# What is a number?

A dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and number mean the same thing, this definition won't do at all.

So, what is a number or quantity?

• If you want to know the meaning of "number" in English, you should ask elsewhere. Mathematicians use this word (and many other words) with their own special meaning, which differs from the common meaning in English. – GEdgar Sep 15 '13 at 22:59
• Surely there must be something that can be said about all numbers such that we can easily differentiate between numbers and something else. For example, is zero a number? Certainly it can be instantiated with such expressions as 'no fabric', 'no heat', 'no gasoline', 'no food' and most importantly 'no water' and the like. So, please tell me, what do all numbers have in common without asserting the definition changes over time as our knowledge of them changes. – Michael Lee Sep 16 '13 at 0:33
• So, please tell me, what do all numbers have in common without asserting the definition changes over time as our knowledge of them changes. But the various different notions of "number" each have different types and extent of conceptual overlap, the only single total commonality is their shared history. It's patently unfair and ignorant to ask someone to answer your question but at the same time explicitly tell them not to tell you the correct answer. Your analogies are false and your "surely" is very false. – anon Sep 16 '13 at 1:54
• This really is a question more suited for philosophy.stackexchange.com – Pax Kivimae Oct 1 '13 at 22:38
• One view is that "number" is just some (undefined) entity that satisfies certain axioms (with some, equally undefined, "operations" and "relations"). Another is that, starting from some undefined entities, you construct some other entities, and the later ones you call "numbers". Somewhere you hit not further defined entities, infinite regress is out in definitions. – vonbrand Jan 4 '16 at 14:52

It depends on the context. The word number could for example mean integer, rational number, real number or complex number, all of which have precise definitions. In some situations it could even mean something like "an element in some particular ring or field", again this is well-defined. There are lots of other "number systems" in use as well, and it's probably impossible to list all of them.

The problem with a dictionary definition is that they don't build the language using undefined terms, axioms and definitions.

• And don't forget about Hartogs numbers, transfinite numbers and the like, which might be considered numbers by some people. – Git Gud Sep 15 '13 at 22:57
• Yes, there are many other "number systems" as well. – mrf Sep 15 '13 at 22:58
• cardinal numbers, ordinal numbers, $p$-adic numbers... – GEdgar Sep 15 '13 at 23:00

What is a number? That is a good question.

If you ask the somewhat educated layman, you might get answers such as "a representation of a physical quantity". You may get some other answers too.

But since mathematicians use words from natural language with a particular meaning, let's cut the foreplay, and skip right down to the mathematician part. However there is no agreed, or even common, definition for "number". The definition I have in mind, and I suspect that many mathematicians would agree with me, is the following one:

We say that $x$ is a number, if it is an element of a number system, which is a system representing and measuring a quantity of some form.

This definition allows for natural numbers, integers, rational numbers, real numbers, complex numbers, ordinals, cardinals, and so on. All these are number system. The only thing they have in common is that they measure some sort of quantity, and they represent it somehow.

Therefore, for me, the context "X numbers" means that "X" is some sort of form of measurement for mathematical objects.

• Please look up in a dictionary of philosophy the word "circular reasoning". I have already said "number" and "quantity" mean the same thing and you use it in the same sense. All you are saying is if such and such is a number, then it corresponds to measurements of reality. – Michael Lee Sep 16 '13 at 3:55
• If you define quantity to be a number, and a number to be a quantity them it is circular. This is not the case, I'm afraid. Quantity, in this context, is simply a form a measurement in mathematics. Numbers are the values of that quantity. However the value and the quantity are two distinct objects. – Asaf Karagila Sep 16 '13 at 5:23
• @Michael: The point in my previous comment, and in my answer after all, is that dictionaries tell you the meaning of a word in the "natural language" sense of the word, and not the in the "mathematical language" sense of the word. Mathematical terms have definitions, and meta-mathematical terms (e.g. number and quantity) have no definition, but they represent a nearly ineffable notion shared by mathematicians. – Asaf Karagila Sep 16 '13 at 6:23

The word "number" means what I wish it to mean, nothing more, nothing less.

Depending upon what I'm doing, things I've wished "number" to mean have included:

• Natural number
• Integer
• Rational number
• Real number
• Projective real number
• Extended real number
• Complex number
• Projective complex number
• Hyperreal number (and the nonstandard versions of all of the above)
• Continuous real-valued function
• Ordinal number
• Cardinal number
• Polynomial over a finite field
• Rational function over a finite field
• Element of a particular ring I'm working with
• Abelian group
• I'm with you on all of these except "abelian group" (but this may just be a parse error on my part). In what context (or perhaps, "for what reason") do you find it useful to think of an abelian group (as opposed to one of its elements) as a number? – Mark S. Dec 29 '13 at 1:49
• @MarkS: Doing computations with homology groups is what I had in mind originally. More generally, objects of any category would count as some sort of number if I'm doing doing algebraic calculations in that category: combinatorial species makes for an excellent example of this! – user14972 Dec 29 '13 at 10:57
• I can see what you're getting at with homology group calculations. I was previously unfamiliar with the language of combinatorial species, but reading up on it makes it clear why you might do that. Thanks! – Mark S. Dec 29 '13 at 14:51

In set theory, we begin by defining $0$ as the number of elements in the set having zero unique elements. Then, for each subsequent number that we wish to define, we say that $i+1$ is the number of elements in the union of the $i$th set and the set containing the $i$th set. If we label this series of sets as $u_i$, then we have $u_0=\varnothing$, $u_{i+1}=u_i\cup \{u_i\}$. Then our first few "numbers" are as follows:

$$u_0=\varnothing$$

$$u_1=\{\varnothing\}$$

$$u_2=\{\varnothing,\{\varnothing\}\}$$

$$u_3=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}$$

...

