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

Re: Logical difficulties in modern mathematics (2012)

#11

Post by fanzhang » Thu, Feb 14, 2019, 1:03 AM UTC

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 3) using applied statistics every day, I can say the naive version is the most useful by far. Yes, there is hand waving, but the hand waving works, and you really don't get into that much trouble unless you try to be pathological.

I admit, it would be great to have a version where the naive intuition (many of which are motivated by empirics, physics, and real world situations) and the theoretical definition matched better. Even still, I don't think the formal treatment is appropriate as an introduction -- it's an unnecessary hazing on learning minds when they can get most of the value with a bit of hand waving.

Re: Logical difficulties in modern mathematics (2012)

#12

Post by btilly » Thu, Feb 14, 2019, 1:12 AM UTC

hnuser355 wrote: post #8

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…

Well I don’t mean to be rude but many logicians and set theorists I know will be sad to find themselves out of a job if what you said about these questions being settled is true

I disbelieve your claim.

In particular I maintain that any competent logician or set theorist should be able to explain to you the sense in which the statements that I made cannot logically be settled, and also explain to you the extent to which the rest of mainstream mathematics accepts these statements as true.

Re: Logical difficulties in modern mathematics (2012)

#13

Post by jacobolus » Thu, Feb 14, 2019, 1:13 AM UTC

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…

> Infinite sets exist, and are described by a known set of axioms called ZFC

This is the assertion of many mathematicians, but the justification for it is “this is convenient” and/or “we take this as an article of faith” (or often “I never really thought about it, but it doesn’t much affect my work day to day one way or the other”).

There is no way to prove that infinite sets “exist” by reasonable definitions of “exist”. Indeed, by a conventional definition of “exist” infinite sets pretty clearly don’t qualify.

Instead, mathematicians have redefined the words “exist” and “true”. In mathematics it now means something like “if we accept a particular set of non-obvious and rather handwavey premises, we will also accept any conclusions that result from symbolic manipulations thereof following our established formal rules.” [This is not a full or precise definition of mathematical existence; folks interested can do a search for those keywords and find piles of material.]

* * *

Personally I am happy to accept ZFC or the like because it is convenient and I can’t be bothered to work up an alternative system from scratch and carefully examine all of the conclusions that might follow from that, and whether ZFC is “true” or not doesn’t really affect me. It seems intuitively wrong to me, but I remain agnostic.

Re: Logical difficulties in modern mathematics (2012)

#14

Post by harf » Thu, Feb 14, 2019, 1:22 AM UTC

jacobolus wrote: post #13

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…

> Infinite sets exist, and are described by a known set of axioms called ZFC This is the assertion of many mathematicians, but the justification for it is “this is convenient” and/or “we take this as an article of faith” (or often “I never really thought about it, but it doesn’t much affect my work day to day one way or the other”). There is no way to prove that infinite sets “exist” by reasonable definitions of “exi…

(i.e. one of the standard set theory axioms amounts to "an infinite set exists")

Re: Logical difficulties in modern mathematics (2012)

#15

Post by spindle » Thu, Feb 14, 2019, 1:31 AM UTC

There are MANY foundations for set theory, some of them very different from each other. It's completely pointless to treat the subject as briefly as this post does.

Re: Logical difficulties in modern mathematics (2012)

#16

Post by [deleted] » Thu, Feb 14, 2019, 1:49 AM UTC

panic wrote: post #10

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?

The amount of paper in the universe is finite, not countably infinite, which means almost all natural numbers can't be written down either. But despite its philosophical dubiousness, the set of natural numbers is still useful. If we can prove things about all natural numbers, it doesn't matter how much paper we have; the things we prove will still be true for any individual natural number we can write. It's the same…

[deleted]

Re: Logical difficulties in modern mathematics (2012)

#17

Post by nokcha » Thu, Feb 14, 2019, 1:52 AM UTC

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…

>He won't accept sqrt(2) as a number because it can't be represented in Hindu-Arabic notation.

You sure of that? Based on another of his other blog posts [1], his objection seems to be about uncomputable real numbers. Very roughly, a real number R is computable iff there exists a Turing machine that, given a natural number n on its initial tape, terminates with the nth digit of R. See [2] for a formal definition. Sqrt(2) and all familiar real numbers are computable. Of course, since there is only a countable infinity of Turing machines, but an uncountable infinity of reals, some reals must be uncomputable. Some versions of constructivist mathematics do differ from standard mathematics by rejecting the uncomputable reals and instead defining "real numbers" in such a way that they are essentially the computable reals.

[1] https://njwildberger.com/2016/01/01/uncomputable-decimals-an...

[2] https://en.wikipedia.org/wiki/Computable_number#Formal_defin...

Re: Logical difficulties in modern mathematics (2012)

#18

Post by zzless » Thu, Feb 14, 2019, 2:08 AM UTC

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 the existence of a set with some very specific properties. His complaints about the undefined nature of a 'property' are also not above criticism: GB theory (equivalent to ZF) eliminates it completely.

Re: Logical difficulties in modern mathematics (2012)

#19

Post by wjn0 » Thu, Feb 14, 2019, 2:08 AM UTC

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…

Interesting that you chose statistics as your example. I've always considered statistics to be the one branch of math I've studied where intuition keeps pace with depth and rigour. The caveat being that I am by definition an amateur when it comes to any other branch of math.

Re: Logical difficulties in modern mathematics (2012)

#20

Post by aeneasmackenzie » Thu, Feb 14, 2019, 2:38 AM UTC

effie wrote: post #2

If real numbers get this guy's goat, I think he will go ballistic when he hears about bra-ket formalism that quantum theory instructors foist upon students of physics.

He hates Dedekind cuts, not real numbers. Real numbers exist all right, here are some: https://arxiv.org/abs/0805.2438