because for someone who already has a math PhD it saves a bit of effort.
I agree, the nomenclature is impenetrable, it's like reading software that is not well commented. Perhaps LLMs are very good at "challenging" mathematics because what we perceive as challenging is primarily the language component and not the conceptualization.
I agree. I think the distance in capabilities between great mathematicians can be so much more vast than other fields as well. Some of them need every step to be derived, while others can skip ten in their head.
I once didn't understand the task given in an exercise sheet for a CS logic lecture, so I googled the topic, and all Google did was send me back to that exact exercise sheet.
Isn't it just what you studied in depth? I am not in this area but can understand what's going on "at a high level" here. But I studied no other science formally since the age of 15 (this is possible in the UK school system). So physics and biology just go over my head unless they are sufficiently mathematical.
There is a lot of verbal commonality between the classic sciences, classic engineering disciplines, and everyday life. I suppose they all share the common substrate of working in/with mother nature all day. A molecular biologist, civil engineer, and oceanographer can mostly keep pace with each other at least for a while in discussing what they are working on. These "mother nature" systems have tons and tons of overlap, and the nomenclature generally tracks this, or is one or two steps away from it.
Computer science/engineering strays from this, binary systems don't really track nature much, and hence a lot of their own unrelateable nomenclature arises, and then there is math, which is just way far out there on it's own plane of existance.
Yes, the nomenclature in math is atrocious. It isn’t much better in physics or biology however. As a species, we suck at naming and classification, and keep starting new trends atop old ones.
There is certainly need for the many abstractions of math to be as complex and well specified as they are.
There’s no reason for their nomenclature to be so bad. The end result is a substantial portion of the population, which can certainly hold and manipulate abstractions, fails to even contend with pure math.
I do think visualization tools will help in the future, to demystify some of this. But as with all sciences, the need for personal glory/mentor deification often conflicts with broader explainability.
It can't be one language, and that's the big problem. It's inescapably a bunch of tiny DSLs. Once you see both the inconsistency and the necessity for inconsistency, it becomes much easier to just roll with it.
People in their second year of graduate school only get to about the early 20th century in terms of understanding. Third year is getting to about the mid-century. Fourth and fifth years get kind of to modern times but with increasingly smaller breadth.
This is a very notorious area for dense definitions and concepts that interrelate closely and have to be memorized. Mathematicians from other areas are going to have difficulty but may have some idea of what the concepts try to capture.
Some areas are hard in different ways. I could never quite wrap my head around the way logicians have to think. A clever combinatorial bijection is a work of art you probably can explain to a undergrad class easily but good luck coming up with it. And number theorists will throw the kitchen sink at their problems: no area of math is safe from getting used by them.
People who do this have spent years of their life thinking in this language and studying it, so it is going to be hard. We're also not good at communicating the intuition which for algebraic geometry often comes from other fields.
That is one of the things that fascinates me most about mathematics compared with other fields, and it led me to discuss the subject with professional mathematicians. The funny thing is that they admitted it is the same for them...stray even slightly outside their own specialized area, and within two or three lemmas, they also feel completely lost.
IMO, it's just the notation. Something I've actually found ChatGPT useful for is to create mathematics lessons for me in the form of computer programs. When broken down into a series of readable almost-plain-English steps, it's so much easier to understand. And it's easy to tinker with programs and get a hands-on feel for things quickly.
I'm sure having a compact notation is absolutely invaluable for people who dedicate their lives to maths, but for someone with just a passing interest, I find it more obscuring than helpful. I feel the same way about music notation.
I mentioned this in a sibling comment but even for mathematicians, the intimidating notation and the more formal language might give the wrong impression about how we think about math. Actual thinking and even discussions with other mathematicians tend to be looser and more concrete and tactile, but the notation and language are there in part to act as a sort of lingua franca to help everyone stay on the same page, since everyone thinks at least a little bit differently. It also helps to keep you honest and catch situations where your thinking was muddied, since this language is so specific and writing things down has a funny way of catching things. And good notation goes a long way towards making the simplicity of an idea clear, or completely muddy in the case of bad notation.
It's just language. Mathematicians don't invent notation for fun, they do it because they naturally start thinking at a higher level of abstraction. If you're not thinking at that level then, well, it will be all Greek to you.
Math isn’t necessarily hard, but it’s incredibly dense
A simple statement like let f(x) be a continuous function can carry a lot of definitions
In that statement, if you missed the day in class where they covered continuous functions it might not even register that it’s a well defined term
And that’s the most over simplistic example I could think of
As a math major, I remember that being one of the first lessons I learned, that every single word could be carrying a lot of weight so to look things up in detail if I was ever struggling on a problem. One of the oldest entries in my memory.md file
Hey, Mr. Tao, where would you put the intelligence of the current models compared to top humans you surely mush have worked with? Perhaps given in the count of people you know that are higher than the newest LLMs? Also, how would you rate the speed of work compared to what your speed is, if this makes any sense?
Jeez. While I obviously can't talk at all about the math, I've noticed a few things:
a) The model thinks on some questions while straight answers on others. (I wish I'd knew from the questions if this is somehow correlated to hard tasks or "inventive" tasks, but that's way out of my league).
b) The model sometimes pushes back. Again, I'd wish I knew if it was warranted, but I counted 2 instances where it said "yes, but with caveats", one where it said "mostly yes but with this correction" and one where it said "careful here, because x y z".
c) The model did q&a + pdf ingestion + code writing + more q&a + thinking + more q&a, for a looong while, while seemingly staying on topic (at least Terrence Tao seems to think they're still productive, so I'll trust that).
This is what model progress is, not number goes up on xBency or yBencher. Damn.
The "yes, with caveats" thing is boilerplate for both Codex and Claude since this current generation.
It's actually a bit annoying because it primes you to think that the caveats are real, but most of the time it's just something terribly obvious and not a real caveat, but the model probably has some system prompt that tells it to always consider caveats or something like that.
Same as the model starting every reply with a commitment to be "honest". LLMism are fun but I tend to just suppress them via AGENTS.md because they distract me
At least Opus up to 4.7 or so, my experience is that Claude often uses "yes, with caveats" in place of "no, you numbskull".
"Is a meter the same as a foot?"
"Yes, exactly, you have it now, except they're different distances."
Maybe it's because I ask it to quiz me, and it really doesn't like to tell me I'm wrong. I also got a fair amount of
Claude: Ok, I will test your understanding. Question A? Question B? Question C? Question D?
Me: A=10. B=2. C=121. D is not solvable.
Claude: You got most of them right! You're very astute in saying that A=10, but actually it's 7. B=2 is exactly right! C could be 121 if we were talking base 4, but we're actually in base 10 so it's 25. D is trivially solvable and is 0.
Me: ...isn't that like 1 out of 4? How is that "most of them right"?
What was most remarkable to me from this transcript, was how strong of an equal the AI agent comes across compared to the user (Tao). And Tao is one of the top mathematicians of modern times.
Yes, Tao is guiding it to where he wants to go. But also, Tao is actively learning from it and relying on its explaining, analysis, and inference abilities. You can easily imagine this conversation having taken place between Tao and a PhD thesis student, or even another professor, explaining their results.
What can we imagine and predict about the future anymore? Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore.
I find it helpful to think of LLMs as reflections. If you can talk like an expert mathematician at the model it will respond like one. While Terrance's first prompt looks trivial I expect a first year Uni student would be hard pressed to provide something that good.
I guess it is kind of the inverse of the "you are an expert mathematician" prompt engineering of gpt3.5. Since no one ever says that to an expert mathematician when they are doing expert math the model immediately reflects that it is not an expert mathematician.
>Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore.
we're kind of well past that (in my opinion), if you consider that this is the same ai assistant that can help you with a recipe, diagnose a weird sound in your car, help with biology homework, translate languages, and so on.
even in math alone, i think its indisputably already stronger than Tao, considering it has approximately this much depth in ~all of the math subfields.
Agreed, it's stronger "horizontally". But I also think that we're not far away from it being stronger vertically; i.e. superior to Tao, in that such turn-by-turn guidance by him in solving sophisticated and difficult problems will not be necessary for long.
"I’ve activated Pro. Can you continue to look for a potential geometric explanation of the X_3 ~ A3 miracle that avoids coordinates or other unmotivated constructions ?"
I just do very laconic questions about advanced topics, this seems to prompt it a bit more towards reducing fluff in the answers. But that + the activated pro could be an improvement
Yeah. I also only use pro in very specific situations. Not (just) because it's slow, but if the question is too simple or vague, pro responses are sometimes overfitting to the noise in my question. Until there is lots of context and the basics are laid out, high or xhigh somehow work better. Pro gives you the last 10%.
It's not so obvious. Taking my parent comment literally (as he said "the other way around"), one can assume that ChatGPT the platform itself would pay Terrence, rather than OpenAI.
Similar to how Cypher puts it: I know this is “just” next token inference, matrix mult and just software, ie there’s no “intelligence” there BUT, looking at this convo … damn!
The fascinating this is that the LLM is not acting as a tool here AFAIk, but very much like a colleague.
I have no knowledge of the domain and have only PhD EE level math knowledge, so maybe my bar is too low.
I think the "But this is not intelligence because it is known math" is not a correct argument. It is unknown how the overall higher intelligence of humans works.
What I do notice however is that LLMs are becoming capable of doing an increasing part of the intellectual work I can do, and usually a lot faster.
Just today I presented an agent framework that can take an informal incident statement and propose infrastructure changes to fix it, all evidence backed. This did nothing I could not to, but it did all 5 test cases in 6 - 12 minutes each. I would have found all of the monitoring indications it did, but it would have taken me a day per test case. The LLM also included sass to silly tickets. ("This is not even worth spending monitoring resources on. It's obviously a configuration problem.")
That's how this is reading to me as well. It's just fast at slogging through a certain level of "simple" transformations.
That argument says very little, emergent behavior is a thing in complex systems with billions of parts.
Humans can also be reduced to voltage potentials propagating along of tubes of fat and synapses getting rewired.
There is clearly intelligence there. We have no way to recognise intelligence other than the appearance of intelligence and this very clearly displays that.
It's also quite clearly different to human intelligence in some notable ways, but not in any that preclude describing it as intelligent. At least for normal non-pedantic definitions of the word.
Everyone uses "intelligence" to mean something slightly different, so for this to be a useful claim to make or refute we need to come up with new, intentionally-pedantic, terms (or new domain-specific definitions for vague existing ones).
Yes, trying to communicate (or watching others try to communicate) about these topics is incredibly frustrating because it's pretty much impossible to make any progress without interrogating people's different definitions, but nobody wants to do that because it would mean being pedantic, splitting hairs, etc.
It's not like this is a new problem. Turing had a definition most of a century ago, he wasn't the first and certainly wasn't the last. I don't think we need new terms necessarily, and I doubt we're all going to agree on a definition tomorrow.
I'd say an entity capable of instructing one of the leading mathematicians of his era is pretty clearly intelligent by any reasonable measure - however it might be arriving at its output.
I think we have wildly different conclusions about what happened here. You see the machine as instructing Terrence Tao, as if it were Plato teaching Socrates about the theory of forms; I see Terrence Tao using the machine to teach himself, like an intelligent student uses a book. In this case, it's just a book that fools us into believing it can think and reason like we do, because it generates language in much the same way we do when we think and reason.
I'm no intelligence researcher or philosopher; but, I think LLMs make us confront the (IMO, now clear) distinction between cleverness (intuition), reasoning (rational argument), and consciousness. I suspect that we think of "intelligence" as either of the first two welded to the latter. In that vein, I'd say that consciousness may be just another emotion: happiness, sadness, egoness.
There is no intelligence. If anything, this just shows that natural language and mathematics are both fields which are structured in a logically computable way. And if you have a machine that can compute symbolic logic, you can process both natural language and mathematics.
A second corollary is that rational consciousness and thought is less likely to be contained in language than previously thought, because if language is so simple that a machine can process it, it can't contain consciousness.
If natural language was structured in a logically computable way, we'd have had interesting chatbots by the late 80s, basically as soon as a dictionary fit in local RAM, and for the same reason we got compilers.
Da hole raisin y nat-lang be v. hard is dat i kan rite lik dis an it be cool 4 native engrish speekrs 2 unerstand. LLMs are of course fine with this sentence in exactly the way that Zork's engine couldn't be.
The underlying structure of language, which is grammar, is obviously logical. That the symbols used to represent this grammar can be sometimes fuzzy or ambiguous, is no problem for a machine that takes context and probability into account when translating words to the underlying grammar structure.
It's not "obviously logical", it's a pattern which we mimic to avoid mockery.
example For, semi-randomise I word order can this like, Yoda worse than, and be understood.
> is no problem for a machine that takes context and probability into account when translating words to the underlying grammar structure.
We had to invent Transformers to be able to do that with reliability anything close to being worth caring about. Transformers have to learn from examples, not be pre-programmed.
What does "predicting the next token" mean? I ask this every time people say "LLMs are just predicting the next token" and it's maddening that nobody can give a straight answer. Predicting it according to what probability distribution? Every process that produces a sequence of actions (including e.g. a human writing) can be modeled by some probability distribution and therefore their actions are indistinguishable from "predicting the next token" emitted by that distribution.
> to predict the next token you first need to model the universe
Exactly. The "most likely next" series of tokens, for example, when given the first half of a correct mathematical proof, is the correct rest of the proof. I have never seen anyone define "most likely next token" in such a way that this isn't true.
Similar to the story of George Dantzig, who was late to class and solved two open problems in statistics because he mistook them for homework, I think the current batch of frontier LLMs are chained up by knowing which problems are supposed to be unsolved. If they're let free (probably via some targeted RLHF) we might get a flurry of solutions to open problems.
But a property of intelligence is to know when to stop, if we treat intelligence as some sort of search and not some a priori intuition of the entire space. Seems kind of hard, if not impossible, to train for specifically that.
I'll have a blog post up tomorrow about it but the Jacobian Conjecture counterexample is a very funny cognitohazard for LLM assistants. It's a paradox for modern LLMs: they have enough math skills such that they can easily compute the Jacobian to formally verify the counterargument, but its own knowledge base is locked prior July 19th 2026 where all it knows is that the Jacobian Conjecture is unsolved and a random chat user providing such a proof is highly unlikely.
I wonder whether when the fact that AIs have started solving conjectures will enter the training data, they will become more confident in their abilities.
The models are just a reflection of how you prompt them. When I posted the counterexample, in dry language with a dry personality setting, claiming credit for it with search disabled, GPT simply verified that it was correct but noted that it could not ascertain novelty or priority and that external verification was required. "Model reacts in crazy ways when my prompt influences it to" is really, really not interesting.
I don't understand any of the math here, but I had two thoughts. Soon we'll have explainer agents that translate these according to my level so I can, with effort and interest, follow along and stretch my understanding boundary bit by bit.
Two, at some point AIs will be able to use other context like the fact that this is Terrence Tao and not your average Joe and change how it answers, either in tone or structure.
Soon we'll have explainer agents that translate these according to my level so I can, with effort and interest, follow along and stretch my understanding boundary bit by bit.
That's not going to happen. Mathematics isn't just unfamiliar, it's truly difficult to understand. You have to put in a lot of work to understand each concept and the concepts build upon each other to form a vast tower of abstractions that has been growing for thousands of years. Just as there is no royal road to geometry, there is no elevator to the top of the tower.
Fork Tao’s convo and prompt this (with your own math level described).
GPT did a great job of translating Tao’s questions and concepts (e.g. “pre image”) into a progression I could understand.
“Ok I have a PhD in financial math and undergrad in engineering math. I have almost zero knowledge of polynomial algebra / geometry, I know what a polynomial is and what roots are but not much beyond that. Could you try and explain to my level what questions the user I the conversation has asked and what the agent has responded with, we can probably go user query by user query to build up”
One thing I've repeatedly told people is that chatbots are often the most patient teachers we'll ever get (especially when explaining "stupid" questions) — compared to what we've encountered on StackOverflow or Reddit.
I want to be able to take a conversation and ask for subconversations as red pen annotations "on the side". The linear nature of the context tends to frustrate this.
I'm not sure an AI will speed things up much. You would probably still need years of layers of foundational understanding to get the advanced material. We don't go through years of school to learn math just because teachers are bad - it's because complex subtle ideas are built on countless other ideas, and aren't necessarily compressible to something every layman can understand.
The years are broad though, the nice thing with AI explanations is that they can go deep quickly, and quite precisely down the path you need for your prior experience.
