Logical difficulties in modern mathematics (2012) • Hacker News Board

Re: Logical difficulties in modern mathematics (2012)

#61

Post by ZoltanAK » Thu, Feb 14, 2019, 6:07 PM UTC

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…

> 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. [...] I wonder why he never talks about this – surely as a mathematician he should be aware of intuitionistic logic? (Or is it that as a computer science and linguistics wannabe, I am aware of it but many mathematicians aren't bothered to take a look?)

Heck, I know professional logicians (mostly model theorists) who have never seen intuitionistic logic, in any context, ever. That said, Norman does know a fair bit about intuitionistic logic and constructive mathematics, and even some type theory. He does not like the underlying philosophy any more than he likes classical mathematics.

> Gentzen style sequent calculus with the only axiom being modus ponens

Sidenote: a calculus where modus ponens is an axiom is definitely not a Gentzen-style sequent calculus.

Re: Logical difficulties in modern mathematics (2012)

#62

Post by School-Cotton » Thu, Feb 14, 2019, 6:36 PM UTC

nimonian wrote: post #60

School-Cotton wrote: post #22

Earlier quoted context omitted.

> I'm very happy to accept than any length in geometry is a number by definition Interestingly, that set of numbers is still very incomplete relative to what we expect to be able to talk about in modern math. It doesn't even include the roots of all polynomials (for example, the unique positive solution to x^3 - 2 = 0 isn't the length of any constructible segment in classical geometry).

What does 'in geometry' mean in this context? There are numbers which are not constructible with compass and straight-edge constructible but which are constructible by other means (e.g. 2^(1/3) is origami constructible).

That's why I said "classical geometry" instead of just "geometry" - I meant Euclidean compass-and-straightedge.

Re: Logical difficulties in modern mathematics (2012)

#63

Post by gchpaco » Thu, Feb 14, 2019, 6:45 PM UTC

zzless wrote: post #18

He is obviously a well trained mathematician so reading his pieces (of which there are many) is akin to watching a tsunami come to shore---you cannot look away yet you know that this is a tragedy unfolding. This said, his complaints about set theory would have been (probably) more convincing if he stated the axioms correctly. The Infinite Set axiom does not simply say that some nebulous infinite set exists it states…

In general his complaints regarding logic are incoherent. A function is not a messy weird object in mathematics; it's a possibly infinite set of ordered pairs where there are no duplicates of the first element of the ordered pair. That's it all it is, formally. Or "class" is a concept that has highly specific meaning in NBG theory—in ZFC it's not a formal distinction and so there's a transformation that must be obeyed where "x \in Class" means "{ x | x satisfies class definition}"; which due to Foundation must be the subset of some other, preexisting, class (and thus is Russell's paradox defeated). Almost all of these things exist and are present and are extremely well defined in the theory, he just doesn't like them for some reason.

Re: Logical difficulties in modern mathematics (2012)

#64

Post by heyitsguay » Thu, Feb 14, 2019, 7:00 PM UTC

psykotic wrote: post #33

heyitsguay wrote: post #32

Earlier quoted context omitted.

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. Th…

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.

Re: Logical difficulties in modern mathematics (2012)

#65

Post by heyitsguay » Thu, Feb 14, 2019, 7:04 PM UTC

jacobolus wrote: post #34

heyitsguay wrote: post #32

Earlier quoted context omitted.

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 mathemati…

> 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".

Re: Logical difficulties in modern mathematics (2012)

#66

Post by heyitsguay » Thu, Feb 14, 2019, 7:06 PM UTC

tnecniv wrote: post #38

heyitsguay wrote: post #32

Earlier quoted context omitted.

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 y…

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.

Re: Logical difficulties in modern mathematics (2012)

#67

Post by btilly » Thu, Feb 14, 2019, 7:54 PM UTC

AlexCoventry wrote: post #30

btilly wrote: post #7

Earlier quoted context omitted.

