# Mathematical definition of number?

1. Jun 12, 2009

### lolgarithms

What is the mathematical definition of "number"?

It seems odd to me that one of the most commonly used mathematical objects has no clear definition. Should the definition of number be "a member of a mathematical structure"? The notion of "sturcture" isn't clearly defined, so I don't like it. Also it will include functions, vectors, tensors, etc.
Or should it be "a member of a magma [which includes group, rings, and fields] that can not be written as an n-tuple"? This will exclude complex numbers, which are isomorphic to R2.

Both of these will exclude cardinals and ordinals, which are not sets, let alone magmas.

IMHO, no definition of "number" can be given such that it encompasses all of the inclusions and the exclusions. Shouldn't we just expel this term from mathematical parlance and just talk about "elements of [put name of set/class/space/vector space/structure here]"?

Last edited: Jun 12, 2009
2. Jun 12, 2009

### qntty

What do you mean by a "number"? A real number? A complex number? Do p-adic numbers count too? How about infinity or transfinites?

3. Jun 12, 2009

### lolgarithms

4. Jun 12, 2009

### HallsofIvy

No, the question was which. Some of those are normally thought of as numbers, some of those are not.

In any case, "undefined terms" are at the heart of mathematics. We certainly don't want to get rid of them! The reason we can apply mathematics to many different fields is that we have undefined terms that can be given definitions from the specific field.

Last edited by a moderator: Jun 13, 2009
5. Jun 12, 2009

### lolgarithms

so the term "number" can be thought as simply a shorthand term to denote an element of a naturals, or an element of reals, or complexes, or whatever, depending on context.

Last edited: Jun 12, 2009
6. Jun 12, 2009

### symbolipoint

HallsOfIvy may wish to clarify what he said; his discussion suggests that "number" is accepted as undefined and is valid and justifiable as a fundamental concept in Mathematics. We just KNOW by ordinary and extraordinary human experience what "number" means and we do not need to define it; in fact "number" is undefined. Basic Notion?

7. Jun 12, 2009

### Hurkyl

Staff Emeritus
I beg to differ -- the fact so many laypeople disagree so strongly about what 'number' means contradicts your assertion.

8. Jun 12, 2009

### lolgarithms

This discussion is complicated by the fact that numbers aren't directly derivable from sets (of naturals, reals, etc); are numbers axiomatized from sets, or sets from numbers? Does structure come from elements, or elements from structures?

Our intuitions are not always at the level of precision that ought to be in the definition of higher mathematical objects such as numbers.

Last edited: Jun 12, 2009
9. Jun 12, 2009

### Dragonfall

What's wrong with set-theoretical definitionof any of those numbers?

10. Jun 13, 2009

### HallsofIvy

I would consideder "set" a more fundamental than "number" but we then define operations on numbers that we do not have on sets.

11. Jun 13, 2009

### Cantab Morgan

I agree, so I'm fascinated that numbers are far more graspable than sets. Almost all humans can count. But almost no humans know anything beyond naive set theory. It would seem that numbers are part of the physical universe, but sets are Platonic. It surprises me that fundamental doesn't imply simple.

12. Jun 13, 2009

### disregardthat

Humans can count as a consequence of how the human brain individualize objects. When we percieve and think about a certain object we think of it as independent of its sorroundings. Therefore is quantification a natural way of ordering the objects we percieve. Numbers, however, are not a part of the physical universe as they are completely human constructs. Objects in themselves are no fundamentally different from their sorroundings. It is only because of us forcing order to nature numbers become a useful concept. In this perspective are sets no different. I would even argue that sets are more fundamental to us than numbers. In order to quantify something, for example count the amount of stones in a pile, we need a concept of what the objects need to have in common in order to be included. We form the 'set of stones' before we count them.

That's at least how i look at it.

13. Jun 13, 2009

### fleem

"number" is an axiom. it doesn't have a definition. Its existence is assumed because of repeated experience that numbers of things are conserved, and thus deserves a name. Some cave man noticed this when he put three apples in a hole and noticed it still contained three apples a moment later. Although i will admit that there is some debate on exactly which concept we should call "axiom", since circular "definitions" can be stated, certainly there has to be axioms in the thought processes somewhere!

14. Jun 13, 2009

### disregardthat

An axiom is a statement which is considered true, either for the sake of discovering its consequences, or because we consider it obviously true. "Number" is not a statement. "Numbers exist" is, however. But, by referring to numbers you automatically proves their existence. Numbers are not physical objects, but an abstract concept.

15. Jun 13, 2009

### junglebeast

The definition of "number" is any member of several specific subsets; counting numbers, integers, reals, etc.

All of these other sets are defined in terms of "counting numbers" which are the most fundamental set. For example, rational numbers is defined by the set of fractions made of counting numbers, real numbers is defined as the set of digits that are counting numbers, integers are defined as the set of counting numbers including negatives, etc.

The set of counting numbers is a specific set of 10 definitions.

0 = none
1 = one countable unit
2 = two countable units
... etc

This representation maps the numerical symbols to observable meaning.

16. Jun 13, 2009

### Hootenanny

Staff Emeritus
According your definition, there are no negative rational numbers. Generally, one defines the set of rationals as the set of all numbers that can be expressed in the form p/q, where $p\in\mathbb{Z}$ and $q \in \mathbb{Z}\setminus 0$.

Last edited: Jun 13, 2009
17. Jun 13, 2009

### junglebeast

Alright, that wasn't intended as a precise formal definition...I was just trying to convey a point that all the number sets can be defined in terms of a small set of digits which have meaningful definitions

18. Jun 13, 2009

### Cantab Morgan

Yes, it's possible to construct the reals out of the rationals (as equivalence classes of sequences, for example), but that's not the only way to make reals. One can simply postulate the existence of the set of reals and list its dozen or so axioms without referencing the rationals. I'm sorry if I'm being a bit pedantic, but I'm just suggesting that one can take take either a constructivist approach or an axiomatic approach to real numbers. And of course, people much smarter than me have shown that these sets are isomorphic.

In either case, we define numbers from sets (whether we start with the Peano postulates and construct from there or not). In this light, it's proper to say that sets are more fundamental than numbers. But I suggest that it's the usefulness of set theory to defining numbers that gives set theory its validity. It's not set theory that makes numbers valid, but the other way around. For example, if tomorrow we found that ZFC led to a contradiction when trying to define numbers, we would not throw away arithmetic, we would look for an alternative to ZFC.

19. Jun 13, 2009

### Cantab Morgan

That is really interesting and thought provoking. However it is not obvious. Let me be clear that I am not necessarily refuting what you have written, just that I'm not on board yet. If there are three rocks on my lawn, there are three rocks on my lawn whether I perceive them or not. I don't see us as forcing order on nature as much as discovering it.

But I don't agree at all that sets are more fundamental to our human cognition than numbers. Maybe lists are, but not sets. Sets have almost no structure, and are extremely abstract. The lack of concreteness is tough on human brains, evinced by humans counting long before we were writing about sets.

20. Jun 13, 2009

### junglebeast

Some specific sets of numbers (eg, rational numbers) are only defined in terms of sets...but the most basic numbers are simply digits, and each individual digit has a definition that does not require the notion of a set. Thus, the most basic type of number is more fundamental than a set...but most numbers that we use are defined with the help of sets.