# What is a number?

D H
Staff Emeritus
Science Advisor
It is mandatory for any scientific system/theory to "clearly define its terms". If we do not do that problems arise
First off, mathematics is not science. Science is constrained in that it must agree with reality. Mathematics isn't constrained in that way. The hypothesis that space and time are continuous is, at least conceptually, a falsifiable hypothesis. Suppose that some future scientific experiment shows this conjecture to be false at some very tiny scale. Will this invalidate the mathematics of the real numbers? No. It won't. It will merely mean that our use of the reals to describe the universe is not quite correct.

Secondly, mathematicians do describe their terms, very precisely -- in the form of undefined terms and axioms. The natural numbers are described by a certain collection of axioms, the integers by another collection of axioms, and the reals by yet another collection of axioms, and so on.

Lay people and scientists typically use the term "number" to mean the a member of the reals. Unless they mean something else. Mathematicians are precise and use terms such as integer, rational number, real number, complex number, p-adic number, quaternion, etc. There is no one concept in mathematics that definitively constitutes "number".

HallsofIvy
Science Advisor
Homework Helper
Lay people and scientists typically use the term "number" to mean the a member of the reals. Unless they mean something else.

Wonderful!

in mathematics is is essential that there exist "undefined terms"- that is, terms that we can treat as "containers" into which we can put whatever meaning we like, as long as the relationships between the terms, given by "axioms" or "postulates" are still true. That is precisely why mathematics is so general.

I was told in another thread that I was confused over the meaning of 'number'. This reply means that I still am!

So my question was 'is there a PF-accepted meaning for the term "numbers" ?'.

So I remain confused: Is it;
A) OK for me to use the term 'number' in a way that *I* go on to qualify (namely, that numbers are representations of value, and there may be different representations of the same value)

or,

B) it is not OK for me to qualify 'number' with the qualification in (A), which therefore implies there are some non-negotiable hard-and-fast attributes of numbers - which I'd like to read some reference material on if it is so

?

This is 'not a contest' I'm pushing with anyone, of whether my proposition is correct or not, I'm just genuinely interested to know if the field of mathematics has already gone over this, and whether material already exists to discuss this, that I can build up my comprehension of.

I think the link Thetes provides, in post #21 is very useful to this end. That was ultimately the type of thing I was expecting to be directed to, and for the thread to go on to discuss.

It is particularly interesting because it completely dissociates the concept of 'number' *away* from both 'representations' AND 'quantities'. I find that interesting and curious, because if the only way to define numbers is in some other way divorced from the original purpose, then I think we are beginning to dig away at some important philosophical aspects of the notion of '[mathematical] concepts' itself.

I'm sure that link is not the only way of defining numbers, and it would be good to see the extensions of that type of mathematical construction to reals and irrationals, so I'd be very grateful if folks have other or further such axiomatic constructions they can post or link to. Maybe we can get on to discuss some of the 'surprising results' Thetes is hinting at?

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HallsofIvy
Science Advisor
Homework Helper
Yes, the link that thetis gives is very good but if you look closely you will see that there us no definition of "number" there. It begins by saying "There exists a set N, whose elements are called natural numbers" so "natural numbers", at least, are whatever is in that set. But there are many different sets that would qualify as satisfying the "axioms" given there.

D H
Staff Emeritus
Science Advisor
I was told in another thread that I was confused over the meaning of 'number'. This reply means that I still am!

So my question was 'is there a PF-accepted meaning for the term "numbers" ?'.
No. There. Is. Not.

So I remain confused: Is it;
A) OK for me to use the term 'number' in a way that *I* go on to qualify (namely, that numbers are representations of value, and there may be different representations of the same value)

or,

B) it is not OK for me to qualify 'number' with the qualification in (A), which therefore implies there are some non-negotiable hard-and-fast attributes of numbers - which I'd like to read some reference material on if it is so

?
I strongly suggest that you don't go reinventing a wheel that took multiple generations of incredibly smart people to develop. The only way to make progress in math and science is to stand on the shoulders of those who have preceded us. Instead, read up on modern algebra. That's the algebra you take after learning calculus, not before.

What you are doing is asking us at PF to write a book, many books in fact. There are books and books on this topic. Asking us to write a book is not a fair question for a discussion forum such as this. You might be able to find some of that information on the internet, but only piecemeal. To find the information as a whole you need to read a book, take a class, or both.

mathwonk
Science Advisor
Homework Helper
2020 Award
come on guys, this is embarrassing

chiro
Science Advisor
Numbers are just ways to capture variation. That is one of the most important central ideas in mathematics: to explain and analyze variation in many different useful ways.

Each different type of number has different properties for variation. Your complex numbers introduce more variation that your real numbers, and your real numbers introduce more variation than your integers.

What that variation corresponds to is another matter. It might be physical, it might not be. We don't care about that, we only care about how the variation can be described, analyzed, and how we can extrapolate useful properties from these things.

The thing that makes mathematics powerful is that we have a lot of results that apply to situations with a great amount of variation.

It is not useful for mathematicians to prove every individual scenario individually. There are potentially infinite numbers of these, even when you constrain the classes of things you wish to prove. The point is to prove properties of something that has a large amount of variation, and the higher the amount of variation, the more powerful the result tends to be.

If we did not focus on variation, then we would be proving every situation individually. A computer can do this, but the practical effect of doing this is, in many situations (not all though, since the state space for some problems might actually be manageable with a computer) is not to be considered.

Mark44
Mentor
I'd love to, but have never heard of a dictionary being devoted to mathematics alone, let alone own one for myself.

Do you have one, and could you post what it says, please?
Dictionary of Mathematics, by C.T. Baker, published by Hart Publishing Co, Inc. The price on the cover is \$2.95. I think I bought it sometime in the late 60s.

It has a definition for Numbers, Cardinal and Ordinal, but doesn't bother to define Number.

The closer you look at the foundation the more wobbly it seems. Set theory is how number systems are defined. But, this is like a shell game. You ask what is a number, so I tell you well it's made of smaller objects. Then you ask what are the smaller objects. So, I quit hiding behind numbers and systems and tell it to you straight, we don't know what they are but they are useful. There has to be a starting point to the definitions. Those are our axioms and postulates which are just assumptions.

This might seem a sad truth that all mathematicians have to face at some point, but it's the best we can do. Unfortunately the problems only increase from there. In the early years of set theory there were a number of http://www.cs.amherst.edu/~djv/pd/help/Russell.html" [Broken] the argument.

So mathematics is a language taken by faith to be correct. Don't worry lies in math are harder to spot than English.

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