# What is Number again

I came here having asked on the math stack exchange site about number. There are several responses to that question or one similar that suggest that here is the best place to ask the question. On the way here, I realized that the are at least two variations of this question. One is: What is a number? Another is: What is number. A common answer is that a number is a quantity. Another questioner noted that a dictionary definition of number was quantity and that the definition of quantity was number, or a number. This is just a circle, of course, and only addresses the first question.

Assuming I use the correct words in the correct order, I could ask most educated people if they can add two numbers together and an answer of, "Yes, of course." This use of the meaning of number is common, but does not answer either question. Years ago, someone made sure I knew the difference between a number and a numeral. Assuming I understood it then and remember it correctly now, that use of the term number is, at most, only slightly different than how I used it in my question about adding two numbers. There are likely multiple uses with slightly different meaning for the concept of number. These meaning might or might not be all there is, but they do not answer either what is a number or what is number. I think, though I have nothing to support it, that an answer to what is number will provide and answer to the second question.

So, from the stand point of philosophy, what is number?

• Basic rule: we cannot define everything. – Mauro ALLEGRANZA Apr 1 '20 at 18:55
• From a mathematical point of view, numbers are the "objects" that satisfy the theory of numbers. – Mauro ALLEGRANZA Apr 1 '20 at 18:56
• From a philosophical point of view, the nature of number as abstract objects has been debated since Plato's time. – Mauro ALLEGRANZA Apr 1 '20 at 18:57
• Let's say "number" means "natural number" (others can be defined in terms of them). You can take "number" as a primitive concept and "define" it implicitly through axioms (e.g. Peano axioms). As a variant, you can "define" it by describing how the concept is used, i.e. give rules for addition, multiplication, use in sentences, etc. Or you need another, more primitive, concept, such as set. Then you can define numbers in terms of that, e.g. by von Neumann's construction. – Conifold Apr 1 '20 at 19:50
• Bertrand Russell expressed the view that we should view Mathematical Induction not as a principle or axiom but as a definition; and as a definition mathematical induction defines the natural numbers. Mathematicians then define all other numbers in terms of the natural numbers. – Nick Apr 1 '20 at 20:02

Maybe this isn't quite general enough to help, but perhaps some material from Mathematical Cognition might be helpful in drawing out a distinction I think you want to be making?

In both live Mathematics research and Mathematics education, we often talk of there being two kinds of numbers.

The first kind of number that we talk about is when we have some collection of things as a quantity. For example, brains fairly quickly in life (and early in development) know to distinguish between one of something, two of something, and many of something. If I have a bunch of sweets in one pile on the table and another bunch in another pile, we can often say that we can tell just by observation and comparison whether there are more sweets in one pile over the other.

This is a broad stroke interpretation of what we mean by the "Cardinality" of a set of things, and our standard measure of cardinality, of "how many", is to use the Cardinal Numbers. Cardinality is abstractly understood - whenever we say that we can take any two sets and can functionally put them in "one-to-one correspondence" with each other, then we say that they have the same cardinality.

The second kind of number is more abstract than that, and takes a bit more teaching for people to grasp. This kind of number is what we are pointing to when we work through a sequence of names of numbers in succession. At school we are taught to "count" by working through those names in order - we go "One, Two, Three, Four, Five...". Each of the points in the sequence is understood to follow after the other, and as we are soon taught, you can "add one" to any element of this sequence to get the next element in that sequence.

This roughly speaking is what we are trying to use to demonstrate our understanding of some ordering of a sequence, and we call this measure of the type of ordering involve the Ordinal Numbers (with each number occupying a position in the sequence). To understand how to get at Ordinal "twenty five thousand, two hundred and thirty six", for example, we don't need to go out into the world and find some set of 25236 things in order to show that we have correctly understood this - we can demonstrate a familiarity with what it means to be the successor of 25235.

Now, there can be some nuance to how we conceptually use ideas of "numbering" and "numbers", because in English (at least), we often assume that measuring "how many" and measuring "how to order" are functionally the same. This is because when we're young, we're taught to use the Cardinality of Ordinal sequences to help us work out exactly how many there are in any given set of objects. This object is "one", this object is "two", this object is "three"... And, of course, through demonstrating little principles like how addition is similar to repeated addings of one, we show how the ordinal sequence of numbers can be used to give us more rich subdivisions of cardinal quantity than our brains naturally jump to themselves ("how many is ten thousand", for example), and also some neat cognitive tricks we can use to get those quantities using ordinal technology.

