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How has mathematics gotten so abstract?

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Re: How has mathematics gotten so abstract?

#151

Post by gaze »

Tazerenix wrote: post #13

>Today, mathematics is regarded as an abstract science. Pure mathematics is regarded as an abstract science, which it is by definition . Arnol'd argued vehemently and much more convincingly for the viewpoint that all mathematics is (and must be) linked to the natural sciences. >On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs…

The only things that are weird in math are things that would not be expected after understanding the definitions. A lot of the early hurdles in mathematics are just learning and gaining comfort with the fact that the object under scrutiny is nothing more than what it's defined to be.

Re: How has mathematics gotten so abstract?

#152

Post by Animats »

Infinity is a convenience that pays off in terseness. There's constructive mathematics, but it's wordy and has lots of cases. You can escape undecidablity if you give up infinity. Most mathematicians consider that a bad trade.

Re: How has mathematics gotten so abstract?

#153

Post by btilly »

tempfile wrote: post #111
btilly wrote: post #104

Earlier quoted context omitted.

When you say "given ZFC", you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean. In particular, you have made sufficient assumptions to prove that almost all real numbers that exist can never be specified in any possible finite description. In what sense do they exist? You also wind up with wei…

You are being very cryptic. Are you trying to say that the existence of uncountable sets requires the axiom of choice? If you are, that's false. If you aren't, I'm not sure what you are trying to say.

I'm definitely not trying to say that the existence of uncountable sets requires the axiom of choice. Cantor's diagonalization argument for the reals demonstrates otherwise.

I'm saying that to go from the uncountability of the reals to the idea that this implies that the infinity of the reals is larger, requires making some important philosophical assumptions. Constructivism demonstrates that uncountable need not mean more.

On the algorithm example, you could have asked what I was referring to.

The result that I was referencing follows from the https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo.... The theorem says that any class of finite graphs which is closed under graph minors, must be completely characterized by a finite set of forbidden minors. Given that set of forbidden minors, we can construct a polynomial time test for membership in the class - just test each forbidden minor in turn.

The problem is that the theorem is nonconstructive. While it classically proves that the set exists, it provides no way to find it. Worse yet, it can be proven that in general there is no way to find or verify the minimal solution. Or even to provide an upper bound on the number of forbidden minors that will be required.

This need not hold in special cases. For example planar graphs are characterized by 2 forbidden minors.

For the toroidal graphs, as https://en.wikipedia.org/wiki/Toroidal_graph will verify, the list of known forbidden minors currently has 17,523 graphs. We have no idea how many more there will be. Nor do we have any reason to believe that it is possible to verify the complete list in ZFC. Therefore the polynomial time algorithm that Robinson-Seymour says must exist, does not seem to exist in any meaningful and useful way. Such as, for example, being findable or provably correct from ZFC.

Re: How has mathematics gotten so abstract?

#154

Post by btilly »

zozbot234 wrote: post #150
btilly wrote: post #147

Earlier quoted context omitted.

Sure, we can choose to work in a set of axioms that says that there exists an oracle that can solve the Halting problem. But the fact that such systems don't create contradictions emphatically *DOES NOT* demonstrate the constructive existence of such an oracle. Doubly not given that in various usual constructivist systems, it is easily provable that nothing that exists can serve as such an oracle.

> emphatically DOES NOT demonstrate the constructive existence of such an oracle Of course, but it shows that you can assume that such an oracle exists whenever you are working under additional conditions where the existence of such a "special case" oracle makes sense to you, even though you can't show its existence in the general case. This outlook generalizes to all non-constructive existence statements (and disjun…

That's like asserting the existence of a bank account in my name with a billion dollars in it that I know nothing about.

I won't ever be able to find a contradiction from that claim, because I have no way to find that bank account if it exists.

But that argument also won't convince me that the bank account exists.

Re: How has mathematics gotten so abstract?

#155

Post by btilly »

ogogmad wrote: post #124
btilly wrote: post #89

Earlier quoted context omitted.

