The usual format that fun mathematics is presented (being talked at by someone who is very well versed in the subject) comes with a heavy cognitive burden - and often I just can't really make it through.

When the author is not an expert the writing is just so much more accessible - it's easier to understand and making it through feels more of an adventure and less of a lecture.

I've never thought previously how much I would enjoy this format though. I'm here to see more amateurs stumbling through mathematics.

Also, wasn't expecting this sort of side-quest from the guy who got me into React.

Knowledge is of 2 kinds: know-that and know-how. Know-that is what LLMs are enabling such as the proof here, while know-how is more useful as that constitutes understanding and puts that knowledge to use.

Counter argument, "historically" is not last year but closer to 2 decades ago.

modern computing has sat atop a mountain of sorcery for ages. You may be able to squint and call a good compiler a "tool of power" more than sorcery but there are plenty of other tech stacks that simply sit upon one opaque box after another.

And it's not unusual for sorcery to produce crap, even if it's production worth crap. Dreamweaver had a WYSIWYG editor ages ago and as someone who was hand crafting HTML in that era it's code was typically a mess.

Now that's the norm, almost nobody makes HTML by hand, some js framework does. And it's often shit, I've seen entire sources of nothing but <div> nested ad-nauseam, conjured up from the depths of "I-don't-care-as-long-as-it-works"

Sorcerers are the vibe coders of D&D magic, which perfectly describes how wizards in-universe feel about them.

For any finite program (eg some LLMs), there is a true math theorem which they cannot prove or disprove (given fixed input of the statement with no other information sources). If that weren’t true, BB would be computable.

Math is beyond computation. Since AI is just bits in bits out, it has this fundamental limitation.

Any magic of AI systems comes from the transformed meaning of its input data. With fixed weights any LLM is just an artifact. For example a human prompting an LLM constitutes an extra information source, which removes the above limitations. In theory any input from the natural world would remove the limitations too. The natural world is a black box and we don't know what kind of meaning or intelligence could underly it.

"Given infinite thinking time a finite number of humans will solve all theorems"

I also love the angle that this was not intelligence just brute force. As if the mathematicians didn't reeaaally want to solve this they were just too lazy to give it a good try.

What does AI have to actually do before you realize these things are actually smart?

Let me first clarify my relationship with mathematics. I think of myself as "an awestruck observer from a distance". I find some parts that I understand beautiful, and I have also tried to understand some of the basics rigorously. However, I generally just can't make my way through any serious paper, as I both lack the prerequisites and struggle with the amount of inference mathematics tends to place on the reader. That's the "from a distance" part.

Now, about picking the problem. I am genuinely "pulled by" surreal numbers themselves. I find them irresistibly beautiful. There is also a bit of bitterness around how they haven't fulfilled their promise (yet?) as Conway hoped they would be able to become a better foundation for some mathematics. But they are a bit too difficult to prove things about so far, and we know too little about them. So what "pulls me" also is a possibility of making enough dents in this that we would be able to use them more broadly, and learn even more things about them.

However, I do not know the details of the latest research. I don't know which problems have actually been solved, which pursue Conway's original vision vs narrower approaches, and which are elegant enough to feel "awestruck" enough about. So this is an invitation from me to LLM to share what it "feels pulled by" (for whatever definition; I think of it as just navigating the languagespace) , and then sifting through that list to see if something it lists makes me feel something. I would assume that with the field currently being so small (serious mathematicians mostly don't care about surreals), it's easy to get the LLM "excited" (again, just a vector in the languagespace) enough that it would give me genuinely interesting candidates. Then it's up to me to sift through them and see if they "speak" to me.

It's like asking a mathrock nerd to share their favorite mathrock albums. Niche enough that you'd likely get good results. Then you can listen and form an opinion.

In this particular example, the "ONAG birthday" and "maybe last Conway's unsolved conjecture about surreals" part spoke to me emotionally, the statement itself amazed me with its simplicity, and I felt "blood in the water" related to the recent results bringing the conjecture closer. So I felt the pull myself and went with it.

Upshot, if there are real errors in published work, I think most mathematicians want to know about them.

You can think of the "surreal numbers" as being built up step by step. We start out with no numbers at all, and then we repeatedly do a construction that makes some new numbers.

A surreal number is made from two sets of (pre-existing) surreal numbers. We typically call them L and R, for "left" and "right", and sometimes write it as L|R or {L|R} or something like that. The "left" numbers have to be smaller than the "right" numbers. The resulting number will turn out to be, in a certain sense, the "simplest" number in between all the left numbers and all the right numbers.

Now, as I said, we start out with no numbers at all. It might seem like that gives us no way to proceed, but it does: even given no numbers at all, we can still make a set of numbers, namely the empty set! So we can use that for both L and R, getting ∅|∅. Empty sets on both sides. We call this 0, and it will turn out to behave in the way you'd expect the number 0 to behave.