It's not a path, it's a tree, and a downward-facing one at that. To understand the root, you have to understand all of the leaves first, and all of the nodes above them, all the way up.
Most people, if they haven't studied mathematics in university, would need to learn hundreds of concepts just to get to the leaves of the tree, and many of these concepts are truly difficult to understand, requiring weeks of study and practice.
Oh for sure, I'm not denying the value of AI as a teacher, just saying that it's not going to speed things up much over a quality personal teacher/tutor.
I’ve had a similar experience using LLMs to have mini personal breakthroughs.
One thing I notice is many models say statements along the lines of “okay we have exhausted this thread it’s diminishing returns from here and we should stop and move on”
It’s funny because I’ve been building a tiny neural network maze solver (23 bytes solves 92.75% of unseen 2D mazes)
When I asked ChatGPT/Fable if we had anymore threads to pull to increase capability and decrease byte size, they both basically said no way - back when I was at ~166 byte models with a ~85% solve rate.
Throughout the experiment I just kept trying different approaches and eventually had 3 mini “breakthroughs” in this particular niche. But if I had listened to the models…
Anyway, these models are amazing to experiment with quickly, but they are dumb as hell and so absolute
probably like a minute or two? im pretty sure someone unironically said stochastic parrot on the HN post with the tweet announcing the counterexample, and ive read several comments with similar sentiments today (including in this thread)
Maybe Terrence is chasing too many red herrings. Maybe he should just have asked it to find a counter example to the conjecture and on success explain how it was found.
Words and sentences to an LLM are like witchcraft. There are certain words, sentences that make LLMs go a certain way and do vastly better. Sometimes its not at all apparent what set of words will work to do what you want it to do. An example I have been using to do design at a high level is to say to claude.
```
A question is salient to the degree that its answer changes what we do next. Operationally, saliency = the product of four things:
- Decision-leverage — would resolving it one way vs another force a different design or invalidate a stated decision? (No leverage → drop, however interesting.)
- Residual uncertainty given current evidence — is it still genuinely open after reading the docs and the code? (Already settled → drop, however deep.)
- Load-bearing-ness — how much rests on the premise.
- Cost of finding out late — architecture-deciding / expensive-to-unwind raises priority; cheap-to-fix-later lowers it.
```
There are a few things to note about this prompt
1. There is no reason from looking at it that it should work, it even has the word load-bearing which people loathe, but it remarkably produces a stable design with questions from claude (atleast from claude Opus 4.8 and even better from Fable5). Otherwise the design document claude likes to really write are implementation level(code or otherwise). I usually pair this with matt pocock's grilling skill to make claude behave.
2. From design -> implementation, its is generally about understanding when claude is trying to trick you into making something sound like a good/easy solution but has tons of untested assumptions. Here you have to read and patiently spot if a how you would get to the solution is not clear. A common error here are when claude makes a big deal based on what it read and interpreted too seriously without questioning the assumptions. There are several more.
But it also comes down to your experience as a SWE, much like a mathematician's. The frustrating thing about it is, it feels tha a skilled mathematician working with AI can make them productive in ways that are more reliable as compared to a SWE (e.g. lean is deterministic and can provide very strong feedback and LLMs are very good at using that feedback). Maybe a mathematician can chime in on that?
I recall several mathematicians (possibly including Terence Tao) mentioning that fields in mathematics have become so specialized and isolated that a conference like the ICM feels more like a collection of mini-conferences. An expert in one area can barely understand a talk in another.
Modern AI feels like a godsend to mathematicians. It helps them break down boundaries and connect concepts in ways a mere mortal couldn't imagine.
I was a chemistry researcher this is true in all science.
I'm not sure LLMs can transform this, the incentive is to get more results in your nich, jumping topics don't help unless you have genuine interest and reason to.
I find it amazing how people can use AI to do things that seem hard but yesterday I could not figure out how to install a package on my system. It kept suggesting dependencies that don't exist, and telling me to use functions that are not in the system. The math does not math...
What's worse is that HN doesn't even try to figure out how such a result could be obtained. They just take it at face value and go into dear diary mode.
> If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function flatten out. [...] What was conjectured is that this local invertibility property everywhere would mean global invertibility.
> it doesn't overturn much. [...] the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.
> From a comment by j2kun https://news.ycombinator.com/item?id=49000833 , someone asked Fable and there was an almost counterexample in 2d but it uses division too. [Instead of f=x^2+7xy they have something like f=x^2+7x/y so it's not a polynomial.] It looks like the new trick was to use a third variable to avoid the division.
Presumably this entire conversation is at least somewhat typical of the way high level mathematicians talk to each other in ego-less fashion? Although, it would be interesting to what extent the model was trained on math conversation as opposed to just analysis and proofs.
The flow of the whole conversation, with Tao guiding and the model calculating, gave me the feel of Tao perhaps talking to himself - just that each of those model responses would have taken him much longer to calculate by hand.
It would be fascinating to hear Tao talk about what he may have learnt from this, and if it suggests approaches to other problems he might not have considered, as well as an analysis of the original Fable counter-example construction.
I'm watching how Tao uses AI, and it's interesting.
Expand the entire expression, then change the representation to find the core axis. You can't see the axis from just one perspective, so you change the representation. In programming terms, it's like applying multiple domain models. Then break it down into small contract units. Why is it a Jacobian monomial? Why does x satisfy a cubic equation? And so on.
Then swap out the modeling under a hypothesis, assemble it all back together, and verify it through the equation.
This feels similar to modeling in programming.
Observe the whole -> explore better modeling -> decompose local problem -> verify independently -> reason about the highre level structure -> integrate back into the original problem.
This feels similar to when I receive work from a client and write a programming proposal
Web clipper is great. just tried the reader mode on this chatgpt transcript, it only shows 1 page of it. Is the purpose of reader mode to enable interactive annotation before saving to notes?
To me, this shows that extremely talented and qualified mathematicians (can) use frontier-level LLMs to automate their personal grind-y workloads that would otherwise (probably) take more time to accomplish with natural intelligence.
By itself, no consequence. But over time, provided we keep pumping out talented and qualified mathematicians and keep subsidizing costs, we could maybe hit a breakthrough... somewhere... that has real impact.
It's an indicator of AI progress. The solutions aren't especially revolutionary, but no person had been able to solve them after decades of collective attempts.
To be fair I don’t think there were too many people really trying to. Symbolically, one could make a parameterization of the Jacobian determinant and then brute force a solution, if one had known such a polynomial existed in only three dimensions.
Oh yes there were. The Jacobian conjecture is "notorious for the large number of published and unpublished false proofs which turned out to contain subtle errors."
It's not quite the Reimann hypothesis, but many prominent mathematicians have spent years working on this problem. Yitang Zhang wrote his PhD thesis on it.
That’s the counterexample. Low integer coefficients, power 7 in three variables. If someone said it was there, couldn’t we all have written a pretty simple brute force solution for the search space, especially with the constraints that the symbolic determinant had to cancel to a constant?
I don't think you even understand the problem. The determinant needs to be a non-zero constant AND you need to prove that particular map is not globally injective, meaning you have to find at least two points mapping to the same value. Of course it looks easy when someone shows you the counterexample.
This is not true at all. The parameter space is absolutely MASSIVE. The counterexample is a degree 7 polynomial in 3 variables, which means 360 coefficients. There's no particular way to bound these coefficients or even the degree or number of variables apriori, but assume you somehow did. Also assume you were confident that it would work with integer coefficients bounded from -12 to 12. Now you have to iterate over 360 degrees of freedom, verify that the Jacobian is a nonzero constant, and somehow show uninvertibility of the transformation, which is not a particularly simple task.
If you searched for coefficients from -12 to 12, this would be 25^360 = 2 * 10^503 different possibilities. A common reference point is that there are 10^80 atoms in the observable universe. Sure you could probably reduce this a bit with clever tricks, but the starting point makes the method completely unviable, even with the knowledge: A) a counterexample exists, B) it's in 3 variables, C) it's in degree 7 or less, D) it's in integer coefficients, E) those coefficients are 12 or lower.
Here the search wouldn't have been chosing the coefficients independently. Note that one intermediate variable is a polynomial in the input variables, and it is used in other polynomials. A search over expressions like the ones in the counterexample would have a much smaller search space.
I learned from poking around that checking the invertibility of a system in C is a much, much harder problem than I thought. Nonetheless if that were no object, let’s say coefficients from -16 to 16 (5 bits) times eight terms times choosing up to cubes (16) times three equations is searchable, especially since you have only the final combination of coefficients in the determinant. It’s not impossible to generate the equations like this Fizzbuzz style.
Some materials are readily available on eMazon and aBay, so I've taken the liberty of ordering those for you. Your credit card bill will be a bit high this month, but it'll be worth it. There weren't any sellers for the advanced EUV lithography machines, so I've hacked into the only place on earth that makes them, changed their records and had them ship it to you. Expect to receive a "pinball machine" from Amsterdam, soon. I've instructed the roomba connected to the local network to start assembling stuff while we wait for the other materials. Oh, and you're gonna need a new toaster.
It's awesome to publish this kind of thing - great PR at least. Even if you don't understand the details, it's interesting to be able to peek into a technical conversation that a world class mathematician is having about their work with a "colleague". It's also the clearest demonstration I've seen of the vision AI people have about a future with truly intelligent copilots in super technical fields.
I expect Centaur Field Medals to be awarded. It's not clear how much steering Levent Alpöge did to get the result. In math it's very common to hide the search pattern and show only a simplified path to reach the final result.
Is ChatGPT's interface always this atrociously jittery? Or is it just because this page is getting an HN hug-of-death right now? Every time I try to scroll the whole page goes blank for a few seconds and then re-renders.
I think that's partly because it's a fairly long conversation and partly because of all the LaTeX. There's enough formulas there to fill a small text book, and I doubt their renderer is all that efficient.
Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point. If any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, then the Jacobian matrix does not allow for an evaluation of the stability.
Yes—for a continuous-time autonomous system
x
˙
=f(x),f(x
∗
)=0,
this is the standard linearization criterion, with J=Df(x
∗
):
If every eigenvalue of J has strictly negative real part, then x
∗
is locally exponentially asymptotically stable.
If at least one eigenvalue has strictly positive real part, then x
∗
is unstable.
If no eigenvalue has positive real part but at least one has real part 0, linearization is generally inconclusive. Nonlinear terms or a center-manifold analysis are needed.
The last case really can go either way. For example, all three scalar equations below have Jacobian J=0 at x=0:
x
˙
=−x
3
,
x
˙
=x
3
,
x
˙
=0.
Yet 0 is respectively asymptotically stable, unstable, and neutrally stable.
A slightly more precise wording is therefore:
If the spectral abscissa
α(J)=
λ∈σ(J)
max
Reλ
is negative, the equilibrium is locally exponentially stable. If α(J)>0, it is unstable. If α(J)=0, the Jacobian test is inconclusive.
This criterion concerns the Jacobian matrix of a dynamical system at an equilibrium; it is unrelated to the “constant Jacobian determinant” condition in the Jacobian conjecture.
It's fun if you ask ChatGPT to guess the identity of its interlocutor :) It will guess math researcher or paper author without hints, but if you give it some additional hints, "this was shared over the internet", "it's someone willing to work with AI", it will guess Terence Tao as the first choice.
/-!
# A counterexample to the Jacobian conjecture in dimension three
We formalize the polynomial map
F : ℂ³ → ℂ³
whose Jacobian determinant is the constant `-2`, but which is not injective.
The final theorem `jacobianConjecture3_false` states the failure of the
polynomial-inverse formulation of the Jacobian conjecture in dimension three.
-/
namespace MvPolynomial
variable {R : Type} {σ : Type}
/-- The formal Jacobian matrix of a family of multivariate polynomials. -/
def jacobianMatrix [CommSemiring R] [DecidableEq σ]
(F : σ → MvPolynomial σ R) :
Matrix σ σ (MvPolynomial σ R) :=
Matrix.of fun i j ↦ pderiv j (F i)
/-- Evaluation of a polynomial map at a point. -/
def evalMap [CommSemiring R]
(F : σ → MvPolynomial σ R) (p : σ → R) : σ → R :=
fun i ↦ eval p (F i)
end MvPolynomial
open MvPolynomial
namespace JacobianCounterexample
variable (K : Type) [Field K]
/--
The three components of the polynomial counterexample.
The variables `X 0`, `X 1`, `X 2` correspond respectively to `x`, `y`, `z`.
-/
def F : Fin 3 → MvPolynomial (Fin 3) K :=
![
(1 + X 0 X 1) ^ 3 * X 2
+ X 1 ^ 2 * (1 + X 0 * X 1)
* (C 4 + C 3 * (X 0 * X 1)),
X 1
+ C 3 * X 0 * (1 + X 0 * X 1) ^ 2 * X 2
+ C 3 * X 0 * X 1 ^ 2
* (C 4 + C 3 * (X 0 * X 1)),
C 2 * X 0
- C 3 * X 0 ^ 2 * X 1
- X 0 ^ 3 * X 2
]
/--
The formal Jacobian determinant of `F` is the constant polynomial `-2`.
-/
theorem jacobianDet_F :
jacobianDet (F K) = C (-2) := by
simp only [
jacobianDet,
jacobianMatrix,
det_fin_three,
of_apply,
F,
cons_val_zero,
cons_val_one,
cons_val_two,
head_cons,
tail_cons,
map_add,
map_sub,
Derivation.map_one_eq_zero,
pderiv_mul,
pderiv_pow,
pderiv_C,
pderiv_X_self,
pderiv_X_of_ne,
ne_eq,
Fin.reduceEq,
not_false_eq_true
]
simp only [map_neg, map_ofNat]
ring
variable {K}
/--
The point `(0, 0, -1/4)` maps to `(-1/4, 0, 0)`.
-/
theorem evalMap_F_p0 :
evalMap (F K) ![0, 0, -(1 / 4)] =
![-(1 / 4), 0, 0] := by
funext i
fin_cases i <;> simp [evalMap, F]
/--
Provided `2 ≠ 0`, the point `(1, -3/2, 13/2)` also maps to
`(-1/4, 0, 0)`.
-/
theorem evalMap_F_p1 (h2 : (2 : K) ≠ 0) :
evalMap (F K) ![1, -(3 / 2), 13 / 2] =
![-(1 / 4), 0, 0] := by
have h4 : (4 : K) ≠ 0 :=
(by norm_num : (2 : K) * 2 = 4) ▸ mul_ne_zero h2 h2
funext i
fin_cases i <;>
simp [evalMap, F] <;>
field_simp [h4] <;>
ring
end JacobianCounterexample
open JacobianCounterexample
/--
The Jacobian determinant of the displayed map over `ℂ` is a unit.
Indeed, it is the nonzero constant `-2`.
-/
theorem F_jacobian_isUnit :
IsUnit (jacobianDet (F ℂ)) := by
rw [jacobianDet_F]
exact
(isUnit_iff_ne_zero.mpr
(by norm_num : (-2 : ℂ) ≠ 0)).map C
/--
The polynomial map `F : ℂ³ → ℂ³` is not injective.
-/
theorem F_not_injective :
¬ Injective (evalMap (F ℂ)) := by
intro hInjective
have hp :
(![0, 0, -(1 / 4)] : Fin 3 → ℂ) =
![1, -(3 / 2), 13 / 2] :=
hInjective
((evalMap_F_p0 (K := ℂ)).trans
(evalMap_F_p1 (K := ℂ) (by norm_num)).symm)
exact zero_ne_one (congrFun hp 0)
/--
The injectivity consequence of the dimension-three Jacobian conjecture
is false over `ℂ`.
-/
theorem unitJacobian_does_not_imply_injective :
¬ ∀ P : Fin 3 → MvPolynomial (Fin 3) ℂ,
IsUnit (jacobianDet P) →
Injective (evalMap P) := by
intro h
exact F_not_injective (h (F ℂ) F_jacobian_isUnit)
/-!
We now formulate the polynomial-inverse version explicitly.
-/
/-- Polynomial self-maps of affine three-space over `ℂ`. -/
abbrev PolyMap3 :=
Fin 3 → MvPolynomial (Fin 3) ℂ
/--
A polynomial map has a polynomial two-sided inverse, viewed as functions
on `ℂ³`.
-/
def HasPolynomialInverse (P : PolyMap3) : Prop :=
∃ Q : PolyMap3,
LeftInverse (evalMap Q) (evalMap P) ∧
RightInverse (evalMap Q) (evalMap P)
/--
The polynomial-inverse formulation of the Jacobian conjecture in
dimension three.
