[umutisik]

With sufficient automation, there shouldn't really be a trade-off between rigor and anything else. The goal should be to automate as much as possible so that whatever well-defined useful thing can come out theory can come out faster and more easily. Formal proofs make sense as part of this goal.

[_alternator_]

Let’s not forget that mathematics is a social construct as much as (and perhaps more than) a true science. It’s about techniques, stories, relationships between ideas, and ultimately, it’s a social endeavor that involves curiosity satisfaction for (somewhat pedantic) people. If we automate ‘all’ of mathematics, then we’ve removed the people from it.

There are things that need to be done by humans to make it meaningful and worthwhile. I’m not saying that automation won’t make us more able to satisfy our intellectual curiosity, but we can’t offload everything and have something of value that we could rightly call ‘mathematics’.

[justonceokay]

> mathematics is a social construct

If you believe Wittgenstein then all of math is more and more complicated stories amounting to 1=1. Like a ribbon that we figure out how to tie in ever more beautiful knots. These stories are extremely valuable and useful, because we find equivalents of these knots in nature—but boiled down that is what we do when we do math

[ianhorn]

I like the Kronecker quote, "Natural numbers were created by god, everything else is the work of men" (translated). I figure that (like programming) it turns out that putting our problems and solutions into precise reusable generalizable language helps us use and reuse them better, and that (like programming language evolution) we're always finding new ways to express problems precisely. Reusability of ideas and solutions is great, but sometimes the "language" gets in the way, whether that's a programming language or a particular shape of the formal expression of something.

[anthk]

More like 1 = 0 + 1.

Read about Lisp, the Computational Beauty of Nature, 64k Lisp from https://t3x.org and how all numbers can be composed of counting nested lists all down.

List of a single item:

     (cons '1 nil)

Nil it's an empty atom, thus, this reads as:

[ 1 | nil ]

List of three items:

    (cons '1 (cons 2 (cons 3 nil)))

Which is the same as

    (list '1 '2 '3)

Internally, it's composed as is, imagine these are domino pieces chained. The right part of the first one points to the second one and so on.

[ 1 | --> [ 2 | -> [ 3 | nil ]

A function is a list, it applies the operation over the rest of the items:

     (plus '1 '2 3') 

Returns '6

Which is like saying:

  (eval '(+ '1 '2 '3))

'(+ '1 '2 '3) it's just a list, not a function, with 4 items.

Eval will just apply the '+' operation to the rest of the list, recursively.

Whis is the the default for every list written in parentheses without the leading ' .

    (+ 1 (+ 2 3))

Will evaluate to 6, while

    (+ '1 '(+ '2 '3)) 

will give you an error as you are adding a number and a list and they are distinct items themselves.

How arithmetic is made from 'nothing':

https://t3x.org/lisp64k/numbers.html

Table of contents:

https://t3x.org/lisp64k/toc.html

Logic, too:

https://t3x.org/lisp64k/logic.html

[_alternator_]

You don’t really have to believe Wittgenstein; any logician will tell you that if your proof is not logically equivalent to 1=1 then it’s not a proof.

[justonceokay]

Sure, I just personally like his distinction between a “true” statement like “I am typing right now” and a “tautological” statement like “3+5=8”.

In other words, declarative statements relate to objects in the world, but mathematical statements categorize possible declarative statements and do not relate directly to the world.

[IsTom]

If you look from far enough, it becomes "Current world ⊨ I am typing right now" which becomes tautological again.

[sesm]

In my view mathematics builds tools that help solve problems in science.

[_alternator_]

This is known as “applied mathematics”.

[nathan_compton]

Sounds lame and boring to me.

[adrianN]

There is a bit about this in Greg Egan‘s Disspora, where a parallel is drawn between maths and art. It is not difficult to automate art in the sense that you can enumerate all possible pictures, but it takes sentient input to find the beautiful areas in the problem space.

[SabrinaJewson]

I do not think this parallel works, because I think you would struggle to find a discipline for which this is not the case. It is trivial to enumerate all the possible scientific or historical hypothesis, or all the possible building blueprints, or all the possible programs, or all the possible recipes, or legal arguments…

The fact that the domain of study is countable and computable is obvious because humans can’t really study uncountable or uncomputable things. The process of doing anything at all can always be thought of as narrowing down a large space, but this doesn’t provide more insight than the view that it’s building things up.

[seanmcdirmid]

Automating proofs is like automating calculations: neither is what math is, they are just things in the way that need to be done in the process of doing math.

Mathematicians will just adopt the tools and use them to get even more math done.

