New Number Systems Seek Their Lost Primes | Hacker News Reader

New Number Systems Seek Their Lost Primes

quantamagazine.org

48 pointsM_Grey12 comments

An interesting comment on this article:

Expanding (semi-)rings is an ancient process: we expand from 0,1,2,… to the integers by adjoining roots – which don't exist in the old structure – of polynomials x + n. The clever symbol '-5' means: 'that what solves x+5'. Our school education has made it into a common place notion but it took quite a while for the notion to be respectable.

It's interesting that negative numbers, square roots, and imaginary numbers were all invented using the same process: write an unsolvable formula, assume there's a solution, and if that doesn't lead to any contradictions, it's now a legitimate part of math.

This is what makes math math :p You can make up any concept you like, and if it doesn't contradict itself its perfectly valid to analyze it. Later someone has a hairy problem and a mathematician points to how the made up thing he studied is isomorphic to the problem and solves it.

> You can make up any concept you like, and if it doesn't contradict itself its perfectly valid to analyze it

Why limit yourself to ideas that don't contradict themselves?

http://www.iep.utm.edu/math-inc/

https://plato.stanford.edu/entries/mathematics-inconsistent/

https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...

Because then you can prove any theorem (by applying modus ponens), which makes the theory uninteresting.

Or, maybe not. From the second link:

  Inconsistent mathematics is the study of the mathematical
  theories that result when classical mathematical axioms are
  asserted within the framework of a (non-classical) logic
  which can tolerate the presence of a contradiction without
  turning every sentence into a theorem.

In paraconsistent logic, no you can't prove any theorem from a contradiction. Inconsistent mathematics is built upon the foundation of paraconsistent logic.

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

Yeah, and this particular case is even more specific than "make up any concept you like" - it's specifically inventing a new category of number that is the solution to formulas expressed in existing notation.

I think I missed a fourth example of the same pattern: fractions. Originally fractions were not considered to be "numbers", they were just a way of comparing numbers with each other. https://nrich.maths.org/2515

That reminds me of the hairy ball theorem: You can't comb a hairy ball smoothly (without at least one cowlick (<- word I just learned...)).

https://en.wikipedia.org/wiki/Hairy_ball_theorem

What you just remarked I've previously found incredibly fascinating;

In fact (shameless plug) I've written a blog post about that progression [0].

[0] https://malynin.ca/numbers-a-brief-history-from-arithmetic-t...

A very interesting read. I hope you keep blogging!

For what it's worth, I think of the natural numbers not as the simplest concept in mathematics, but as a concept built on top of concepts like "existence vs nonexistence" and perhaps "singular vs plural", which seem like they are pre-mathematics in a sense, built into the grammar of our regular spoken language, perhaps even comprehensible by animals.

Not sure if you are interested in taking your blog in that sort of philosophical direction but I think there might be even more interesting math history before the start of your post.

I have had trouble understanding class numbers. Usually they measure the "failure" of unique factorization. So how does these failures become a special type of number, that multiply to form a group?

Here is an article explaining why exp(pi*sqrt(163)) = is near an integer: http://mathoverflow.net/questions/4775/why-are-powers-of-exp... http://math.stackexchange.com/questions/4544/why-is-e-pi-sqr...

I have met all three of the mathematicians pictured.

>I have had trouble understanding class numbers. Usually they measure the "failure" of unique factorization. So how does these failures become a special type of number, that multiply to form a group?

That is a very good question. A priori the "ideal class group" is just a set. Why do you get a multiplication rule?

I don't think there is any particularly simple explanation. You might read, say, Cox's Primes of the form x^2 + ny^2 for more, but nothing comes to mind now which I can explain in a short amount of space.

But please know, at least, that you have identified a very important question!

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