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I don’t understand this argument. Just because functional extensionality is undecidable for arbitrary functions doesn’t mean that it is undecidable for every class of functions.

In the specific situation, let’s say that by an array we mean a finite, ordered list whose entries are indexed by the numbers 0, 1, …, n - 1 for some natural number n. Let’s also say that two arrays are equal if they have the same length and the same value at each position (in other words, they have “the same elements in the same order”).

If we now want to represent a function f as an array arr such that f(i) = arr[i] for every possible input i of f, then this will only be possible for some very specific functions: those whose domain are the set {0, 1, …, n - 1} for some natural number n. But for any two such functions f, g : {0, 1, …, n - 1} → t, their extensional equality is logically equivalent to the equality of the corresponding arrays: you really can check that f and g are extensionally equal by checking that they are represented by equal arrays.

It’s also the standard notation for zero objects, i.e., for terminal objects that are also initial. This entails all abelian/additive/preadditive categories, such as categories of modules, vector spaces, or abelian groups. (But there are also counterexamples, such as the categories of groups and of pointed sets.)

But I’d agree that it’s not standard notation to use 0 for a terminal object in an arbitrary category. I’d guess that most people use 1 instead, so that for example 1 × X ≅ X. (The post talks about group objects in the category of algebraic varieties (over some field), in which case 1 seems to be more appropriate than 0.)

> every finite set of vectors, with your favorite metric, can be embedded in euclidean space with at most ~41% relative error

I’ve never heard of this before. Do you have a reference?

I think these are two different questions: - Why care about radicals? - Why try to solve polynomial equations in terms of radicals?

For the first question:

Taking Nth powers is a fairly basic operation, which occurs all the time in mathematics. Taking Nth roots is simply the inverse operation, so it is fairly natural to be interested in it/having to deal with it.

For the second question:

Let’s pretend for a moment that we didn’t know how the quadratic formula looked like. Could we nevertheless say anything about it?

The quadratic formula is supposed to give us the solutions to the equation a x^2 + b x + c = 0. A special case of this general quadratic equation is x^2 - p = 0. There are two ways of solving this specialized equation: either by taking a square root, giving us the two solutions ±√p, or by using the general quadratic formula (with a = 1, b = 0, c = -p). Both of these approaches need to give us the same results, since they are both correct.

This tells us that if we simplify the quadratic formula with a = 1, b = 0, c = -p, then a square root needs to appear. How can this happen? Well, the most basic guess is that the quadratic formula contained at least one square root to begin with.

Looking at the actual quadratic formula tells us that this guess is correct: the formula uses the four basic arithmetic operations (addition, subtraction, multiplication, division) and a square root.

We can repeat the same thought experiment for cubic equations, and we find that the cubic formula should probably contain third roots. Looking up the formula confirms this suspicion. However, it should be noted that the cubic equation does not only contain third roots, but also square roots.

The situation for the quartic equation is similar: we suspect that the quartic formula contains fourth roots. And thanks to our experience with the cubic formula, we may also suspect that the quartic formula contains third roots and square roots. Looking up the formula, we see that it contains both third roots and square roots, but not (directly) any fourth roots. (Our original idea breaks down a bit because fourth roots can be expressed as iterated square roots. This makes it possible that the general quartic formula does not contain fourth roots, even though its simplified version will contain them.)

So what about a general polynomial equations of degree N >= 5? Our original observation tells us that a solution formula needs to contain some sort of operation(s) that, when the formula is applied to certain special cases, gives us Nth roots. Just as before, the most basic guess is that the formula will contain Kth roots, and the previous examples suggest that one should expect K = 2, ..., N to occur.

Summary: To find a formula for polynomials equations of degree N >= 2, we are forced to use additional operations apart from the four basic arithmetic operations. In certain special cases, these additional operations need to simplify to roots. This suggests using roots in the formula, and the cases N = 2, 3, 4 support this idea.