I'm curious. Did either of you ever notice the implicit philosophical assumptions that you have to make to come to the conclusion that one infinity can be larger than another? Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way.

It might be relevant to look at this: https://home.sandiego.edu/~shulman/papers/jmm2022-complement... Also this: https://arxiv.org/pdf/1212.6543 Assuming you haven't looked at these already, of course.

I had already read the second. I'm not so enthused about the first.

Re: How has mathematics gotten so abstract?

#156

Post by nextaccountic »

Etherlord87 wrote: post #102
btilly wrote: post #89

Earlier quoted context omitted.

I'm curious. Did either of you ever notice the implicit philosophical assumptions that you have to make to come to the conclusion that one infinity can be larger than another? Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way.

You don't need an implicit philosophical assumption, you just need to define what an infinity is and the comparison method.

This looks like a philosophical stance in the philosophy of mathematics actually, and it's called formalism

Re: How has mathematics gotten so abstract?

#157

Post by zozbot234 »

btilly wrote: post #154
zozbot234 wrote: post #150

Earlier quoted context omitted.

> emphatically DOES NOT demonstrate the constructive existence of such an oracle Of course, but it shows that you can assume that such an oracle exists whenever you are working under additional conditions where the existence of such a "special case" oracle makes sense to you, even though you can't show its existence in the general case. This outlook generalizes to all non-constructive existence statements (and disjun…

That's like asserting the existence of a bank account in my name with a billion dollars in it that I know nothing about. I won't ever be able to find a contradiction from that claim, because I have no way to find that bank account if it exists. But that argument also won't convince me that the bank account exists.

That argument ought to convince you that there's a mere "possible world" where that bank account turns out to exist. Sometimes we are implicitly interested in these special-cased "possible worlds", even though they'll involve conditions that we aren't quite sure about. Non-constructive existence is nothing more than a handy way of talking about such things, compared to the constructively correct "it's not the case that the existence of X is always falsified".

Re: How has mathematics gotten so abstract?

#158

Post by nitwit005 »

goatlover wrote: post #146
nitwit005 wrote: post #130

Earlier quoted context omitted.

A proof is just an argument that something is true. Ideally, you've made an extremely strong argument, but it's still a human making a claim something is true. Plenty of published proofs have been shown to be false. Math is scientific in the sense that you've proposed a hypothesis, and others can test it.

Difference is mathematical arguments can be shown to be provably true when exhaustively checked (which is straight forward with simpler proofs). Something you don't get with the empirical sciences. Also the empirical part means natural phenomena needs to be involved. Math can be purely abstract.

You're making a strong argument if you believe you checked every possibility, but it's still just an argument.

If you want to escape human fallibility, I'm afraid you're going to need divine intervention. Works checked as carefully as possible still seem to frequently feature corrections.

Re: How has mathematics gotten so abstract?

#159

Post by charlieyu1 »

thadt wrote: post #2

This reminds of of that one time when I was on a date with a girl from the history department who somehow bemusedly sat through my entire mini-lecture on comparing infinite sets. Twenty years and three kids later, she'll still occasionally look me straight in the eye and declare "my infinity is bigger than your infinity."

I taught my wife simplex algorithm for linear programming and she forgot all of it

Turns out I’m neither good in maths nor teaching

Re: How has mathematics gotten so abstract?

#160

Post by pfdietz »

thadt wrote: post #2

This reminds of of that one time when I was on a date with a girl from the history department who somehow bemusedly sat through my entire mini-lecture on comparing infinite sets. Twenty years and three kids later, she'll still occasionally look me straight in the eye and declare "my infinity is bigger than your infinity."

Way back then, calculus was a culture war battleground. Bishop Berkeley famously argued the foundations of calculus weren't any better that those of theology. This sort of thing motivated much work into shoring them up, getting rid of infinitesimals and the like (or, later, making infinitesimals rigorous in nonstandard analysis).

https://en.wikipedia.org/wiki/The_Analyst

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