(Epistemic status: I consider the following quite obvious and self-evident, but decided to post anyways.^[1])

Mathematics is a social activity done by mathematicians.

— Paul Erdős, probably

There’ve been a few attempts to create mathematical models of math. The examples that come to my mind are Gödelian Numbering (GN) and Logical Induction (LI). Feel free to suggest more in the comments, but I’ll use those as my primary reference points. In this post, I want to contrast them with the way human mathematicians do math by noticing a few features of their process, the ones that are hard to describe with the language of math itself. Those features overlap a lot and reinforce each other, so the distinction I make is subjective. There’s also probably more of them, those are just what I was able to think of. What unites them is that they make mathematical progress more tractable.

Theorem Selection

The way in which Kurt Gödel proved his incompleteness theorems was by embedding math into the language of a mathematical theory (number theory in that particular case, but the trick can be done with any theory that’s expressive enough). But this way of describing mathematics is very eternalistic: it treats math as one monolith. It does not give advice on how to make progress in math. How could we approach it in a systematic way?

Fighting the LEAN compiler

What if we just try to prove all statements we can find proofs for?

Let’s do some back-of-the-envelope Fermi estimations. Here’s a LEAN proof of the statement “if and if , then ” (sorry for JavaScript highlighting):

example (a : ℕ → ℝ) (t : ℝ) (h : TendsTo a t) (c : ℝ) (hc : 0 < c) :
    TendsTo (fun n ↦ c * a n) (c * t) := by
    simp [TendsTo] at *
    intro ε' hε'
    specialize h (ε'/c ) (by exact div_pos hε' hc)


    obtain ⟨B, hB⟩ := h
    use B
    intro N hN
    specialize hB N hN
    /-theorem (abs_c : |c| = c) := by exact?-/


    calc
    |c * a N - c * t| = |c*(a N - t)| := by ring
    _ = |c| * |a N - t| := by exact abs_mul c (a N - t)
    _ = c * |a N - t| := by rw [abs_of_pos hc]
    _ < ε' := by exact (lt_div_iff₀' hc).mp hB

It’s 558 bits long in its current form. I didn’t optimize it for shortness, but let’s say that if I did we could achieve 200 bits. Let’s say that we run a search process that just checks every possible bitstring starting from short ones for whether it is a valid LEAN proof. There are possible bitstrings shorter than this proof. So if the search process checks proofs a second, we will reach this particular proof in years. Not great.

That marks the first and most important unmathematical feature of math: the selection of theorems. We do not prove, nor do we strive to prove, every possible theorem. That would be slow and boring. GN enumerates every statement regardless of its importance. LI prioritizes short sentences, which is an improvement, as it does allow us to create a natural ordering in which we can try to prove theorems and therefore make progress over time. But it’s still very inefficient.

Naming

The way we name theorems and concepts is important. Most of the time we name them after a person (though most of the time it’s not even the person who discovered it), but if you think about it, the Pythagorean theorem is actually called “the Pythagorean theorem about right triangles.” Each time we need to prove something about right triangles, we remember Pythagoras.

LI and GN all name sentences by their entire specification, and that shouldn’t come as a surprise. There wouldn’t be enough short handles because, as described above, they try to talk about all sentences.

Naming allows us to build associations between mathematical concepts, which helps mathematicians think of a limited set of tools for making progress in a specific area.

Step Importance

When we teach math, we do not go through literally every step of a proof. We skip over obvious algebraic transformations; we do not pay much attention when we treat an element of a smaller set as an element of a larger set with all properties conserved (when doing 3D geometry and using 2D theorems, for example); we skip parts of a proof that are symmetrical to the already proven ones (“without loss of generality, let X be the first…”).

We do that because we want to emphasize the non-trivial parts. And the feeling of non-triviality is a human feeling, not identifiable from a step’s description alone. This same feeling is also what guides mathematicians to prove more useful lemmas.

GN doesn’t do that — it checks every part of the proof. I’m not as sure about LI; there might be traders that do glance over obvious steps but check more carefully for less trivial ones.

Lemma Selection

Some theorems are more useful and more important than others because they help prove more theorems. This score could hypothetically be recovered from some graph of mathematics, but it is usually just estimated by math professors creating the curriculum. This taste is then passed on to the next generation of mathematicians, helping them find more useful lemmas.

GN doesn’t try to do that. LI might do that implicitly via selecting for rich traders.

Real-world Phenomena

The reason humans started doing math was that they noticed similar structures across the real world. The way you add up sheep is the same way you add up apples. Pattern-matching allowed mathematicians to form conjectures and target their mathematical efforts. (“Hmm, when I use 3 sticks to form a triangle, I end up with the same triangle. What if that’s always true?”)

GN and LI do not do that because they do not have access to outside world. Though there is a mathematical theory that attempts to do precisely that, which is Solomonoff Induction.

Categorising

This is very similar to Naming: we separate math into topics and when we need to prove some statement we know where to look for tools. GN and LI do not attempt to do that.

An important caveat, applicable to most of the features above: there should be a balance. If you stick too much within a topic, you will never discover fruitful analogies (algebraic geometry being helpful for proving Fermat’s Last Theorem is a great example). Too much reliance on any one feature and you lose creativity.

Curiosity/Beauty

There isn’t much I can add about this one, but it’s arguably the most important. It both guides the formation of conjectures and helps with intermediate steps.

GN and LI definitely lack it.

Conclusion

All of this is to support the point that math is invented rather than discovered. I agree that there is a surprising amount of connection between the different types of math humans find interesting, and there is probably more to learn about this phenomenon. But I wouldn’t treat it as a signal that we are touching some universal metaphysical phenomenon: this is just human senses of beauty and curiosity, along with real-world utility and patterns echoing each other (partly because human intelligence and the senses were shaped to seek usefulness and real-world patterns).

1. ^{^}

   Because of this and this.

Discuss