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Logical difficulties in modern mathematics (2012)

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Re: Logical difficulties in modern mathematics (2012)

#71

Post by btilly »

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…

The thing is that the useful things that we can prove about natural numbers are ones that can be proven about practical ones. We can, at least in principle, follow the construction and wind up with whatever exists.

Now compare with the kinds of results that classical mathematics gives us. The Robertson-Seymour theorem (see https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo... for the theorem) says that certain classes of graphs are characterized by a finite forbidden set. This means that membership can be tested by a polynomial time algorithm. However the construction provides no way to actually find that finite set. It also provides no way to find how many members it has. It not only provides no way to prove that you actually have all of them for a given class of graphs, but there are classes of graphs which it is impossible for us to prove that we actually have a complete list. Not only in practice, but in principle.

So the theorem asserts the existence of a finite set. But in what meaningful way does it exist, or is it finite?

Re: Logical difficulties in modern mathematics (2012)

#72

Post by ginnungagap »

btilly wrote: post #70
ginnungagap wrote: post #43

Earlier quoted context omitted.

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.

You said "acccording to classical mathematics", which is not constructive so I assumed you were working in ZFC or some similar set theory.

Also if we want to be precise I'd like to hear your definition of "finite definition", the only definition of "definability" I'm familiar with is relative to a model of a theory and if ZFC has models at all then it has countable pointwise definable models, but I guess that won't satisfy your idea of "finite definition"

Re: Logical difficulties in modern mathematics (2012)

#73

Post by btilly »

ginnungagap wrote: post #72
btilly wrote: post #70

Earlier quoted context omitted.

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.

You said "acccording to classical mathematics", which is not constructive so I assumed you were working in ZFC or some similar set theory. Also if we want to be precise I'd like to hear your definition of "finite definition", the only definition of "definability" I'm familiar with is relative to a model of a theory and if ZFC has models at all then it has countable pointwise definable models, but I guess that won't s…

When I said "according to classical mathematics" I did mean according to the normal orthodoxy, which does indeed mean ZFC. By "finite definition" I mean a finite sequence of statements in first or second order logic which can be proven to define a unique real number. (Even if, as with Chaitin's Constant, we can't figure out what that number is.)

The fact that this may not be a definition from my point of view is irrelevant - I was making a statement about what classical mathematics implies, and from the point of view of classical mathematics this is a perfectly reasonable definition.

Since the number of such statements is countable, and only some of them define real numbers, the set of such real numbers is countable. Being countable it is a set of measure 0, and therefore the "almost all" that I stated follows immediately.

Re: Logical difficulties in modern mathematics (2012)

#74

Post by gizmo686 »

School-Cotton wrote: post #68
gizmo686 wrote: post #57

Earlier quoted context omitted.

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.

You generally define them as infinite sequences with a finite number of non-zero terms. Since we only care about a specific quotient field you could force it to work by working with only polynomials of bounded degree (You could also ignore polynomials entirely. Since Q(sqrt(2)) is just a two dimensional vector space over Q, we could define it as the ordered pairs Q^2 with an appropriate definition of multiplication and division (like we often define the complex numbers to highshoolers), but this also gets ugly.

I guess the moral of this post is 99% of the time a finitist or constroctivist complains you can rework your theory into a more ugly one that avoids the complaint.

Re: Logical difficulties in modern mathematics (2012)

#75

Post by fanzhang »

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…

Correct on your first point. I do not mean I learned the specific version published by Bourbaki, but rather in the modern mathematical style (hence the "a la", as opposed to 19th century hand waving style that nearly all undergraduate first-courses do).

And by the modern style, I meant starting out with analysis, defining the axioms of what measures are, demonstrating the existence of nonmeasurable sets with the axiom of choice, etc. IIRC, the course did rely on Borel algebras for its buildup, but did not openly buildup from sigma algebra machinery.

Re: Logical difficulties in modern mathematics (2012)

#76

Post by hyperpallium »

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…

He did at least do the MathFoundations youtube videos he mentioned (pity he didn't edit a link in) https://youtube.com/playlist?list=PL5A714C94D40392AB

Re: Logical difficulties in modern mathematics (2012)

#77

Post by MaxBarraclough »

pron wrote: post #49

When discussing a certain discipline, be it mathematics or any other, the issue of rigor and foundation are completely separate. Rigor is established if you are careful to always follow the axioms determined by the foundation. But the foundation, i.e. the choice of the axioms can only be established in another, lower level, discipline, which, in the case of mathematics is philosophy. In the early decades of the 20th…

But axioms aren't assertions of truth. They're not declarations from on high about universal truths, they're how mathematicians ensure they're talking about the same things.

Mathematics is about exploring the consequences of your axioms. Axioms are chosen, not dictated by the universe. If your axioms turn out to be inconsistent, well, congratulations on successfully showing that consequence of those axioms.

