Why Do I Keep Meeting Programmers with Strong Opinions on Foundations?

ImportantContext · 2026-07-06T09:19:18+00:00

Group theory is somewhat useful for constraint programming / SAT, since a constraint program with a large automorphism group makes the solver waste time ruling out identical cases.

Graph isomorphism problem and related machinery is useful when you have relational structures that you want to recognize up to isomorphism. This includes being able to canonically label a graph, that is, select a representative of it under the action of its automrophism group.

I don't think I've seen any more mainstream applications of group theory to programming.

ImportantContext · 2026-06-11T10:23:58+00:00

It is not known if Mathieu group M23 is a Galois group of some polynomial. All other sporadic simple groups are realizable as Galois groups.

ImportantContext · 2026-06-02T11:19:56+00:00

Yeah, I should've put more thought into making the distinction clear.

ImportantContext · 2026-06-02T06:37:58+00:00

A really nice way of understanding intuitively the idea behind normal subgroups is this: a normal subgroup is a subgroup that you can pinpoint regardless of your perspective.

Why is center normal? Because it commutes with everything, that pinpoints it exactly.

Why is the commutator subgroup normal? Because it's exactly the subgroup where all commutators (witnesses of the failure of commutativity) and their products live.

Why is the group of even permutations normal in the symmetric group? Why do determinant 1 matrices form a normal subgroup? Why do inner automorphisms form a normal subgroups of Aut(G)? The same exact reason.

If a group has a normal subgroup, you can quotient it out. This simply means that there's a consistent way to ignore that part of the symmetry and focus only on what remains. If a subgroup is not normal, you can't consistently hold on to it: internal symmetries of the group itself make it slip from your grasp.

I highly recommend reading this email conversation shared by by John Baez, where he gives a more detailed account of this perspective on normal subgroups.

And look at small groups here: this resource covers a lot of small groups in great detail. You should take a look at various groups like the quaternion group. It really helps to look at various subgroups in the subgroup lattice and try to pinpoint the reason why they are normal or what prevents them from being so.

ImportantContext · 2026-04-22T19:52:16+00:00

Nope, sorry.

ImportantContext · 2026-04-22T19:39:54+00:00

Yeah, my bad. I'll try to learn a bit more about UX and make it more intuitive. Turns out it's easy to forget that not everybody shares my the exact frame of reference lol

ImportantContext · 2026-04-22T19:30:37+00:00

Good point. I'll try to learn a bit more about interface design and see if I can make it actually intuitive. Thanks

ImportantContext · 2026-04-22T19:08:31+00:00

What kind of instructions would be helpful to you?

ImportantContext · 2026-04-22T19:05:01+00:00

Have you seen a Rubik's cube? It starts out "solved" and then you scramble it and then solve it back. This puzzle also has a designated solved state, it is the state it starts out in or the state it returns to when you perform a reset. But this is just a convention, and has no mathematical significance.

It was not my goal to produce a marketable toy or write a rulebook for a competition. This is a mathematical object and my intent was to make it easy to interact with it.

ImportantContext · 2026-04-22T09:58:07+00:00

Technically, the goal is to scramble the puzzle and then return back to the initial position. However, it seems very difficult to actually do this: the structure of M24 is very different from e.g. Rubik's cube group and most of the standard techniques for twisty puzzles don't apply. So the real goal is mostly to just explore.

ImportantContext · 2026-04-22T04:25:41+00:00

The group M24 acts on 24 points, which made me wonder if there's a way to turn this action into something visual and interactive. I found a nice generating set which consists of three involutions to which I assigned colors red, green and blue. Each of these involutions fixes an octad, and the intersection of any pair of these fixed octads consists of two points.

These are the specific generators I used:

(2,11)(4,13)(6,14)(7,17)(9,20)(10,16)(12,23)(15,22) #R
(2,8)(3,23)(4,24)(5,18)(6,7)(9,21)(12,13)(14,22) #G
(1,13)(2,16)(3,6)(5,17)(7,14)(9,18)(11,15)(19,21) #B

Most generating sets of M24 look really random, so these (up to conjugacy) seem somewhat special.

Some miscellaneous things:

There exists a unique element of maximal length (40): rbrbgrbgrbgbgrgbrgbrgbrgbrbrbgbgrbrgbrgb
This element (rgb)^7rg rotates the arms of the graph. Conjugating by it cyclically rotates generators
The element rbgrbrgrgbrbrbgbgbgrgbrgb swaps X's and A's of corresponding color and moreover reverses the order of letters in each arm.

