"Indeed, persistently trying to relate the foundations of math to reality has become the calling card of online cranks." <-- Hm??? I'm getting self-conscious. Details?
How has mathematics gotten so abstract?
121–130 of 220 posts
Re: How has mathematics gotten so abstract?
#122This 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."
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Re: How has mathematics gotten so abstract?
#123Earlier quoted context omitted.
Of course I can. I frequently use numbers which are great abstraction. I can use same number five to describe apples, bananas and everything countable.
> to describe apples, bananas and everything countable An apple is an abstraction over the particles/waves that comprise it, as is a banana. Euclid is no more abstract than the day to day existence of a normal person, hence to claim that it is unusually abstract is to ignore, as you did, the abstraction inherent in day to day life. As I pointed out it's very possible to create formal reasoning systems which are not s…
> An apple is an abstraction over the particles/waves that comprise it, as is a banana.
No, you don’t understand what abstraction is. Apple is exactly arrangement of particles, it’s not abstraction over them.
> hence to claim that it is unusually abstract
Who talks about him being unusually abstract (and not just abstract)?
> is to ignore, as you did, the abstraction inherent in day to day life.
How am I ignoring this abstraction when I’ve provided you exactly that (numbers are abstraction inherent in day to day life). I’m sorry but you seem to be discussing in bad faith.
Re: How has mathematics gotten so abstract?
#124This 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'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.
Re: How has mathematics gotten so abstract?
#125I found it a bit ironic that the author introduced C code there as an aid, but didn't incorporate it into their argument. As I see it, code is exactly the bridge between abstract math and the empirical world - the process of writing code to implement your mathematical structure and then seeing if it gives you the output you expect (or better yet, with Lean, if it proves your proposition) essentially makes math a natu…
No, the correctness of your implementation is a mathematical statement about a computation running a particular computational environment, and can be reasoned about from first principles without ever invoking a computer. Whether your computation gives reasonable outputs on certain inputs says nothing (in general) about the original mathematics.
Re: How has mathematics gotten so abstract?
#126Earlier quoted context omitted.
Mathematics arose from ancient humans need to count and measure. Even the invention\discovery of Calculus was in service to physics. It has probably only been 300 years or so since Mathematics has been symbolic, before that it was more geometric and more attached to the physical world. Leibniz (late 1600s) helped to popularize negative numbers. At the time most mathematicians thought they were "absurd" and "fictitiou…
Sorry what? Ancient humans invented symbols to count. How is that not symbolic? Geometry is “attached” to the physical world… but in an abstract way… but you can point to the thing your measuring maybe so it doesn’t count… Abstraction was perfected if not invented by mathematics.
Symbolic here refers of doing math with place holders, be it letters or something. Ancient world had notations for recording numbers. But much less so to do math with them. Say like long division.
Re: How has mathematics gotten so abstract?
#127Earlier 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.
He never mentioned the Axiom of Choice. I think he articulated his opinion clearly enough. It's his own subjective value judgement.
Re: How has mathematics gotten so abstract?
#128Earlier quoted context omitted.
> 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. That’s not at all true at the level where you are dealing with different infinities, usually, which tends to come after the (usually, fairly early) part dealing with proofs and the fact that all mathematics is dealing with “unprovable intellec…
I guarantee that a naive presentation doesn't actually include the axioms, and doesn't address the philosophical questions dividing formalism from constructivism. Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.
> I guarantee that a naive presentation doesn't actually include the axioms
> Uncountable need not mean more.
If you're trying to find mistakes in the logic does it not make sense to push it at its bounds? Look at the Banach-Tarski Paradox. Sure, normal people hear about it and go "oh wow, cool." But when it was presented in my math course it was used as a discussion of why we might want to question the Axiom of Choice, but that removing it creates new concerns. Really the "paradox" was explored to push the bounds of the axiom of choice in the first place. They asked "can this axiom be abused?" And the answer is yes. Now the question is "does this matter, since infinity is non-physical? Or does it despite infinity being non-physics?"
You seem to think mathematicians, physicists, and scientists in general believe infinities are physical. As one of those people, I'm not sure why you think that. We don't. I mean math is a language. A language used because it is pedantic and precise. Much the same way we use programming languages. I'm not so sure why you're upset that people are trying to push the bounds of the language and find out what works and doesn't work. Or are you upset that non-professionals misunderstand the nuances of a field? Well... that's a whole other conversation, isn't it...
Re: How has mathematics gotten so abstract?
#129Earlier quoted context omitted.
> to describe apples, bananas and everything countable An apple is an abstraction over the particles/waves that comprise it, as is a banana. Euclid is no more abstract than the day to day existence of a normal person, hence to claim that it is unusually abstract is to ignore, as you did, the abstraction inherent in day to day life. As I pointed out it's very possible to create formal reasoning systems which are not s…
> An apple is an abstraction over the particles/waves that comprise it, as is a banana. No, you don’t understand what abstraction is. Apple is exactly arrangement of particles, it’s not abstraction over them. > hence to claim that it is unusually abstract Who talks about him being unusually abstract (and not just abstract)? > is to ignore, as you did, the abstraction inherent in day to day life. How am I ignoring thi…
> Apple is exactly arrangement of particles, it’s not abstraction over them.
No. You can do things to that apple, such as bite it, and it is still an apple, despite it now having a different set of particles. It is the abstract concept of appleness (which we define . . . somehow) applied to that arrangement of particles.
> I’m sorry but you seem to be discussing in bad faith.
Really?
> No, you don’t understand what abstraction is.
Re: How has mathematics gotten so abstract?
#130>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…
> Pure mathematics is regarded as an abstract science, which it is by definition. I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.
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.