Comment by psykotic | Hacker News Reader

Comment by psykotic | Hacker News Reader

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>Classical measure theory lives in a model divorced from physical reality.

Everything in mathematics is divorced from reality. Unbounded integers are divorced from reality. (Once you move beyond naive realism, bounded integers are divorced from reality, but that's a deeper philosophical debate.) The only question is whether these models are more or less effective for their various theoretical and applied purposes.

> There are better approaches to measure theory which live in different "foundations". For example, you can build measure and probability theory based on the locale of valuations on a locale instead of a sigma-algebra on a topological space.

Better by what definition? According to the practical needs of students, pure and applied mathematicians, etc? I've studied some topos theory and know a little bit about locales from the Topology Via Logic book, but it's hard for me to see that as anything more than a fun curiosity when considering the practical needs of mathematics as a whole. In my mind that kind of thing is much closer to navel-gazing than something like measure theory.

> It is really not the most straightforward approach.

The onus is on critics to do better. Dieudonne/Bourbaki made a valiant and elegant attempt even if they intentionally snubbed the needs of probability theory. And "better" will obviously be judged by the broader community.

> Everything in mathematics is divorced from reality.

Mathematics is an abstraction, but it is still useful for talking about concrete problems. Your mathematical assumptions can be either close or far away from your problem domain. Sometimes we introduce idealized objects, such as unbounded integers, in order to abstract further and simplify our reasoning.

These ideal objects can then either be "compiled away" in specific instances, or really do ignore corner cases which might invalidate your results.

For an example of the former, you can assume that there is an algebraically closed field containing a specific field, give an argument in terms of this closure and then translate this argument to one which does not construct the closure explicitly. The translation is mechanical and does not represent additional assumptions you made.

The second kind of ideal object is something like the real numbers applied to physics. We can think of a real number as an arbitrarily good approximate result. In practice we can only ever work with finite approximations. At the scales we are operating on the difference is usually not relevant, but there might, for example, be unstable equilibria in your solutions which are not physically realizable.

> Better by what definition?

Informally, better because it is "simpler". There are fewer corner cases to consider, theorems are more inclusive, constructions are more direct.

Formally, the theory has more models and is therefore more widely applicable. Theorems have fewer assumptions (but talk about a different and incompatible type of objects).

> The onus is on critics to do better. Dieudonne/Bourbaki made a valiant and elegant attempt even if they intentionally snubbed the needs of probability theory. And "better" will obviously be judged by the broader community.

Oh, sure, but that's not what I want to argue about.

I can tell you with certainty that classical measure theory is complicated by the interplay of excluded middle and the axiom of choice. This is a technical result. You can see this yourself in textbooks every time the author presents an intuitive "proof idea" which then has to be refined because of problems with the definitions. In alternative models, or in a metatheory with alternative assumptions, the simple proof idea usually works out fine.

> Everything in mathematics is divorced from reality. Unbounded integers are divorced from reality. (Once you move beyond naive realism, bounded integers are divorced from reality, but that's a deeper philosophical debate.) The only question is whether these models are more or less effective for their various theoretical and applied purposes.

It's a matter of degree. You can't point to a particular large integer as being "too large" to matter in real life, but there are plenty of objects in measure theory (like unmeasurable sets) that blatantly violate physical intuitions and don't seem to exist in any sense in real life.

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