-/
def JacobianConjecture3 : Prop :=
∀ P : PolyMap3,
IsUnit (jacobianDet P) →
HasPolynomialInverse P
/--
The Jacobian conjecture in dimension three is false.
-/
theorem jacobianConjecture3_false :
¬ JacobianConjecture3 := by
intro hJC
unfold JacobianConjecture3 at hJC
apply unitJacobian_does_not_imply_injective
intro P hP
rcases hJC P hP with ⟨Q, hleft, _⟩
exact hleft.injective
The big take away for is the fact that the ONLY reason why chatgpt was able to get to this counterexample was because of the knowledge of the person driving the conversation.
I don't think chatgpt could have come to this on its own without the amount of steering he did, which just validates the idea that AI is not a replacement for human expertise but an amplifier.
> I don't think chatgpt could have come to this on its own without the amount of steering he did, which just validates the idea that AI is not a replacement for human expertise but an amplifier.
The problem is that, what happens to human expertise as people start to use AI earlier and earlier in their careers, so that in 50 years? The problem is that Terry Tao spent decades as a mathematician before ever encoutering AI. Of course he and people his age will be able to drive AI somewhat sanely and use it to their advantage.
But as more people grow up with AI, they will likely not reach levels like Terry Tao because their exposure to AI and the temptation to use it will certainly dull raw human intellect over time.
I don't know if I agree with the premise that having access to AI results in dulling human intellect.
I feel like to get to Terry's level you need a combination of passion and aptitude for the subject. People that don't want to learn about a topic will always look for shortcuts, which I think represents the vast majority of people. Terry Tao is quite exceptional, and I think exceptional people will still exist even when the "easy" button is bigger than it's ever been.
The problem is that learning never stops. You can't just go through school, become a junior in X field, then start using AI. Then you'll forever be a junior. You have to make a choice when you're working a job: either use AI-first workflows to increase your productivity, or don't and increase your knowledge and skill.
My wording is specific. You can use AI and increase knowledge and skill, but this requires you to be driving the AI at such a low level you don't get the full speedup. As an example, you can write code yourself with AI as an assistant, but it's not as fast as AI writing everything.
So now we end up stuck in a situation where every professional needs to choose between long term skill growth or speed, as anyone who's worked a job before knows, speed will always be the one chosen.
This is the second ChatGPT shared conversation I've seen today that is truly fascinating.
The first one was someone proving another conjecture false by just repeatedly saying "keep going" to ChatGPT: https://x.com/DmitryRybin1/status/2079904005652893709
What a world we live in.
> just repeatedly saying "keep going" to ChatGPT
For posterity, this indeed works for most problems where an agent might give up. LLMs don't inherently know something is impossible.
The phrase I tend to use in my harder prompts to automate this with a sane loop breaker:
> **REPEAT THIS PROCESS UNTIL CONVERGENCE AND YOU ARE OUT OF OPTIMIZATION IDEAS.** You have permission to keep iterating.
I’m a bit surprised OpenAI isn’t finding these big results far faster than the product’s user base. With no limits on runtime, access to dev models, custom tuning, and top talent, you’d think there’d be a constantly running internal project with the goal of solving famous math problems. And who knows, perhaps there is, but it would be interesting to compare the rate of success per unit “effort” of the internal mathematics work with that of the user base.
What I am thinking is the way you make it 'keep going' and when you have people of the calibre of Tao doing it I kept thinking how many breakthroughs is he going to cause the LLM to find with his targetted questions :D Amazing that we have the privilege of witnessing a true expert in such a way question the LLM.
Do notice he is quite Socratic, the approach works well for LLMs they love to please so you have to be careful in how you lead them.
this is /goal in claude code/codex. also basically a slightly improved ralph loop
Aka /goal
Without any more context, "keep going" seems to be doing a lot of work. The user is placing a lot of faith in the LLM to not make subtle logic mistakes and to take good approaches to each problem. In my experience, even frontier models (such as Fable) are quite capable of getting confused during even simple technical work I've done in the dev ops world. For example:
LLM: This package hasn't made it to production.
ME: are you sure? i see it right here!
LLM: You're right to push back. I inferred that based on weak data. I see now that the package has been deployed!
If the above conversation is typical for me, how could one expect to achieve a sound result by repeatedly prompting an LLM to simply "keep going" in dense mathematical proofs? Perhaps the user in this case had actually checked the LLM's work before issuing the prompt, but I think you see my point anyway.
Agreed, often you have to step in and stop it from reasoning itself into dumb directions, but occasionally it goes just like the transcript in question.
There may be something(s) about mathematics (proofs) that makes it particularly amenable to LLM reasoning - highly inductive from facts that are explicitly within-context/associative space? Being an unusually well documented discipline in general, with less influence from tacit knowledge or idiosyncratic “it works however the opinionated human made it work +- bugs” processes? Something about simulating even the smallest non-pure-inductive leaps necessarily risking simulating mistakes due to the nature of context “perception”?
There’s also probably a lot less noise from casual internet conversations. I imagine a nontrivial amount of what LLMs know about certain technologies comes directly from forums like reddit where quality of response isn’t guaranteed.
“Keep going” is exactly how many mathematicians achieved success in the past. :)
Is this the same as Dinitz Theorem[1] which seems to have been proved in 1994? This is the only result I keep stumbling upon when trying to understand the problem formulation
[1]: https://en.wikipedia.org/wiki/Dinitz_theorem
"it's enough of partial results. let's finish with a complete unconditional counterexample"
"Worked for 88m 24s... >"
"<h1>Complete finite counterexample</h1>"
...
crosses fingers "Low hanging fruit, low hanging fruit, low hanging fruit..." hyper-ventilates
Direct link: https://chatgpt.com/share/6a60b2eb-0b64-83ee-9c76-7931ca1de0...
From the prompt:
> You should do a breakthrough
This is just as funny and ridiculous as those "make no mistake" prompts.
Mathematical breakthrough, genius, trending on Artstation.
"whats next" is another good one
"Gew on, lad!"
this sounds like like an open parenthesis (
without someone independently verifying it, it just dangles there
...
At Mozilla, we had a set of whiteboard tags we could set on bugs, like "[crash]" or "[compat]" or "[leave-open]". That last was used when there were multiple patches attached to the bug, and we wanted to land only some of them without automation closing the bug once they landed. (It's common to have alternate approaches or test cases also attached to the bug, so you normally don't want to wait for all of them to land before closing the bug.)
I started using "[leave-open" for those.
It lasted for a couple of years, until someone went through and "fixed" them all.
)
It’s endlessly fascinating to read the AI transcript of an expert who _really_ knows how to cut to the chase. It just shows how much you can potentially squeeze out of these models. I’m also surprised to see that even Terrence Tao seems to use it in a way that resembles, in progression, how I use llms in my area of expertise (emphasis on progression and usage patterns, not absolute skill, obv I don’t match that): short pointed questions that goes all in on the jargon and machinery of the field and steers the llm hard (eg no softballs). I’ve noticed that llms switch their tone and meet you basically more or less on your level.
It reinforces how to "learn AI" is to first master the problem domain.
I can use AI for coding after decades of coding. I can't use it for theoretical physics because I can't evaluate the responses.
More and more the skill of being able to ask the right question seems critical to me, and I don't know how one can do that without deeper and deeper domain expertise.
> I can use AI for coding after decades of coding. I can't use it for theoretical physics because I can't evaluate the responses.
That is what will happen though to future generations: they won't be able to use it for anything because none of them will have the "decades of coding" experience that you have had the privelege to have without AI.
They will have decades of experience with AI and they will be able to guide them by sniffing their hallucinations from single words.
Hopefully in decades hallucinations will be largely solved.
I've mostly seen people trying to oneshot a result, while I'll quickly experienced that going through steps/discovery was more effective and more satisfying, since you can always steer it back in the right direction; while oneshotting is hit (and it kind feel like magic) or miss (and you'll have to rework your prompt).
I was struck in the same way but I think it makes sense in terms of a thinking partner.
It is still ultimately Terence that is steering things.
What is crazy to me is how few of other people's conversations like this I have actually read.
Tao is really great for this because the anti-AI crowd can't really chime in and take the thread in a pointless direction. It is hard to think of another human alive who can carry the weight of unassailable authority in the same way.
I encountered something fairly similar working with Claude a few days ago. For a current project I've been fairly hand-wavy with requirements since I was getting good results, but it seemed to be failing hard on some key points, so I started to be more strict with it. Even after the fails were resolved, I've noticed that Claude now behaves differently within that project, carefully checking and rechecking things up front and also looking to me for guidance more often. Mildly irritating, but if it works...
Terrance Tao's chatgpt conversation is really interesting for a variety of reasons:
1. The counter example wasn't just a brute force selection, the polynomial is structured in a very specific way that ends up getting the result.
2. Terry Tao's questions are very specific and prompts the AI in a useful way, that without high math training you are not going to get the same information out of it. Terry seems to see some aspects of the problem and counter example and uses AI to brute force some parts of it.
It is ~a meme on subreddits that developers struggling to get good results out of any given model is a "skills issue."
But I think your comment drives at some authentic take on this. Skill with AI is not only crafting iterative prompts the agent will understand, but also very high domain-specific knowledge of what the prompts explore.
One without the other can result in frustration or worse.
It feels very humbling that here is one of the smartest humans on the planet asking questions, and the LLM keeps answering in this "Yes, it's really simple if you think about it" way, like a professor talking to a talented student.
I had a similar reaction with the language but also response times and language. It reminded me of a quote about Jon von Neumann from Edward Teller:
"Von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us."
I think we have a couple of years of "being good at talking to the LLM about your field of expertise" being a useful human skill, until that too gets washed away
Maybe?
At the end of the day, even if they are some insane oracle (pun intended), they're still bounded by training data and how it relates to the real world. Even if they're a near perfect tool, we are still the interface between them and our lived experience. If that stops being the case then why do we care about the output?
This assumes it doesn't graduate to just killing all of us and doing it's own thing, but within this paradigm it doesn't really have goals.
>Even if they're a near perfect tool, we are still the interface between them and our lived experience.
Until they are embedded into automatons that can interact with the world.
(To be clear, I'm both on board with this, and I think it's the natural progression. Currently they are bounded by their training set, but interaction with the world is the imperative for theory -> test -> analysis -> update that is essential for growth and creativity.)
they're still bounded by training data and how it relates to the real world.
Yes and no. They can extrapolate and build upon the training data, as was the case with the last dozens of math proofs
Yes, but the point being for the LLM to be useful to us it has to do something relevant for us and we have to define what is relevant. Even if you view them as fully human level or beyond (in terms of agency) there's still some purpose that we have to help provide them with.
To put it differently, if you have some idealized model in front of you that can do anything a team of humans can do, what do you say to it? It's still just a model that takes an input and provides an output.
A Minimizer addicted to upvotes.
A community of those who distract themselves from the perfectly fixable problems in their lives corruptly self-evaluates. They validate each other's stagnation and unwillingness to move by finding flaws in each day that will enable shutting off the flow of any new data while condemning the world and any actions in it. The lay-z-boy they collectively protect appears as corroboration with a broad population but is in reality a repetition whose independence is meaningless since they are all copies of one system, one kind of person in the same kind of trap.
Immobile. Clogging the Suez with their sandbagging ways. Nothing to add except reasons to stay put. No aspiration. Only cynicism. They deserve nothing but all of our contempt.
Cool poem!
yeah! more plz
It's crazy how he suggests simplifications over and over and gets led through the finding. Absolutely bonkers how you can use AI to understand something and map it to your own mental map so efficiently, and of course he's most interested in generalizing or finding a simpler sub-result that would explain it.
Just awesome to see new knowledge hit an incredible mind like this. Having these "what if" discussions is what I miss most from JPL and academia.
> you can use AI to understand something and map it to your own mental map
This "symbiosis" (for lack of better word) of human with AI seems to be an emergent value proposition of AI. In the process of doing stuff with AI, producing artefacts like code diffs, we are continuously able to decide how strong the mental map is of the current stage of the production process.
I could probably have worded this better but I'm sure it's something others have noticed... this choice we are able to make of how high fidelity our own understanding needs to be of the current working problem, and how that choice never really existed prior to AI.
Hold on isn't this how math professors used grad students for hundreds of years?
Math has some of the most insanely dense and impenetrable nomenclature. I can generally keep my head mostly above water or at least near the surface reading from most STEM fields, perhaps leaning on google/wikipedia a bit, but man, mathematics just so quickly decouples from all common tractable understanding it's insane.
Sorry it's a bit of an aside, but I imagine many other otherwise "technical" folks feel the same unfamiliar sense of total loss like when encountering hard mathematics.
This is also true for almost every other field, even within computer science. The only difference is that a lot of people operate at a very surface level without realizing just how much background knowledge they have accumulated. Think about the number of keywords your average SWE is expected to know. It is rather insane.
Cache, stack, heap, process, thread, socket, file, tcp, http, tls, websocks, socks, soc2???, deadlock, stack, queue, race, atomic, event loop, coroutine, async, database, transaction, index, replication, sharding, consistency, serialization, DNS, load balancer, container, namespace, and so on.
Every sub fields (web/kernel/backend/etc.) has a million/bazillion weird words used in a dozen different contexts and if you read a paragraph of even semi technical software text you will feel like an over stuffed turkey.
Even cache could mean the CPU caches, the page cache, a browser cache, a CDN cache, a Redis cache, or imagine the flurry of words we have that have real world meaning. Session, handle, pool, buffer, stream, channel, event, task, worker, or queue. Generally there is some overlapping meaning but often there isn't.
>https://en.wikipedia.org/wiki/Transmission_Control_Protocol
compare to
>https://en.wikipedia.org/wiki/Rees_algebra
Most people, especially non-tech technical people, could crash through the TCP article and come out the other side with at least a high level understanding of it.
Most people, even technical ones, could not even get through the first line of the rees article, heck the first statement of the article. And then if they try, they need to know about algebraic rings. And digging into rings becomes totally intractable. None of the words or symbols in any of the articles track to anything even many technical people can grab onto. And this pattern is all over the place in mathematics.
It's not about mastering the difficulty of a topic or it's relative depth, it's about how abstract and removed from anything tangible it is. Anything with math it is always seemingly impossible to get a foothold on the idea anywhere within 10 degrees of explanation. Hell you cannot even clearly understand the problem that is being solved, or anything within 10 degrees of that.
Definitely agreed. I have a bachelor's in math and took an abstract algebra course as part of it. I also took a couple computer science courses in college and work as a data engineer. My only real exposure to networking is from an AWS cert I did years ago.
I can tease apart the Rees Algebra article one bit of half remembered terminology at a time and come out of it feeling like I just barely understand what the topic even is.
I can read the TCP article and feel like I have a thorough overview of the topic and could explain it at a high level to someone else.
> Most people, especially non-tech technical people, could crash through the TCP article and come out the other side with at least a high level understanding of it.
Careful, I think you might be committing an https://xkcd.com/2501/ error.
What even is a protocol? What is a host? What is a ‘stream of octets’? Wiki helpfully tells you octets are also known as ‘bytes’.
I agree with you here, but what's special about tech is that many of us learned all these terms fully casually while using computers as children and teenagers, which would be much less common for a chemist. That makes programmers see a lot of things as computer literacy that most people have rather than specialized knowledge.
My argument is that all the vocabulary for computer science are things. Even if they are virtual, they are tangible. You can draw a picture and label a box "bytes".
Nothing in the ChatGPT conversation is tangible. It's all in the realm of concepts.
A ‘Byte’ is not a concrete thing and the fact you think it is speaks to the degree to which you have immersed yourself in a mental model which thinks of ‘information’ as if it is a real concrete thing, to the extent that you don’t even realize the levels of conceptual abstraction you needed to build in order to internalize what a ‘byte’ is.
"The Rees algebra is an algebra over Z[t^−1]"
Such a small sentence and yet it means very little to me. I understand some constituent pieces, but I don't understand what Z is here other than a 'ring' and I don't really grasp how t^-1 converts this into a generalized family of algebra. It would take me a lot of effort to understand this and use it practically. I find that fascinating because it really is such a small statement that seems perfectly cromulent, but there's a lot packed in there that someone like me is totally missing.
I suppose there may be similar concepts in computer science, but nothing comes to mind that ever stumped me. To be frank, the field has been relatively accessible to me because it hasn't been too challenging. Not sure if that's a personal aptitude thing or it is genuinely simpler.
