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
a structuralist definition, in that numbers are whatever it takes to get certain number-like operations - like those that preserve certain global symmetries, such as commutativity or associativity. — apokrisis
There is an advantage to this approach. Mathematicians are not constrained by a definition of number, which allows them to discover new types of numbers all the time. — fishfry
If you don't like the fact that numbers are defined in terms of set theory — tom
I guess you won't like the fact that numbers are also defined in terms of field axioms. — tom
And no, new types of number are not "discovered all the time". — tom
Good idea and a very natural attempt; but arithmetic properties aren't sufficient. Weirder still, there are numbers that lose associativity as well, such as the octonions. — fishfry
In short though, mathematical structuralism is more subtle than just listing arithmetic properties like associativity.The kinds of properties that they use in category theory are ... well, they're kind of weird and nonintuitive when you first see them. The structural relations they have in mind are various types of universal mapping properties. It's hard to do justice to what this means in a simplified format but I might take a run at it once I get into responding in detail to your earlier post on structuralism. — fishfry
I did not write the quote you attributed to me. What is your attitude problem? — fishfry
Nobody has provided a counterexample and you now seem to agree. — fishfry
If you choose category theory as your foundation, there's still no general definition of number. — fishfry
Your inability to discuss the foundations of maths is noted. — apokrisis
Your complete misunderstanding and lack of comprehension of category theory and mathematical structuralism was evident to several other posters the last time we discussed this. — fishfry
I'm done responding to your posts on this site. — fishfry
The question then arises, 'what is the nature of number?' Conjecturally, one might say, number is a series of equal values (quantity). Hence, Pythagoras' and other ancient mathematicians' inclination to render number as equal, whole values. If this is an accurate description of number, then it follows, the concept of number is tied to the idea of a 'unity' value (unit measure). — cruffyd
Temporal construction is such that inequality defines its nature. Equality, on the other hand, can only be outside of temporality. One might say: 'equality can only exist eternally.' — cruffyd
Yeah. The historical view is a good way to get at it. There was a reason why the Greeks were so horrified by the notion of an irrational number. That very reaction betrays the underlying belief about what a definition might be. — apokrisis
He made a grammatical point, and in this he was correct.Bill Clinton made that very same argument to try to wiggle out of a sex scandal. In the end he lost his license to practice law and was impeached (but not convicted). — fishfry
This criticism might have some merit if that were what we were doing. But we weren't, so it doesn't.You and apokrisis seem to feel that "a number is anything that's number-like" and "quantities is whatever can be quantified" represent valid definitions. What happens if the biologists get hold of this trick? A fish is whatever is fish-like. A cat is whatever is cat-like. A virus is whatever is virus-like. And the deepest question of all: life is whatever is life-like. — fishfry
I should think not; keep in mind I did not offer a definition of "quantity." You asked what quantity is, and I answered. I thought it was a pretty good answer - to the question asked!So my definition is, a quantity is anything that can be quantified. But that's no help! — fishfry
Question: does i exist in some, or any, sense or way that is different, in any way, from the way that other numbers exist? Question: Where did you see an i?To be sure, I have no idea what you mean by existence. I would say that i exists because it exists in math according to the formal rules; and also because we see many instantiations of the i in the physical world. — fishfry
To sum up, or rather to get back to basics, you claimed that numbers represent quantities. The number i represents a phase angle in electromagnetism or a quarter turn if you're in the plane. But I don't see those as quantities. So I have to ask again, what is a quantity? Are you claiming that the number i represents a quantity? That I do not agree with. I don't see it. — fishfry
He made a grammatical point, and in this he was correct. — tim wood
This criticism might have some merit if that were what we were doing. But we weren't, so it doesn't. — tim wood
I should think not; keep in mind I did not offer a definition of "quantity." You asked what quantity is, and I answered. I thought it was a pretty good answer - to the question asked! — tim wood
Question: does i exist in some, or any, sense or way that is different, in any way, from the way that other numbers exist? Question: Where did you see an i? — tim wood
If i is not a number, then what is it? — tim wood
If numbers do not represent quantities, then what do they represent? — tim wood
If i is not a number, then what is it?
You're free to agree or disagree with whatever you like; in this case, you might have done some research. I did. Mathematicians appear to classify i as a number. — tim wood
This is a perfect example of where we're bumping into each other. What I said was that Clinton made a grammatical point, about which point he was correct. (Whether he was correct to make the point, or any other "whether correct" you care to adduce, is a different question). I made no mention of dependence or of anything that "depended" on it.But when you say it depends on what the meaning of "is" is, — fishfry
The difficulty here is that you're thinking (I think) I defined the term. I didn't.Ok. So when you say that a quantity is that which can be quantified — fishfry
This is a proposition. Among you're options are to agree with it or disagree. Which, If you care to say? By the way, to suppose that my proposition says,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
is to misread the plain English of it to the extent that I must suppose that perhaps you're writing in a second or third language. Credit to you if you are! But you're misreading/misunderstanding the proposition....that a quantity is that which can be quantified. — fishfry
Now you have me saying that i is a number (yes), that number is quantity (no), and therefore i is a quantity. What you're stumbling over is the distinction between representing something and being something. Which means our real discussion/problem is understanding each other.Fair enough, but this only means that, as you say, there is no definition. That neither stops us - anyone - from offering one, or relieves us from the obligation to try, if we want to have any sort of reasonable discussion. It might start out, "For the purposes of this discussion, in which I intend to make points a, b, and c, I provisionally define number as.... And I'll try one: number is that which has neither extension, substance, nor quality, but that expresses/represents quantity. — tim wood
As to the quantity i: Question: is i ever an answer, in any form, to any question of how many? — tim wood
Really all this says is that 'what is a number' and 'do numbers exist' are to some extent independent from the concerns of doing mathematics. — fdrake
The reals (excluding weird stuff about 0) under multiplication and addition in the usual sense satisfy modern intuitions about what it means to be a number. When those intuitions are formalised, in turns out that there are other structures which aren't commensurate with folk intuitions that nevertheless satisfy the axiomatisation of a field inspired by those folk intuitions. — fdrake
I noted that there is no general definition of number in mathematics. A well-known and true observation. For whatever reason, this simple and harmless statement triggered several people. I still don't understand why. — fishfry
Shapiro and Resnik hold that all mathematical theories, even non-algebraic ones, describe structures. This position is known as structuralism (Shapiro 1997; Resnik 1997). Structures consists of places that stand in structural relations to each other. Thus, derivatively, mathematical theories describe places or positions in structures. But they do not describe objects. The number three, for instance, will on this view not be an object but a place in the structure of the natural numbers.
https://plato.stanford.edu/entries/philosophy-mathematics/#WhaNumCouNot
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