5 ms·
> It is a practical question to ask about existence questions that require an axiom (axiom of choice often) which gives no constructive way of doing something.
by BoiledCabbage 9y ago
> It is a practical question to ask about existence questions that require an axiom (axiom of choice often) which gives no constructive way of doing something. To what extent is that actually useful?
I'm by no means a mathematician, but I do sympathize somewhat with this. To put it a bit controversially, the axiom of choice feels to me almost like the string theory of mathematics.
There's a lot of really cool stuff you can do with it, but we've left the realm of even potentially useful, and are off approaching a form of rigorous philosophy.
I was looking into Banach–Tarski recently. Is it actually that useful to discuss something which almost certainly can never physically exist in our universe?
Now of course you could play devil's advocate and say as some point people claimed you could never in the universe have a negative amount of something, so integers aren't useful, or a square root of a negative isn't useful so no imaginary numbers. Things are created all the type in math that we don't find practical applications for for decades or centuries.
Yes that's true, but it's not a question of absolute, it's a matter of degree. Has math swung too far into the world non-relevant "play" at the expense of other discoveries?
Recently we've just come across one of the most incredible tools of the past 1000 years. Computers! As a programmer aware of Hindley-Milner it boggles my mind the amount of programming people have been doing doing centuries without the benefit of computers to run them on. Now that we have them we can course correct. Previously we shooting in the dark, and were surprisingly accurate. Now that we have computers this utilize them to focus more and prioritize the amazing things that come from powerful computability.
Let's explore the systems that can be constructed using these new amazing tools. Infinity in itself is ok as a model common real world applications (pi, e, sqrt 2).
Getting out of my knowledge area, here, but what tools would we discover if we were focusing on a math without the Axiom of Choice? What proofs would be reformulated and what tools would we develop to do this work.
Again, everyone should be free to think about whatever they like, but as a field it should also be cautious of how it spends it's resources. Particularly it's most valuable resource of intellectual capital.
- fmap 9y agoOne thing which the author of this essay gets right is that contemporary mathematics is a bit dogmatic when it comes to the logical foundations. We should be looking at the Banach-Tarski theorem as evidence that a particular logical framework for reasoning about volumes/probability is more complicated than strictly necessary. Presumably no non-measurable sets exist in the "real world". At this point there are two possibilities: - A theory with non-measurable sets is much simpler/more expressive than the alternatives and thus is still a useful tool. - There is a simpler theory without this defect, which is at least as expressive when reasoning about real world phenomena. In this case, the latter possibility turns out to be true. It is just unbelievably difficult to convince mathematicians to change the rules of the game, even if you can point to concrete gains.
- BoiledCabbage 9y ago> There is a simpler theory without this defect, which is at least as expressive when reasoning about real world phenomena. This is really interesting to know, and was exactly the type of think that I was wondering about. And I assume the author felt similarly although didn't draw identical conclusions. Out of curiosity could you describe or link me to this other theory?
- fmap 9y agoI was thinking about categorical probability theory or measure theory based on locales. Unfortunately there are (to the best of my knowledge) no textbooks or good writeups available for either, merely long lines of research papers. As said, it's a bit of a niche area, since most mathematicians don't want to think about reworking foundations.
- rayuela 9y agoAre there any papers you would recommend as good starting points?
- yequalsx 9y agoWithout the axiom of choice you get similarly bizarre results. For instance wothout choice there exists a surjection from the reals to a set of greater cardinality.
- BeetleB 9y agoThat's because the reals don't exist, which is Doron's point. We need the Axiom of Choice because we invented the reals.
- yequalsx 9y agoThe existence of the reals is independent of the axiom of choice. The collection of mathematicians that don't believe infinite sets exist has maybe a few members. Constructive mathematicians believe the reals exists. They can be constructed.
- empath75 9y agoYou should read about constructivism. https://en.m.wikipedia.org/wiki/Constructivism_(mathematics) https://en.m.wikipedia.org/wiki/Constructivism_(mathematics)
- GregBuchholz 9y agoBetter yet, read the book Zeilberger mentions at the beginning of his essay: "The Mathematical Experience" by Phillip J. Davis, and Reuben Hersh http://isbn.nu/9780395929681 http://isbn.nu/9780395929681 ...which is a great book, composed of a bunch of different essays comparing and contrasting three philosophies of mathematics (Platonism, Formalism, Intuitionism).
- tel 9y agoThere is a whole body of mathematics which takes these things seriously (constructive math, intuitionistic foundations) and it's, if anything, even more heavily formalized. I think the author has some technical points, but his desire for "make math more like science" is outmoded in many areas. Mathematicians went toward formalism for good reasons. They might not have made all the right decisions, but that doesn't mean that upending the table and starting fresh makes sense.