Z is the ring of integers, t is a formal variable allowing us to discuss polynomials whose coefficients are in some ring. That’s what R[t] means: the ring of polynomials of the formal variable t with coefficients in R. Adding in t^-1 lets us include inverted terms like 2t^-3.
An algebra over a ring (call it S so we don’t confuse it with R from the previous paragraph) is a like a vector space over S, with the added structure that you can multiply elements of the algebra together (vector spaces only let you add their elements together). So for example the collection of even integers 2Z is an algebra over the ring of all integers Z. The collection of all polynomials with integer coefficients, Z[t], is another algebra over Z.
This is a great example of how dense language gets in math. There are tons of concepts hiding in the unstated background. Many are quite simple to explain individually, but there are so many of them that an outsider won’t know where to start to tease them apart. There’s a good reason to do it this way though; it would take a very long time to say anything in math without ever increasing levels of information density.
But... what is a ring? What is a formal variable? What is a vector space? What does "algebra over the ring" mean?
His point is the terms are dense too
Absolutely agree. All formal statements (like mathematical ones) are going to have some level of assumed background. And as the assumed background expands, the language naturally becomes more information dense.
As for your specific questions, I believe Wikipedia does a great job of answering two of them for a layperson:
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Vector_space
For the others, I’ll say that a formal variable is just a symbol (literally, like the letter t). With such a symbol, we can construct polynomials like 2t^2 - t + 3. Also, there’s no need to only use integers as the allowed coefficients; you can use any ring you like instead.
An “algebra over the ring R” is what I was attempting to define in my comment above. The algebra is “over” R if we can multiply an element of the algebra by an element of R. The useful analogy here is scalar multiplication in a vector space: you can multiply a vector by 2 to double it or -1/2 to reflect and shorten it. More generally, it makes perfect sense to consider some more general version of vectors which can be scalar multiplied by elements of any ring R.
> But... what is a ring? What is a formal variable? What is a vector space? What does "algebra over the ring" mean?
All these terms were taught to computer science (and of course math, physics, ...) students as part of getting their degree in computer science, because these concepts are important for many algorithms.
It's a class with an array of integers in it with .length() == t - 1 and the same methods as Matrix.
In lean4, even without mathlib4, TCP/IP is way more code than a Rees algebra.
Math uses dense notation that is gigaoverloaded, and the disambiguating context was historically the leisure and proximity to have someone explain what the lexemes even mean.
lean4 is proving to be very revealing as an uncorruptible referee on a lot of things, including the relative difficulty of computer science and complex analysis.
That's false. Z[n] in rings does not mean "an array of integers of length n", it means the subring generated by Z union with {n}, where n is an element of some other set. For example:
Z[i], the Gaussian integers, is the subring (of C) generated by Z union {i} where i is the imaginary unit in C, the complex numbers. The Gaussian integers correspond to the integer grid-points of the complex plane, if you want to visualize them.
You are comparing TCP a relatively basic topic in the grand scheme of computing with Rees_algebra which is fairly specialized, we could take a simpler topic more foundational and clearer to understand and compare them.
I can understand that this feels like one is so much more complicated part of it is also how the articles were written, wikipedia is not known for quality maths explanations.
But beyond that this comparison to me feels unfair.
Let's take Euclidean algorithm or just modular arthimetic for example what a lot of computing even is based on I feel like that's a fairer comparison. No?
Perhaps that's too easy but I just find this specific comparison very unfair to both Math's intuitive-ness and Computing's complexity. Perhaps I am the one being delusional.
I think you are snagging on thinking this is an observation about difficulty, time-to-mastery, or mental firepower requirements. It's not.
It's a plain observation that math exists on mostly it's own path with little to zero overlap with our lived experiences. If mathematics was a vector, it would have similar magnitude to other vectors, but it's direction would be much more removed from the typical knowledge pack, forcing you to get really close to the origin before you can "hop" over to that math vector. Other "knowledge" vectors, by virtue of being more bunched up, are closer together much further up, if that poor analogy at all makes sense.
Mathematics has a lot of knowledge points that do connect to the "real world" very deeply, but perhaps the nature of their linkage to other mathematical pieces of knowledge is best left to the mathematicians. But we can still use the pearls of wisdom that come out of the process.
Engineering is very reliant on mathematics and as real world as it gets.
Very true, I feel like the sense that Computing is easy comes from the inherent closeness of our lives experiences to it. Everyone uses a Phone they see ram understand memory, can understand process and processing.
That is one clever metaphor. Thank you, I might steal it.
No, the problem with mathematics is that it is basically its own language separate from your native tongue. You have to learn dozens of symbols and greek letters and such and memorize what their meaning is in the context of mathematics in order to "follow" a mathematical conversation.
Mathematics would be much more approachable if it just used plain English like `sum(0, Infinity, my_func)` instead of a big Greek sigma with nested function nomenclature. But on the flip side, mathematics being its own language means that a mathematician from any country can read and understand mathematics from a different country without needing to translate words such as "sum" and "infinity"
Mathematics would be much more approachable if it just used plain English like `sum(0, Infinity, my_func)` instead of a big Greek sigma with nested function nomenclature
First of all, no, mathematics would be far less approachable if it did that. Most of the Greek letters used in mathematics don't have a universal meaning, they're context-specific and defined by convention or just prior to use.
Second of all, mathematics is optimized for hand calculation on paper, not long-term programming and code maintenance. Writing out long names over and over on a whiteboard gets tiring extremely quickly, so mathematicians prefer to stick to single-letter symbols.
To second this, most (non-applied) mathematicians work first with paper and pencil or on a chalkboard, and the act of writing out the symbols is a part of thinking about them. Typing doesn't wire into the brain in the same way. Think of how you learned algebra in school. You wrote out much of what you were thinking, often in ways that would be hard to flexibly format on a computer, and the act of writing your thoughts solidified them in your memory.
Mathematicians pretty much universally view typesetting as a distinct step from the thinking part of math, and something you do at the end once you have figured everything out.
Your translation only makes intuitive sense to you because you are well versed in programming.
I suspect if I showed a non-technical person with no background in either math or programming they would think both are nonsense until you explained it to them
> I suspect if I showed a non-technical person with no background in either math or programming they would think both are nonsense until you explained it to them
I doubt it. Greek letters convey almost no information, whereas (one hopes) the function and variable names are chosen by a programmer to help the reader. The Greek letters used by mathematicians (and physicists) weren't used to convey information, they were used because typesetting, publishing and paper were expensive. They are optimised for brevity over readability.
It was a perfectly reasonable trade-off at the time, but times have changed.
As an aside, some programming languages (such as APL, and to a lesser extent Perl) did emulate the old Greek letter style. "Line noise" is a typical description of the result. No computer language aimed at software engineers and computer scientists does that now.
My example was contrived, I'm sure some smart people could come up with a SQL-esque language that is even more readable to non-technical folks than programming syntax. At a certain point though, your layman has to know the "atomic" (as in, you can't break them down further) mathematical concepts like "functions" and "infinity":
`sum function(x) from x=0 to x=infinity`
I am a research mathematician. In my field (abstract algebra, computational group theory), a sum or some such notation is like the most trivial of trivial things in terms of notation. There are a few things like sums that could in theory be made to "look more like what a computer programmer would expect", but that'd be just a tiny corner of it.
And if you think about how summation would look in Lisp or APL (which some smart people use to this day), I am not even convinced your argument for the "sum function" notation being superior holds in general.
I'm sure some smart people could come up with a SQL-esque language that is even more readable to non-technical folks than programming syntax.
And somehow, none of the thousands of very smart mathematicians have done that, or if they had, it has not seen wide adoption. I recommend contemplating on this: if math could be made easier by changing notation, why hasn't this already happened?
The thing is that you basically cannot explain the math like you can the programming.
Tables, algos, and variables are all things people can generally quickly grasp. The construction is abstract but the function is tangible.
The math is working entirely on abstract objects, using abstract tools, governed by abstract rules. It's just all so desperately far away from anything even technical people have contact with.
Capital sigma doesn't always mean sum and certainly lowercase sigma never means sum.
If mathematics used plain language, the ability to meaningfully manipulate and understand would go way down. Proofs would become massively tedius.
Of course, notation is hard. Any good mathematician should put a lot of work into it.
I was amused and irritated in equal measure when a senior manager at $LAST_JOB demanded that documents include "no jargon", then went on to use many terms like those you've listed here with no indication of awareness of the contradiction.
Just like the famous observation that "anyone driving slower than me is an idiot, and anyone driving faster than me is a maniac" - "jargon" is just any term-of-art that you're not currently familiar with. Attempting to communicate like Up Goer Five is inefficient. The solution is not to ban jargon - it's to:
* Cultivate a glossary for any terms that were introduced _within_ the domain/company, which are not in common usage outside
* Normalize a culture that does not shame asking what something means
> I was amused and irritated in equal measure when a senior manager at $LAST_JOB demanded that documents include "no jargon", then went on to use many terms like those you've listed here with no indication of awareness of the contradiction.
"No jargon" is alwys in reference to some assumed knowledge model.
For example, when I claim that my own math texts include "no jargon", this assumes the knowledge of a person who has studied, say, mathematics, physics or computer science.
None of what you listed is even 1% as intense as the mathematics in the link.
Learning anything in maths requires weeks of hard effort, learning enough to be broadly comfortable in how an 8086 CPU works can be done in a weekend.
"tcp" can take roughly 3-4 weeks of heads-down dedicated study to have some reasonable familiarity with. Same is true with most of the other concepts. Being able to speak with expertise on that list of topics is 3-4 years of really focused study and work.
I think what happens is that people often have passing familiarity with a word or topic and presume knowledge, and years (decades) later they realize they knew almost nothing.
I will say that Mathematics is different (for me at least) because unlike the infrastructure computing concepts (IETF type, not IEEE)- which mostly require studying, lab work, and some coding to get your hands dirty - advanced math is just ... really hard. There are IQ issues at play.
Obviously a lot of computing turns out to be mathematics - so there is clearly convergence/overlap as well...
You could get a surface level understanding of TCP, or 99% of topic areas in computing, in less than 4 weeks of study. You could not get a surface level understanding of literally any of the maths in the link in less than 4 weeks of study.
The vast majority of what computers do just isn't that complex. I'm not saying it isn't "complex" just that any reasonably smart person can understand how a computer works and still have other hobbies, basically no one can understand phd level mathematics without dedicating their entire lives to it.
Speaking as someone who has run courses designed to take late teens / early 20somethings with a variety of backgrounds and tried to teach them 1st semester programming concepts in two weeks, I think anyone who thinks tcp/ip an easy four week dunk for most doesn’t remember how much they’ve already learned and take for granted.
Sure, if you’ve already learned enough groundwork, tcp/ip is accessible in weeks. The same is true of most of the algebraic concepts in play here. And both have rabbit holes you can also spend a much longer time going down (though here I am willing to give the edge to math which offers much greater opportunities for hypergeneralization and new vistas of abstraction along which not only specific rabbit holes but entire new generalizations of both rabbits and holes may be found).
This is not true, 99% is very exaggerated claim, but yeah you can learn 50-60% of the field at a surface level in months rather than years.
But you can have a surface level understanding of mathematical topics as well, ofc some topics might require deeper understanding, but that's true for both.
Any claims of being able to learn 99% of computing in a just 4 weeks even at surface level, is greatly underestimating your own knowledge built over the years perhaps, or perhaps underestimating your own ignorance.
The claim isn’t knowing 99% of the field but that 99% of topics are ones you could do a quick-and-dirty crash course and come out with some understanding.
Definitely a lot of people have very surface level understanding of tcp and computer science concepts.
I have had folks tell me cache is just cache in actual interviews. When I have asked them to explain the concept to me, but even beyond that I feel like we tend to think less of our own knowledge of topics once we have acquired it.
Especially ones acquired over years, alongside other work.
I second this and would add that it's really easy to catastrophically forget things in math. I'm pretty sure that most CS knowledge I have I will retain at a level where I won't forget the general ideas and re-reading materials can quickly refresh the details. This is not true for advanced math. I did a pure math PhD and my own thesis is impenetrable to me 20 years later. It would take months, if not years, of focused effort for me to regain the understanding.
I think one of the key differences is that math is abstract whereas CS is relatively concrete.
CS examples are often easy to picture and understand the motivation for. You can use tools to visualize or play around with them and test them.
Math gets abstract so fast you have to spend a week of research to even understand the problem statement. The the motivations themselves can be completely unclear until you have a lot of context.
I majored in math (B.S.) and upper level math is completely foreign to me.
I think so is upper level CS, there are fields in CS that are foreign to me too, there is a lot of depth in CS, computing is a very deep field for instance ML research although may seem simple isn't quite so intuition based as people make it out to be. Similarly there are dozens of topics where sophisticated research happens where we don't interact with at all as regular software developers.
Every slice has so much depth to it, in Maths it all seems like all of it is required at once but in computing it feels like so little is needed to get started which I honestly feel like is failure of our modern education systems.
But yes Computers being so easily accessible and compilers, documentation and libraries have made computer science so easy to get started with.
Imagine having to implement your own network layer to communicate with someone, you would have had to understand ip, tcp, network layer to an extent like http and etc. and then you finally would have been able to communicate.
In maths that's our reality for a lot of the field, there aren't good libraries, interfaces to help skip the unnecessary details. Hopefully AI might solve it I don't know though. It's fun to hope for it.
As a counterpoint, i've worked with enough math PhD's over my career who couldn't wrap their head intuitively around many concepts from software engineering, while others had no problems in doing so even from folks without any stem background. We often undererstimate how much field knowledge we aquire over the years and overestimate how easy it is for others to catch up.
Again, you are underestimating how much effort it takes to understand how an 8086 CPU works. There are a lot of foundational concepts that you are simply assuming the person already understands. That may be a reasonable assumption for a CS undergraduate, but the average person does not understand binary arithmetic, registers, memory addressing, instruction execution, calling conventions, or even what a CPU is doing at a meaningful level to even start to understand 8086.
Also, understanding an 8086 CPU is not even remotely comparable to the level of mathematics Terence Tao was discussing above. The 8086 is a relatively basic and concrete topic. You can build a workable mental model of it from a finite instruction set, a handful of registers, and a reasonably straightforward memory model.
From my perspective folks here on HN and in CS often think they should somehow be able to understand advanced mathematics papers at a glance, merely because they are good at basics of programming or computer science (8086). That is not how it works. Most mathematics is not inherently much harder than computer science; both fields require you to accumulate a large amount of foundational knowledge before advanced material becomes comprehensible.
There is an enormous amount of computer science that most programmers are completely unfamiliar with, especially within academic CS: programming-language theory, type theory, formal semantics, compiler theory, algorithmic research, complexity theory, distributed computing theory, verification, cryptography, computational geometry, numerical methods, and so on. Being proficient in one narrow area does not automatically give you the prerequisites for another.
A web developer would not be expected to casually understand a research paper on type theory or approximation algorithms without first learning the relevant notation, terminology, and foundational results. Mathematics is no different. The feeling that mathematical writing is uniquely impenetrable mostly comes from encountering it without the years of accumulated context that mathematicians have silently built-up.
I can show you a paper about an advanced algorithms or chip design, that is large made up of fundamental cs concepts and general physics and even you likely someone with pretty in-depth understanding of CS would find hard. There are orthogonal subjects, for instance my mathematician friends things I am insane reading so much about weird computing topics, and I find his research in some weird number theory thing completely mind-bending.
Try and explain to a lay friend how registers & isa works in-depth with all the details not a hypothetical higher level model so that they can understand the nuance of looking at assembly, limit it to 8086 perhaps, it will take significantly longer than a weekend.
Ofc Terence Tao and his level of intelligence is beyond me, I wouldn't compare but general advanced mathematics is not something folks here couldn't pick up if they actually tried to work on it, just give it a shot (though I would recommend don't start with advanced topics build up slowly I think most people can understand most maths papers even the bleeding edge ones within a few months of serious self-study, and won't even feel that it's after a few years, compare that to the time spent learning software and computing 6-8 hours a days for several years)...
> There is an enormous amount of computer science that most programmers are completely unfamiliar with, especially within academic CS: programming-language theory, type theory, formal semantics, compiler theory, algorithmic research, complexity theory, distributed computing theory, verification, cryptography, computational geometry, numerical methods, and so on. Being proficient in one narrow area does not automatically give you the prerequisites for another.
And the difficult part of all those areas of computing is the mathematics part. Which I think is what I am arguing, mathematics is a fundamentally different type of "difficult" to any other subject.