But the two ideas do importantly come apart. For example, even if we think there is some evolutionarily advantageous basic "number sense" of cardinality across human cognition, different cultures and societies form different models of ordinal sequences, and different people seem to have a better or worse time learning to grasp and use these sequences well. One plausible suggestion is that there are similar brain functions used in both concepts of number, but that the Ordinal one shares more in common with the language processing mechanisms of our brain, while the cardinal one ties closer to our image perception and object recognition.

If that is the case, then while it seems like Cardinal numbers can roughly reduce to being patterns of human brains, Ordinal numbers might be more of a social protocol - an abstract pattern of thought that is built up across a community of mathematics practitioners rather than just for any one of us. But that is just a theory (...), and certainly wouldn't be considered philosophical canon; if anything, a philosopher of science should be leaving this question to the psychologists to answer, rather than trying to make specific headway on it themselves.

• I think you have given me insight into what is a number, or, what are two kinds of numbers. There is still the question of what is number. Water exists without human "intervention". Does number? I have reached a point where I know that I do not have the capacity to talk well about this. Perhaps it is enough that I know which pile of sweets to take and where to get in line. – user45970 Apr 2 '20 at 12:37

What is a number?

I assume that you are familiar with numbers, that they are used for counting and for quantifying; and that you are not asking about the genus of number, that is what is counted as a number - since mathematicians have discovered many other number systems and mathematical systems which could be justifiably called numbers - for example complex numbers, octonions and the like and are asking about the ontological status of numbers.

There are two positions on them: nominalism and mathematical platonism. The former states that they are ways of speaking without an underlying reality; for example the number two denotes the set of all pairs of objects; the latter states that they are ideas in a timeless and spaceless sphere that is only accessible by the rational faculty in man; it's related to platonism, since the world is mathematical, the universe itself is said to have a kind of rationative faculty; thus in platonism the two are related.

Newton, for example, was a Christian Platonist; in his short work On Gravity, he wrote:

And by an equal reason it is conceded that God by the sole action of thinking or willing could embrace any defined space by certain limits that some bodies not advance [penetrate] into it ... [thus] impervious to bodies, and thus light and all pressing things would stop or would rebound; it seems impossible that with the aid of our senses (which should be constituted judges in this matter only) we will disclose this space not actually to be body; it were indeed tangible on account of the impenetrability, and visibly opaque and coloured on account of the reflection of light, and a blow would resonate for the reason that the neighbouring air would be moved by the blow.

Thus atoms, according to him, are mere nothings and are created and sustained by God by the idea of an ideal sphere impressed on the nothingness of the world; numbers likewise, in the mind of man.

In its most abstract sense, number is a property of our conception of multiple things which we associate together. Number doesn't exist outside of our intellects: number is part of our perception of what exists, it is one way we classify things.

• Does that mean that if, or maybe when, we no longer exist that numbers will not exist either? That is a little tough. I am more of a math guy than a philosopher, so it may well be that the concept of number in this realm is beyond me. I would very much like to think that number exists without our intellect. – user45970 Apr 8 '20 at 2:25
• Think about it this way: we perceive number in things which exist; number is not that which exists, but merely our understanding of something that exists. The idea of a certain number can be communicated through word and writing, and in that sense, the communicated number exists without our intellect. – user96931 Apr 8 '20 at 22:09
• If I understand correctly, I pick up several rocks and put them on a picnic table and the quantity of rocks is something that we identify as a number. The quantity does not go away, but without our intellect the entity I named as "a number" does. I find this difficult to talk about. Two questions, what is water and what is number should elicit isomorphic answers. One answer would use the word "wet" and the other would use the word "quantity". Does this seem close to what you are saying? – user45970 Apr 8 '20 at 23:24
• Yep, you've got it. – user96931 Apr 10 '20 at 1:12

Short answer : numbers are not additional entities along concrete ones, they are higher level entities, namely, sets of sets.

• One can consider distinct ontological levels.

• First level : concrete objects.

• Second level: sets of concrete objects.

• Third level : sets of sets.

• Suppose you want to classify sets according to this criterion : is set A equinumerous to set B, that is , can a one to one correspondance be established between the elements of A and the elements of B? Doing this, you operate a partition ( corresponding to an equivalence relation , namely " equinumerosity").

Note : you do not need to know any number do do this, you do not even neeed to possess the concept of number ; hence, no circularity.

• The classes of sets that are generated by this partition are what we call numbers. For example, the number 5 is simply the class of all sets that can be put in one to one correspondance with the set having as elements the fingers of your right hand.

• hence , saying that a set has 5 elements amounts to saying that this set belongs to the class called " number 5".

Reference : Bertrand Russell, Introduction To mathematical Philosophy.