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Logical difficulties in modern mathematics (2012)

njwildberger.com

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Re: Logical difficulties in modern mathematics (2012)

#31

Post by School-Cotton »

roenxi wrote: post #24
btilly wrote: post #3

There is a distinction between analytically correct expositions, and ones which build on naive intuition to teach students. It is inappropriate to expect beginning students to follow a logically rigorous exposition. That said, there are philosophical questions about truth, infinity, unknowable statements and the like. Mathematicians have by and large settled on a set of answers to these. Every statement is true or fa…

> Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way. (In what sense do they exist again?) A profound question to reflect on; and interesting to contrast with modern science as a philosophical foundation that there might not be any continuous objects present in reality. Might be discrete all the way down. In a sense, it seems like the uncomputable reals…

I don't think "between any two numbers located on a line there must be more numbers" is the property you're looking for, since it holds for the rationals (all of which are, of course, computable).

Maybe "connectedness" is the notion you're trying to get at -- the real numbers are topologically connected, but the rationals aren't. If "A" is the set of rational numbers x with x^2 2, then the rationals are the union of A and B, and there is a "hole where sqrt(2) should be", so the rationals are disconnected. It's possible to define the word "connected" in a way that makes this notion precise.

A related notion is what's called "(sequential) completeness". The infinite sequence whose terms are (2, 2 + 1/2, 2 + 1/2 + 1/6, 2 + 1/2 + 1/6 + 1/24, ...), where the nth term is obtained by adding 1/(n!) to the previous term, intuitively "should" converge, since its elements get arbitrarily close together as n gets arbitrarily large. Any such sequence converges to a real value (this one converges to the exponential constant "e"). But if our number system is only countably infinite, there must be some sequences that get arbitrarily close together but don't converge. For example, if we restrict ourselves to rational numbers, this is a valid infinite sequence (every element is rational), and its terms get arbitrarily close together as "n" is large, but it doesn't converge to anything.

Re: Logical difficulties in modern mathematics (2012)

#32

Post by heyitsguay »

psykotic wrote: post #25
fanzhang wrote: post #11

Did the guy ever deliver on his promise to create a better system? I thought the problems he was pointing out was motivation for a better system (that united the naive understanding and rigor better?), but the further into the article I got, the more it just seemed like a list of complaints. Having 1) studied "naive statistics" when I was young and 2) learning the full-on measure-theory version a la Bourbaki, and now…

"Measure theory a la Bourbaki" is probably not what you meant if you did probability theory and statistics. Bourbaki famously sidestepped classical measure theory with sigma algebras by constructing Radon measures as functionals via functional analysis. This is sufficient for many purposes but not for probability theory. First and foremost, sigma algebras in probability theory are not just a technical device to avoid…

As someone who moved from pure mathematics -> applied mathematics -> machine learning, perhaps I can offer a similar perspective to GP that explains why the measure theory stuff might not seem so useful to some people. Basically, in application you never need to worry about anything but your simplest case - that of a discrete-time, finite-valued process. All of the subtleties of measure theory, which occupy most of what mathematical measure theorists/analysts work on, are introduced when you assume you've got an infinity or a continuum somewhere in a model, and that assumption is never borne out in practice. Unless you really want to do theoretical math, you're better off learning only the aspects that relate to finite functions, i.e. almost none of what a real mathematician would consider to be "math". And that's not just out of laziness or disinterest - once you've restricted yourself to finite functions for empirical applications, you encounter a whole new realm of engineering difficulties to deal with, that theoretical mathematics doesn't touch. You've got to learn how to solve a different class of problems that aren't as purely logical, but are every bit as challenging.

To put it another way - think about all the pathologies you may encounter working with even a "nice" function space like L^2(R). You'll never deal with those pathologies in reality, because every empirical function is much better behaved - finite domain, finite range, and even if you assume a continuous domain, you can choose a model that only has a finite number of discontinuities, is Lipschitz continuous in between them, has finite total variation, etc. And that's why the hand-wavey, "intuitive" approach works so well if you're not a theorist, at least in my opinion.

Re: Logical difficulties in modern mathematics (2012)

#33

Post by psykotic »

heyitsguay wrote: post #32
psykotic wrote: post #25

Earlier quoted context omitted.

