Can it? Not counting any geometric proof. Or do we half to accept it as an axiom?
How about the distributive law?
What properties of multiplication can I start with?, or do I have to go through all of real analysis up to commutation and distribution of certain operations?
Yes.
It depends on the approach you take. If you start by defining the integers as a ring with certain properties, then commutativity and distributivity are just some of the properties assumed in that definition. On the other hand, if you start by defining the counting numbers with the Peano Axioms, multiplication must first be defined in terms of addition, which is in turn defined in terms of the successorship operation. Then it must be proved that multiplication, so defined, is commutative and distributive. All these proofs and definitions are completely formal, and rely on no geometrical reasoning at all.
Eeek!! have not half :smack:
Please, can you elaborate more or point me to some place that does? Every book I have seen just lists those properties as axioms to be accepted, and I know that they are true, but I am still being driven nutty by the idea that I don’t really understand why.
I am very embarresed to be asking this question.
I think it’s a good question. But you realize that group “multiplication” is not necessarily commutative, right? For instance, matrix multiplication is not commutative. So are we talking about the integers here?
Yes. Just plain numbers. But what is “group” multiplication?
A group is a non-empty set G together with a binary operation * on G satisfying
(G1) ( Closure) xy is in G whenever x and y are ( in fact, this is included in the words “binary operation” but it is worth mentioning explicitly)
(G2) ( Associativity) For every x,y and z in G (xy)z = x(yz)
(G3) ( Identity) There is an element e of G such that xe = ex = x for every x in G
(G4) ( Inverse) For every x in G there is a y in G such that xy = y*x = e.
The operation * is usually referred to as multiplication, so “group multiplication” simply means applying the binary operation in some group. It need not be commutative: in the group of invertible n-by-n matrices under matrix multiplication, multiplication is not commutative.
To return to your original question, abstract algebra studies axiomatic systems where commutativity is a matter of definition and in analysis the familiar properties of real numbers are assumed. To see these properties derived you would need to look in a book of set theory. For example, in Axiomatic Set Theory by Patrick Suppes, Theorem 6.32 states that addition and multiplication of rational numbers are commutative and Theorem 6.46 extends this to real numbers.
Of course, to get started on such proofs you need to make some other more basic assumptions.
Jabba’s got it. The whole point is that in analysis, the property of multiplication is such that the commutative property works. You have to give a rigorous proof of exactly what the operation of multiplication on the real numbers is in order to prove inherent properties of the operation. This is set theorgy (or possibly graph theory, but that’s debatable as to whether operational definitions are axiomatic for graph theory). As I’m not sure where you wish to start from, I’m not sure what I have to work with.
For example, if we start with set theory, there are automatic properties of commutativity and distributivity of set operations that arise from the inherent nature of sets. You can prove them by contradiction (say that an element of a set union or interesect with another set is not an element of the commutated set union or intersect with another set, you will find that such is a contradiction, therefore the set operations of intersection and union must be commutative). However, operational defintions are a bit more subtle than this.
Given the associative property, the identity property, and the commutativity property, you can prove the distributive property. (I leave it as an exercise for the reader.)
The classic in the field is Edmund Landau’s Foundations of Analysis, which starts with the axioms of set theory and works up to the complex numbers. Be warned that the proofs can be difficult–the commutative property of multiplication for natural numbers is actually not that simple, and I doubt anyone will offer to reproduce it here.
There are very few axioms in math that don’t reduce to theorems from other, more basic axioms. However, the proofs from those axioms are often very long.
Jabba: I’m not sure if I’m reading you correctly, but I want to clarify that abstract algebra studies non-commutative and non-associative systems as well.
Multiplication from graph theory? This is strange; please elaborate.
Yes, I worded my statement badly. I meant that, for example, multiplication is associative in a group because that is part of the definition of what a group is. I did not mean to suggest that all systems studied in algebra have these properties.
I’m afraid I don’t quite understand this. Could you elaborate?
OK, just so we’re all on the same page.
As an aside, the commutative property does not hold for multiplication of transfinite cardinals (aleph-null, aleph-one …)
Rapidly trying to get the lid back on this can of worms…
Every math theory starts out with some undefined objects and some axioms that are assumed (without proof) that the objects obey. If we are talking about number theory, numbers are our undefined objects. There is an axiom that says that our object obey the commutative property. This way the results can be applied to any system of objects that obey the axioms. Such a system of objects is called a model of the specific theory.
In set theory, the undefined objects are sets. The only thing we know about our sets is that they obey the axioms of set theory. Using set theory, we can define certain sets as numbers and define multiplication on these sets. We can prove that these sets obey the axioms of number theory. What this shows is that number theory is consistent provided that the simpler <snicker> set theory is consistent.
So the commutative property is an axiom of number theory (and as such, is not proven, but must be assumed), and using set theory, it is a theorem that certain objects form a model of number theory.
No. Addition and multiplication are commutative for transfinite cardinals. For any two cardinals if one of them is not finite, the sum and the product are both equal to the larger of the two. Addition and multiplication over the transfinite ordinals (omega-null, omega-one, . . . ) are non-commutative operations.
Thanks - always get ordinals and cardinals confused.
Not necessarily. The outer product is a binary operation, but sets are not, in general, closed under outer product.
[hijack] I’m curious, are you familiar with Rudin’s Analysis, and if so, how do they compare? I studied from Rudin in my ill-fated attempt to double-major in math and physics and it was one of the few books on this subject that I found even slightly approachable.
I love reading these threads, even if they drive me a little nuts. It still chafes that the one thing that I really enjoyed in school is the one thing that consistently kicked my butt.[/hijack]
Chronos: This was a slip on my part. I meant a binary product. By definition a binary product on S is a function from the Cartesian product S[sup]2[/sup] to S and so implies closure.