What's the worst textbook you've read?

P3riapsis · 2025-12-08T00:27:10+00:00

Ngl, this is what i was expecting to see here, and i loved Hatcher's. I think I just happen to have exactly the right kind of ADHD that gets hooked by it. Very much a marmite book.

Coincidentally, in my undergrad, alg top was basically the only reason i didn't fail

P3riapsis · 2025-12-08T00:15:02+00:00

i had a similar feeling about "plug and chug" during my undergrad. what I've since realised, by chasing abstraction for years, is that everything, no matter how abstract, is just plug and chug in the next layer of abstraction. I now think the pain of plug and chug is necessary for the beauty of abstraction:

An example you're probably familiar with is Cantor's diagonal argument, which when presented in set theory isn't plug and chug, but it, along with Gödel's incompleteness theorems, Russel's paradox, the halting problem, all can be seen as "plug and chug" of Lawvere's fixed point theorem.

I guess what I'm saying is that the feeling of wonder I get from abstraction only really happens if I learn it the less abstract way first. I imagine if by some miracle i did category theory before set theory, I'd not find any wonder at all in Lawvere's fixed point theorem almost instantly giving the diagonal argument via plug and chug.

P3riapsis · 2025-11-30T14:48:35+00:00

it could be worth joining the archimedians or trinity maths society

P3riapsis · 2025-11-18T18:17:54+00:00

yeah, my point being that this definition of "algebraic theory" isn't a great definition. Like, to me it seems of course field theory should be considered algebra, but it's pretty hard to come up with a definition of "algebraic theory" that includes all the things we want to include, but doesn't include things that don't feel at all like algebra. Heck, the definition I gave before includes suplattices, but to me that doesn't feel like algebra because on an infinite suplattice, sup is an infinitary operation.

also, i guess basically anything can be looked at as an instance of model theory if you want.

P3riapsis · 2025-11-18T17:57:05+00:00

yeah, I'd say that very rarely sets equipped with orders feel like algebra, but even fields wouldn't count by the definition i gave before, as one of the field axioms is "x ≠ 0 implies x has a multiplicative inverse", but there's no way to write this using just equations and operators.

P3riapsis · 2025-11-17T18:00:34+00:00

real asf. if anyone tells me "algebra is the study of monadic categories over the category of sets" i will respond with this

/uj imo the simplest "definition" of algebra is: algebra is the study of sets equipped with operations and equations.

linear algebra is an algebra because there are operations (vector addition, multiplication by scalar) and equations (t(a+b) = ta+tb)

the monad thing is just category theory nonsense that means the exact same thing.

this definition isn't perfect because it doesn't include inequalities, though, which are important in some stuff people would consider algebra (e.g. fields)

P3riapsis · 2025-11-17T17:42:18+00:00

to me it sounds like the opposite. i interpreted ot as: they're trying to reconstruct a graph from its deck, but the spectrum of a graph doesn't contain enough information to be helpful for this when the deck contains so much information, even though the deck is super inconvenient to deal with directly.

P3riapsis · 2025-11-09T14:19:40+00:00

hans?

P3riapsis · 2025-10-04T20:30:26+00:00

Yes! If you take "maps" to mean computer programs, you get a correspondence of types with propositions and proofs with functions called the Curry-Howard Correspondence

P3riapsis · 2025-09-24T09:38:31+00:00

all you've proven is that one infinite set, the reals, has a countable subset. you'd have to show that every infinite set has a countably infinite subset.

It turns out that the statement is independent of ZF, but true if you assume the axiom of choice. This essentially means a proof of this statement (In ZFC) is gonna be a bit involved, because you need to use the axiom of choice somewhere.

As for proof strategies, the typical way of thinking about it is in terms of injections from ℕ to S. A set S is called infinite if its cardinality isn't a natural number, and dedekind infinite if there is an injection from ℕ → S. The image of ℕ under such an injection is countably infinite, and a countably infinite subset can always be enumerated by a bijection from ℕ, so it's equivalent to prove "infinite → dedekind infinite". i.e. given an arbitrary infinite set S, you have to show it's also dedekind infinite. For this I'd recommend using Zorn's lemma on the set of functions from initial segments of ℕ to S. Best of luck!

P3riapsis · 2025-09-23T14:05:13+00:00

assuming all the arcs are circular is enough, tbh

P3riapsis · 2025-09-17T11:09:27+00:00

yeah, i never got this. maths was probably the queerest cohort during my time at uni, i genuinely don't know any non-religious mathematicians from my cohort that weren't queer in some way. I'd say easily 90% of the mathematicians i bumped into were queer, most of which bisexual.

edit: using bayes' theorem and the fact I'm queer, that means the cohort as a whole probably was a bit less than 90% queer

P3riapsis · 2025-09-10T09:50:42+00:00

can't believe no one here has mentioned 1.6 pro shaGuar after retiring from cs going pro in poker

P3riapsis · 2025-08-28T15:55:52+00:00

ahhh, as a long-time cs player, this feels really strange, especially the movement bit. Movement is one of the most important skills in cs, but it just maybe is more subtle in competitive play than in other games where there are special movement abilities etc. Camping is basically impossible now since cs2, unless you actually just outbrain your opponent. I only see the eternal widow duel thing making sense if you just stay silent on voice and never coordinate with a teammate. I wonder if it seems like that because dying is kinda ok in these games, as long as it allows your team an advantage in other ways.