"Measure theory a la Bourbaki" is probably not what you meant if you did probability theory and statistics. Bourbaki famously sidestepped classical measure theory with sigma algebras by constructing Radon measures as functionals via functional analysis. This is sufficient for many purposes but not for probability theory. First and foremost, sigma algebras in probability theory are not just a technical device to avoid…

As someone who moved from pure mathematics -> applied mathematics -> machine learning, perhaps I can offer a similar perspective to GP that explains why the measure theory stuff might not seem so useful to some people. Basically, in application you never need to worry about anything but your simplest case - that of a discrete-time, finite-valued process. All of the subtleties of measure theory, which occupy most of w…

For what it's worth, my professional background is entirely in game development and systems programming. I'm not a professional academic. All the professional work I've done with mathematics is applied.

I think you're overstating some things, but I mostly agree. My main disagreement is with your implicit premise that the best practical theory should exist at the same level of abstraction as practical applications. The real numbers have been a really successful practical theory. Physicists and applied mathematicians know they don't "really" exist, but more "realistic" alternatives are awkward and messy. The same applies to more extravagant theoretical constructions like Hilbert spaces. They're an extremely nice mathematical setting for applications (e.g. optimal control, approximation theory, finite element methods, quantum mechanics). No-one should be losing much sleep over their ubiquity in applications. If your point is that we shouldn't belabor some of their technical details when teaching them to practitioners, sure, but that's already the case.

Re: Logical difficulties in modern mathematics (2012)

#34

Post by jacobolus »

heyitsguay wrote: post #32
psykotic wrote: post #25

Earlier quoted context omitted.

"Measure theory a la Bourbaki" is probably not what you meant if you did probability theory and statistics. Bourbaki famously sidestepped classical measure theory with sigma algebras by constructing Radon measures as functionals via functional analysis. This is sufficient for many purposes but not for probability theory. First and foremost, sigma algebras in probability theory are not just a technical device to avoid…

As someone who moved from pure mathematics -> applied mathematics -> machine learning, perhaps I can offer a similar perspective to GP that explains why the measure theory stuff might not seem so useful to some people. Basically, in application you never need to worry about anything but your simplest case - that of a discrete-time, finite-valued process. All of the subtleties of measure theory, which occupy most of w…

For folks developing or analyzing new applied math techniques (for solving differential equations or function approximation or whatever), it is helpful to make formal proofs about their behavior and bounding their error etc., and from the papers I have looked at those are often (usually?) done on top of measure-theoretic models.

It might be possible to develop alternative proofs on purely finite/approximate mathematics, but for a working applied mathematician who already went through standard math grad school curriculum that is probably more trouble than it’s worth.

The users of those mathematical tools (whether software implementors or people just calling some software library) usually don’t need to care about the details of the proofs.

This is similar for other kinds of science/engineering.

Re: Logical difficulties in modern mathematics (2012)

#35

Post by jacobolus »

aragnoil wrote: post #29

I'm not sure if he's a finitist or ultrafinitist [1] but the uncountable reals are clearly a problem for him. I'd like to argue for their usage from a "soft" point of view as opposed to just giving axioms. Mathematics and physics both began with the advent of astronomy: Babylonians and others were curious to trace star patterns and from there both physics and math developed in tandem and influenced each other greatly…

> Mathematics and physics both began with the advent of astronomy

Arguably mathematics began with accounting and surveying and trade, and more generally with counting and measurement.

Re: Logical difficulties in modern mathematics (2012)

#36

Post by tnecniv »

psykotic wrote: post #25
fanzhang wrote: post #11

Did the guy ever deliver on his promise to create a better system? I thought the problems he was pointing out was motivation for a better system (that united the naive understanding and rigor better?), but the further into the article I got, the more it just seemed like a list of complaints. Having 1) studied "naive statistics" when I was young and 2) learning the full-on measure-theory version a la Bourbaki, and now…

"Measure theory a la Bourbaki" is probably not what you meant if you did probability theory and statistics. Bourbaki famously sidestepped classical measure theory with sigma algebras by constructing Radon measures as functionals via functional analysis. This is sufficient for many purposes but not for probability theory. First and foremost, sigma algebras in probability theory are not just a technical device to avoid…

I've only studied measure theory in a probability context. Can you talk a little bit about why the Radon measure isn't sufficient there, and what it is useful for outside of probability?