[quietbritishjim]

I don't think that's true. Often, to come up with a proof of a particular theorem of interest, it's necessary to invent a whole new branch of mathematics that is interesting in its own right e.g. Galois theory for finding roots of polynomials. If the proof is automated then it might not be decomposed in a way that makes some new theory apparent. That's not true of a simple calculation.

[seanmcdirmid]

> I don't think that's true. Often, to come up with a proof of a particular theorem of interest, it's necessary to invent a whole new branch of mathematics that is interesting in its own right e.g. Galois theory for finding roots of polynomials. If the proof is automated then it might not be decomposed in a way that makes some new theory apparent. That's not true of a simple calculation.

Ya, so? Even if automation is only going to work well on the well understood stuff, mathematicians can still work on mysteries, they will simply have more time and resources to do so.

[ndriscoll]

This is literally the same thing as having the model write well factored, readable code. You can tell it to do things like avoid mixing abstraction levels within a function/proof, create interfaces (definitions/axioms) for useful ideas, etc. You can also work with it interactively (this is how I work with programming), so you can ask it to factor things in the way you prefer on the fly.

[integralid]

>This is literally the same thing as

No.

>You can

Not right now, right? I don't think current AI automated proofs are smart enough to introduce nontrivial abstractions.

Anyway I think you're missing the point of parent's posts. Math is not proofs. Back then some time ago four color theorem "proof" was very controversial, because it was a computer assisted exhaustive check of every possibility, impossible to verify by a human. It didn't bring any insight.

In general, on some level, proofs like not that important for mathematicians. I mean, for example, Riemann hypothesis or P?=NP proofs would be groundbreaking not because anyone has doubts that P=NP, but because we expect the proofs will be enlightening and will use some novel technique

[ndriscoll]

Right, in the same way that programs are not opcodes. They're written to be read and understood by people. Language models can deal with this.

I'm not sure what your threshold for "trivial" is (e.g. would inventing groups from nothing be trivial? Would figuring out what various definitions in condensed mathematics "must be" to establish a correspondence with existing theory be trivial?), but I see LLMs come up with their own reasonable abstractions/interfaces just fine.

[jhanschoo]

There are areas of mathematics where the standard proofs are very interesting and require insight, often new statements and definitions and theorems for their sake, but the theorems and definitions are banal. For an extreme example, consider Fermat's Last Theorem.

Note on the other hand that proving standard properties of many computer programs are frequently just tedious and should be automated.

[seanmcdirmid]

Yes, but > 90% of the proof work to be done is not that interesting insightful stuff. It is rather pattern matching from existing proofs to find what works for the proof you are currently working on.

If you've ever worked on a proof for formal verification, then its...work...and the nature of the proof probably (most probably) is not going to be something new and interesting for other people to read about, it is just work that you have to do.

[jhanschoo]

You're right, I misread your comment. Apologies.

[storus]

There are still many major oversimplifications in the core of math, making it weirdly corresponding with the real world. For example, if you want to model human reasoning you need to step away from binary logic that uses "weird" material implication that is a neat shortcut for math to allow its formalization but doesn't map well to reasoning. Then you might find out that e.g. medicine uses counterfactuals instead of material implication. Logics that tried to make implication more "reasonable" like relevance logic are too weak to allow formalization of math. So you either decide to treat material implication as correct (getting incompleteness theorem in the end), making you sound autistic among other humans, or you can't really do rigorous math.

[jojomodding]

People keep getting hung up on material implication but it can not understand why. It's more than an encoding hack--falsity (i.e. the atomic logical statement equivalent to 0=1) indicates that a particular case is unreachable and falsity elimination (aka "from falsity follows everything") expresses that you have reached such a case as part of the case distinctions happening in every proof.

Or more poetically, "if my grandmother had wheels she would have been a bike[1]" is a folk wisdom precisely because it makes so much sense.

1: https://www.youtube.com/watch?v=A-RfHC91Ewc

[storus]

Material implication was not the default implication historically; it came as a useful hack by people who hoped that by enforcing it they could formalize the whole math and knowledge and have a sort of a "single source of truth" for any statement, and evaluate all statements purely syntactically. This proved to be futile as incompleteness theorem showed, and which material implication directly enabled by allowing self-referential non-sense as valid statements. There were many attempts to reconcile this with different logics but they all ended up weaker and unable to formalize all statements. We are now entering the next phase of this attempt, by using hugely complex reasoning function approximators as our "single source of truth" in the form of AI/LLMs.

I used to do a lot of proofs coming all the way from Peano arithmetics, successor operators and first-order tableaux method.

[YetAnotherNick]

The thing is if something is proved by checking million different cases automatically, it makes it hard to factor in learning for other proofs.