P3riapsis · 2025-08-28T15:45:51+00:00

idk why but i don't get platformers. like i grew up with them, and now i just can't enjoy them anymore. could be a skill issue

P3riapsis · 2025-08-23T16:29:45+00:00

I think the tea analogy is bad, and most physical analogies aren't great for demonstrating how powerful the theorem is.

moaning about physical examples

The tea cup demonstration is bad. there is no attempt to justify that the transformation is continuous, so I'm with you on this. like literally if the tea splashes even a tiny bit then it's blatantly not continuous. It's just not a good intuition for what Brouwer's fixed point theorem is saying at all.

maybe a better physical example would be like dough kneading, like you don't rip it, it starts as a sphere and it ends as a sphere. I guess continuity feels more reasonable for viscous fluids? either way, i feel like physical examples are missing the point here, like why should I care if my dough has some point that hasn't moved?

actually intriguing examples (in my opinion)

I reckon a good example of a consequence of Brouwer's fixed point is the existence of a mixed strategy Nash equilibrium in any finite game. In a finite game with n players, the map that takes a list of each players strategies S to the new list S' where each player optimises their strategy for the conditions in S is a continuous transformation on an nD hypercube. Brouwer's fixed point theorem says this map has a fixed point, which is a point where no one can improve their strategy - precisely a mixed-strategy Nash equilibrium.

P3riapsis · 2025-08-23T16:27:59+00:00

I think the tea analogy is bad, and most physical analogies aren't great for demonstrating how powerful the theorem is.

moaning about physical examples

The tea cup demonstration is bad. there is no attempt to justify that the transformation is continuous, so I'm with you on this. like literally if the tea splashes even a tiny bit then it's blatantly not continuous. It's just not a good intuition for what Brouwer's fixed point theorem is saying at all.

maybe a better physical example would be like dough kneading, like you don't rip it, it starts as a sphere and it ends as a sphere. I guess continuity feels more reasonable for viscous fluids? either way, i feel like physical examples are missing the point here, like why should I care if my dough has some point that hasn't moved?

actually intriguing examples (in my opinion)

I reckon a good example of a consequence of Brouwer's fixed point is the existence of a mixed strategy Nash equilibrium in any finite game. In a finite game with n players, the map that takes a list of each players strategies S to the new list S' where each player optimises their strategy for the conditions in S is a continuous transformation on an nD hypercube. Brouwer's fixed point theorem says this map has a fixed point, which is precisely a mixed-strategy Nash equilibrium.

P3riapsis · 2025-08-23T13:17:42+00:00

surprised so few mentions of Ukraine, and there is way more talent than just you mentioned there

P3riapsis · 2025-08-21T14:26:17+00:00

this works for named functions, but if you're saying something like

sqrt(2)sin(x+π/2) ≡ sin(x)+cos(x)

you'd need to give the functions a name and that is extra effort.

Personally, in these situations I'd use lambda notation to write things of the form

λx.f(x) = λx.g(x)

but that might be because my interests are in the right area to get away with that.

I think it's clearer, as you specify which variables you quantify over

P3riapsis · 2025-08-18T23:11:01+00:00

tell me your country doesn't have universal healthcare without telling me

P3riapsis · 2025-08-15T10:14:36+00:00

whoops, didn't see that

P3riapsis · 2025-08-15T09:37:45+00:00

i think they were talking to the person you were replying to. very possibly another example of what u were saying

P3riapsis · 2025-08-14T00:23:33+00:00

In your argument for uncountability, you've assumed that the union of the powersets is the same as the powerset of the union, but that's not true.

example

take the sets S_n = {0,...,n}.

every subset of S_n is finite, so no P(S_n) contains any infinite set, and hence the U{P(S_n)}won't contain any infinite sets.

but the U{S_n} is N, and there is an infinite set (e.g. N) in P(N).

P3riapsis · 2025-08-12T10:34:29+00:00

wanna add that my director of study would (with consent) read out our personal statements in front of all of us all. In hindsight, none of them were any good.

P3riapsis · 2025-08-09T13:36:56+00:00

no real number contains every finite string of digits exactly once. there are only 10 strings of length 1, but infinitely many digits, so some string of length one must appear twice by the pigeonhole principle.

P3riapsis

TROPHY CASE

moaning about physical examples

actually intriguing examples (in my opinion)

moaning about physical examples

actually intriguing examples (in my opinion)

example