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One 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 lo
by fmap 9y ago
One 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.
- GregBuchholz 9y ago>Constructive mathematicians believe the reals exists. I think when people mention the "reals" they are usually referring to the uncountable reals, which you can't construct, compute, name, or know: https://arxiv.org/abs/math/0404335 https://arxiv.org/abs/math/0404335 ...(maybe skip to chapter 5 to get to the meat of it). So constructivists don't believe in those types of numbers, and so get "choice" as a theorem instead of an axiom.
- yequalsx 9y agoConstructive mathematicians definitely do believe the reals exist and are uncountable. Not all reals are computable or definable. The set of computable reals is countable. There are different varieties of constructive mathematics. A very large majority of constructive mathematicians believe the reals are uncountable. In intuitionistic mathematics the reals are uncountable. In some constructive versions of math you get weird things like there being an injection from R to N but there not being a bijection due to diagonalization. I think you are confusing computable/nameable with constructive as that term is used by most mathematicians. Note that Chaitan's Constant is not computable.