Eh, you don’t “know” what comes after one. We give a name to “that thing that comes after one;” in this case, two. There is nothing more mystical about “two” than “apple” or “xenophobia.” Just words.

I am also wondering about “multiplicative.” You understand that multiplication is defined as repeated addition, right?

oh no… you used the D-word…

Mathmatics is a measuring system for how we define the universe. If it didnt exist, it would be necessary to invent it. Mathmatics itself is not made up. Our system for defining Mathmatics is.

Am I reading this correctly? Are you sure you meant to put the “not” in there?
I suppose mathematics must be tied to the real world at some point. 1 is 1 because that is a representation of a single object. But purely in the abstract (which a goodly portion of mathematics still is), it doesn’t make much sense. Why must 1 be 1? Why can’t it be…
I can’t even think of a counter example. The idea of OUR numbers has been so ingrained in me, it’s hard to seperate it out and think about what would happen if I knew absolutely nothing about math or numbers or counting and tried to create a seperate system.

I am of the opinion though that the basic functions (±*/) are the constants rather than the numbers themselves. Numbers can be anything based upon, as The Ryan said, what your definition of “1” is, or even what your base system is.

It’s a basic property of the field. Take any field, and “multiplication” is the second operation associated with it. There’s no way to know what multiplication is until you’re told what the field is.

Vinnie Virginslayer

Huh? They’re both representing the same concept: the multiplicative identity. Since they’re the same thing, of course they’re equal.

erislover

[quote]
Eh, you don’t “know” what comes after one. We give a name to “that thing that comes after one;” in this case, two. quote]
That’s my point. We don’t know what comes after one until we have defined “what comes after”. “Two” depends on the definition of “plus”, not vice versa.

No, I wasn’t aware of that. I’m aware that it is explained that way, but not that it is defined that way. Can you elaborate? Suppose A, B and C are given as below. How wopuld you get C by repeated addition?

A= 01 B= 01 C=20
10 20 01

AB=C

Enderw24, I think you’re getting too hung up on the symbols we use to represent numbers, as opposed to the numbers themselves. This whole issue of different base systems, for example, is pretty meaningless: hexadecimal, duodecimal, binary, and so on are just different names. What matters isn’t what symbol we use to represent sixteen, be it “sixteen”, “16”, “10”, “14”, “10000”, or even “XVI”, what matters is that sixteen is what I get if I add one to itself, then add the result to itself, and then do that two more times. Even The Ryan’s five-by-five matrices are just different names. What matters is how the numbers behave. No matter what kind of system of arithmetic you might try to invent, either it will have an object I can call “one”, and another that I can call “two”, and a notion of “plus” such that “one plus one” equals “two”…or else it won’t describe the way real-world objects behave.

Your questions of “Why must 1 be 1?” and “Why must 1 be a whole number?”…well, I’m not sure the first question is even meaningful, and the second is like asking “Why must apples be fruit?”. The answer is, “We don’t have a choice in the matter”.

For the integers (or to be strictly accurate, for the natural numbers), multiplication is defined as repeated addition. For other systems which are built up from the integers (like the rationals, the reals, and matrices of same) multiplication is defined in terms of the sums and products of integers.

You example of matrices as an alternative form of arithmetic is kind of misleading. If you stick to diagonal integer matrices, then all you get is different symbols for the usual integers. Then you’d have no problems defining multiplication as repeated addition.

If you start to use non-diagonal matrices like you have above, then you get a system which behaves in fundamentally different ways from the integers. In your example above, for instance, AB doesn’t equal BA.

Precisely. I want to start with integers. Whilst it doesn’t matter what you call them, it is a feature of integers that multiplication is defined in terms of addition.

Ryan - what you’re saying is correct. But I don’t think that it is the “beginning” of maths. You wish to start with multiplication and an identity. But then where are the other integers coming from? I wish to start with the integers and define addition as the mapping that connects them. I can then create, IIRC, modulus arithmetic and show that it has the properties of a group (and a ring? don’t remember. Too many years ago).