Heuristically speaking, we are not trying to use roots because we want to, but because they seem to be the bare minimum required to even hope of finding a formula.

> Lamport says that either pronunciation is acceptable […].

This fits what Knuth said about the issue in 2006: “[…] lah-tek, lay-tek, the author never has decided how to pronounce it […].” [1]

[1]: https://www.youtube.com/watch?v=8HuwiBPLV3A&t=41s

Barany published his findings as a short article in Nature [1]. He also talked about the topic in an episode of the podcast “My Favorite Theorem” [2].

[1]: https://www.nature.com/articles/d41586-018-00513-8 [2]: https://kpknudson.com/my-favorite-theorem/2020/11/12/episode... (from 21:40 to 30:44).

MathJax 3 currently only supports the default font, as more general font support is still under development [1]. According to the relevant GitHub issue, font support is planned to be part of the next major release, which is supposed to arrive sometime this year [2]. This should then include Gyre Pagella out of the box (as was the case for MathJax 2), but also make it possible to use custom fonts (which apparently wasn’t possible with MathJax 2) [3].

[1]: http://docs.mathjax.org/en/latest/output/fonts.html [2]: https://github.com/mathjax/MathJax/issues/2503#issuecomment-... [3]: https://github.com/mathjax/MathJax/issues/2503#issuecomment-...

Variable names in mathematics have no intrinsic meaning (as they have, for example, in physics). In a mathematical text, every occuring variable must be properly defined. This is most commonly done before they are used, with formulations like “let x be …” or “x := …”, or immediately after they have just been used in a formula, with something like “where x denotes …”. Failing to do so is just as much of a mistake in mathematics as it is in programming. (In an homework assignment or exam, doing so will lose you points.)

In praxis one should be aware of the following points:

  - In programming, the computer will complain if an undefined variable is used. In mathematics, this is sadly missing. (The next best things are other proofreaders, i.e. other mathematicians.)

  - Variable names aren’t just picked at random (or as a, b, c, …), but nearly always follow sensible patterns. (Natural numbers are n, m, k, l, …; vectors are v, w, u, …; indices are i, j, k, l, …; radius is r; …) Different authors may use different conventions, but they still allow mathematicians to kind of understand what the variable means just from looking at its name.

  - Every area of mathematics has certain keywords which the reader has to be aware of. Again, some authors may use (slightly) different conventions, but there are typically only few conventions out there, and they often don’t differ much. (Example: The space of homomorphisms/linear maps between two vector spaces V and W is commonly denoted by Hom(V, W), hom(V,W), ℒ(V, W) or Lin(V, W).) One can oftentimes tell what a keyword means just from it’s name, its signature, and its usage. Keywords also often consist of more than one letter or are typeset in a special way to distinguish them from regular variables.)

Good mathematical writers will oftentimes go out of their way to explain their notation at the beginning of their text, just to be sure.

> Are mathematicians ever frustrated that they don't understand what the variables mean?

So to answer the question: if a mathematician doesn’t understand what a variable or a notation means, then one of the following has happend:

  - The variable was already introduced beforehand, but the reader forgot about it. (This is the most common scenario.)

  - The variable is explained in the upcoming line. (Also very common. The reader will—of course—only notice this after going through the previous part of the text multiple times in seach of just this explanation.)

  - It is a standard notation that the reader is not familiar with. (Often happens if the reader is missing the background knowledge assumed by the author, or if the author uses some outdated notation (e.g. because they have been dead for over 50 years).)

  - The author made a simple mistake while writing. (Typo; forgot to change a variable name after shuffeling things around).

  - The author actually forgot to define the variable: a mistake that is hopefully catched by their peers.

  - The explanatory text was left out for time reasons (giving a talk, writing some rough/informal lecture notes, quickly scribbling down homework in the morning).

There is also “Mathematical Writing” by Knuth: https://jmlr.csail.mit.edu/reviewing-papers/knuth_mathematic...