Re: Logical difficulties in modern mathematics (2012)

#37

Post by [deleted] »

School-Cotton wrote: post #31
roenxi wrote: post #24

Earlier quoted context omitted.

> Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way. (In what sense do they exist again?) A profound question to reflect on; and interesting to contrast with modern science as a philosophical foundation that there might not be any continuous objects present in reality. Might be discrete all the way down. In a sense, it seems like the uncomputable reals…

I don't think "between any two numbers located on a line there must be more numbers" is the property you're looking for, since it holds for the rationals (all of which are, of course, computable). Maybe "connectedness" is the notion you're trying to get at -- the real numbers are topologically connected, but the rationals aren't. If "A" is the set of rational numbers x with x^2 2, then the rationals are the union of…

[deleted]

Re: Logical difficulties in modern mathematics (2012)

#38

Post by tnecniv »

heyitsguay wrote: post #32
psykotic wrote: post #25

Earlier quoted context omitted.

"Measure theory a la Bourbaki" is probably not what you meant if you did probability theory and statistics. Bourbaki famously sidestepped classical measure theory with sigma algebras by constructing Radon measures as functionals via functional analysis. This is sufficient for many purposes but not for probability theory. First and foremost, sigma algebras in probability theory are not just a technical device to avoid…

As someone who moved from pure mathematics -> applied mathematics -> machine learning, perhaps I can offer a similar perspective to GP that explains why the measure theory stuff might not seem so useful to some people. Basically, in application you never need to worry about anything but your simplest case - that of a discrete-time, finite-valued process. All of the subtleties of measure theory, which occupy most of w…

> All of the subtleties of measure theory, which occupy most of what mathematical measure theorists/analysts work on, are introduced when you assume you've got an infinity or a continuum somewhere in a model, and that assumption is never borne out in practice.

That's not entirely fair. People doing controls work with continuous-time models often enough, so there are some practical benefits.

I agree with the rest of your comment though.

Re: Logical difficulties in modern mathematics (2012)

#39

Post by psykotic »

tnecniv wrote: post #36
psykotic wrote: post #25

Earlier quoted context omitted.

"Measure theory a la Bourbaki" is probably not what you meant if you did probability theory and statistics. Bourbaki famously sidestepped classical measure theory with sigma algebras by constructing Radon measures as functionals via functional analysis. This is sufficient for many purposes but not for probability theory. First and foremost, sigma algebras in probability theory are not just a technical device to avoid…

I've only studied measure theory in a probability context. Can you talk a little bit about why the Radon measure isn't sufficient there, and what it is useful for outside of probability?

It's been forever since I looked at this stuff, sorry. But I think Bourbaki's shortcut to integration theory via continuous linear functionals only works in locally compact spaces. It doesn't let you construct the Wiener measure on a path space corresponding to Brownian motion or other continuous-time stochastic processes. And the example I gave of the conceptual, information-theoretic role played by sigma algebras in stochastic processes shows that sidestepping them is the wrong move for probability theory, even if they weren't used for some of the advanced technical constructions like Wiener measure.

Bourbaki's shortcut to Radon measures is very elegant but it's noteworthy that unlike many other Bourbaki innovations I don't think it was picked up by other textbook authors. Already at that point there was a mathematical consensus that measure theory was a valuable part of the foundations of modern mathematics and shouldn't be eliminated or minimized.

Outside probability theory, measure theory is primarily used as a foundation for integration ("expectation"). There are also more specialist subjects like geometric measure theory; there's an excellent introductory textbook called Measure Theory and Fine Properties of Functions, and if you look at its table of contents you can get an idea of the breadth of topics.

Re: Logical difficulties in modern mathematics (2012)

#40

Post by theaeolist »

GolDDranks wrote: post #28

After listening to Wildberger's rants – some of which are very educational and some of which are just rants, I keep thinking that his actual problem is that he doesn't seem to believe in the implications of the axiom of the excluded middle. He goes on about infinities and such, but the deeper issue seems to be that he thinks in a constructive, intuitionistic way whereas the majority of mathematics uses classical logi…

While reading his rant I kept thinking it's a pitch for HOTT or Univalent Math. Many of the motivations are similar.
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