Most people are uninformed. This question strikes at the heart of the debate between Constructivism and Formalism. A debate about what it means for things to exist, statements to be true, and so on. This is very much a matter of philosophy. To a Constructivist, most of classical mathematics is nonsense. And Constructivism is at least as logically consistent as classical mathematics. More precisely any contradiction f…

I think you needed to be more precise. Most people, would read "almost all" in the context of the reals as dependent on the existence of an uncountable set, to make sense.

I was perfectly precise.

In real analysis you learn that "almost all" means that the exceptions are a set of measure 0. Since all countable sets have measure 0, the result is trivially true in classical mathematics.

In the constructible universe, you again have measure theory. Almost all still has a perfectly well-defined meaning. And all sets with enumerations again have measure zero, just like in classical mathematics. But "uncountable" now is a statement about self-referential complexity, not size. Next, "the set of all numbers with finite definitions" is not a well-defined set. And numbers without concrete definitions do not exist.

Re: Logical difficulties in modern mathematics (2012)

#68

Post by School-Cotton » Thu, Feb 14, 2019, 7:59 PM UTC

gizmo686 wrote: post #57

roenxi wrote: post #9

I know Norman; not personally. His real complaint as far as I've ever been able to determine is that he is a highly symbolic thinker; and because of that won't accept certain assumptions that everyone else takes as a given - usually what the Reals are. I'm very happy to accept than any length in geometry is a number by definition (hence sqrt(2), constructed by a 1-1-sqrt(2) triangle, is clearly a number corresponding…

Sqrt(2) is algebraic, and can therefore be constructed fairly explocitly. Starting with integers, you can construct the rationals as an equivelence class of ordered pairs of integers with a particular definition of addition and multiplication. From their you can define polynomials with rational coeficients, and from there you can define quotient fields, Q[x]/ which contains two elements whose square is 2, and is isom…

Can't you just define polynomials as finite sequences of rationals with nonzero last term? E.g. encode x^2 + 2x -1/2 as (-1/2, 2, 1), and define evaluation at a given point in the obvious way.

Re: Logical difficulties in modern mathematics (2012)

#69

Post by btilly » Thu, Feb 14, 2019, 8:01 PM UTC

taejo wrote: post #46

fmap wrote: post #42

Earlier quoted context omitted.

> “I never really thought about it, but it doesn’t much affect my work day to day one way or the other” In my experience, people won't come out and say it, but this seems to be what everyone is thinking. :) The problem with this is that it is wrong. Classical ZFC in particular is a very strong and specific set of assumptions* with a very tenuous link to any practical application. If you actually want to develop a use…

> The problem with this is that it is wrong. > You can translate between different logical theories by building a model of one in another, so it's not like you loose anything. But it's cooky to insist that we should start with ZFC of all things. I don't see how these two are consistent. Almost everything most mathematicians do can be done both in ZFC and your favourite non-kooky axiom system. Certain Powers That Be s…

Almost everything most mathematicians do can be done both in ZFC and your favourite non-kooky axiom system.

You would be amazed at how many uniqueness and existence theorems in how many areas of mathematics require Zorn's Lemma. Which is, of course, equivalent to the axiom of choice. For example, "Every vector space has a (possibly infinite) basis." Or, "Every Hilbert space has a (possibly infinite) orthonormal basis."

It is rare for mathematicians to think much about choice. But it underpins key results in a surprising number of fields.

Re: Logical difficulties in modern mathematics (2012)

#70

Post by btilly » Thu, Feb 14, 2019, 8:04 PM UTC

ginnungagap wrote: post #43

btilly wrote: post #6

Earlier quoted context omitted.

According to classical mathematics, only a countable number of finite definitions of numbers exist. And an uncountable number of real numbers exist. Therefore almost all real numbers that exist do not correspond to any possible finite definition of a number. Tell me. In what sense does an abstract concept exist that has no possible definition or unique description?

So you're arguing that a set can exist even if its elements don't? Because we can surely give a unique description of the set of real numbers (the unique complete ordered field up to iso) and this description forces it to be uncountable

No, I am not.

Also the definition that you gave is complete nonsense to a Constructivist. And the reasoning that forces it to be uncountable in classical set theory requires reasoning that also makes implicit assumptions you are probably not aware of.