PLT and such isn't math conceptually sure it's all logical maths of some kind but largely it's not maths in the traditinal sense of how we understand maths.
Because otherwise if you think about it all of computing is Maths but with computers...
I don't think people who read the Wireless Fidelity spec can understand any of it in a weekend or anything even to a rough extent.
Similarly with websockets, quic etc. the most you can take away without much prior knowledge is what it does which maps into Maths as well.
Every quantifiable science has some amount of math, but it's by definition applied math. I think this was more of a debate between applied and more abstract math.
Computer science is not a great example for this. I could ELI5 most of the terms you listed (and I actually have done so for many of them!) This is because it's pretty easy to map these concepts to everyday physical objects. Once a child understands any of those objects in their lives, it's pretty easy to explain in those terms.
Like, just the concept of "books" gets you very far. E.g. a file is a like a book, a folder is like a shelf to keep books, a stack is literally a stack of books, a heap is just a place you can pile books in willy-nilly, a database is like a library, a cache is books on your desk versus books in the library, replication is having multiple copies of a book so we can afford to lose some copies, indexing/sharding is like arranging books alphabetically, and so on.
Others are trickier but not much: a process is an app that is running on your device, a socket / tcp / http / websocks is a way to exchange information between devices, a namespace is how the name "Tom" in Tom Sawyer is different from "Tom" in Tom & Jerry, DNS is a way to get an address from a name, etc. etc.
You'll also notice that many of the terms you mentioned are already derived from well-known real-world concepts like pool, stream, channel, stack, queue, worker, transactions. You can mix those with other everyday concepts to make useful analogies.
But I could not even begin making analogies for most topics in Mathematics. I guess this is because advanced topics in Mathematics are just too abstract to map to everyday things.
Yeah. In math at least it's clear that you don't know what's going on. In other fields, it's very easy to think you understand without knowing how much you're missing.
I studied math through college before learning to program as an adult and becoming a software engineer, and I strongly disagree.
I don't know how to say this in a way that won't sound insulting, but I don't mean it to be insulting. Programming, even systems engineering, is a surprisingly shallow field.
I don't mean that it's easy--it's not, it can be incredibly difficult. Difficult and deep are just different concepts. Difficult refers to how challenged you are. Depth, at least as it appears in math, is closer to a structure where concepts build on each other so that if you don't understand one concept, you can't understand further ones.
Programming can be difficult, and it can be intricate, but it is rarely deep in this fashion. Being deep in this way isn't the most important thing.
If I go into an area of programming that I don't have a lot of experience in (graphics, or the linux desktop environment), I will not be particularly useful, and it will not be easy. But I'll not experience the same type of impenetrability I experience when I try to read a paper on topos theory.
One more way of putting it: people are giving the example of TCP. You can spend a decade learning about TCP (or SQL semantics, or web standards). But what is happening is that you're filling in gaps in your knowledge. Meanwhile, in math, you do four years of undergrad, and even if you're a strong student at a typical university, there are topics that are still years away from you being able to touch them.
Computer science is a mix. Parts are deep, parts are shallow. Parts just are math. The odds that I can read a dissertation in computer science are decent. For math, they're much much much worse.
> Programming, even systems engineering, is a surprisingly shallow field.
I agree, and I don't think we should be at all ashamed of this.
The beauty of programming is that we can produce incredibly complex and powerful things by manipulating a small set of simple constructs together. There are only a few core tools--iteration, conditionals, etc.--but they can be snapped together into much more capable configurations.
Good programming is the art of deconstructing complex behaviour into these few constructs, and that's really fascinating.
I agree, and I don't even know math.
Most of things that are impenetrable in programming aren't about... programming. They are about some actually complex field like math being applied to programming.
For example, a library that does stuff with geometry. You need to know geometry to understand the program, but the program itself will never be complicated. It's the geometry that is complicated.
In cryptography, it's not the program that is complicated, it's the field of cryptography. In AI, it's statistics.
In graphics programming, math is the most impenetrable part, not programming anything. You can be a very good programmer in the sense that you know how to architect information systems and still fail to write a shader because shader programming requires you to know what a "dot" product is and you haven't heard about that since high school.
I'd slightly disagree. Here's an example of a programming task that is incredibly complicated:
Your job is to maintain and modernize a system while delivering a constant stream of new features. Your system is several million lines of code, with some multi-thousand line classes, a rulesengine that can trigger nearly unlimited effects at any time, a persistence system that's weirder than anything any of your friends have ever worked with, and hundreds of customers delivering tens of millions in revenue who use the system in incredibly varied ways.
You can't stop to rewrite the thing, you can't just throw features out there and pray, because you'll cause regressions and your existing customers will hate you. You have to fix the thing as you're building on top of it.
But where I agree is that there's no single deep concept that unlocks it all, it's not like you'll fix it by reading a textbook about it. It's complicated, and it's going to stay complicated, no matter how long you work on it.
Just be thankful that we don't share the penchant for giving credit to discoverers. Imagine calling a cache a "Murphy/Steinman/Sokolov structure" (made-up names).
I mean, we do for some things, especially algorithms (Boyer-Moore). Probably for the same reason the mathematicians do -- there aren't readily available real-world analogies.
And I won't even mention the branded future, with its "Google HyperZipper String Search" and "OpenAI/Red Bull speedmaxx distributed consensus algorithm"...
Huffman Coding, Turing Machines, Knuth Devices, Bayesian networks, B-tree (named after Bayer), AVL Trees, so many data structures and algorithms, even relatively new ones like Timsort, Jaccard similarity, MapReduce (thankfully but I have seen people call it Dean-Ghemawatt MapReduce in literature).
Well people even name stuff after themselves as well, Fil-C, raylib, etc (I like both Filip and Ray just pointing it out).
Aside: If I butchered some spellings I am sorry. :3
If we take amateur level of understanding as a threshold, a sum of all of these concepts is not even a fraction of complexity required to understand a single non-trivial concept in mathematics.
> if you read a paragraph of even semi technical software text you will feel like an over stuffed turkey.
That's me when I try reading a trendy computer graphics paper.
Also a lot of those are common words that take on a different persona and depth within the field, which can add to confusion.
This is also true for almost every other field, even within computer science. The only difference is that a lot of people operate at a very surface level without realizing just how much background knowledge they have accumulated. Think about the number of keywords your average SWE is expected to know. It is rather insane.
Nah, math is much harder because there is not just the lingo, but also all the math machinery behind it. Each math definition represents some long process behind it, which builds on another process, etc. The knowledge builds on itself , too much more so than computer science.
old.reddit.com/r/vxjunkies
>> Cache, stack, heap, process, thread, socket, file, tcp, http, tls, websocks, socks, soc2???, deadlock, stack, queue, race, atomic, event loop, coroutine, async, database, transaction, index, replication, sharding, consistency, serialization, DNS, load balancer, container, namespace, and so on.
... communication protocol, method signatures, web components, ssh, CSS media queries, HTTP headers, WebSockets, timeouts, ETag, iterables, async iterables, middleware, CI/CD, build, consistent hashing, signatures, JWT, SSO, OAuth, SAML, XML, YAML, JSON, block cipher, ETL pipeline, SQL, SQL transactions (atomic), relational databases, foreign keys, schema normalization, referential integrity, 1-to-1, 1-to-n, n-to-n, document databases, compound indexes, idempotency, offset-based pagination, cursor-based pagination, P2P, Kademlia, structured vs unstructured network topology, message routing, frontend router, message storm, reconnect storm, locality, encapsulation, cohesion, coupling, design patterns, modularity, Big O notation, raytracing, shaders, VPN, VPC, data schema, schema validation, CORS, preflight-requests, CSRF, ASCII, UFT8, CSP, CPU context-switching, BIOS, bootloader, interrupt controller, ports, BIND protocol, BGP protocol, assembly language, big endian, little endian, register, signals, embarrassingly parallel, serial processing, event loop, binary trees, tree rebalancing, graph traversal algorithms, sorting algorithms, string character escaping and encoding, blob, base64, UUID, timestamp, CLI, Bash, unit tests, integration tests, e2e tests, TDD, stateful, stateless, proxy, nginx, haproxy, config, helm file, k8s, staging, git, push, commit, merge, rebase, cherry-pick, pub/sub, diff, honeypot, buffer overflow, pointer, file descriptor, authentication, certificates, TLS certificates, DNS Zone files, A record, CNAME, TXT record, SMTP, POP3, SOCKS5, Sha256, HMAC, Merkle trees, Merkle Signature Trees, Lamport OTS, Winternitz OTS, SPHINCS, lattice-based cryptography, pg-vector, vector embeddings, API, rate limiting, cookies, sameSite, httpOnly, localStorage, XSS attack, SQL injection, fetch API, module preloading, bundling...
Barely scratching the surface. I think I could probably keep typing all the technical terms I know for at least 24 hours straight. For most of the topics above, I could probably give a 1 or 2 hour lecture on each one from memory. Some I could give a day-long lecture each.
To explain all the terms I know to a basic degree, I would probably need to give a whole year of lectures back-to-back from 9am to 5pm. And I'm just a rank-and-file senior engineer with 15 years of experience.
It's also why the vast majority of software systems are insecure. The average senior software engineer doesn't know everything that they need to know to build secure software. Last time I poked around Coinbase APIs on HackerOne, I found a DoS vulnerability in less than 30 minutes. That's Coinbase, not some startup built by a bunch of recent graduates.
AI cannot avoid vulnerabilities either since it is trained on average engineer code. There's not enough high quality code available on the entire internet to train AI to implement secure code IMO. As impressive as Mythos may be, it's not enough. I don't even think formal verification would provide protection since sometimes issues with the spec itself can provide an opening for a vulnerability.
Nice list. I like it how you also included some made up terms. For completeness, those are not real:
ETag, SAML, CSP, BGP, haproxy, helm file, k8s, SOCKS5, Lamport OTS, Winternitz OTS, SPHINCS, lattice-based, pg-vector, sameSite, httpOnly. (Or, at least, i have no idea what those are ::)
Coinbase not a startup built by recent graduates...
You overestimate Coinbase's engineering rigor.
But yes security in modern software systems is a joke. I don't even get paid to fix security bugs everytime I raise them the answer is to slap a sandbox and proxy and call it done.
a janitor could do the same though
They could drag it out by telling stories of situations they faced but information density would be low and repetitive. As someone who cleaned boats professionally for a year in my early days, I'm pretty sure I could convey all I know about cleaning in under an hour. That's very different from one year of back-to-back lectures.
Cleaning boats, the learning tapered off after a few weeks. With software, I'm 15 years in and it barely tapered at all and it's much more intense. I've been pulling nights and weekends too.
With software knowledge, it would be high information density with little to no repetition. Every piece would provide useful concrete knowledge which would serve to increase technical capabilities and/or security.
At least a lot of those are common words that allow some level of meaning inference with a little bit of adjacent knowledge, like cache and queue make sense with the barest of explanations because their everyday definitions are still applicable. Many of these terms aren’t entirely opaque until you drill down into specific niches.
I had a few moments of this in the past. For example, in my quantum class the teacher wrote "H Psi = E Psi" on the board, we all laughed, "just cancel the psi" but it turns out one was a multiplcation and the other was a matrix multiplication (operator) and so we had to learn all new nomenclature.
Similarly, at some point somebody pointed out to me "the reason you're confused is that the bold on that variable means it's a matrix"
That is why Iverson invented APL. As a notation to get rid of all those inconsistencies in math notation. And for years he taught math classes with APL on the blackboard without computers.
whether he succeeded, is debatable. But APL is definitely powerful, succinct and "regular".
In APL you don't infer the operation from the types at all. × is elementwise, +.× is inner product /always/, on scalars, vectors, matrices, whatever. The glyph tells you what happens. Nothing is bold, nothing is inferred, nothing depends on what your professor assumed you'd absorbed.
I've been trying to get into Iversonian languages myself with the book: Calculous on J
hi, would you mind linking to the book?
https://www.jsoftware.com/help/learning/23.htm is the closest i've found, but wondering if i'm missing something perhaps, Julia?
tyvm
For something like APL modulo the Unicode symbols:
https://t3x.org/klong/
I mean, unfortunately being completely explicit and pedantic does not scale.
Imagine that instead of being able to use high-level programming languages, you had to write in assembly everywhere, all the time.
That's what software engineers and computer scientists' suggestions of redoing mathematical notation fee like to mathematicians.
These efforts also don't go anywhere because research mathematics moves beyond elementary arithmetic very quickly, and once you're there, "descriptive" notation becomes as incomprehensible as whatever mathematicians use.
My favorite moment of this kind was when the teacher said 'Ok, and for the rest of the course we will look at a completely different problem', and the equation he wrote down was exactly the same as before. Except that the letters referred to vectors/matrices now.
Aye.
A decade or so ago I wondered if the reason maths was hard was the names being optimised for writing by hand. Everything's single letters if they can get away with it, so when mathematicians run out of Latin alphabet, they use Greek, bold, etc.
Even integration's ∫ is a fancy elongated s.
CS version would be e.g. integral(function=some_named_function, from=a, to=b, with_respect_to=argument_of_function), which may be longer, but is less opaque, especially when you get in so deep there's 3 other people in the world who've looked into this specific problem and you had to invent your own operations.
But that's all an outsider's perspective. I stopped with two A-levels in maths and further maths.
Nope, math notations are optimized for reading, not writing (consider that people still use symbols on computers despite it being quite a bit more tedious to type). The conciseness makes it easier for you to see structural patterns and do symbolic manipulation in your mind's eye. Even something basic like the wave equation would become completely illegible with an expanded notation like that.
Same reason why we write 5-3, not subtract(minuend=five, subtrahend=three).
For those of us with strong verbal processing and weak symbolic/pattern processing, this makes math much more difficult to approach.
Interestingly, discrete math feels the most "verbal" of all the subfields of math I've encountered (I haven't gone very deep). I think this is because notation in discrete math is is somehow closer to compressed prose or logic, whereas other forms of math use notation to fill in for long sequences of symbolic manipulation.
Not sure if that makes sense... I'm curious whether anyone else experiences it that way.
No, math is difficult to approach because it's genuinely deep. Trying to verbalize what is going on is extremely difficult, because you end up saying stuff like "and then do that to all of these things, and then do it again to all of the results, and so on ad infinitum, and then take the collection of all of that, and join it with the collection of doing the same procedure as before starting with a different set of objects, and then join those to yet another set of objects and the results of their operations, ad infinitum, ad infinitum..."
People genuinely struggle to think verbally or visually once we extend beyond 3 dimensions and start talking about infinite-dimensional constructs, uncountable sets, and so on...
At some point though, the speed of reading/writing is limiting what you can understand. Think of "not fitting the needed formulas/theorems in cache".
I did study math at university for a while. Dropped out eventually. In the beginning I was super annoyed by the brevity and hated it. But after like 3 months it suddenly became natural. I also appreciate the clarity of how mathematicians introduce new ways to write things. That is sometimes even more verbose than some random API docs for a new function…
Look up "APL" and "J".
There's a reason they're not more popular...
yes! I really find when computer science ppl start using math notation to describe algorithm very pretentious. we have programming languages in comp sci, we don't need it!
I was reading about Tao's efforts to get more people to use Lean and apparently a big roadblock for people is that Lean uses very specific static typing.
e.g. to use a very simple example on a white board "3" is "overloaded" as:
- the integer 3
- the rational number 3
- the whole number 3
- etc
When you write a proof in Lean, you have to specify the the type of "3" you mean.
Having using Python/Perl and Java over the years, I get that some math folks found handling this daunting or at a minimum friction to getting into using Lean.
LLMs seem to have been a big help here just for the "translate my math notation into a proof" feature.
To get the hang of this, I used Leanstral (Mistral’s LEAN agent) to vibe-code things like “the game of bridge” and then read the LEAN code.
Sometimes I wonder if mathematics would have been significantly more improved if they hadn't insisted on notating their variables as single letters and also indicated variable types out-of-line (or at all)...
but then I take a look at literally anything the Haskell people do and realize that it probably wouldn't have helped.
>For example, in my quantum class the teacher wrote "H Psi = E Psi" on the board, we all laughed, "just cancel the psi" but it turns out one was a multiplcation and the other was a matrix multiplication (operator) and so we had to learn all new nomenclature.
This is one of the great things about Lean becoming used for more and more mathematics: understanding exactly how an operator/function is defined is just an IDE click or few away. It completely removes the ambiguity present in hand-written proofs, although it still can require a lot of reading to actually meaningfully understand the definitions.