This is rather like getting something from nothing, but it definitely gives uniqueness to each integer.

The concept of "number" according to mathematicians has expanded dramatically over the centuries. It's by now very hard to give an all-encompassing definition.

There is a book Numbers written by a group of German mathematicians, with some historical notes, and broadly speaking in three parts. Under their Part A, we have Chapter 1: Natural numbers, integers, rational numbers. Chapter 2: Real numbers. Chapter 3: Complex Numbers. (...) Discussion of algebraic numbers. (...) Chapter 6: The $p$-adic numbers (and I suppose we ought to include local fields in the sense of Weil, and also adeles).

Under Part B, we head off into real division algebras; we have Chapter 7 on Hamiltonian numbers or quaternions. Chapter 9: Cayley's numbers or octonions. One could go on to composition algebras (chapter 10).

Under Part C, we head off into more set-theoretic territory. There are the models of nonstandard real numbers inaugurated by Abraham Robinson (chapter 12). Then there are Conway numbers or surreal numbers (chapter 13). Chapter 14 discusses cardinal numbers and ordinal numbers.

In most of these systems, there are operations of addition and multipication so that most of these number systems can at least be described as rings or algebras of one sort or another, in the senses that mathematicians give these terms.

• Please give a citation for the book. – vonbrand Jan 4 '16 at 14:55
• @vonbrand I'm pretty sure it's Zahlen, by Ebbinghaus et. al. Published by Springer a long time ago. No idea whether it was translated to English (or any other language for that matter). It was. – Daniel Fischer Jan 4 '16 at 14:59
• @vonbrand Yes, it's the one mentioned by Daniel Fischer. My own edition is in English. – user43208 Jan 4 '16 at 15:44

This is a never interesting question and is remarkably complicated. The reason for the complexity is two-fold, there are many different number systems, and constructing them is a lot of work. For example, the most popular way to define natural numbers defines $2=\{\{ \}, \{\{ \} \} \}$. The number $\frac{1}{2}$ is a set containing $\{(1, 2),(2, 4), (3, 6) ...\}$ (It's all the numbers $p/q$ such that $2p=q$, but then you have to define multiplication (This is done with the Peano Axioms)). Real numbers are even more complicated, and can be defined as all the rational number before the real number itself (A dedekind cut). In the end, if you want a satisfying answer to this question, you need to learn a lot of math. Then again, there are multiple definitions of each, and few mathematicians really ever use them unless there doing a proof involving the construction itself.

If your interested in learning, Naive Set Theory by Paul Halmos and Real Analysis by Pugh are a good start.

To my view and knowledge with referring some Frege's arguments: A number is a least common property that objects/things of the same "quantity"(in a primitive sense) have to share.

• Yes, for example (one one), how many numbers appear in the brackets? There are two numbers, namely one. Or, there is one number, namely one. Or, as David Armstrong would say, "there are two tokens of the same type." – Michael Lee Sep 19 '13 at 0:32

I love this question.

Most people's definitions of number seem to involve the idea of quantity. I guess this derives from what we presume to be the origins of numbers - the counting of things like goats. But I think this is misleading and ultimately a dead end...

Quantity seems to be left behind quite early on in the development of numbers. As soon as we start using irrational numbers we have things regarded as numbers which do not relate directly to quantity - you can't have Pi of something or the square root of 2 of something. When we get to complex numbers the connection to quantity seems more remote still. If we had started with complex numbers instead of natural numbers the idea that numbers related to quantity would seem far fetched indeed.

Having given it some thought the only consistent theme I can find between all the generally accepted uses of the word number is that of 'constancy': a number is the name we give to a constant; that is why Pi is a number: Pi is the name we give to the constant ratio between the area of a circle and the square of its radius. Any value of any numbering system expresses some quality of unchangingness.

So my definition of a number system would be a 'symbolic system of constants' and a number would be a member of such a system. Natural numbers are thus the set of symbols which represent all possible states of discreet existence(or something like that).

• But $c$ is a constant (speed of light), which isn't a "number". Similarly, $\aleph_0$ is the cardinality of $\mathbb{N}$, $\varnothing$ is the empty set, ... there are lots and lots of entities with "names" that just aren't "numbers". – vonbrand Jan 4 '16 at 15:57
• My definition was a member of a "system of constants". but even so - c is a member of the system of real numbers; why is ∅ the empty set not a number? What characteristic of 'numberness' does it lack? It can certainly be used in symbolic statements of logic in very much the same way that '0' can be: it just belongs to a different system of constants. – Jumblegreen Jan 5 '16 at 11:12
• The same is true for ℵ0 its just that these constants do not belong to systems of constants which were designed to describe characteristics having to do with quantity – Jumblegreen Jan 5 '16 at 11:16
• I take back the comment about c being a real number. Clearly the speed of light can be expressed using the real numbering system but the letter c is not itself a member of that system. C is an arbitrary constant as opposed to a 'member of system of constants' – Jumblegreen Jan 5 '16 at 11:22
• you are the calling anything that can be used in "mathematical statement" number, which is definitely overreaching. And just pushes back the question to "what is a mathematical statement"... – vonbrand Jan 5 '16 at 12:46