Case in point: the axiom of choice. There is no objective truth value to it. The physical world gives us no 'answer'. Mathematicians can choose to adopt it, or not, and then explore the consequences. ( https://en.wikipedia.org/wiki/Axiom_of_choice#Statements_con... )

Re: Logical difficulties in modern mathematics (2012)

#78

Post by pron »

MaxBarraclough wrote: post #77
pron wrote: post #49

When discussing a certain discipline, be it mathematics or any other, the issue of rigor and foundation are completely separate. Rigor is established if you are careful to always follow the axioms determined by the foundation. But the foundation, i.e. the choice of the axioms can only be established in another, lower level, discipline, which, in the case of mathematics is philosophy. In the early decades of the 20th…

But axioms aren't assertions of truth. They're not declarations from on high about universal truths, they're how mathematicians ensure they're talking about the same things. Mathematics is about exploring the consequences of your axioms. Axioms are chosen, not dictated by the universe. If your axioms turn out to be inconsistent, well, congratulations on successfully showing that consequence of those axioms. Case in p…

Well, that certainly was not the common perception in the philosophy of mathematics in the first half of the 20th century. Sure, they believed mathematics is an exploration of the axioms, but also that the axioms must be true in some deep philosophical sense. Brouwer wrote: [A]n incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is none the less incorrect,[1] and Russell believed that all of mathematics could be deduced from laws of logic that are true in the most absolute sense.

It is also doubtful that this is the perception today. If we want to use mathematics to derive any result about the physical world -- and we most certainly do -- our axioms must be consistent with it. How do you ensure that that is the case?

But such questions are not part of mathematics itself, just as the scientific method is not part of physics, but rather belong in a more fundamental discipline, called foundations of mathematics, or the philosophy of mathematics, which is usually studied by philosophers/logicians.[2]

[1]: On the Significance of the Principle of Excluded Middle in Mathematics, 1923

[2]: https://plato.stanford.edu/entries/philosophy-mathematics/

Re: Logical difficulties in modern mathematics (2012)

#79

Post by MaxBarraclough »

pron wrote: post #78
MaxBarraclough wrote: post #77

Earlier quoted context omitted.

But axioms aren't assertions of truth. They're not declarations from on high about universal truths, they're how mathematicians ensure they're talking about the same things. Mathematics is about exploring the consequences of your axioms. Axioms are chosen, not dictated by the universe. If your axioms turn out to be inconsistent, well, congratulations on successfully showing that consequence of those axioms. Case in p…

Well, that certainly was not the common perception in the philosophy of mathematics in the first half of the 20th century. Sure, they believed mathematics is an exploration of the axioms, but also that the axioms must be true in some deep philosophical sense. Brouwer wrote: [A]n incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is none the less incorrect ,[1] and Russell beli…

> they believed mathematics is an exploration of the axioms, but also that the axioms must be true in some deep philosophical sense

That's a contradiction, and it's easy enough to show it: a mathematician can research the consequences of the axiom of choice, and can research the consequences of its negation. It would be silly to deny that such research is legitimate mathematics.

Brouwer's statement strikes me as circular. Beyond that, it seems to me that the law of excluded middle, is patently false. I already gave a counterexample: the axiom of choice. Neither the axiom, nor its negation, has a derivable truth value to settle upon.

Ah, I see I'm behind the times [0]

As to Russell, he of course turned out to be wrong. I refer of course to Gödel. I'm not sure I see your point in mentioning him, or for that matter Brouwer - do explain.

> our axioms must be consistent with it

They don't. We hope that we have the mathematics to advance physics, but we needn't hope that all mathematics is applicable to physics.

It's perfectly legitimate for a mathematician to explore the consequences of denying the axiom of choice. (I already linked to such.) Such work would presumably never have application in physics, but it's still legitimate mathematics.

> How do you ensure that that is the case?

As you later allude to, that isn't a question of mathematics, it's a question of how and why mathematical discoveries map onto observable realities like physics and statistics. This is a completely legitimate line of question. The canonical article about how remarkable this is: Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".

> such questions are not part of mathematics itself

Agreed.

[0] https://en.wikipedia.org/wiki/Law_of_excluded_middle#In_math...

Re: Logical difficulties in modern mathematics (2012)

#80

Post by pron »

MaxBarraclough wrote: post #79
pron wrote: post #78

Earlier quoted context omitted.

Well, that certainly was not the common perception in the philosophy of mathematics in the first half of the 20th century. Sure, they believed mathematics is an exploration of the axioms, but also that the axioms must be true in some deep philosophical sense. Brouwer wrote: [A]n incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is none the less incorrect ,[1] and Russell beli…

> they believed mathematics is an exploration of the axioms, but also that the axioms must be true in some deep philosophical sense That's a contradiction, and it's easy enough to show it: a mathematician can research the consequences of the axiom of choice, and can research the consequences of its negation. It would be silly to deny that such research is legitimate mathematics. Brouwer's statement strikes me as circ…

It's not a matter of legitimacy (which is a social construct) but of foundation. Russell, Brouwer and Hilbert believed that mathematics must be based on a solid philosophical foundation so that it leads to some "Truth." I am not saying this is the only way to think about the philosophy of mathematics -- in fact, my original comment said just the opposite -- but it was very much at the center of the mathematical world in the early decades of the 20th century, and is still a matter of debate among logicians. The philosophy of mathematics is a deep and complex subject. It does not seek to mathematically derive theorems from mathematical axioms, but to derive mathematical axioms from philosophical underpinnings -- whether they are physical reality, some Platonic reality, or even common sense. Logicians don't think the subject is silly at all, and if you're interested in learning more about it, the SEP link I provided above is a great place to start.
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