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How has mathematics gotten so abstract?

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Re: How has mathematics gotten so abstract?

#81

Post by fidotron »

empath75 wrote: post #78
fidotron wrote: post #48

Earlier quoted context omitted.

> 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.

I love how you lot just redefine words to suit your purpose:

https://www.etymonline.com/word/formal "late 14c., "pertaining to form or arrangement;" also, in philosophy and theology, "pertaining to the form or essence of a thing," from Old French formal, formel "formal, constituent" (13c.) and directly from Latin formalis, from forma "a form, figure, shape" (see form (n.)). From early 15c. as "in due or proper form, according to recognized form," As a noun, c. 1600 (plural) "things that are formal;" as a short way to say formal dance, recorded by 1906 among U.S. college students."

There's not a much better description of what Euclid was doing.

Re: How has mathematics gotten so abstract?

#82

Post by ubj »

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.

Science involves both deductive and inductive reasoning. I would in turn argue that mathematics is a science that focuses heavily (but not entirely) on deductive reasoning.

Re: How has mathematics gotten so abstract?

#83

Post by empath75 »

fidotron wrote: post #81
empath75 wrote: post #78

Earlier quoted context omitted.

> 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.

I love how you lot just redefine words to suit your purpose: https://www.etymonline.com/word/formal "late 14c., "pertaining to form or arrangement;" also, in philosophy and theology, "pertaining to the form or essence of a thing," from Old French formal, formel "formal, constituent" (13c.) and directly from Latin formalis, from forma "a form, figure, shape" (see form (n.)). From early 15c. as "in due or proper form,…

I am not, this is what formal logic and formal reasoning means:

https://plato.stanford.edu/entries/logic-classical/

"Formal" in logic has a very precise technical meaning.

Re: How has mathematics gotten so abstract?

#84

Post by [deleted] »

empath75 wrote: post #83
fidotron wrote: post #81

Earlier quoted context omitted.

I love how you lot just redefine words to suit your purpose: https://www.etymonline.com/word/formal "late 14c., "pertaining to form or arrangement;" also, in philosophy and theology, "pertaining to the form or essence of a thing," from Old French formal, formel "formal, constituent" (13c.) and directly from Latin formalis, from forma "a form, figure, shape" (see form (n.)). From early 15c. as "in due or proper form,…

I am not, this is what formal logic and formal reasoning means: https://plato.stanford.edu/entries/logic-classical/ "Formal" in logic has a very precise technical meaning.

[deleted]

Re: How has mathematics gotten so abstract?

#85

Post by TuringTest »

taeric wrote: post #65
TuringTest wrote: post #44

Earlier quoted context omitted.

> My understanding was that mathematics was abstract from the very beginning. It wasn't; but that's a common misunderstanding from hundreds of centuries of common practice. So, how has maths gotten so abstract? Easy, it has been taken over by abstraction astronauts(1), which have existed throghout all eras (and not just for software engineering). Mathematics was created by unofficial engineers as a way to better acco…

I think it is fair to say that it was always an abstraction. But, crucially, it was built on language as much as it was empiricism. That is, numbers were specifically used to abstract over how other things behave using simple and strict rules. No?

> That is, numbers were specifically used to abstract over how other things behave using simple and strict rules. No?

Agree that math is built on language. But math is not any specific set of abstractions; time and again mathematicians have found out that if you change the definitions and axioms, you achieve a quite different set of abstractions (different numbers, geometries, infinity sets...). Does it mean that the previous math ceases to exist when you find a contradiction on it? No, it's just that you start talking about new objects, because you have gained new knowledge.

The math is not in the specific objects you find, it's in the process to find them. Rationalism consider on thinking one step at a time with rigor. Math is the language by which you explain rational thought in a very precise, unambiguous way. You can express many different thoughts, even inconsistent ones, with the same precise language of mathematics.

Re: How has mathematics gotten so abstract?

#86

Post by fidotron »

empath75 wrote: post #83
fidotron wrote: post #81

Earlier quoted context omitted.

I love how you lot just redefine words to suit your purpose: https://www.etymonline.com/word/formal "late 14c., "pertaining to form or arrangement;" also, in philosophy and theology, "pertaining to the form or essence of a thing," from Old French formal, formel "formal, constituent" (13c.) and directly from Latin formalis, from forma "a form, figure, shape" (see form (n.)). From early 15c. as "in due or proper form,…

I am not, this is what formal logic and formal reasoning means: https://plato.stanford.edu/entries/logic-classical/ "Formal" in logic has a very precise technical meaning.

What you mean is someone has redefined the word to suit their purpose, which is precisely what I pointed out at the top.

Edit to add: this comment had a sibling, that was suggesting that given a specific proof assistant requires all input to be formal logic perhaps the word formal could be redefined to mean that which is accepted by the proof assistant. Sadly this fine example of my point has been deleted.

Re: How has mathematics gotten so abstract?

#87

Post by abdullahkhalids »

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 proofs are checked by noisy finite computational machines (humans). Even computer proofs' inputs-outputs are interpreted by humans. Your uncertainty in a theorem is lower bounded by the inherent error rate of human brains.

Re: How has mathematics gotten so abstract?

#88

Post by btilly »

Proposed rule: People writing about the history of mathematics, should learn something about the history of mathematics.

Mathematicians didn't just randomly decide to go to abstraction and the foundations of mathematics. They were forced there by a series of crises where the mathematics that they knew fell apart. For example Joseph Fourier came up with a way to add up a bunch of well-behaved functions - sin and cos - and came up to something that wasn't considered a function - a square wave.

The focus on abstraction and axiomatization came after decades of trying to repair mathematics over and over again. Trying to retell the story in terms of the resulting mathematical flow of the ideas, completely mangles the actual flow of events.

Re: How has mathematics gotten so abstract?

#89

Post by btilly »

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'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.

Re: How has mathematics gotten so abstract?

#90

Post by compressedgas »

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…

> Leibniz (late 1600s) helped to popularize negative numbers.

Wasn't that imaginary numbers?

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