clintonc

I agree with many of the sentiments here. But an open letter signed exclusively by Fields Medalists that purport to define precisely what the "mathematical community" (who is inside and outside) and what their goals are raises my hackles for some reason.

It is true that the manufacturing of "true/false" statements is not the same as gaining understanding of a problem. However, for many mathematicians, true/false statements are already manufactured by others. Think of a student who is given a conjecture to investigate, with their advisor describing it as "it must be true". Most exercises in a textbook are stated such that the outcome is known before you begin. That's not really a problem -- investigating the conjecture/exercise yields its own dividends, whether or not the outcome is known. It is also the case that defining new directions involve understanding and synthesizing related problems, asking the right questions, and deciding upon the right directions, and it's not clear that AI can do that at all.

The real risk, I think, its the public's (and funding agencies') perception of the importance of "human" mathematics, but that's already a struggle. For example, it's tough to explain to the lay person why it's still important to research group theory -- the main example people cite is RSA encryption, which was invented almost 50 years ago.

I wish this were organized according to suggested/logical reading order. For example, the paper introducing the attention mechanism probably ought to precede "attention is all you need".

My mind went to Java's checked exceptions -- not sure if anyone today believes that coloring is still a good idea.

I love this. Thank you.

I have a Ph.D. in a field of mathematics in which complex numbers are fundamental, but I have a real philosophical problem with complex numbers. In particular, they arose historically as a tool for solving polynomial equations. Is this the shadow of something natural that we just couldn't see, or just a convenience?

As the "evidence" piles up, in further mathematics, physics, and the interactions of the two, I still never got to the point at the core where I thought complex numbers were a certain fundamental concept, or just a convenient tool for expressing and calculating a variety of things. It's more than just a coincidence, for sure, but the philosophical part of my mind is not at ease with it.

I doubt anyone could make a reply to this comment that would make me feel any better about it. Indeed, I believe real numbers to be completely natural, but far greater mathematicians than I found them objectionable only a hundred years ago, and demonstrated that mathematics is rich and nuanced even when you assume that they don't exist in the form we think of them today.

Curious to know what makes this "a proper VT100 implementation in the browser, not a JavaScript approximation of one" -- isn't Ghostty also an approximation, just implemented in a different language? Seems unnecessarily pejorative to me.

Whale oil and solar panels both being signs of high status.

The initial board position is certainly reachable (and reached in every game!), but there are only 20 legal moves available: the 16 legal pawn moves for White, and the 4 legal knight moves for White.

Then use 1/3 instead of 1/2 for a combined length of 2/3 -- the total length of the intervals can be as small as you like. This hints at the fact that any countable subset of the real numbers is Lebesgue measure zero.

Even using 1/2, the set that remains is nonempty due to the Cantor intersection theorem. The total length of the intervals is 1, which means that the remainder has no "interior" (i.e., contains no open interval), but the converse is not true: removing intervals whose lengths sum to less than one does not mean that the remainder will contain any interval. This is the consideration that allows you to create what are called "fat Cantor sets" -- the middle thirds Cantor set has Lebesgue measure zero, but by removing smaller intervals you can get other, homeomorphic sets that have positive measure.

There are a couple of strategies for understanding the real numbers. One is to write down a definition of real numbers, for example using rational numbers and Dedekind cuts, hoping that what you're describing is really what you mean. The other is to write down the properties of real numbers as you understand them as "axioms", and go from there. An important property of real numbers that always comes up (either as a consequence of Dedekind cuts or as an axiom itself) is the least upper bound property -- every set which has an upper bound has a least upper bound. That's what gives you the "completeness" of the real numbers, from which you can prove facts like the completeness of the real numbers (i.e., Cauchy sequences always converge), the Heine-Borel theorem (closed and bounded subsets of the reals are "compact", and vice-versa), and Cantor's intersection theorem (that the nested intersection of a sequence of non-empty compact sets is also compact).

The diagonalization argument is an intuitive tool, IMHO. It is great if it convinces you, but it's difficult to make rigorous in a way that everyone accepts due to the use of a decimal expansion for every real number. One way to avoid that is to prove a little fact: the union of a finite number of intervals can be written as the finite union of disjoint intervals, and that the total length of those intervals is at most the total length of the original intervals. (Prove it by induction.)

THEOREM: [0, 1] is uncountable. Proof: By way of contradiction, let f be the surjection that shows [0, 1] is countable. Let U_i be the interval of length 1/2*i centered on f(i). The union V_n = U_1 + U_2 + ... + U_n has combined length 1 - 1/2*n < 1, so it can't contain [0, 1]. Another way to state that is that K_n = [0, 1] - V_n is non-empty. K_n also compact, as it's closed (complement of V_n) and bounded (subset of [0, 1]). By Cantor's intersection theorem, there is some x in all K_n, which means it's in [0,1] but none of the U_i; in particular, it can't be f(i) for any i. That contradicts our assumption that f is surjective.

Through the right lens, this is precisely the idea of the diagonalization argument, with our intervals of length 2*-n (centered at points in the sequence) replacing intervals replacing intervals of length 10*-n (not centered at points in the sequence) implicit in the "diagonal" construction.

You can get that every integral domain is a field with fewer words by using a higher powered set theory result -- injections on finite sets are also surjections. The cancellation property says multiplication by any element is an injection, so it is also a surjection, i.e., 1 is in the range, so that gives you the multiplicative inverse.

From the page about dwm:

> Because dwm is customized through editing its source code, it's pointless to make binary packages of it. This keeps its userbase small and elitist. No novices asking stupid questions.

...sucks less than what? :) Simple is good, but simpler does not necessarily mean better.