How can we recognize if something is a number?

Question

There are formal definitions of various types of numbers; natural numbers, real numbers, ordinal numbers, cardinals etc. And we all regard them as some type of number.

The definition given in Wikipedia "A number is a mathematical object used to count, measure, and label". However this doesn't answer my question as it doesn't explain what common properties that counting, measuring and labelling have that makes them number like.

Are there properties that are universal across all things we call numbers that allow us to recognise them as being numbers and exclude non-numbers.

Answer 1

This is an extended comment rather than an answer.

Maybe it's noteworthy, first of all, that we are in a significantly different situation from some other cases in mathematics. For example, in algebra there is a very clear and common definition of what a "ring" is, and then there are Artinian rings, Noetherian rings, semisimple rings, commutative rings, primitive rings, reduced rings etc. etc., all with precise definitions which are specialisations of the general but well-defined mathematical concept of "ring".

Here it is different. Mathematicians have precise definitions for: natural numbers; complex numbers; $p$-adic numbers; ordinal numbers; real numbers; hyperreal numbers etc. However, many of these have sort of evolved independently historically, and it was very different people who gave them those names for various reasons. Unlike in the "ring" example, these are not special cases of some commonly agreed definition of "numbers". That concept, "number", is not really a mathematical one, but more of a common-language and/or philosophical one, whose name mathematicians have used for different concepts they investigate.

That being said, I do want to challenge the idea underlying the question and your comment (from Feb 25, 2016), that such a concept, like "number", is necessarily given by some common definition, i.e. all the things we call "number" have to have some clear thing in common.

Actually, this very notion has been famously discussed by Wittgenstein in his Philosophical Investigations, especially paragraphs 65--77, where instead he proposes the idea that some concepts are given by "family resemblance":

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

(This WP article seems to be a good first introduction, and it discusses similar ideas other philosophers have or have had, and (of course) also criticisms of the idea; but I really recommend reading at least the above paragraphs in Wittgenstein's original text. Then again, I recommend reading the entire Philosophical Investigations whenever I can.)

Actually, it turns out that "numbers" are the next example (after his famous introductory example "games") that Wittgenstein explicitly talks about:

And for instance the kinds of number form a family in the same way. Why do we call something a "number"? Well, perhaps because it has a direct relationship with several things that have hitherto been called number; and this can be said to give it an indirect relationship to other things we call the same name. And we extend our concept of number as in spinning a thread we twist fibre on fibre. And the strength of the thread does not reside in the fact that some one fibre runs through its whole length, but in the overlapping of many fibres.

-- loc. cit., §67

Answer 2

Perhaps I have not thought about this enough for this to be a necessarily reliable rule of thumb for what a "number" ought to be, but from the prime examples I have of numbers being

• natural numbers
• complex numbers
• ordinal numbers
• cardinal numbers
• $p$-adic numbers (this is a prime example, get it?)

the common denominator I can see among these that I can see are that

• they contain the natural numbers
• they (mostly) extend the arithmetic operations that are well-defined over the naturals in some way
• (loosely) the system is "generated" by the natural numbers by some philosophy of extending them

To illustrate what I mean, it's clear that complex numbers extend many of the arithmetic operations over the naturals (addition and multiplication are fine, exponentiation is mostly okay, though requiring a bit of care). As for being generated by $\Bbb N$, I suppose you can do this in stages:

• $\Bbb Z$ extends $\Bbb N$ by allowing addition to always be undone (closure under subtraction)
• similarly, $\Bbb Q$ extends $\Bbb Z$ by allowing multiplication to (almost) always be undone
• then $\Bbb R$ extends $\Bbb Q$ by "filling in the gaps" that rational numbers leave behind (being a Cauchy completion of the rationals). If you fill in the gaps a different way, this would lead to $p$-adic numbers $\Bbb Q_p$
• finally $\Bbb C$ extends $\Bbb R$ by including all solutions to polynomials (algebraic closure)

all the while, the arithmetic operations are also extended. Similarly, ordinal numbers also have their own kind of addition, multiplication, and exponentiation, and they can be thought of as being generated by $\Bbb N$ by "allowing you to count for infinitely (transfinitely) many steps" unlike $\Bbb N$ which only allows finitely many successions. On the other hand, cardinal numbers extend $\Bbb N$ under the philosophy that "you can determine the size of infinitely large sets" and so is "generated" this way.

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On the other side of the coin, some examples of "number-like" systems I didn't consider to actually be "number systems" were

• integers modulo $n$
• vector spaces
• arbitrary rings
• something like $\Bbb N\cup\{\spadesuit\}$, where any arithmetic operation involving $\spadesuit$ yields $6$

the first example is eliminated as it does not contain $\Bbb N$ in some canonical way, the second because it only generally preserves the addition operation (and to some extent the multiplication operation), and the third example because rings typically do not have an exponential operator. The final example is intentionally contrived, and was meant to address why I had my third criterion: technically $\Bbb N\cup\{\spadesuit\}$ meets all the previous criteria, but the $\spadesuit$ element is quite... arbitrary. Perhaps this is also a commentary about how any doctrine you might have for what a "number" ought to be will always welcome some absurd example that technically fits the bill, but morally should not be a number system. This likely can't be circumvented without one of your criterion being particularly imprecise.

