How has mathematics gotten so abstract? • Hacker News Board

Re: How has mathematics gotten so abstract?

#71

Post by masklinn » Tue, Sep 30, 2025, 2:59 PM UTC

nivter wrote: post #61

I believe that abstraction is recursive in nature which creates multiple layers of abstract ideas leading to new areas or insights. For instance our understanding of continuity and limit led to calculus, which when tied to the (abstract) idea of linearity led to the idea of linear operator which explains various phenomena in the real world surprisingly well.

You could say that abstraction is a step or a ladder: by climbing on an abstraction you can see new goals and opportunities, possibly out of reach until you build yet new steps.

Re: How has mathematics gotten so abstract?

#72

Post by elliotec » Tue, Sep 30, 2025, 3:01 PM UTC

stonemetal12 wrote: post #60

rob74 wrote: post #24

How has mathematics gotten so abstract? My understanding was that mathematics was abstract from the very beginning . Sure, you can say that two cows plus two more cows makes four cows, but that already is an abstraction - someone who has no knowledge of math might object that one cow is rarely exactly the same as another cow, so just assigning the value "1" to any cow you see is an oversimplification. Of course, simp…

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.

Re: How has mathematics gotten so abstract?

#73

Post by daxfohl » Tue, Sep 30, 2025, 3:04 PM UTC

One could also say the opposite. It's not abstract at all, just a set of rules and their implications. Plausibly the least abstract thing there is.

On the other hand, two cookies plus three cookies, what even is a cookie? What if they're different sizes? Do sandwich cookies count as one or two? If you cut one in half, does you count it as two cookies now? All very abstract. Just give me some concrete definitions and rules and I'll give you a concrete answer.

Re: How has mathematics gotten so abstract?

#74

Post by kannanvijayan » Tue, Sep 30, 2025, 3:04 PM UTC

empath75 wrote: post #68

nkrisc wrote: post #56

Earlier quoted context omitted.

> The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way. I may be off-base as an outsider to mathematics, but Euclid’s Elements, per my understanding, is very mu…

Quite the opposite, Plato, several hundred years before Euclid was already talking about geometry as abstract, and indeed the world of ideas and mathematics as being _more real_ than the physical world, and Euclid is very much in that tradition. I am going to quote from the _very beginning_ of the elements: Definition 1. A point is that which has no part. Definition 2. A line is breadthless length. Both of these two…

Another point to keep in mind is that a lot of mathematics that's not considered abstract _now_ was definitely considered "hopelessly" abstract at the time of its conception.

The complex number system started being explored by the greeks long before any notion of the value of complex spaces existed, and could be mapped to something in reality.

Re: How has mathematics gotten so abstract?

#75

Post by currymj » Tue, Sep 30, 2025, 3:08 PM UTC

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.

two interesting cases: convex analysis and linear algebra are both relatively easy, concrete areas of mathematics. also beautiful and unbelievably useful. yet they didn't develop until the 19th century and didn't mature until the 20th.

Re: How has mathematics gotten so abstract?

#76

Post by empath75 » Tue, Sep 30, 2025, 3:09 PM UTC

stonemetal12 wrote: post #60

rob74 wrote: post #24

How has mathematics gotten so abstract? My understanding was that mathematics was abstract from the very beginning . Sure, you can say that two cows plus two more cows makes four cows, but that already is an abstraction - someone who has no knowledge of math might object that one cow is rarely exactly the same as another cow, so just assigning the value "1" to any cow you see is an oversimplification. Of course, simp…

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…

Almost from the first time people started writing about mathematics, they were writing about it in an abstract way. The Egyptians and the Babylonians kept things relatively concrete and mostly stuck to word problems (although lists of pythagorean triples is evidence for very early "number theory"), but Greece, China and India were all working in abstractions relatively early.

Re: How has mathematics gotten so abstract?

#77

Post by dominicrose » Tue, Sep 30, 2025, 3:09 PM UTC

wvlia5 wrote: post #47

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."

Once I taught the binomial coefficient formula to a girl after sex

that's not too abstract, I can see how this formula applies to sex

I tried using if for this: https://adventofcode.com/2023/day/12 but computer said no

Re: How has mathematics gotten so abstract?

#78

Post by empath75 » Tue, Sep 30, 2025, 3:10 PM UTC

fidotron wrote: post #48

azan_ wrote: post #31

Earlier quoted context omitted.

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

> because abstraction is the point. Formal reasoning is the point, which is not by itself abstraction. Someone else in this discussion is saying Euclid's Elements is abstract, which is near complete nonsense. If that is abstract our perception of everything except for the fundamental [whatever] we are formed of is an abstraction.

> Formal reasoning is the point, which is not by itself abstraction.

What do you think "formal" means in that sentence.

It means "formal" from the word "form". It is reasoning through pure manipulation of symbols, with no relation to the external world required.

Re: How has mathematics gotten so abstract?

#79

Post by hollerith » Tue, Sep 30, 2025, 3:13 PM UTC

empath75 wrote: post #76

stonemetal12 wrote: post #60

Earlier 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…

Almost from the first time people started writing about mathematics, they were writing about it in an abstract way. The Egyptians and the Babylonians kept things relatively concrete and mostly stuck to word problems (although lists of pythagorean triples is evidence for very early "number theory"), but Greece, China and India were all working in abstractions relatively early.

In particular, ancient Greek geometry at least after 300 BC proceeded from axioms, which is a central component of the abstract approach.

Re: How has mathematics gotten so abstract?

#80

Post by The_suffocated » Tue, Sep 30, 2025, 3:14 PM UTC

Discussions of this sort can easily get chaotic, because people tend to conflate intuitiveness and concreteness. Sometimes the whole point of abstraction is to make a concept clearer and more intuitive. The distinction between polynomial function and polynomial is an example.