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I wonder which characteristics of a (structured) set $X$ let us consider it a set of numbers.

Does the existence of two operations $+: X \times X \rightarrow X$ and $\times: X \times X \rightarrow X$ suffice which obey some laws, e.g. the laws of a ring? So would we consider each and every ring a set of "numbers"? Or has there to be more? Which structured sets are far from being a ring but considered a set of numbers nevertheless?

For example, the set of natural numbers doesn't form a ring, but is considered a set of numbers. And the set of prime numbers doesn't form a ring, but is considered a set of numbers. So maybe being a subset of a ring does suffice? But each and every subset? What are the minimal requirements?

Note, that there are different kinds of numbers:

  • counting (cardinal) numbers

  • ordering (ordinal) numbers

  • relating (rational) numbers

  • constructible (algebraic) numbers

  • measuring (real) numbers

Some kinds of numbers don't fit into this scheme, e.g. Gaussian integers, complex numbers or quaternions, or do they?

What do these kinds of numbers have in common?

[This question has to do with the question What is (a) geometry? which may be rephrased: What makes a (structured) set a set of points?]

Hans-Peter Stricker
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  • Numbers are usually thought of as "scalars" (= one-dimensional entities). If you allow complex numbers (= two-dimensional entities) as numbers, why shouldn't you quaternions (= four-dimensional entities)? – Hans-Peter Stricker Sep 18 '18 at 15:52
  • The other way around: Why do people consider complex numbers as numbers, but not so likely quaternions? What's the difference between the two? Is it just a matter of acquaintance? – Hans-Peter Stricker Sep 18 '18 at 16:53

1 Answers1

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There was a time when each of $-1, 0, $ and $1$ were not considered numbers.

The modern use of structures, $\{[\mathbf S, +, \times] + \text{axioms}\}$, for example, allows us to abstract away nonessential details and create useful theorems. Structures give names to a thing, but they are more concerned with proving theorems about that thing.

Feynman on the difference between knowing a thing and knowing about a thing

I think that, because of mathematical structures, the exact definition of number will always be a fuzzy and controversial thing. I don't plan on losing any sleep over that.

Steven Alexis Gregory
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