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As a counterpoint to myself, you could argue that I was being unfair excluding arbitrary rings simply on the basis of not having a notion of exponentiation since, for instance, $\Bbb Q$ is not closed under exponentiation as $2^{1/2}\notin\Bbb Q$. However, $\Bbb Q\subset\Bbb C$ and exponentiation is (mostly) well-defined in the latter encompassing number system. I'll argue this is a fair response to that counterpoint because, for instance, the even numbers are numbers despite the fact that the set of even numbers does not contain $\Bbb N$, but this is irrelevant bc the even numbers are a subset of a number system which does meet my criteria (namely, $\Bbb N$).

Of course, I wouldn't dare claim that my criteria should be law for when "thou shalt be a number" but these are my two cents for what roughly a number ought to be. At the end of the day, there isn't an agreed-upon definition of the word "number" in mathematics (as far as I know), so there will always be leeway. The ultimate guide is that "a number is something that behaves like a number."

Answer 3

tl;dr: We have axiomatic systems that tell us when a set of objects is a number. Sets of objects that comply with those axioms can be seen as numbers.

Axiomatically we can define the natural numbers in terms of set theory via Peano's axioms. Let $N$ be a set and a $S:N\rightarrow N$ a function. Then we say $N$ is a set of

P1. $\exists n_0$ such that $S(n)\neq n_0$, for all $n\in N$.

P2. $S$ is injective.

P3. If $K$ is a set such that:

1. $n_0\in K$, and

2. For all $n\in N$, $n\in K\Rightarrow S(n)\in K$,

then $K=N$.

In this way, we can associate "natural numbers" or "counting numbers" with the set $\mathbb{N}=\{1,2,\ldots\}$. Any such set of objects that adhere to these properties can be considered counting numbers. Thus the set $\{\{1\}, \{1,2\}, \{1,2,3\},\ldots\}$ can be seen as a set of counting numbers since it adheres to these axioms under the function $S(A)=A\cup\{\max{(A)}+1\}$.

From here, we can define $+$ on $N$ recursively by:

1. $n+n_0 =: S(n)$
2. $n+S(m) =: S(n+m)$

Furthermore, we can define the integers via an equivalence relation $\sim$ on $N\times N$: $$(m,n)\sim (p,q) \equiv m+q=n+p.$$ It is fairly straightforward to verify this is an equivalence relation. We then define the integers by $$Z=: N\times N/\sim.$$

We also see that $$\{[(n,n_0)]:n\in N\}$$ adheres to the Peano axioms with $$S([(n,n_0)])=:[(S(n),n_0)].$$ So we identify $N\subseteq Z$.

For the integers, we may define + and $\cdot$ by:

1. $[(m,n)]+[(p,q)]=[(m+p,n+q)]$
2. $[(m,n)]\cdot [(p,q)]=[(mp+nq,np+mq)]$

We can show that these operations are well-defined and confidently use the notation $$\mathbb{Z}=\{0,1,-1,2,-2,\ldots\},$$ where

1. $n=:[(n,0)]$
2. $-n=:[(0,n)]$
3. $0=:[(n,n)]$

So here we now see that any set of counting numbers (i.e., a set that adheres to the Peano axioms) can be extended to a set which behaves like the integers, where we may talk about the "negative" of a number.

But then we may wish to think of the rationals, which similarly, we can define an equivalence relation $\sim$ on $Z\times N$: $$(n,m)\sim (p,q) \equiv nq=mp$$

As above, we now form the rationals via equivalence classes $$Q=:Z\times N/\sim.$$

This matches our usual understanding of rationals as numbers of the form $\frac{p}{q}$, where $q\neq 0$. To see this, we associate $$\frac{p}{q}=:[(p,q)]$$ So we can now write $$\mathbb{Q}=\left\{\frac{p}{q}:p\in\mathbb{Z},q\in\mathbb{N}\right\}$$ We can define concepts like addition, subtraction, and absolute value for this set, though I won't do that here.

Thus far, the construction of these numbers have all been based on logic, set theory, and algebraic concepts. But what about the reals? The reals are different because they have a topological structure. For this we need the concept of Cauchy sequences.

Define a rational sequence as a function $q:N\rightarrow Q$, where $q(n)=:q_n$ and we often write $q=\{q_n\}$. We say a sequence $q$ is Cauchy if for every $\epsilon>0$, where $\epsilon\in Q$, there exists $n\in N$ such that: $$|q_k - q_m|<\epsilon, \forall k,m\geq n$$ The intuition behind this is that elements of the sequence can be pushed arbitrarily close together eventually. We then define: $$C_Q = \{q:N\rightarrow Q:q \text{ is Cauchy}\}$$ This is the set of all rational Cauchy sequences. On this set we define $\sim$ an equivalence relation: $$q\sim p \equiv q_n-p_n\rightarrow 0,$$ where the right hand side of this equation means, for every rational $\epsilon>0$ there exists $m\in N$ such that $$|q_n-p_n|<\epsilon, \forall n\geq m.$$ We can thus define the reals as $R=C_Q/\sim$. Similar to before, we can define + and $\cdot$ as well-defined operation and confidently associate every real number with an equivalence class to obtain our standard notation and write $\mathbb{R}=:\mathbb{C_Q}/\sim$.

So what we have is that from the Peano axioms we can axiomatically characterize counting numbers, introduce equivalence relations to extend these numbers to integers (negatives and 0) and rationals (ratios of integers). If we go further and endow the rationals with topology (sequences and concepts of convergence), we can use equivalence classes once again to arrive at a conceptual definition for the reals. We can do so with any set of objects that adheres to the original Peano axioms, the set $\{1,2,\ldots\}$ is but one example of this.