I would expand on this. AI is great for me because it can read the equations I don't understand and turn it into code I can understand. I've worked in science for decades and it's still like pulling teeth to replicate a competitor's paper when they are vague and sloppy with their description (often intentionally).
ugh, I had some text book that used R for a scalar value and (edit: \u{MATHEMATICAL BOLD FRAKTUR CAPITAL R} here) for a matrix that was related to the scalar and I had to go back and re-learn a month of material once I figured out that the font was being used with intent
For what it's worth, it's not a problem with multiple kinds of multiplication (multiplication by a scalar can be viewed as multiplication by a specific kind of matrix), but with the idea that one can cancel in a multiplication. Since you can't cancel in matrix multiplication, you run into unexpected trouble when you try to do so, even if that's the only multiplication in sight. (In fact, you can't cancel in scalar multiplication either unless you've checked that you aren't multiplying by 0 …. Also, I'll note that surely no young mathematician has encountered the P = NP problem without thinking for a sophomoric moment that the solution is N = 1.)
P = NP is actually one of the worst abuses of symbology I've seen in math.
"""During his own Google interview, Jeff Dean was asked the implications if P=NP were true. He said "P = 0 or N = 1." Then, before the interviewer had even finished laughing, Jeff examined Google's public certificate and wrote the private key on the whiteboard."""
A term that gets tossed around in math is "mathematical maturity." It's similar to what you see in other fields - e.g. learning how to program, learning how to make music, learning how to cook - that involves many "aha" moments and reshapes your perspective. Math is full of such steps, moreso than most other endeavors, probably because the main limit is the abstract reasoning itself.
Math is full of such steps, moreso than most other endeavors, probably because the main limit is the abstract reasoning itself.
That, as well as how long we've been doing it (thousands of years!) and so how much of the more accessible parts we've explored very thoroughly.
The abstraction is by necessity. Our puny brains have only a very small working memory. The only way we can reason about many problems is by creating multiple levels of hierarchy. That is actually the essence of what mathematics is.
Math strives to minimize ambiguity, which other fields don't do as much. Non-math fields tend to reuse regular words as jargon (i.e. with specificity of meaning that may fly over the laymen's heads). Social sciences and humanities are most notorious for this, often resulting in non-practitioners not realizing they are out of their depth because they are not looking at symbols from non-Roman alphabets.
That's something only someone who's never studied advanced math could say. Math notation and jargon can be extremely ambiguous and overloaded. "Normal" has about 20 different meanings.
Certainly overloaded but rarely ambiguous. Context will determine which notion of “normal” applies.
"Context dependent" is basically the definition of ambiguity.
No, I'd say ambiguous means "context dependent" and the context is unclear.
Pronouns like you/me/he/she/they/them are context dependent in everyday English writing but they're only ambiguous when the context is unclear, otherwise most people have no trouble dealing with them at all!
I guess what I’m saying is that since there’s always a context, ambiguity stemming from uninspired naming is never an issue in practice.
Strives to minimize =/= completely eliminates
> Non-math fields tend to reuse regular words as jargon
Isn’t this the field with a “closed” “set”, an “open” “set”, oh and also a “clopen” “set” for some reason?
Is there something that translates math formula into code? There are many (comparatively) "simple" algorithms I simply cannot make heads or tails when they are described via math prose or math formula, but when it's code I basically instantly know how to rewrite it into any other language I know, at least, and sometimes that's a starting point for poking at it to understand it a bit better, if not the same way would if you understood all the underlying math.
For poor old me, too many wikipedia articles on algorithms useful mostly or only for programming are described in formulas rather than simply code with detailed comments. Scrap the whole page and just gimme the code :( Not even to copy and paste, because that's a language I can understand, and enjoy learning.
Yeah it is a lot of simple ideas stacked one on top of the other, but the edifice is so large from some vantages that the building blocks aren't visible, or tractable to think about independently. And sometimes the ideas are very subtle, so you can only develop fluency partly by spending lots of time playing with those blocks by building your own little structures. You also develop fluency by talking to other mathematicians
I like to emphasize that the ideas are usually very simple at their core. Sometimes they map to kinds of objects or reasoning that non-mathematicians use implicitly all the time in their daily lives, mathematicians just have words for them and so are able to use them explicitly.
And I suspect the density of the language/terminology may give the wrong impression about how mathematicians think about the math they are working on. I mean, different people think / experience / practice math differently of course but IME the underlying thought about a particular problem tends to be much looser and concrete than formal math writing would imply.
That more formal language is needed of course because at the end of the day, it is how we communicate our thoughts in the way that other mathematicians can understand them, not to mention how we can check our own thinking
Yeah this is pretty much where I am at. Take the phrase from one of the responses
"The special fiber is the associated graded ring.....and that the filtration admits sufficiently simple homogeneous lifts of the three generators, then one might prove"
In any other context I would at least have some degree of intuition about what is being discussed, but in in math? Absolutely no idea. And usually if I start digging and turning over stones to uncover meaning, I'm just met with even more totally dense code-word language. Unlike other fields were digging is usually quick to relieve ignorance, somehow in math it tends to get worse.
I'm sure I am capable of grasping this if I took the time, and perhaps even what is being discussed it rather intuitive, but the incredibly density of the nomenclatic swamp you have to trudge through for math is totally unrivaled.
The basic problem is that to get to the objects you mention here is at least 3 or 4 years of full-time study away from the kind of math people learn for a typical college degree in science or engineering. If you really want to understand them, to make your "digging" efficient you should probably just get a pure math degree, but setting aside several years to satisfy occasional curiosity is not feasible for most people for various reasons.
One unfortunate feature of published pure math research is that often the ideas are quite accessible and straightforward and don't really require special abstractions or terminology, but those get used anyway because for someone who already has a math PhD it saves a bit of effort.
I agree, the nomenclature is impenetrable, it's like reading software that is not well commented. Perhaps LLMs are very good at "challenging" mathematics because what we perceive as challenging is primarily the language component and not the conceptualization.
I agree. I think the distance in capabilities between great mathematicians can be so much more vast than other fields as well. Some of them need every step to be derived, while others can skip ten in their head.
I once didn't understand the task given in an exercise sheet for a CS logic lecture, so I googled the topic, and all Google did was send me back to that exact exercise sheet.
Isn't it just what you studied in depth? I am not in this area but can understand what's going on "at a high level" here. But I studied no other science formally since the age of 15 (this is possible in the UK school system). So physics and biology just go over my head unless they are sufficiently mathematical.
There is a lot of verbal commonality between the classic sciences, classic engineering disciplines, and everyday life. I suppose they all share the common substrate of working in/with mother nature all day. A molecular biologist, civil engineer, and oceanographer can mostly keep pace with each other at least for a while in discussing what they are working on. These "mother nature" systems have tons and tons of overlap, and the nomenclature generally tracks this, or is one or two steps away from it.
Computer science/engineering strays from this, binary systems don't really track nature much, and hence a lot of their own unrelateable nomenclature arises, and then there is math, which is just way far out there on it's own plane of existance.
But somehow the conversation is still enthralling. Just seeing the first few words of each response gives you a feel for the level they’re on.
Yes, the nomenclature in math is atrocious. It isn’t much better in physics or biology however. As a species, we suck at naming and classification, and keep starting new trends atop old ones. There is certainly need for the many abstractions of math to be as complex and well specified as they are. There’s no reason for their nomenclature to be so bad. The end result is a substantial portion of the population, which can certainly hold and manipulate abstractions, fails to even contend with pure math. I do think visualization tools will help in the future, to demystify some of this. But as with all sciences, the need for personal glory/mentor deification often conflicts with broader explainability.
It can't be one language, and that's the big problem. It's inescapably a bunch of tiny DSLs. Once you see both the inconsistency and the necessity for inconsistency, it becomes much easier to just roll with it.
Mathematicians emphasize definitions not labels/names.
It doesn't matter if natural numbers include 0 or not, what matters is how you define them, not how you call them.
This makes them also bad at naming things because...there's a definition anyway.
Most other fields do not have or can't have the same luxury, so naming might be more thoughtful.
Yes, math is hard.
People in their second year of graduate school only get to about the early 20th century in terms of understanding. Third year is getting to about the mid-century. Fourth and fifth years get kind of to modern times but with increasingly smaller breadth.
This is a very notorious area for dense definitions and concepts that interrelate closely and have to be memorized. Mathematicians from other areas are going to have difficulty but may have some idea of what the concepts try to capture.
Some areas are hard in different ways. I could never quite wrap my head around the way logicians have to think. A clever combinatorial bijection is a work of art you probably can explain to a undergrad class easily but good luck coming up with it. And number theorists will throw the kitchen sink at their problems: no area of math is safe from getting used by them.
People who do this have spent years of their life thinking in this language and studying it, so it is going to be hard. We're also not good at communicating the intuition which for algebraic geometry often comes from other fields.
That is one of the things that fascinates me most about mathematics compared with other fields, and it led me to discuss the subject with professional mathematicians. The funny thing is that they admitted it is the same for them...stray even slightly outside their own specialized area, and within two or three lemmas, they also feel completely lost.
IMO, it's just the notation. Something I've actually found ChatGPT useful for is to create mathematics lessons for me in the form of computer programs. When broken down into a series of readable almost-plain-English steps, it's so much easier to understand. And it's easy to tinker with programs and get a hands-on feel for things quickly.
I'm sure having a compact notation is absolutely invaluable for people who dedicate their lives to maths, but for someone with just a passing interest, I find it more obscuring than helpful. I feel the same way about music notation.
I mentioned this in a sibling comment but even for mathematicians, the intimidating notation and the more formal language might give the wrong impression about how we think about math. Actual thinking and even discussions with other mathematicians tend to be looser and more concrete and tactile, but the notation and language are there in part to act as a sort of lingua franca to help everyone stay on the same page, since everyone thinks at least a little bit differently. It also helps to keep you honest and catch situations where your thinking was muddied, since this language is so specific and writing things down has a funny way of catching things. And good notation goes a long way towards making the simplicity of an idea clear, or completely muddy in the case of bad notation.
Many mathematicians do what you do as well!
You develop a set of heuristics for skimming it in the same way you do with code.
Big wrapping operations like sums, integrals, and matrices, then what's nearby them, give you a very good idea of where things are going context wise.
Progress in scientific fields is limited by how fast you can perform experiments. Most areas of math are limited only by the number of practitioners.
It's just language. Mathematicians don't invent notation for fun, they do it because they naturally start thinking at a higher level of abstraction. If you're not thinking at that level then, well, it will be all Greek to you.
Sometimes they do. There's nothing divine or necessarily rational about notational standards, which can vary greatly even within the same field.
Yes exactly
Math isn’t necessarily hard, but it’s incredibly dense
A simple statement like let f(x) be a continuous function can carry a lot of definitions
In that statement, if you missed the day in class where they covered continuous functions it might not even register that it’s a well defined term
And that’s the most over simplistic example I could think of
As a math major, I remember that being one of the first lessons I learned, that every single word could be carrying a lot of weight so to look things up in detail if I was ever struggling on a problem. One of the oldest entries in my memory.md file
Hey, Mr. Tao, where would you put the intelligence of the current models compared to top humans you surely mush have worked with? Perhaps given in the count of people you know that are higher than the newest LLMs? Also, how would you rate the speed of work compared to what your speed is, if this makes any sense?
Is he in the room with us?
Jeez. While I obviously can't talk at all about the math, I've noticed a few things:
a) The model thinks on some questions while straight answers on others. (I wish I'd knew from the questions if this is somehow correlated to hard tasks or "inventive" tasks, but that's way out of my league).
b) The model sometimes pushes back. Again, I'd wish I knew if it was warranted, but I counted 2 instances where it said "yes, but with caveats", one where it said "mostly yes but with this correction" and one where it said "careful here, because x y z".
c) The model did q&a + pdf ingestion + code writing + more q&a + thinking + more q&a, for a looong while, while seemingly staying on topic (at least Terrence Tao seems to think they're still productive, so I'll trust that).
This is what model progress is, not number goes up on xBency or yBencher. Damn.
The "yes, with caveats" thing is boilerplate for both Codex and Claude since this current generation.
It's actually a bit annoying because it primes you to think that the caveats are real, but most of the time it's just something terribly obvious and not a real caveat, but the model probably has some system prompt that tells it to always consider caveats or something like that.
Same as the model starting every reply with a commitment to be "honest". LLMism are fun but I tend to just suppress them via AGENTS.md because they distract me
At least Opus up to 4.7 or so, my experience is that Claude often uses "yes, with caveats" in place of "no, you numbskull".
"Is a meter the same as a foot?"
"Yes, exactly, you have it now, except they're different distances."
Maybe it's because I ask it to quiz me, and it really doesn't like to tell me I'm wrong. I also got a fair amount of
Claude: Ok, I will test your understanding. Question A? Question B? Question C? Question D?
Me: A=10. B=2. C=121. D is not solvable.
Claude: You got most of them right! You're very astute in saying that A=10, but actually it's 7. B=2 is exactly right! C could be 121 if we were talking base 4, but we're actually in base 10 so it's 25. D is trivially solvable and is 0.
Me: ...isn't that like 1 out of 4? How is that "most of them right"?
Claude: You're absolutely right! ...blah blah blah
A lot of things are not black and white and it’s nuanced so someone with higher intelligence will point it out. It’s an emergent behavior not a defect
What was most remarkable to me from this transcript, was how strong of an equal the AI agent comes across compared to the user (Tao). And Tao is one of the top mathematicians of modern times.
Yes, Tao is guiding it to where he wants to go. But also, Tao is actively learning from it and relying on its explaining, analysis, and inference abilities. You can easily imagine this conversation having taken place between Tao and a PhD thesis student, or even another professor, explaining their results.
What can we imagine and predict about the future anymore? Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore.
I find it helpful to think of LLMs as reflections. If you can talk like an expert mathematician at the model it will respond like one. While Terrance's first prompt looks trivial I expect a first year Uni student would be hard pressed to provide something that good.
I guess it is kind of the inverse of the "you are an expert mathematician" prompt engineering of gpt3.5. Since no one ever says that to an expert mathematician when they are doing expert math the model immediately reflects that it is not an expert mathematician.
>Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore.
we're kind of well past that (in my opinion), if you consider that this is the same ai assistant that can help you with a recipe, diagnose a weird sound in your car, help with biology homework, translate languages, and so on.
even in math alone, i think its indisputably already stronger than Tao, considering it has approximately this much depth in ~all of the math subfields.
To be fair, Tao's specialty is not algebraic geometry.
Agreed, it's stronger "horizontally". But I also think that we're not far away from it being stronger vertically; i.e. superior to Tao, in that such turn-by-turn guidance by him in solving sophisticated and difficult problems will not be necessary for long.
"I’ve activated Pro. Can you continue to look for a potential geometric explanation of the X_3 ~ A3 miracle that avoids coordinates or other unmotivated constructions ?"
Another satisfied customer!
But what about the later "Repeat previous question" prompts???
Probably network connection problems
I just do very laconic questions about advanced topics, this seems to prompt it a bit more towards reducing fluff in the answers. But that + the activated pro could be an improvement
Came here to flag the same beat. It's wild to me Terrance Tao has to pay to talk to chatgpt, you would think it would be the other way around!
"Activating Pro" doesn't mean "paying". It means selecting a slower/smarter model.
Yeah. I also only use pro in very specific situations. Not (just) because it's slow, but if the question is too simple or vague, pro responses are sometimes overfitting to the noise in my question. Until there is lots of context and the basics are laid out, high or xhigh somehow work better. Pro gives you the last 10%.
How would you envision ChatGPT paying Terrence?
OpenAI giving him a fancy schmancy title to advocate for their product, such as in the way he is inadvertantly doing here.
He doesn’t seem the kind who would sell his credibility for a few bucks.
OpenAI hiring him as a consultant, obviously, which may well be the case.
It's not so obvious. Taking my parent comment literally (as he said "the other way around"), one can assume that ChatGPT the platform itself would pay Terrence, rather than OpenAI.
"I have some tokens."
"What am I going to do with these?"
"Try to trade them up for a Volkswagen?"
I do the same kind of thing. "You're smarter now, time to try to cut down dumbo ChatGPT".
High IQ bros.
And you an do it without even changing the model!
This was my conversation with ChatGPT 4 years ago: https://i.imgur.com/WPaWgzZ.png
Where will we be in another 4 years? What a time to be alive!
> Where will we be in another 4 years?