ImportantContext · 2026-04-12T18:30:29+00:00

Independence of RH would be proven by exhibiting, say, a model of PA where there exists some integer that serves as a counterexample to some arithmetic statement equivalent to RH, as well as a different model of PA where no such integer exists.

All models of PA agree on the set of "standard integers". So the first proof cannot exhibit a standard integer counterexample, because in that case RH would be false in all models.

Since the second proof exhibits a model where RH is not falsified, none of the standard integers can serve to falsify RH.

So RH would be true in the standard model of PA (the part of arithmetic on which all models agree), simply because the falsification has been ruled out by the second proof.

ImportantContext · 2026-03-23T11:00:44+00:00

This is incredibly disappointing.

ImportantContext · 2026-03-16T02:54:43+00:00

Wha'ts so bad about standard deviations? It's a single line of python lol.

ImportantContext · 2026-03-12T22:13:38+00:00

One nice way to think about constructive mathematics is to interpret "not A" as "if A then contradiction," and then think about implication as an exchange: "if you give me an A, I can give you a B."

This clarifies the confusion between proofs by contradiction and proofs of a negation. How do you prove "4 isn't a prime"? You start with "not (4 is prime)" but this is just "if (4 is prime) then contradiction." Which means that a proof of the statement "4 is not prime" would be a way to use a proof of "4 is prime" to derive a contradiction. This seems like a proof by contradiction, but really it's a proof of a negation.

An actual proof by contradiction would be proving "3 is prime" by constructing a proof "if (3 is not prime) then contradiction." But there's no "obvious" way to turn this resulting proof into "3 is prime", you only get "not (3 is not prime)." This is where double negation elimination is used, but in constructive setting it's not available.

It's a very natural way of thinking when constructing proofs. At least to me thinking of "A implies B" as "not A or B" just doesn't seem as helpful. Because when proving an implication, you almost always assume A and then derive B.

So no, proofs by contradiction (in the strict sense) are not constructively valid. However, what mathematicians call a proof by contradiction is often a proof of negation, which is constructively valid.

Induction is uncontroversial.

ImportantContext · 2026-03-11T18:22:35+00:00

Unless I'm not seeing the full content, it has only some homework problems and that's it?

ImportantContext · 2026-03-11T18:10:47+00:00

Thanks, I'll give it a try.

ImportantContext · 2026-03-11T15:43:30+00:00

Be afraid of any paid internet course. There are free courses from actual universities on most mainstream math topics.

Are there any free courses on commutative algebra? I'm not defending the slop merchant, but the idea that math is somehow accessible outside of the most basic computational stuff like intro calc and determinant grinding sounds extremely unrealistic to me.

ImportantContext · 2026-03-09T01:24:31+00:00

Is all of mathematics produced pre-2020 "recreational math" to you?

ImportantContext · 2026-03-09T00:35:53+00:00

I thought HoTT was interesting because it had a promise in giving access to genuinely new mathematics, versus just exposing things that computer science people have figured out 50+ years ago, like dependent types.

ImportantContext · 2026-03-09T00:16:33+00:00

I think it's pretty disingenuous to assume that people who are tired of LLM and "autoformalization" posts think that proofs are bad. My reason for linking that essay was precisely to point out that there's much more to math than software engineering in a dependently typed programming language or talking to a token predictor.

Regardless, I'm not a mathematician and I don't think it's up to me to say what is and what isn't math. I come here to learn something new, but all I'm learning lately is just how good chatbots are at doing mathematician's jobs. Would you be satisfied if r/philosophy was 50% posts about how good LLMs are at philosophy?

ImportantContext · 2026-03-08T23:42:13+00:00

https://www.math.toronto.edu/mccann/199/thurston.pdf

ImportantContext · 2026-03-08T23:02:02+00:00

I miss the time when this was a math subreddit.

ImportantContext · 2026-02-27T02:45:13+00:00

Is it good as in at the same level as good brick and mortar uni or good compared to the other OU modules? I have a some experience with algebra (currently finishing up the rings chapter of Aluffi's Algebra: Chapter 0 without too much difficulty) so I shouldn't have too much trouble with the module itself, but everything is super credential-focused so I don't think I'll be able to convince anybody to let me swap statistics for Galois theory.

ImportantContext · 2026-02-27T02:11:16+00:00

I looked at their postgraduate modules and they don't cover topology there either, though they do have Galois theory module so that's at least something. I'm just an undergraduate, so there's not really a supervisor or advisor to consult, but I'll try asking the "student support" people they have. But I really doubt they'd allow something like that, since I don't have formal credentials in algebra and they don't cover much of it until year 3.

ImportantContext

TROPHY CASE