Now we suddenly have another set available, namely {0}, the set containing only zero. Which means that instead of being able to make one number, maybe we can make four: ∅|∅, ∅|{0}, {0}|∅, {0}|{0}. The first of these we already knew about. The last isn't actually admissible -- remember that the "left" numbers have to be smaller than the "right" numbers, which is "vacuously" true when one of those sets is empty (it means "if you have a number x in the left set, and a number y in the right set, then x<y", and if there are no numbers in the left set or no numbers in the right set then that's trivially true) but isn't true when both sets contain 0 because 0<0 is false.

So actually we get two new numbers: ∅|{0} and {0}|∅. The first fits into what OP calls the gap "between nothing and zero". The second first into what OP calls "the gap between zero and nothing". In both cases, "zero" means a number and "nothing" means a space where we don't yet have any numbers.

The number ∅|{0} is called -1 (it has to lie to the left of 0, and there's no constraint on its left, and -1 is "the simplest number less than 0") and the number {0}|∅ is called +1 (it has to lie to the right of 0, and there's no constraint on its right, and +1 is "the simplest number greater than 0").

I should explicitly acknowledge that I haven't defined what "less than" and "greater than" actually mean for these numbers, nor anything else about how they relate to one another that could possibly justify giving these things the specific names 0, -1, and +1. But there are definitions for "less than" and "greater than" and "plus" and "minus" and so forth, and the whole thing does turn out to work very nicely.

Anyway, once we've got these numbers we have eight possible sets that can go on the left or on the right. The requirement for left-things to be smaller than right-things reduces the possibilities somewhat, and the actual new numbers we get next time around are: ∅|{-1}, which turns out to be -2; {-1}|{0} which turns out to be -1/2; {0}|{+1} which turns out to be +1/2; {+1}|∅ which turns out to be +2. We also get some already-existing numbers in new ways; for instance, {-1}|{+1} is actually equal to 0 ("0 is the simplest number between -1 and +1"). Again, I should explicitly acknowlege that I haven't said anything about how you determine when two of these things are actually equal; again, it does all turn out to work properly.

If you keep going with this construction, you produce all the integers, two at a time, and also all the "dyadic rationals", meaning fractions where the denominator is a power of 2. And then, once you've got all those, at the next stage of construction you abruptly get all the real numbers -- e.g., the square root of 2 is L|R where L = {dyadic rational numbers that are negative or have a square smaller than 2} and R = {dyadic rational numbers that are positive and have a square larger than 2} -- and you also get {0,1,2,3,4,...}|∅, conventionally written as a lower-case Greek letter omega, which is an infinite number, larger than all the integers. (And its negation.) And {0}|{1,1/2,1/3,1/4,...} which is an infinitesimal number, positive but smaller than any ratio of positive integers. And you can then proceed further and construct a vast infinitude of numbers, including all the real numbers (which we've already made) and all of the so-called infinite ordinals (which you can kinda think of as being a sort of "infinite positive integer", though there's more to them than that) and much more, all in a system that lets you do arithmetic and suchlike. It's very elegant, if your brain has been twisted into the mathematician-y shape that finds such things elegant.

The numbers don't matter and you could replace -1 and 1 with anything. It's just easier to begin your new fake number at - 1 and 1. Because position does matter.

Sorry it was confusing.

Edit: the picture is now edited into the article.

That's a mistake. They should've written "empty set of surreal numbers" and not "nothing".

It is a constructive theory, like sets/ordinals. For ordinals you can use ∅, { }, ∪ and you construct

  ∅, {∅}, {{∅},∅}, ... (von Neumann ordinals). 

You build up surreal numbers as

  {∅|∅}, {∅|{∅|∅}}, {{∅|∅}|∅}

---

Edit: A happy accident: I denoted the "empty set of surreal numbers" with the symbol ∅. It works, and it gives you the surreal numbers. But if you think of ∅ as the empty set in set theory, then the same construction (using an ordered-pair construction) gives you the surreal numbers as sets!

> (crucially, “to the left of all” and “to the right of all” also count as “gaps”)

So there are two "nothings" here, left of "all" - i.e. the zero - and right of it.

Though I'm not quite sure how you'd get infinite or irrational numbers by this procedure. Wouldn't you simply get the rational numbers by this?

(unless the "put a number" step is doing more work here than it seems. He doesn't really say which number to put there. In the examples, he mostly did "new number = (left number + right number) / 2", with special cases if any number is "nothing" - but he never actually wrote what the rules are here.)

I'm hoping someone develops an interactive tutor that can teach any subject to any depth.

The tutor should optimize its pedagogy. It should use online RL to adapt to a learner's ideal learning style, model what the student understands and to what degree, and understand what the gaps and next steps are.

I'd subscribe in a heartbeat.

It’s a terrible explanation. A surreal number is defined as a pair of sets of surreal numbers (where you fiddle around the recursion in that definition by defining them in waves, so strictly speaking you’re defining “the surreal numbers born at time T” for each individual T given access to the surreal numbers born at all earlier times, and then you “take the union across all times”, scare quotes because there are too many times for this to result in a set). Zero is a surreal number but the LLM is using the word “zero” to mean “the set containing just the surreal number 0”; “nothing” here is the LLM’s obtuse word for the empty set. Wikipedia may actually be easier to follow.