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How has mathematics gotten so abstract?

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Re: How has mathematics gotten so abstract?

#31

Post by azan_ »

fidotron wrote: post #30

Unlike Zeno's famous example the paradox which does better at explaining the problem is https://en.wikipedia.org/wiki/Coastline_paradox which Mandelbrot seemed particularly keen on. The tendency towards excessive abstraction is the same as the use of jargon in other fields: it just serves to gatekeep everything. The history of mathematics (and science) is actually full of amateurs, priests and bored aristocrats that…

Theirs no such thing as excessive abstraction in math, because abstraction is the point. Is category theory “excessive abstraction” in your opinion?

Re: How has mathematics gotten so abstract?

#32

Post by elAhmo »

Isn't this true for many other fields of study?

Given the collective time put into it, easier stuff was already solved thousands of years ago, and people are not really left with something trivial to work on. Hence focusing on more and more abstract things as those are the only things left to do something novel.

Re: How has mathematics gotten so abstract?

#33

Post by iamwil »

It's always been abstract. They'll say to me, "Give me a concrete example with numbers!"

I get what they're saying in practice. But numbers are abstract. They only seem concrete because you'd internalized the abstract concept.

Re: How has mathematics gotten so abstract?

#34

Post by [deleted] »

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."

[deleted]

Re: How has mathematics gotten so abstract?

#35

Post by The_suffocated »

pdpi wrote: post #21
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…

> 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.

Somewhat tangential to the discussion: I have once read that Richard Feynman was opposed to the idea (originally due to Karl Popper) that falsifiability is central to physics, but I haven't read any explanation.

Re: How has mathematics gotten so abstract?

#36

Post by falcor84 »

I 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 natural science again.

Re: How has mathematics gotten so abstract?

#37

Post by TimPC »

myrmidon wrote: post #29
pdpi wrote: post #21

Earlier quoted context omitted.

> 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.

You could turn the argument around and say that math must be a science because it builds on falsifiable hypotheses and makes testable predictions. In the end arguing about whether mathematics is a science or not makes no more sense than bickering about tomates being fruit; can be answered both yes and no using reasonable definitions.

In general you aren't testing as an empiricist though, you are looking for a rational argument to prove or disprove something.

Re: How has mathematics gotten so abstract?

#38

Post by AlexandrB »

pdpi wrote: post #21
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…

> 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.

Mathematical "truth" all depends on what axioms you start with. So, in a sense, it doesn't prove "truth" either - just systemic consistency[1] given those starting axioms. Science at least grapples with observable phenomena in the universe.

[1] And even this has limits: https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...

Re: How has mathematics gotten so abstract?

#39

Post by dist-epoch »

elAhmo wrote: post #32

Isn't this true for many other fields of study? Given the collective time put into it, easier stuff was already solved thousands of years ago, and people are not really left with something trivial to work on. Hence focusing on more and more abstract things as those are the only things left to do something novel.

You are right, the low hanging fruits were picked a long time ago.

But also wrong, the easier stuff was solved INCORRECTLY thousands of years ago. But it takes advanced math to understand what was incorrect about it.

Re: How has mathematics gotten so abstract?

#40

Post by wizzwizz4 »

jlongr wrote: post #18
roel_v wrote: post #17

Earlier quoted context omitted.

Did you bother to google his handle? While I don't know his pure mathematics credentials, he's nerd-famous enough to not warrant an introduction. In fact, you not recognizing it says something about you.

>he's nerd-famous enough to not warrant an introduction What is nerd-famous supposed to be. He's at the center of some subjective in-group that exists in your head?

He wrote the American Fuzzy Lop fuzzer, which was extremely influential – pretty much put fuzzing on the map.
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