What is the true definition of a number?

[mathwonk]

come on guys, this is embarrassing

 

[chiro]

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]

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.

 

[Thetes]

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" 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.