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This is going to sound insane but... would it not be nice, if there was no need for numbers (except for counting like in subscripts)? Mathematicians talk about eliminating the Axiom of Choice, because of the craziness of the Banach-Tarski Paradox. But the paradoxes numbers create are even crazier.

  1. It is possible to create a numerical function called the Busy Beaver function that lets you prove the Goldbach conjecture (a statement about an infinite set of numbers) by testing it on every number from 1..BB(n) (a finite, although very large, set of numbers).
  2. Gödel's theorems require arithmetic in order to construct the true-but-unprovable sentences. No numbers and such sentences cannot be made.
  3. Numbers give rise to Berry's paradox (there is nothing analogous to it without numbers), which in turn gives rise to the undefinability of an information compression algorithm, even though humans rely on being able to calculate the Kolmogorov complexity of information in day-to-day life.

Has some eccentric mathematician talked about moving maths beyond numbers just like how some mathematicians talked about moving beyond sets in order to define the field of one element?

Alex2006
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Fomalhaut
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    Slightly confused about this - do you mean "Math without the specific symbolic representation we now call 'numbers'" (i.e. could we have come up with a different symbolic representation), "Math without any symbolic representation", or "math without symbolic manipulation"? – LinkBerest Aug 20 '19 at 03:52
  • Also do you mean "high-level" mathematics or everyday math like counting (which we definitely did way before symbols were introduced)? – LinkBerest Aug 20 '19 at 03:58
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    Finally, how is this involved in building a world? If it is not you might be better off asking on Math.SE - though expect more comments like mine seeking clarification. – LinkBerest Aug 20 '19 at 04:04
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    This sound more about math philosophy than world building. And it would be nice if you provided some explanation of each thing you mentioned, for those who didn't grow up with bread and math. – L.Dutch Aug 20 '19 at 04:28
  • I am not fully convinced that I understand the question. By and large, mathematics is a field of study concerned with finding what is provably true. One cannot "eliminate" anything from mathematics; if people can think of something, whatever that something is, and if their thoughts can be formalized in such a way as to allow strict reasoning, then that something is by definition part of mathematics. (Oh, and "counting" is really all that's needed. The entire edifice of arithmetic sits on the foundation of counting. Once you have counting you have everything about numbers.) – AlexP Aug 20 '19 at 05:40
  • P.S. (1) "Gödel's theorems require arithmetic" -- they are about arithmetic. The entire point of Gödel's theorems is to show that our knowledge of arithmetic, which is the simplest part of mathematics (remember that all arithmetic is little more than counting), cannot be at the same time complete and self-consistent. (2) Proving statements about infinite sets of numbers by proving them on a finite set and reasoning from there is called mathematical induction and it is a very very common technique. (3) You don't need Berry's paradox to prove that Kolmogorov complexity is not computable. – AlexP Aug 20 '19 at 05:50
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    Urdula K. Le Guin is one of the greatest scifi writers ever and she describes an alien race that did just that in one of her books. Also this may be relevant, though not a duplicate: https://worldbuilding.stackexchange.com/a/52386/21222 – The Square-Cube Law Aug 20 '19 at 14:26
  • See also:https://worldbuilding.stackexchange.com/questions/69434/how-sophisticated-could-an-illiterate-society-become/ – nzaman Aug 20 '19 at 16:38

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Doing that would cripple mathematics and sciences.

Imagine a world where accountants and engineers do arithmetic, and mathematician do theoretical work on geometry and the like. But something keeps the two groups from talking to each other in a systematic way.

No calculus for engineers. No statistics for epidemiologists. No trigonometry for surveyors.

o.m.
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  • Perhaps there's a better concept that is superior to numbers. The concept of average is outclassed by the concept of distribution, for instance. – Fomalhaut Aug 20 '19 at 20:00
  • @BalancedTryteOperators, I don't see that. You need numbers to talk about statistics. – o.m. Aug 21 '19 at 04:33