What is best in science?

Aug 26 2023, 23:51

It was foretold of a legendary scientist, one who would master all of mathematics, all of physics, all of chemstiry, all of biology, some of engineering, bits of psychology, and none of economics. Truly, they were a Master of Science, complete with an M.Sc. and all that.

One night, drinking around a campfire with their students, one of them asked: "Master, what is best in science?"

Everything Wrong With The Princess Bride

Sep 05 2022, 00:38

Nothing. Nothing is wrong with that movie. If you haven't watched it in the last couple of weeks, go and watch it tonight.

Multiple Choice

Aug 13 2022, 09:16

Which answer is the most correct?

1. All of the above.
2. All of the above.
3. All of the above.
4. All of the above.
5. None of the above.

Guest comic: Foundations of Forcing

Jul 28 2020, 13:29

I recently posted an answer on MathOverflow where I explained a bit about the approaches to forcing in the literature, at least as I experienced them.

Hanul Jeon took some of these words, and made them into a real nice comic. Originally appearing on his Twitter account. He was kind enough to let me post it here with his permission.

What can you learn about writing papers from cooking pasta?

May 25 2020, 21:40

Those who know me in real life will know that I enjoy cooking. I particularly enjoy cooking pasta. Most authentic Italian recipes are so simple (algio e olio anyone?) that it's just wonderful.

I was cooking dinner today, bucatini alla matriciana if you must know, and I realised that cooking pasta and writing papers have nothing in common. Exactly one of those things is enjoyable, and it is not the writing part of papers (I do love the research part, of course).

Choose your own adventure (redux)

Apr 16 2020, 12:39

I've decided to design a small "choose your own adventure". For fatalists. You can also play the interactive version here!.

This was originally "in-blog", but I decided that the interactive version is a bit more interesting. Enjoy it while it lasts!

Bonus questions

Mar 31 2020, 12:25

With most people under quarantine, I spent some time going over older files in my computer. Exercise sheets, notes, whatever. Several years ago, when I was a teaching assistant for Itay Kaplan on Mathematical Logic 2 (incompleteness and basic model theory), I had put "bonus questions" in most homework sheets. Here are a few, translated from Hebrew.

(Week 2, arithmetic hierarchy, \(\beta\)-function)

In a single paragraph, analyse the squire's approach to the existential crisis of the knight, Antonius Block, in Ingmar Bergman's "The Seventh Seal". What is the role of the knight's wife in support and contrast to the squire's?

Pickles!

Nov 16 2018, 10:03

Those who know me, also know my strong liking of the amazing Spreewaldhof Get One! pickles (go, get one!). I got one as a present from a friend who visited Germany, and after that, I started obsessing over them. I found them in Vienna on my first visit (with the help of Jakob Kellner), and they became the standard Viennese gift from visitors to Jerusalem for me. Mozart chocolate balls for the rest, a bunch of pickles for me. Thanks, Viennese people!

So naturally I could not skip on this video.

Five Star Theorems

Aug 24 2018, 16:35

Talks. Giving talks. We usually don't give talks about past research. Talks are meant to present recent research, things you've just finished, that you're finishing right now, that you've found out!

So often times, it seems, it is very tempting to talk about theorems that you haven't finished writing their proofs in full. Usaully, we put "work in progress" to indicate that this is something not fully verified, not fully vetted (at the very least by ourselves).

Constructive proof that large cardinals are consistent

Jul 06 2016, 13:30

I am not a Platonist, as I keep pointing out. Existence, even not in mathematics, is relative and confusing to begin with, so I don't pretend to try and understand it in a meaningful way.

However, we have a proof, a constructive proof that large cardinals are consistent. And they exist in an inner model of our universe.

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?

Banach-Tarski Banach-T-Rex

Oct 21 2015, 01:10

I had already written about anti-anti-Banach-Tarski arguments. But now the Mathematical T-Rex has something to say too.

Mathematical T-Rex

Aug 28 2015, 22:53

Ever explained to a "working mathematician" about the undecidability of the continuum hypothesis? I bet you too had felt like this T-Rex.

Blurbs!