And you are defining multiplication as “a basic property of the field”… but we don’t have a field yet! We don’t even have a ring! We certainly won’t get a field until we have fractions, irrationals and complex numbers. Christ - even rigorously defining fractions is a pain in the arse. I don’t think this is the fundamental axiom. I still think that the fundamental axiom is that “the integers exist”.

pan

YES!!! I thought that I was the only one who seriously thought it was.
I mean, really, people…what makes a person sit down and come up with this crazaaay stuff?! Then, somehow they make it all legit and then I gotta learn it? why??? I believe it’s sorta like the Bible and philiosophy. It’s all a matter of interpretation. There is no actual proof anywhere that 1 + 1 = 2 Sez who?! I don’t care what the square of PI or whatever the hell it is! How do you know that when you move that damn decimal point 3 places that it becomes a smaller number? ARGHHHHHH!!!

Can you explain how that works?

No, the point was to show that the standard conception of multiplicatrion is not generally valid; multiplication is a much more abstract concept than simply “just keep adding stuff together”.

[nitpick]Actually, they’re a larger set, since not all diagonal matrices are scalar multiples of the identity[/nitpick]

I already told you. They’re defined recursively/ inductively. For instance,
7=6+1
6=3+3
3=2+1
2=1+1

But part of what the integers are is the addition properties. The integers are not just a set. They are a set with a structure on them.

We don’t have any fields? Surely we have some field. And once we have a field, we can take a subset to be the integers (well, we have to have an infinite order field).

Oh, those last three quotes were from kabbes.

It’s a recursive definition. Two of the Peano axioms (which define the natural numbers) are “There exists 0” and “For any natural number x there exists another natural number Sx, the successor to x”. Then addition is defined recursively in terms of the successor operation, and multiplication is defined recursively in terms of addition. Specifically, we say “+” has to satisfy two rules: 0+x=x for any natural number x, and (Sx)+y=S(x+y) for any x and y. Similarly "" has to satisfy two rules: 0x=0 for any natural number x, and (Sx)y=(xy)+y for any x and y. These together define “+” and "" for any two natural numbers, for example if we let “1” denote S0, then 1+1=(S0)+1=S(0+1)=S1, and 11=(S0)1=(01)+1=0+1=1.

**

Whoops, you’re right. I should have said “diagonal matrices with equal entries along the diagonal”.

But note that if you include diagonal matrices with unequal entries along the diagonal, then once again you get a system that doesn’t behave like the integers. For example you can find matrices A and B such that A*B=0, but neither A nor B equal 0.

Sorry if this has already been mentioned, but I’m kind of in a hurry and couldn’t read the whole post, but:

Civilizations from different corners of the globe have indepentently “discovered” mathematics, and all count 1, 2, 3, 4, etc. Of course, the systems weren’t flawless (the ancient Greeks believed pi to be exactly 3), but they had the same basic counting system and system of logic.

math geek:

So once you’ve defined * for two cases, you can define it everywhere. But you still have to have two cases that aren’t repeated additions. It’s an interesting formulation, but it doesn’t change my position that numbers are defined in terms of multiplication, not multiplication in terms of numbers.

I disagree. Have you looked at a book that describes the construction of the natural numbers from the axioms of set theory? In that situation (which I don’t believe anyone disagrees with, although I could easily be wrong), it’s very obvious that numbers come first, and then operations on them.

um, just so people don’t think I’m ignoring this thread, I’m not. But the math gurus of this board have taken the debate past my ability to intelligently weigh in. Which is fine, I’m still reading…and trying to learn…

Godel’s Incompleteness Theorem

Basically, this theorem proves that some things are unprovable. You might even say that everything is made up at some level.

Actually, technically it’s only one case, the case of 0x=0. It’s true that that case is not defined in terms of repeated addition, but all the others are, even the case of x0=0.

The Peano axioms that I mentioned before define the natural numbers completely without mentioning multiplication at all. They don’t even mention addition. Those two operations are defined by showing how to compute them recursively for any pair of natural numbers, but only after the natural numbers are themselves defined. It’s simply false to say that the natural numbers are defined in terms of multiplication.