In praise of Replacement

Mar 06 2019, 22:20

I have often seen people complain about Replacement axioms. For example, this MathOverflow question, or this one, or that one, and also this one. This technical-looking schema of axioms state that if \(\varphi\) defines a function on a set \(x\), then the image of \(x\) under that function is a set. And this axiom schema is a powerhouse! It is one of the three component that give \(\ZF\) its power (the others being power set and infinity, of course).

You'd think that people in category theory would like it, from a foundational point of view, it literally tells you that functions exists if you can define them! And category theory is all about the functions (yes, I know it's not, but I'm trying to make a point).

Syntactic T-Rex: Irregularized

Jun 20 2016, 20:39

One of my huge pet peeves is with people who think that writing \(1+2+3+\ldots=-\frac1{12}\) is a reasonable thing without context. Convention dictates that when no context is set, we interpret infinite summation as the usual convergence of a series, namely the limit of the partial sums, if it exists (and of course that \(1+2+3+\ldots\) does not converge to any real number). However, a lot of people who are [probably] not mathematicians per se, insist that just because you can set up a context in which the above equality holds, e.g., Ramanujan summation or zeta regularization, then it is automatically perfectly fine to write this out of nowhere without context and being treated as wrong.

But those people forget that \(0=1\) is also very true in the ring with a single element; or you know, just in any structure for a language including the two constant symbols \(0\) and \(1\), where both constants are interpreted to be the same object. And hey, who even said that \(0\) and \(1\) have to denote constants? Why not ternary relations, or some other thing?

Don't worry about it

Nov 09 2015, 16:37

In a recent Math.SE question about the foundations of category theory without set theory, someone made a claim that \(\ZF\) makes it hard to learn mathematics, because in \(\ZF\) the questions "is \(\RR\subseteq\pi\)?" and "is \(\RR\in\pi\)?" can be phrased. They continued to argue that there are questions like whether or not hom-sets are disjoint or not, which are hard to explain to people who are "drunk on ZF's kool-aid".

So I raised a question in the comment, and got replies from two other people who kept repeating the age old silly arguments of what are the elements of \(\RR\times\RR\) or what are these or that elements. And supposedly the correct pedagogical answer is "It does not matter what are the elements of \(\RR\times\RR\)." With that I strongly agree, and when I taught my students about ordered pairs on the very first class of the semester, I made it very clear that there are other ways to define ordered pairs and that we only do that because we want to show that there is at least one way in which ordered pairs can be realized as sets; but ultimately we couldn't care less about what way they encode ordered pairs into sets, as long it is a "legal" way.

How do you read a paper?

Nov 01 2015, 10:29

Some time ago I was talking to some people about how they read a paper. And I learned that I am somewhat significantly different from a lot of people. I spent some time thinking about it, and I arrived at some interesting conclusions.

So here is how I read a paper, and I'd like to ask you to think about how you read a paper, and why you read it this way.

Is the Continuum Hypothesis a definite problem?

Aug 07 2015, 04:44

I am not a Platonist.

In general, while I do find it entertaining to think about god, afterlife, or a concrete mathematical universe, I find more comfort in the uncertainty of existence than I do in the likelihood that my belief is wrong, or in the terrifying conviction that comes along with believing in something (and everyone else is wrong).

The rules of research

Jul 08 2015, 03:27

Here are the rules of research. Feel free to add your own.

1. If it seems obvious, it's probably false as stated.
2. If it seems obvious and true, it's probably false without additional hypotheses.
3. If you think that you wrote a proof, you probably missed something obvious. See (1) and/or (2).
4. You missed something obvious, see (1).
5. When you go to see your advisor, suddenly all your thoughts align, and you find the solution.
6. Two hours after finally talking with your advisor, you realize that your solution is obvious, therefore (1) or (2) apply.
7. If you use forcing to prove the argument, then you probably missed some object being encoded generically.
8. If you use forcing, and you didn't miss some crucial object, then you missed some other crucial object not being coded by the generic.
9. When the truth is found to be lies, and all the joy within you dies...
10. It's not false if you can force it.
11. It's not true if you used the axiom of choice more than three times in the proof.
12. It's not cheating if you asked a visitor to the university whose visit did not span longer than two weeks from the moment you asked them.
13. If your question was about inner models, you may extend the above timespan to a month. Equally, if the question is about the axiom of choice, it should be shortened to a week.
14. It's not considered unethical to make sacrifice in order to appease Mayan and Aztec gods. Just in case we got it wrong, and they're in charge of the mathematical universe.
15. If it still seems obvious, you're probably right. It's still false, though.
16. If you need six technical lemmas, whose proof is reduced to a single line (or just one lemma with an actual proof), then it's probably obvious. Unfortunately, see (1) and (2).
17. If by some chance something is obvious, but you wrote out the proof, and it checks out, then it wasn't obvious at all.
18. Remember what the dormouse said: feed your head.
19. If you haven't watched Futurama in a while, then you're doing something wrong.
20. Whatever happens, it's the other guy's fault. Also, see (1).
21. I just work here, you know? I don't.
22. Rolling a D20 die to determine the truth value of a statement is the original algorithm behind proof verification software.
23. When you hit the wall, and you're about to give up and decide that whatever you're trying to prove is false, see (4).
24. The only proofs that write themselves are obvious proofs. If your proof is obvious, see (2) and (3).
25. To be honest, it needs more cowbell.
26. Seriously, you're gonna want that cowbell in your proof.
27. See (1), (2) and (4).

I need your help!

May 08 2015, 15:51

The account has been suspended, I'd like to thank everyone who helped! I have removed the comments posted by "Isa Bria" after the real Isa Bria has contacted me and asked to remove them.

We have verified, in the meantime, that the same person impersonating me on Quora is the one who used Isa's name in those comments.

A problem and a possible solution

May 06 2015, 09:41

So closing in on my third year, and in theory I should finish my dissertation by next summer. This means that I should probably start the writing process around April (I'm a fast writer, what with having a quality keyboard and knowing LaTeX quite well).

But if I want to be sure that I can finish next year, I should probably omit one of the problems I originally wanted to solve; and keep that for later, unless it turns out to be particularly simple when I finish the rest.

Anti-anti Banach-Tarski arguments

Sep 22 2014, 13:31

Many people, more often than not these are people from analysis or worse (read: physicists, which in general are not bad, but I am bothered when they think they have a say in how theoretical mathematics should be done), pseudo-mathematical, non-mathematical, philosophical communities, and from time to time actual mathematicians, would say ridiculous things like "We need to omit the axiom of choice, and keep only Dependent Choice, since the axiom of choice is a source for constant bookkeeping in the form of non-measurable sets".

People often like to cite the paradoxical decomposition of the unit sphere given by Banach-Tarski. "Yes, it doesn't make any sense, therefore the axiom of choice needs to be omitted".