Jul 29 2015, 22:08

I don't like social media very much. I never really subscribed to the whole Friendster, MySpace, Facebook, Twitter, Google Wave, Google+, and what have you social network sort of approach that you need to have "friends" and "followers" and "follow" other people.

I always preferred to be the master of my domain. The king of my castle. But literally, not the Seinfeld euphemisms sense. In any case. I've been thinking about a page where I can post short thoughts about math, life and otherwise. The blog is not suitable, since I'm not going to add a post each time I have a new thought. So instead I've started a blurbs page. Each blurb has a number, and an anchored link that you can use in case you want to share it.

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).

The Torture of Mathematical Research

Apr 22 2015, 08:12

In a manner more befitting to Edgar Allan Poe, Mathematics is a cruel and unforgiving mistress.

Mathematics will often dangle in front of you some ideas, and you will work them out, to find a mistake. Then you will go back to the beginning, find new ideas that she had in store, work those out and proceed only to find a mistake much later. Then you go back to the beginning, and you find yet another minor idea that was missing, and now when everything works you continue. But then you find another gap, and you have to go back to the beginning and hope to find yet another idea. And don't get me started on those ideas that you find not to work during all these searches.

Turning Green!

Mar 17 2015, 00:16

Well, it's that special day of the year again. The holiest of days. The day we celebrate the patron of alcohol enthusiasts, Saint Patrick. So raise your whiskey glasses (my recommendation is Jameson 12 for those with deep pockets; Kilbeggan for those with shallow pockets), your Guinness pints, and wear green. Because tomorrow is all about soaking your brains in ethanol while listening to Irish folk songs, Irish punk rock (Thin Lizzy and Flogging Molly, for example) and other drinking songs.

Much needed terminology, that isn't going to happen any time soon

Feb 02 2015, 20:29

One of the reasons I love set theory so much, and specifically choice related research, is that this is an extremely fertile ground for amusing terminology. We have forcing, cardinals, collapsing, we have all sort of gems and rodents at our disposal... we even have a swamp thing.

Here are a few terminological ideas that I doubt are going to be developed by anyone. But if you plan on doing something similar (or if my terminology inspires some proof) feel free to use these terms, and please let me know!

To Colloops a cardinal

Oct 05 2014, 21:24

This is nothing new, but it's a choice-y way of thinking about it. Which is really what I enjoy doing.

Definition. Let \(V\) be a model of \(\ZFC\), and \(\PP\in V\) be a notion of forcing. We say that a cardinal \(\kappa\) is "colloopsed" by \(\PP\) (to \(\mu\)) if every \(V\)-generic filter \(G\) adds a bijection from \(\mu\) onto \(\kappa\), but there is an intermediate \(N\subseteq V[G]\) satisfying \(\ZF\) in which there is no such bijection, but there is one for each \(\lambda\lt\kappa\).

Ramsey cardinals are large large small large cardinals

Sep 19 2014, 22:03

There is no well defined notion for what is a large cardinal. In some contexts those are inaccessibles, in others those are critical points of elementary embeddings, and sometimes \(\aleph_\omega\) is a large cardinal.

But we can clearly see some various degrees of largeness by how much structure the existence of the cardinal imposes. Inaccessible cardinals prove there is a model for second-order \(\ZFC\), and Ramsey cardinals imply \(V\neq L\). Strongly compact cardinals even imply that \(\forall A(V\neq L[A])\).

My love-hate relationship with forcing

Sep 10 2014, 08:27

Forcing is great. Forcing is an amazing method. If you can think about it, then you can probably force to make it happen. All it requires is some creativity and rudimentary understanding of the objects that you are working with.

Forcing is horrible. If you can think about it, you can encode it into generic objects. If you can't think about it, you can encode it into generic objects. If you think that you can't encode it into generic objects, then you are probably wrong, and you can still encode it into generic objects.

How Fields Became "Nobel"

Aug 13 2014, 11:18

Here is some interesting piece of mathematical history: How the Fields medal went from "Soviet award" to "Mathematical Nobel".

How Math Got Its 'Nobel'.

This is not a blog post.

Aug 12 2014, 08:35

This is not a blog post.

Debates About The Climate

May 12 2014, 15:13

John Oliver (and his team of writers, I suppose) makes a particularly sharp point about the role of the media in the debate about climate changes.