Proof and Mathematics

[Monkey]

Proof if for maths and alcohol, so they say. I understand quite well that we don't prove things in science, especially after 2 and some years at university, but what constitutes a "proof" in Mathematics?

I had a friend tell me that you can't prove a negative number. My bull*&$% detector went off but I wasn't really sure why. Can anyone enlighten me please? :)

[Undercurrent]

Originally posted by Monkey
Proof if for maths and alcohol, so they say. I understand quite well that we don't prove things in science, especially after 2 and some years at university, but what constitutes a "proof" in Mathematics?

Informally, a proof of a statement is a series of statements, starting with one (or a few) that is obvious or assumed, and ending with the statement in question, where each statement in the series is clearly true based on the previous statements and a few "obvious" rules of inference.

More formally, a logical system is a set of "well-formed" strings of symbols together with a subset of well-formed strings called axioms, and a set of transformation rules that convert well-formed strings to other well-formed strings. A proof of a well-formed string, S, is a sequence of the transformation rules that transform the axiom strings into S.

I had a friend tell me that you can't prove a negative number. My bull*&$% detector went off but I wasn't really sure why. Can anyone enlighten me please? :)

Your friend is confused. The common assertion is "you can't prove a negative statement", meaning a statement of the form "X does not exist". This is often followed by bleating about how you can't prove that the asserter's particular deity doesn't exist, so it takes faith to be an atheist, so the asserter can feel justified.

The common assertion is also false, but at least it makes sense as a statement. How would you conceivbly prove a number?

[Monkey]

Informally, a proof of a statement is a series of statements, starting with one (or a few) that is obvious or assumed, and ending with the statement in question, where each statement in the series is clearly true based on the previous statements and a few "obvious" rules of inference.

I understand this so far. So an example of this in Mathematics would be?

The common assertion is "you can't prove a negative statement", meaning a statement of the form "X does not exist".

I'm quite aware of how this pertains to theism, but I'm after how you "prove" things in Maths.

How would you conceivbly prove a number?

So you don't prove things in maths? Or am I completely off track with my understanding of this. Examples are good.

[Sven]

Originally posted by Monkey
I understand this so far. So an example of this in Mathematics would be?

A very, very simple example: We "prove" the third "binomi" (that's the way we called it in school in Germany)

That is, we want to show (a+b)*(a-b) = a^2 - b^2
The assumptions we need herein is that the common rules of algebra hold, that is the commutative law and the distributive law. OK, let's start.

(a+b)*(a-b) = a^2 + a*b - b*a - b^2 = a^2 - b^2

That's it. A proof. :)

Sorry, guys, that I took a so simplistic example. But I wanted to keep it really, really simple.

More "advanced" proofs can be found when searching for "proof by contradiction" and "complete induction". I have examples in mind for both - if you like, I can post these, too.

Edited to add: My "proof" above actually nicely demonstrates what Gooch's Dad says in the post below.

[Gooch's dad]

Bertrand Russell pointed out, and I'm sure he wasn't the first, that mathematical proofs, and most mathematical statements, are tautologies.

So, as others have pointed out so far, you make some definitions, and then use those definitions to make some statements that appear to be new and different, but are just restatements of the original definitions.

[Aethari]

...but what constitutes a "proof" in Mathematics?

A proof in mathematics is some sort of relation derived from axioms. Here's a really easy proof, just using the axioms of the reals:

"THM: If z and a are elements in R with z + a = a, then z =0.

PROOF: Using the axioms that guarentee the existence of a zero element, the existence of negative numbers, and the associative property of addition, the hypothesis z + a = a, we get z = z+0 = z+ (a + (-a)) = (z+a) + (-a) = a + (-a) = 0."

[Introduction to Real Analysis, 3rd Ed. Bartle and Sherbert]

Also, as someone else mentions, you can also do proofs by induction, where you prove that a relation is true for the n=1 case, demonstrate that the truth of the k+1 cases follows from the truth of the k case, and then by induction state that the relation is true for all cases where n is a member of set of natural numbers.

