That's one reason a definition for number is good to have before talking about them. — tim wood
Yet there is not one single definition of number. It's an amorphous concept. Mathematicians "know one when they see one." I don't know if this has caught the attention of philosophers. But there is no definition of number. — fishfry
And I'll try one: number is that which has neither extension, substance, nor quality, but that expresses/represents quantity. — tim wood
Yet there is not one single definition of number. It's an amorphous concept. Mathematicians "know one when they see one." I don't know if this has caught the attention of philosophers. But there is no definition of number. — fishfry
The theme of mathematical structuralism is that what matters to a mathematical theory is not the internal nature of its objects, such as its numbers, functions, sets, or points, but how those objects relate to each other. In a sense, the thesis is that mathematical objects (if there are such objects) simply have no intrinsic nature. The structuralist theme grew most notably from developments within mathematics toward the end of the nineteenth century and on through to the present, particularly, but not exclusively, in the program of providing a categorical foundation to mathematics.
Mathematical structuralism is similar, in some ways, to functionalist views in, for example, philosophy of mind. A functional definition is, in effect, a structural one, since it, too, focuses on relations that the defined items have to each other.
A structure is the abstract form of a system, which ignores or abstracts away from any features of the objects that do not bear on the relations. So, the natural number structure is the form common to all of the natural number systems. And this structure is the subject matter of arithmetic.
http://www.iep.utm.edu/m-struct/
And I'll try one: number is that which has neither extension, substance, nor quality, but that expresses/represents quantity.
— tim wood
Ok I'll play. Four questions.
* What is quantity?
* The imaginary unit i with i^2 = -1 ... what quantity does it represent?
* Do you regard i as a number?
* Does i exist? — fishfry
It's a philosophical curiosity that there is no definition of number in mathematics. In other words if you major in math, get a Ph.D. spend a career as a professional mathematician, you will never encounter a book or a paper that says, "A number is such and so." — fishfry
* What is quantity?
* The imaginary unit i with i^2 = -1 ... what quantity does it represent?
* Do you regard i as a number?
* Does i exist? — fishfry
In a sense, the thesis is that mathematical objects (if there are such objects) simply have no intrinsic nature.
A structure is the abstract form of a system, which ignores or abstracts away from any features of the objects that do not bear on the relations.
Numbers do number-like things. What are the important things left unsaid? — apokrisis
Saying that a number is anything that's number-like is a circular definition. No better than the poster above who said that a quantity is anything that's quantitative. You are defining a thing in terms of itself. It's not a definition. — fishfry
Matrices can be added, subtracted, multiplied, and sometimes divided. In fact the set of nxn matrices for fixed n forms a ring, an important algebraic structure. But matrices are not regarded as numbers. — fishfry
I was surprised at the, let's say, passion of some of the responses to this tame and factual assertion. — fishfry
It's surprisingly tricky to give a good definition of number. I hope my examples bear that out. — fishfry
Really? What is number theory about? What is Principia Mathematica about? — tom
Sorry but you are bullshitting to an extraordinary level. — tom
You think numbers are defined in terms of quantity? — tom
i represents the square root of -1. — tom
i is a number. — tom
i exists. — tom
Simples. — tom
There is no general definition of number anywhere in mathematics. Of course there are perfectly clear definitions of particular types of numbers. Integers, reals, quaternions, p-adics, transfinite ordinals, and so on. But nowhere in mathematical literature will you find anyone who ever says: "A number is defined as such and so." — fishfry
Post that reference and my thesis stands refuted. — fishfry
At least Frege, Russell, and Whitehead defined what a number is. There are probably several others. — tom
Hang on, there's even a Wikipedia page:
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers — tom
My remark is entirely agnostic of foundational approach, a point apokrisis does not sufficiently appreciate.
There is no general definition in math that tells us what a number is. — fishfry
There is no general definition in math that tells us what a number is. — fishfry
"Is" can be a tricky word. — tim wood
You ask "what is quantity?" Quantity is the general name for an idea that is always particular, and that refers to anything that can be quantified. — tim wood
Now I think you are confused in that you think a definition somehow "is" what something is. — tim wood
Or you apparently think that the definition of number, or quantity, will tell you what these things are. — tim wood
This approach or understanding - actually utilization - gets a lot of the world's work done, but it isn't remotely true. A definition is simple an agreed description, for some purpose. — tim wood
As to i, it's the square root of -1, it's a number, and it exists (keeping in mind you probably have at best a partial idea of what "existence" means, and of what I mean by it). — tim wood
Definitions, then, are functional. — tim wood
And if any thing is going to be discussed in terms of its definition, or any understanding of what that something is, then it's best to start with some explicit expression of that definition or understanding. That's just good navigation. And of course it's negotiable, if that's appropriate. — tim wood
Why wouldn't ZFC count? — Akanthinos
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