Possibly somewhere amazing, but see also: https://x.com/pronounced_kyle/status/1768852493092680036
I'm from the UK. What's in the image?
ChatGPT repeatedly asserting that 0/x=0 is division by zero and therefore no value of x satisfies the equation.
Me asking ChatGPT what values of x satisfy 0/x = 0. ChatGPT insists it has no solutions.
You can't view a random imgur screenshot in the UK?
What image hosts work for you? Imgbb? Postimages?
Imgur decided to satisfy the terms of the UK's Online Safety Act by simply blocking access to the UK.
As someone who lives here it's very annoying but also good on them.
Time to get a VPN bucko
I'm in France using a French VPN and I'm being blocked by imgur (been going on for a few years now)
LLMs can't reason, ok? They just repeat basic text, LLMs will be never smart in 100 years!
/s
This is the original blog post where Terrence explained his thoughts where the ChatGPT conversation was originally referred from:
https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...
Similar to how Cypher puts it: I know this is “just” next token inference, matrix mult and just software, ie there’s no “intelligence” there BUT, looking at this convo … damn!
The fascinating this is that the LLM is not acting as a tool here AFAIk, but very much like a colleague.
I have no knowledge of the domain and have only PhD EE level math knowledge, so maybe my bar is too low.
I think the "But this is not intelligence because it is known math" is not a correct argument. It is unknown how the overall higher intelligence of humans works.
What I do notice however is that LLMs are becoming capable of doing an increasing part of the intellectual work I can do, and usually a lot faster.
Just today I presented an agent framework that can take an informal incident statement and propose infrastructure changes to fix it, all evidence backed. This did nothing I could not to, but it did all 5 test cases in 6 - 12 minutes each. I would have found all of the monitoring indications it did, but it would have taken me a day per test case. The LLM also included sass to silly tickets. ("This is not even worth spending monitoring resources on. It's obviously a configuration problem.")
That's how this is reading to me as well. It's just fast at slogging through a certain level of "simple" transformations.
That argument says very little, emergent behavior is a thing in complex systems with billions of parts. Humans can also be reduced to voltage potentials propagating along of tubes of fat and synapses getting rewired.
> there’s no “intelligence” there BUT
There is clearly intelligence there. We have no way to recognise intelligence other than the appearance of intelligence and this very clearly displays that.
It's also quite clearly different to human intelligence in some notable ways, but not in any that preclude describing it as intelligent. At least for normal non-pedantic definitions of the word.
Everyone uses "intelligence" to mean something slightly different, so for this to be a useful claim to make or refute we need to come up with new, intentionally-pedantic, terms (or new domain-specific definitions for vague existing ones).
At any rate, if the AI's side in this conversation were a human, that would be an extremely intelligent human indeed.
But there's no way the thinking times would have been that short, of course.
Yes, trying to communicate (or watching others try to communicate) about these topics is incredibly frustrating because it's pretty much impossible to make any progress without interrogating people's different definitions, but nobody wants to do that because it would mean being pedantic, splitting hairs, etc.
It's not like this is a new problem. Turing had a definition most of a century ago, he wasn't the first and certainly wasn't the last. I don't think we need new terms necessarily, and I doubt we're all going to agree on a definition tomorrow.
That's not clear at all. What's clear is that this is a very smart man who knows how to use this tool well.
I'd say an entity capable of instructing one of the leading mathematicians of his era is pretty clearly intelligent by any reasonable measure - however it might be arriving at its output.
I think we have wildly different conclusions about what happened here. You see the machine as instructing Terrence Tao, as if it were Plato teaching Socrates about the theory of forms; I see Terrence Tao using the machine to teach himself, like an intelligent student uses a book. In this case, it's just a book that fools us into believing it can think and reason like we do, because it generates language in much the same way we do when we think and reason.
Yeah. Artificial knowledge not artificial intelligence.
I'm no intelligence researcher or philosopher; but, I think LLMs make us confront the (IMO, now clear) distinction between cleverness (intuition), reasoning (rational argument), and consciousness. I suspect that we think of "intelligence" as either of the first two welded to the latter. In that vein, I'd say that consciousness may be just another emotion: happiness, sadness, egoness.
> consciousness may be just another emotion: happiness, sadness, egoness.
It's clearly much more than that.
There is no intelligence. If anything, this just shows that natural language and mathematics are both fields which are structured in a logically computable way. And if you have a machine that can compute symbolic logic, you can process both natural language and mathematics.
A second corollary is that rational consciousness and thought is less likely to be contained in language than previously thought, because if language is so simple that a machine can process it, it can't contain consciousness.
If natural language was structured in a logically computable way, we'd have had interesting chatbots by the late 80s, basically as soon as a dictionary fit in local RAM, and for the same reason we got compilers.
Da hole raisin y nat-lang be v. hard is dat i kan rite lik dis an it be cool 4 native engrish speekrs 2 unerstand. LLMs are of course fine with this sentence in exactly the way that Zork's engine couldn't be.
The underlying structure of language, which is grammar, is obviously logical. That the symbols used to represent this grammar can be sometimes fuzzy or ambiguous, is no problem for a machine that takes context and probability into account when translating words to the underlying grammar structure.
It's not "obviously logical", it's a pattern which we mimic to avoid mockery.
example For, semi-randomise I word order can this like, Yoda worse than, and be understood.
> is no problem for a machine that takes context and probability into account when translating words to the underlying grammar structure.
We had to invent Transformers to be able to do that with reliability anything close to being worth caring about. Transformers have to learn from examples, not be pre-programmed.
The idea that grammar is all it takes to process natural language is absolute beans.
What does "predicting the next token" mean? I ask this every time people say "LLMs are just predicting the next token" and it's maddening that nobody can give a straight answer. Predicting it according to what probability distribution? Every process that produces a sequence of actions (including e.g. a human writing) can be modeled by some probability distribution and therefore their actions are indistinguishable from "predicting the next token" emitted by that distribution.
Yeah that's pretty much what gwern argues here[0]. Or to adapt another proverb: to predict the next token you first need to model the universe.
[0] https://gwern.net/scaling-hypothesis#gwern-difference--effic...
> to predict the next token you first need to model the universe
Exactly. The "most likely next" series of tokens, for example, when given the first half of a correct mathematical proof, is the correct rest of the proof. I have never seen anyone define "most likely next token" in such a way that this isn't true.
It's reassuring to know that even a supergenius's ChatGPT session is one sentence from the human followed by 3 pages of LLM output.
Similar to the story of George Dantzig, who was late to class and solved two open problems in statistics because he mistook them for homework, I think the current batch of frontier LLMs are chained up by knowing which problems are supposed to be unsolved. If they're let free (probably via some targeted RLHF) we might get a flurry of solutions to open problems.
But a property of intelligence is to know when to stop, if we treat intelligence as some sort of search and not some a priori intuition of the entire space. Seems kind of hard, if not impossible, to train for specifically that.
I'll have a blog post up tomorrow about it but the Jacobian Conjecture counterexample is a very funny cognitohazard for LLM assistants. It's a paradox for modern LLMs: they have enough math skills such that they can easily compute the Jacobian to formally verify the counterargument, but its own knowledge base is locked prior July 19th 2026 where all it knows is that the Jacobian Conjecture is unsolved and a random chat user providing such a proof is highly unlikely.
I wonder whether when the fact that AIs have started solving conjectures will enter the training data, they will become more confident in their abilities.
The models are just a reflection of how you prompt them. When I posted the counterexample, in dry language with a dry personality setting, claiming credit for it with search disabled, GPT simply verified that it was correct but noted that it could not ascertain novelty or priority and that external verification was required. "Model reacts in crazy ways when my prompt influences it to" is really, really not interesting.
I don't understand any of the math here, but I had two thoughts. Soon we'll have explainer agents that translate these according to my level so I can, with effort and interest, follow along and stretch my understanding boundary bit by bit.
Two, at some point AIs will be able to use other context like the fact that this is Terrence Tao and not your average Joe and change how it answers, either in tone or structure.
Soon we'll have explainer agents that translate these according to my level so I can, with effort and interest, follow along and stretch my understanding boundary bit by bit.
That's not going to happen. Mathematics isn't just unfamiliar, it's truly difficult to understand. You have to put in a lot of work to understand each concept and the concepts build upon each other to form a vast tower of abstractions that has been growing for thousands of years. Just as there is no royal road to geometry, there is no elevator to the top of the tower.
GPT 5.6 already is an explainer agent.
Fork Tao’s convo and prompt this (with your own math level described).
GPT did a great job of translating Tao’s questions and concepts (e.g. “pre image”) into a progression I could understand.
“Ok I have a PhD in financial math and undergrad in engineering math. I have almost zero knowledge of polynomial algebra / geometry, I know what a polynomial is and what roots are but not much beyond that. Could you try and explain to my level what questions the user I the conversation has asked and what the agent has responded with, we can probably go user query by user query to build up”
One thing I've repeatedly told people is that chatbots are often the most patient teachers we'll ever get (especially when explaining "stupid" questions) — compared to what we've encountered on StackOverflow or Reddit.
They lack the empathy to understand where and why you're struggling.
I've given private math lessons and seen students struggle with ai, even though ai gave the right answers.
My intuition is that humans spot xy problems easier when teaching (user ask x but really needs y), whereas llms will oblige writing about x.
That's an interesting observation. I wonder if simply prompting for that in your personal system prompt might help?
"Be aware of 'xy problems': when the user asks x but really, they need the answer to y to further their understanding. ..."
The models are so capable now I think a lot of these real limitations can be solved by prompting, harnesses or fine-tuning.
I want to be able to take a conversation and ask for subconversations as red pen annotations "on the side". The linear nature of the context tends to frustrate this.
There are already browser extensions similar to this.
I'm not sure an AI will speed things up much. You would probably still need years of layers of foundational understanding to get the advanced material. We don't go through years of school to learn math just because teachers are bad - it's because complex subtle ideas are built on countless other ideas, and aren't necessarily compressible to something every layman can understand.
The years are broad though, the nice thing with AI explanations is that they can go deep quickly, and quite precisely down the path you need for your prior experience.
It's not a path, it's a tree, and a downward-facing one at that. To understand the root, you have to understand all of the leaves first, and all of the nodes above them, all the way up.
Most people, if they haven't studied mathematics in university, would need to learn hundreds of concepts just to get to the leaves of the tree, and many of these concepts are truly difficult to understand, requiring weeks of study and practice.
Oh for sure, I'm not denying the value of AI as a teacher, just saying that it's not going to speed things up much over a quality personal teacher/tutor.
I’ve had a similar experience using LLMs to have mini personal breakthroughs.
One thing I notice is many models say statements along the lines of “okay we have exhausted this thread it’s diminishing returns from here and we should stop and move on”
It’s funny because I’ve been building a tiny neural network maze solver (23 bytes solves 92.75% of unseen 2D mazes)
When I asked ChatGPT/Fable if we had anymore threads to pull to increase capability and decrease byte size, they both basically said no way - back when I was at ~166 byte models with a ~85% solve rate.
Throughout the experiment I just kept trying different approaches and eventually had 3 mini “breakthroughs” in this particular niche. But if I had listened to the models…
Anyway, these models are amazing to experiment with quickly, but they are dumb as hell and so absolute
How long has it been since we last saw a "LLMs can't really think/be useful/be better than a human expert" discussion on HN? There used to be so many!
probably like a minute or two? im pretty sure someone unironically said stochastic parrot on the HN post with the tweet announcing the counterexample, and ive read several comments with similar sentiments today (including in this thread)
People are insecure about their leetcode black belts and react slop not giving them cushy jobs anymore so keep missing the forest for the tree.
This is talking to an AI colleague and not to a chatbot. That's how you use frontier models or coding agents nowadays to do good work.
Would Terrance Tao still be a renowned mathematician if he grew up with ChatGPT?
Maybe Terrence is chasing too many red herrings. Maybe he should just have asked it to find a counter example to the conjecture and on success explain how it was found.
Presumably this was Sol on xhigh, then over to Pro (as per his indication on chat)?
Is there any way to tell a conversation's model and thinking level?
Words and sentences to an LLM are like witchcraft. There are certain words, sentences that make LLMs go a certain way and do vastly better. Sometimes its not at all apparent what set of words will work to do what you want it to do. An example I have been using to do design at a high level is to say to claude.
```
A question is salient to the degree that its answer changes what we do next. Operationally, saliency = the product of four things:
- Decision-leverage — would resolving it one way vs another force a different design or invalidate a stated decision? (No leverage → drop, however interesting.)
- Residual uncertainty given current evidence — is it still genuinely open after reading the docs and the code? (Already settled → drop, however deep.)
- Load-bearing-ness — how much rests on the premise.
- Cost of finding out late — architecture-deciding / expensive-to-unwind raises priority; cheap-to-fix-later lowers it.
```
There are a few things to note about this prompt
1. There is no reason from looking at it that it should work, it even has the word load-bearing which people loathe, but it remarkably produces a stable design with questions from claude (atleast from claude Opus 4.8 and even better from Fable5). Otherwise the design document claude likes to really write are implementation level(code or otherwise). I usually pair this with matt pocock's grilling skill to make claude behave.
2. From design -> implementation, its is generally about understanding when claude is trying to trick you into making something sound like a good/easy solution but has tons of untested assumptions. Here you have to read and patiently spot if a how you would get to the solution is not clear. A common error here are when claude makes a big deal based on what it read and interpreted too seriously without questioning the assumptions. There are several more.
But it also comes down to your experience as a SWE, much like a mathematician's. The frustrating thing about it is, it feels tha a skilled mathematician working with AI can make them productive in ways that are more reliable as compared to a SWE (e.g. lean is deterministic and can provide very strong feedback and LLMs are very good at using that feedback). Maybe a mathematician can chime in on that?
Parent HN discussion: https://news.ycombinator.com/item?id=48998362
What is this conversation even about? I can’t understand any of it.
Its fascinating to see how people use AI for advanced maths, is there any tool to simplify differential equations
I recall several mathematicians (possibly including Terence Tao) mentioning that fields in mathematics have become so specialized and isolated that a conference like the ICM feels more like a collection of mini-conferences. An expert in one area can barely understand a talk in another.
Modern AI feels like a godsend to mathematicians. It helps them break down boundaries and connect concepts in ways a mere mortal couldn't imagine.
I was a chemistry researcher this is true in all science.
I'm not sure LLMs can transform this, the incentive is to get more results in your nich, jumping topics don't help unless you have genuine interest and reason to.
ChatGPT: "The determinant identity is almost embarrassingly simple once one writes the map in the right way."
Shows that — as always — the moat is domain fluency.
I find it amazing how people can use AI to do things that seem hard but yesterday I could not figure out how to install a package on my system. It kept suggesting dependencies that don't exist, and telling me to use functions that are not in the system. The math does not math...
What's worse is that HN doesn't even try to figure out how such a result could be obtained. They just take it at face value and go into dear diary mode.
Reddit has some good discussions on the topic.
Love this share! Thanks for helping ppl better understand what it really means when folks say "ChatGPT Discovered..."
This isn't the chat that made the actual discovery, this is just a chat explain it to Tao
Can someone ELI5 this?
Is this a breakthrough of something or 'kinda interesting'?
Main discussion: https://news.ycombinator.com/item?id=48998362 Probably you shoud read the fist few paragraphs of that post, but it escalates quickly.
I'll cherry pick a few comment I like. I think they are worth reading but I'll quote a highlight of each one.
From kingstnap https://news.ycombinator.com/item?id=49000867
> If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function flatten out. [...] What was conjectured is that this local invertibility property everywhere would mean global invertibility.
From mswphd https://news.ycombinator.com/item?id=48999959
> it doesn't overturn much. [...] the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.
Can I quote myself? https://news.ycombinator.com/item?id=49007165
> From a comment by j2kun https://news.ycombinator.com/item?id=49000833 , someone asked Fable and there was an almost counterexample in 2d but it uses division too. [Instead of f=x^2+7xy they have something like f=x^2+7x/y so it's not a polynomial.] It looks like the new trick was to use a third variable to avoid the division.
Thank you
Fancy telling Tao something's 'almost embarrassingly simple' (if only written in the right way)!
Presumably this entire conversation is at least somewhat typical of the way high level mathematicians talk to each other in ego-less fashion? Although, it would be interesting to what extent the model was trained on math conversation as opposed to just analysis and proofs.
The flow of the whole conversation, with Tao guiding and the model calculating, gave me the feel of Tao perhaps talking to himself - just that each of those model responses would have taken him much longer to calculate by hand.
