Sunday, July 03, 2005

Definition of mathematical proof

mathematical proof (noun): a mathematical argument that begins with a truth and proceeds by logical steps to a conclusion which then must be true.

Updating the page to add a traditionally formatted definition at the top of the page. After is the original post from 2005 without edits.  James Harris, February 6th, 2012.

A mathematical proof begins with a truth and proceeds by logical steps to a conclusion which then must be true.

I call that the functional definition of a mathematical proof, which I came up with a couple of years ago and posted on Usenet, as part of extreme mathematics as I practice it, is open research meaning you publicize your work.

An absolute definition would be:

A mathematical proof is a mathematical argument that begins with a truth and proceeds by logical steps to a conclusion which then must be true.

I like "logical steps" while others might prefer "valid mathematical steps".

By convention, the conclusion of a math proof is often called a theorem, and students are taught to state it up front, basically just saying the conclusion at the outset.

That convention is just that, a convention, of saying the conclusion up front, and then going through the steps to prove that conclusion.

So, a theorem is actually the conclusion of a math proof, which includes the theorem itself, as well as the argument proving it.

For some reason (surprise, surprise) you will not find my definitions of mathematical proof in the mainstream literature, as in my experience, mathematicians willfully ignore information that they don't like.

So why are my definitions worth mentioning?

Well, for some reason, the idea has become part of the mainstream thinking on proofs that they are creations of mathematicians that are "fragile" or delicate, with the idea of broken proofs, and failed proofs, etc., which comes about when failed arguments are called proofs.

But, consider the word "proof", as, for an example from another domain, if prosecutors claim to have proof that a suspect committed a crime, but it turns out their "proof" is not proof at all--but maybe a flight of fancy or reach by the prosecutors--it's not called failed proof!!!

They just didn't have proof.

There is a direct contradiction in declaring a flawed argument to be proof or a proof, and what I don't like about that being routine is it gives the wrong impression to the public where people don't realize that proofs are solid in a way not known in other areas--they are absolutely true.

Failed arguments are failed arguments and not proofs, no matter what anybody calls them.



SSP Lab said...

Hi James,

I am wondering why you have the condition that a proof starts with "a truth". Typically, a truth begins with a "hypothesis", something we *assume* or *claim* to be true. Not to split hairs, but you can understand the distinction when you consider the results of Euclidean and non-Euclidean geometry. You would also run into this sticky problem of how one determines what is "true".

James said...

Good question! My answer is that if it is valid, a mathematical argument will start with something that is true and build to another truth by logical steps.

So it's like a structure thing, like if someone says, that's a healthy, living tree, you note that it must have roots.

With the Parallel Postulate we can simply say it was conditional, and not a truth. So if two parallel lines never meet, you have Euclidean geometry, but if they do you have non-Euclidean.

Notice the "if" of the conditional.

Truth is not a slippery concept mathematically either as if you read my post on mathematical consistency, you'll see I simply related it to tautology.

So 1=1 is an example of a mathematical truth, from which you can build proofs!

And 1=1 is not something that is debatable.

And I built an entire set of tools from tautologies which I call tautological spaces, with which I was able to do a LOT including figuring out a new way to generally reduce binary quadratic Diophantine equations!

Truth can be defined mathematically, by tautologies. And then you can prove that mathematics is a subset of logic, as well.

Thanks for the question!