I had a friend tell me that you can't prove a negative number.

Well, the existence of negative numbers is axiomatic in the set of real numbers-- you can't 'proove' it in the sense that, being axiomatic, it isn't derived from any more fundamental principle. You just say something like 'the set of the real numbers is that which has properties blah, blah2, blah3...blahN...etc.'

~Aethari

[Demosthenes]

Originally posted by Gooch's dad
Bertrand Russell pointed out, and I'm sure he wasn't the first, that mathematical proofs, and most mathematical statements, are tautologies.

So, as others have pointed out so far, you make some definitions, and then use those definitions to make some statements that appear to be new and different, but are just restatements of the original definitions.

I disagree strongly with that statement. I don't feel that it does proofs justice to call them mere restatement of original definitions. A proof is a chain of logical inferences from axioms or already known true statements to the desired result. It's designed to verify that something is true and to convince other people too. Definitions are the supporting pillars in a proof. After all how does one prove anything if he doesn't know what he's proving? A lot of mathematical insights can be gleamed by proving something since we can gain a much better understanding of the abstract concepts we're employing in our proofs.

You make it sound like there's something useless about proofs but otherwise how could we make any progress and find our way through the mathematical maze if we didn't know what's true or not?

[LVLLN]

I believe others have pointed out already the standard method of proving, where you take a bunch of true statements and show than another statement is true. My personal favorite is proof by contradiction, where you assume the negative of what you're trying to prove, and then show how if that were true, there would be a contradiction (like A and not A). Reductio ad Absurdum. Induction is pretty cool as well, involving series, where you show that something is true for n=1, and show also that if something is true for any n, then it must be true for n+1, thus creating a chain which proves it for all (positive) n.

Oh, and I don't really think the statement "you can't prove a negative number" makes any sense. A negative number is just a thing, not a statement. Are numbers undefined?

[Undercurrent]

Originally posted by Monkey
So you don't prove things in maths? Or am I completely off track with my understanding of this. Examples are good.

You definitely prove things in math. The things you prove are called "theorems". (Well technically the things you prove are "postulates" or "hypotheses" or "conjectures", which become theorems once they are proved.) One can no more "prove a number" than one can "prove a bowling ball". :)

[Jet Black]

Bear in mind Godel's Incompleteness Theorem as well. It doesn't impact directly on what anyone here has said so far, but put really roughly it states that any consistent, formal axiomatic system cannot prove it's own consistency.

[wade-w]

Several people have mentioned proof by contradiction. The basic idea is that in order to prove P, show that ~P leads to a contradiction. In mathematical philosophy, the intuitionist (also called the constructivist) school does not accept this as valid reasoning. I don't agree with them, but I think it's important to note that not all mathematicians agree on what constitutes a valid proof.

A famous example of this method is the standard proof that the square root of 2 is irrational:

Suppose that sqrt (2) is rational. Then there exists integers p and q, such that p and q are relatively prime (they have no common factors), q is non-zero, and p/q =sqrt(2) => p^2/q^2 = 2 => p^2 = 2q^2 => p^2 is even, and thus p is even. Therefore there exists an integer n such that 2n = p, hence (2n)^2 = 2q^2 => 2n^2 = q^2 => q^2 is even, and thus q is even. But p and q cannot both be even since they are relatively prime, a contradiction. Therefore sqrt (2) is irrational.

[Undercurrent]

Originally posted by Jet Black
Bear in mind Godel's Incompleteness Theorem as well. It doesn't impact directly on what anyone here has said so far, but put really roughly it states that any consistent, formal axiomatic system cannot prove it's own consistency.

Just to be a very broad pedant, it actaully says that any formal axiomatic system that is sophisticated enough to encode arithmetic either admits a statement, S, for which neither S nor not S is provable, or it admits a statement S for which both S and not S is provable.