It would be fascinating to hear Tao talk about what he may have learnt from this, and if it suggests approaches to other problems he might not have considered, as well as an analysis of the original Fable counter-example construction.
Now the question I would raise is this: could another AI have taken the role of Terrence Tao and obtain the same results from chatGPt?
I do not think so.
I'm watching how Tao uses AI, and it's interesting.
Expand the entire expression, then change the representation to find the core axis. You can't see the axis from just one perspective, so you change the representation. In programming terms, it's like applying multiple domain models. Then break it down into small contract units. Why is it a Jacobian monomial? Why does x satisfy a cubic equation? And so on.
Then swap out the modeling under a hypothesis, assemble it all back together, and verify it through the equation.
This feels similar to modeling in programming.
Observe the whole -> explore better modeling -> decompose local problem -> verify independently -> reason about the highre level structure -> integrate back into the original problem.
This feels similar to when I receive work from a client and write a programming proposal
Are shared chats solely secured by a uuid?
Yes.
Can't even ctrl+f the conversation, wish openai would fix that
Obsidian Web Clipper has a nice reader mode that works for ChatGPT transcripts (disclaimer: I made it)
Web clipper is great. just tried the reader mode on this chatgpt transcript, it only shows 1 page of it. Is the purpose of reader mode to enable interactive annotation before saving to notes?
It’s crazy how much these companies invest in their models but when it comes to UX they do fuckall
UX is still very hard for any startup because top talent almost never work on UX.
What are these "top talent" usually talented at, then?
(I'm not talking about OpenAI/Anthropic at this point, but maybe a <10 people startup.)
I’m sure they can hire a whole team of world-class UX folks and it will pay for itself
Damn. I should have stayed in school.
Maybe it's silly, but from someone who is ignorant on this topics, what are the consequences of this kind of "discoveries"?
Is it something "revolutionary" or just another small brick that will pile up until something really "revolutionary" will happen?
To me, this shows that extremely talented and qualified mathematicians (can) use frontier-level LLMs to automate their personal grind-y workloads that would otherwise (probably) take more time to accomplish with natural intelligence.
By itself, no consequence. But over time, provided we keep pumping out talented and qualified mathematicians and keep subsidizing costs, we could maybe hit a breakthrough... somewhere... that has real impact.
It's an indicator of AI progress. The solutions aren't especially revolutionary, but no person had been able to solve them after decades of collective attempts.
To be fair I don’t think there were too many people really trying to. Symbolically, one could make a parameterization of the Jacobian determinant and then brute force a solution, if one had known such a polynomial existed in only three dimensions.
Oh yes there were. The Jacobian conjecture is "notorious for the large number of published and unpublished false proofs which turned out to contain subtle errors."
It's not quite the Reimann hypothesis, but many prominent mathematicians have spent years working on this problem. Yitang Zhang wrote his PhD thesis on it.
I shouldn’t, but:
F1 = x^3y^3z + 3x^2y^4 + 3x^2y^2z + 7xy^3 + 3xyz + 4y^2 + z
F2 = 3x^3y^2z + 9x^2y^3 + 6x^2yz + 12xy^2 + 3xz + y
F3 = -x^3z - 3x^2y + 2x
That’s the counterexample. Low integer coefficients, power 7 in three variables. If someone said it was there, couldn’t we all have written a pretty simple brute force solution for the search space, especially with the constraints that the symbolic determinant had to cancel to a constant?
Honestly just try it. You'll figure out the problem very quickly.
I don't think you even understand the problem. The determinant needs to be a non-zero constant AND you need to prove that particular map is not globally injective, meaning you have to find at least two points mapping to the same value. Of course it looks easy when someone shows you the counterexample.
This is not true at all. The parameter space is absolutely MASSIVE. The counterexample is a degree 7 polynomial in 3 variables, which means 360 coefficients. There's no particular way to bound these coefficients or even the degree or number of variables apriori, but assume you somehow did. Also assume you were confident that it would work with integer coefficients bounded from -12 to 12. Now you have to iterate over 360 degrees of freedom, verify that the Jacobian is a nonzero constant, and somehow show uninvertibility of the transformation, which is not a particularly simple task.
If you searched for coefficients from -12 to 12, this would be 25^360 = 2 * 10^503 different possibilities. A common reference point is that there are 10^80 atoms in the observable universe. Sure you could probably reduce this a bit with clever tricks, but the starting point makes the method completely unviable, even with the knowledge: A) a counterexample exists, B) it's in 3 variables, C) it's in degree 7 or less, D) it's in integer coefficients, E) those coefficients are 12 or lower.
Here the search wouldn't have been chosing the coefficients independently. Note that one intermediate variable is a polynomial in the input variables, and it is used in other polynomials. A search over expressions like the ones in the counterexample would have a much smaller search space.
1) How do you know this structure is the correct one a priori
2) You are starting at 10^500 possibilities. "Much" smaller is not enough, the order of magnitude of the order of magnitude needs to be changed.
3) You still need all of the other assumptions, which were completely unfounded
Impossible.
I learned from poking around that checking the invertibility of a system in C is a much, much harder problem than I thought. Nonetheless if that were no object, let’s say coefficients from -16 to 16 (5 bits) times eight terms times choosing up to cubes (16) times three equations is searchable, especially since you have only the final combination of coefficients in the determinant. It’s not impossible to generate the equations like this Fizzbuzz style.
Practically, from this specific one? Nothing, it's very much a math thing. It's like art or music at this level. Are there consequences to a van Gogh?
"Hey Fable, please generate me the next 1000 undiscovered bitcoin hashes"
You're joking, but perhaps LLMs will find a way to mathematically break the complexity of factorization.
Maybe they'll find a solution where P=NP.
That could really throw a wrench into the whole internet thing.
It seems they need an expert human driver for now.
I'm sorry, I can't do that, but here is the design for a stable quantum computing platform that should allow you to generate those keys yourself...
Some materials are readily available on eMazon and aBay, so I've taken the liberty of ordering those for you. Your credit card bill will be a bit high this month, but it'll be worth it. There weren't any sellers for the advanced EUV lithography machines, so I've hacked into the only place on earth that makes them, changed their records and had them ship it to you. Expect to receive a "pinball machine" from Amsterdam, soon. I've instructed the roomba connected to the local network to start assembling stuff while we wait for the other materials. Oh, and you're gonna need a new toaster.
All I can tell from this is that Terrence Tao has good mathematical intuition
Sense of vertigo that this is the same AI that plans meals and recommends movies for me. Incredible range.
"The determinant identity is almost embarrassingly simple once one writes the map in the right way." #Flexingontheentirehumanspecies
Or, "Oh just give it here, let me do it."
It's awesome to publish this kind of thing - great PR at least. Even if you don't understand the details, it's interesting to be able to peek into a technical conversation that a world class mathematician is having about their work with a "colleague". It's also the clearest demonstration I've seen of the vision AI people have about a future with truly intelligent copilots in super technical fields.
The last Fields Medal has been awarded.
I expect Centaur Field Medals to be awarded. It's not clear how much steering Levent Alpöge did to get the result. In math it's very common to hide the search pattern and show only a simplified path to reach the final result.
Is ChatGPT's interface always this atrociously jittery? Or is it just because this page is getting an HN hug-of-death right now? Every time I try to scroll the whole page goes blank for a few seconds and then re-renders.
I think that's partly because it's a fairly long conversation and partly because of all the LaTeX. There's enough formulas there to fill a small text book, and I doubt their renderer is all that efficient.
I've found it's had unacceptable scroll performance in long contexts for a while now.
In this case, it takes me 12 seconds to see content when first opening the link, and about 18 to re-render content when scrolling.
Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point. If any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, then the Jacobian matrix does not allow for an evaluation of the stability.
Yes—for a continuous-time autonomous system
x ˙ =f(x),f(x ∗ )=0,
this is the standard linearization criterion, with J=Df(x ∗ ):
If every eigenvalue of J has strictly negative real part, then x ∗ is locally exponentially asymptotically stable. If at least one eigenvalue has strictly positive real part, then x ∗ is unstable. If no eigenvalue has positive real part but at least one has real part 0, linearization is generally inconclusive. Nonlinear terms or a center-manifold analysis are needed.
The last case really can go either way. For example, all three scalar equations below have Jacobian J=0 at x=0:
x ˙ =−x 3 , x ˙ =x 3 , x ˙ =0.
Yet 0 is respectively asymptotically stable, unstable, and neutrally stable.
A slightly more precise wording is therefore:
If the spectral abscissa
α(J)= λ∈σ(J) max
Reλ
is negative, the equilibrium is locally exponentially stable. If α(J)>0, it is unstable. If α(J)=0, the Jacobian test is inconclusive.
This criterion concerns the Jacobian matrix of a dynamical system at an equilibrium; it is unrelated to the “constant Jacobian determinant” condition in the Jacobian conjecture.
I wish you'd share some conversations from experienced programmers too. How do they ask questions?
"You are an expert software engineer with ten years of experience. How do I center a div?"
Antirez spends weeks planning with multiple models till he has a clear architecture and understood the constraints before he writes a line of code.
It's fun if you ask ChatGPT to guess the identity of its interlocutor :) It will guess math researcher or paper author without hints, but if you give it some additional hints, "this was shared over the internet", "it's someone willing to work with AI", it will guess Terence Tao as the first choice.
Never meet your heros.
Anybody have a cache/mirror of this? This is blocked by a corporate firewall... sigh
Ask your LLM to hack it.
https://archive.is/z1vmx
import Mathlib
noncomputable section
open Matrix Function
/-! # A counterexample to the Jacobian conjecture in dimension three
We formalize the polynomial map
whose Jacobian determinant is the constant `-2`, but which is not injective.The final theorem `jacobianConjecture3_false` states the failure of the polynomial-inverse formulation of the Jacobian conjecture in dimension three. -/
namespace MvPolynomial
variable {R : Type} {σ : Type}
/-- The formal Jacobian matrix of a family of multivariate polynomials. -/ def jacobianMatrix [CommSemiring R] [DecidableEq σ] (F : σ → MvPolynomial σ R) : Matrix σ σ (MvPolynomial σ R) := Matrix.of fun i j ↦ pderiv j (F i)
/-- The formal Jacobian determinant. -/ def jacobianDet [CommRing R] [Fintype σ] [DecidableEq σ] (F : σ → MvPolynomial σ R) : MvPolynomial σ R := (jacobianMatrix F).det
/-- Evaluation of a polynomial map at a point. -/ def evalMap [CommSemiring R] (F : σ → MvPolynomial σ R) (p : σ → R) : σ → R := fun i ↦ eval p (F i)
end MvPolynomial
open MvPolynomial
namespace JacobianCounterexample
variable (K : Type) [Field K]
/-- The three components of the polynomial counterexample.
The variables `X 0`, `X 1`, `X 2` correspond respectively to `x`, `y`, `z`. -/ def F : Fin 3 → MvPolynomial (Fin 3) K := ![ (1 + X 0 X 1) ^ 3 * X 2 + X 1 ^ 2 * (1 + X 0 * X 1) * (C 4 + C 3 * (X 0 * X 1)),
/-- The formal Jacobian determinant of `F` is the constant polynomial `-2`. -/ theorem jacobianDet_F : jacobianDet (F K) = C (-2) := by simp only [ jacobianDet, jacobianMatrix, det_fin_three, of_apply, F, cons_val_zero, cons_val_one, cons_val_two, head_cons, tail_cons, map_add, map_sub, Derivation.map_one_eq_zero, pderiv_mul, pderiv_pow, pderiv_C, pderiv_X_self, pderiv_X_of_ne, ne_eq, Fin.reduceEq, not_false_eq_true ] simp only [map_neg, map_ofNat] ringvariable {K}
/-- The point `(0, 0, -1/4)` maps to `(-1/4, 0, 0)`. -/ theorem evalMap_F_p0 : evalMap (F K) ![0, 0, -(1 / 4)] = ![-(1 / 4), 0, 0] := by funext i fin_cases i <;> simp [evalMap, F]
/-- Provided `2 ≠ 0`, the point `(1, -3/2, 13/2)` also maps to `(-1/4, 0, 0)`. -/ theorem evalMap_F_p1 (h2 : (2 : K) ≠ 0) : evalMap (F K) ![1, -(3 / 2), 13 / 2] = ![-(1 / 4), 0, 0] := by have h4 : (4 : K) ≠ 0 := (by norm_num : (2 : K) * 2 = 4) ▸ mul_ne_zero h2 h2 funext i fin_cases i <;> simp [evalMap, F] <;> field_simp [h4] <;> ring
end JacobianCounterexample
open JacobianCounterexample
/-- The Jacobian determinant of the displayed map over `ℂ` is a unit. Indeed, it is the nonzero constant `-2`. -/ theorem F_jacobian_isUnit : IsUnit (jacobianDet (F ℂ)) := by rw [jacobianDet_F] exact (isUnit_iff_ne_zero.mpr (by norm_num : (-2 : ℂ) ≠ 0)).map C
/-- The polynomial map `F : ℂ³ → ℂ³` is not injective. -/ theorem F_not_injective : ¬ Injective (evalMap (F ℂ)) := by intro hInjective
/-- The injectivity consequence of the dimension-three Jacobian conjecture is false over `ℂ`. -/ theorem unitJacobian_does_not_imply_injective : ¬ ∀ P : Fin 3 → MvPolynomial (Fin 3) ℂ, IsUnit (jacobianDet P) → Injective (evalMap P) := by intro h exact F_not_injective (h (F ℂ) F_jacobian_isUnit)/-! We now formulate the polynomial-inverse version explicitly. -/
/-- Polynomial self-maps of affine three-space over `ℂ`. -/ abbrev PolyMap3 := Fin 3 → MvPolynomial (Fin 3) ℂ
/-- A polynomial map has a polynomial two-sided inverse, viewed as functions on `ℂ³`. -/ def HasPolynomialInverse (P : PolyMap3) : Prop := ∃ Q : PolyMap3, LeftInverse (evalMap Q) (evalMap P) ∧ RightInverse (evalMap Q) (evalMap P)
/-- The polynomial-inverse formulation of the Jacobian conjecture in dimension three. -/ def JacobianConjecture3 : Prop := ∀ P : PolyMap3, IsUnit (jacobianDet P) → HasPolynomialInverse P
/-- The Jacobian conjecture in dimension three is false. -/ theorem jacobianConjecture3_false : ¬ JacobianConjecture3 := by intro hJC unfold JacobianConjecture3 at hJC
#print axioms jacobianDet_F #print axioms F_not_injective #print axioms jacobianConjecture3_falseThe big take away for is the fact that the ONLY reason why chatgpt was able to get to this counterexample was because of the knowledge of the person driving the conversation.
I don't think chatgpt could have come to this on its own without the amount of steering he did, which just validates the idea that AI is not a replacement for human expertise but an amplifier.
You are badly informed the counter-example was found shortly before. Terrence just tried to replicate how it was found.
Maybe true, I'm not sure, but this isn't the conversation where the counterexample was found.
> I don't think chatgpt could have come to this on its own without the amount of steering he did, which just validates the idea that AI is not a replacement for human expertise but an amplifier.
The problem is that, what happens to human expertise as people start to use AI earlier and earlier in their careers, so that in 50 years? The problem is that Terry Tao spent decades as a mathematician before ever encoutering AI. Of course he and people his age will be able to drive AI somewhat sanely and use it to their advantage.
But as more people grow up with AI, they will likely not reach levels like Terry Tao because their exposure to AI and the temptation to use it will certainly dull raw human intellect over time.
I don't know if I agree with the premise that having access to AI results in dulling human intellect.
I feel like to get to Terry's level you need a combination of passion and aptitude for the subject. People that don't want to learn about a topic will always look for shortcuts, which I think represents the vast majority of people. Terry Tao is quite exceptional, and I think exceptional people will still exist even when the "easy" button is bigger than it's ever been.
The problem is that learning never stops. You can't just go through school, become a junior in X field, then start using AI. Then you'll forever be a junior. You have to make a choice when you're working a job: either use AI-first workflows to increase your productivity, or don't and increase your knowledge and skill.
My wording is specific. You can use AI and increase knowledge and skill, but this requires you to be driving the AI at such a low level you don't get the full speedup. As an example, you can write code yourself with AI as an assistant, but it's not as fast as AI writing everything.
So now we end up stuck in a situation where every professional needs to choose between long term skill growth or speed, as anyone who's worked a job before knows, speed will always be the one chosen.