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Is Mathematics Real? (2020)

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Re: Is Mathematics Real? (2020)

#11

Post by teucris »

bluepoint wrote: post #9
teucris wrote: post #2

“God made the integers, all else is the work of man” I’m not sure if even that is true. Integer counts and succession seem like human constructs to me. An apple isn’t fundamentally a single unit. If I take a bite out of it, is it still one apple?

I think this is a really good comment. Are you implying that the concept of integer is a consequence of the abstraction of an apple, an abstraction which is entirely human? I really like this point of view.

That’s exactly what I’m saying. I’m glad you like it!

Re: Is Mathematics Real? (2020)

#12

Post by teucris »

LambdaTrain wrote: post #4
teucris wrote: post #2

“God made the integers, all else is the work of man” I’m not sure if even that is true. Integer counts and succession seem like human constructs to me. An apple isn’t fundamentally a single unit. If I take a bite out of it, is it still one apple?

This quote is neither true nor false. The author called it a "sentiment", but I think when Kronecker made this statement, he took it as some kind of principle to work out the foundation of modern mathematics. Basically the mathematical notion of integer is trying to make sense of our idea of a discrete unit. Taking a bite of apple makes it non-unit, but I think you would agree that the notion of "one apple" or "two d…

> assuming the idea of increment by one is solid.

Therein lies the rub! Succession is a human concept. All math is a magnificent city built upon that one stone.

Re: Is Mathematics Real? (2020)

#13

Post by klyrs »

MikeBVaughn wrote: post #6

I've always been interested by the Platonist-vs-anti-Platonist debate. I generally disagree with finitist mathematicians, but I really do find their position interesting, and as a computer scientist I'm not unsympathetic to their perspective. At the end of the day, though, I always go back to an old joke my undergrad philsophy professor told me: Analytic Philosopher A: "Hey, do you believe in baptism?" Analytic Phils…

Me: I believe in any number that I can represent with my computer

Professor: I am suddenly concerned that my PhD student does not believe in induction

Me: I am suddenly concerned that my PhD supervisor does not know the difference between converse and contrapositive

Re: Is Mathematics Real? (2020)

#14

Post by ChainOfFools »

victor106 wrote: post #8

what are some good resources/books to learn the thoughts/debates/ideas that forced us to create/discover say for example concepts like complex numbers or logarithms or calculus etc.,

try "a tour of the calculus" which dives into the genesis of many mathematical premises in order to explore why calculus had to come into being, before explaining what calculus does.

Re: Is Mathematics Real? (2020)

#15

Post by xelxebar »

victor106 wrote: post #8

what are some good resources/books to learn the thoughts/debates/ideas that forced us to create/discover say for example concepts like complex numbers or logarithms or calculus etc.,

Wikipedia often has a "History" section on these large/broad math topics. Though sometimes it requires you to be relatively familiar with the lingo to follow along (See the History of Complex Numbers [1]).

Specifically, for complex numbers, the problem of finding roots of polynomials is what led us there. Cubic polynomials (ones of the form ax^3 + bx^2 + cx + d) gave us the first hints of something like an imaginary unit. Even if the root is just a real number, expressing that number in terms of radicals (n-th roots) usually contains some square root of -1 that isn't eliminable. After lots of debate about the "existence" of such things, we eventually discovered the Fundamental Theorem of Algebra.

For Calculus, you just need to look at Newton, Leibniz, etc. This subject was strongly motivated by physicsists trying to solve hard problems of the day.

Logarithms essentially encde a relationship between addition and multiplication, something that has been recognized for quite some time! [0]

[0]:https://en.wikipedia.org/wiki/History_of_logarithms

[1]:https://en.wikipedia.org/wiki/Complex_number#History

Re: Is Mathematics Real? (2020)

#16

Post by sethc2 »

This kind of reminds me of a question a theistic friend of mine with a masters in math asked me once. Was the fundamental theorem of calculus something discovered by man (i.e created by God) or something created by man?

He thought it was both something created by man, and yet also something discovered.

Our mathematical constructs seem to me to describe things as they are, like discoveries, but it’s like we could only understand the discovery by creating a new way to think about things. What’s most fascinating is they’re not just constructs that are true in that they’re predictable and can be tested, but they have a. absoluteness to them that is this is true or nothing is true.

Re: Is Mathematics Real? (2020)

#18

Post by sethc2 »

teucris wrote: post #2

“God made the integers, all else is the work of man” I’m not sure if even that is true. Integer counts and succession seem like human constructs to me. An apple isn’t fundamentally a single unit. If I take a bite out of it, is it still one apple?

An apple is a single unit though.

An apple is made up of constituent parts, but it is a singular thing.

If you argue against an Apple being singular then you are essentially professing that nothing is singular.

When you take a bite out of an apple you now have a subset of an apple “an apple with a bite out of it”.

To me saying God made the integers is more akin to saying, there is and there isn’t, and all else is built upon the difference.

Re: Is Mathematics Real? (2020)

#20

Post by smokel »

teucris wrote: post #11
bluepoint wrote: post #9

Earlier quoted context omitted.

I think this is a really good comment. Are you implying that the concept of integer is a consequence of the abstraction of an apple, an abstraction which is entirely human? I really like this point of view.

That’s exactly what I’m saying. I’m glad you like it!

I have been entertaining this idea for years. First time I hear someone else voice it.

Do you have an idea what the philosophical term for this idea is? It seems related to connectionism.

One might assume that abstraction and pattern matching in the brain are required to determine that two "things" are the same. It is perhaps a small step to go from comparing things to comparing integers.

Unfortunately, this line of reasoning quickly goes awry, because one cannot assume even the existence of things to describe brains or pattern matching.

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