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As someone who moved from pure mathematics -> applied mathematics -> machine learning, perhaps I can offer a similar perspective to GP that explains why the mea

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.

psykotic 8y ago
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.
- heyitsguay 8y ago
  Yeah, my point is exactly what you say at the end - for practical applications, those theoretical details (which are enormously challenging) generally aren't necessary, and the problems they address aren't the ones application practitioners will encounter. I definitely understand the motivation for theoretical structures within math itself, and i have no issue with e.g. the axiom of choice for people doing mathematics itself.
jacobolus 8y ago
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.
- heyitsguay 8y ago
  > 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 Oh for sure, and perhaps this is just a confusion of terms but i think that's what the thread parent was speaking with "applied statistics". In academia, "applied math/statistics" can mean "I'm doing theoretical math with an eye towards applications but it still requires heavy mathematical machinery", but it can also mean "I'm using mathematical tools to solve empirical problems, and I'm never going to need to worry about Lebesgue measures".
tnecniv 8y ago
> 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.
- heyitsguay 8y ago
  Oh yeah agreed, and also in other applications as well, and i tried to speak to that with the end of my comment - even with continuous time/space models, you're never going to encounter the truly pathological functions that require, e.g., the careful definitions that go into defining Lp spaces. Everything physical is much better behaved.
fmap 8y ago
You're right. Classical measure theory lives in a model divorced from physical reality. You can show that all of the fancy counterexamples which necessitate the complicated constructions of measure theory are artificial (e.g., the characteristic function of a non-measurable set is uncomputable). 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. You can do even better by starting in a constructive metatheory and adding some anti-classical assumptions which are modeled by all computable functions. The reason we are teaching classical measure theory as the foundation of probability theory is historical and because there are no good expositions available for most alternative approaches. It is really not the most straightforward approach. --- Before you accuse me of being overly negative: classical measure theory offers a consistent approach to probability theory which is well understood and for which carefully written textbooks are available. If you really need to go back to the definitions to derive something then you need to know at least one consistent set of definitions. So it is useful to teach measure theory, even if it is more complicated than it has to be...
- psykotic 8y ago
  >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.

psykotic 8y ago

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.

heyitsguay 8y ago
Yeah, my point is exactly what you say at the end - for practical applications, those theoretical details (which are enormously challenging) generally aren't necessary, and the problems they address aren't the ones application practitioners will encounter. I definitely understand the motivation for theoretical structures within math itself, and i have no issue with e.g. the axiom of choice for people doing mathematics itself.

jacobolus 8y ago

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.

heyitsguay 8y ago
> 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 Oh for sure, and perhaps this is just a confusion of terms but i think that's what the thread parent was speaking with "applied statistics". In academia, "applied math/statistics" can mean "I'm doing theoretical math with an eye towards applications but it still requires heavy mathematical machinery", but it can also mean "I'm using mathematical tools to solve empirical problems, and I'm never going to need to worry about Lebesgue measures".

tnecniv 8y ago

> 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.

heyitsguay 8y ago
Oh yeah agreed, and also in other applications as well, and i tried to speak to that with the end of my comment - even with continuous time/space models, you're never going to encounter the truly pathological functions that require, e.g., the careful definitions that go into defining Lp spaces. Everything physical is much better behaved.

fmap 8y ago

You're right. Classical measure theory lives in a model divorced from physical reality. You can show that all of the fancy counterexamples which necessitate the complicated constructions of measure theory are artificial (e.g., the characteristic function of a non-measurable set is uncomputable). 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. You can do even better by starting in a constructive metatheory and adding some anti-classical assumptions which are modeled by all computable functions. The reason we are teaching classical measure theory as the foundation of probability theory is historical and because there are no good expositions available for most alternative approaches. It is really not the most straightforward approach. --- Before you accuse me of being overly negative: classical measure theory offers a consistent approach to probability theory which is well understood and for which carefully written textbooks are available. If you really need to go back to the definitions to derive something then you need to know at least one consistent set of definitions. So it is useful to teach measure theory, even if it is more complicated than it has to be...

psykotic 8y ago
>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.