Re: Mathematical proofs

jstevh_at_msn.com
Date: 02/14/05 

Date: 13 Feb 2005 16:22:03 -0800

Beth wrote:
> In article <1108322762.984404.250320@l41g2000cwc.googlegroups.com>,
> <jstevh@msn.com> wrote:
>
> > 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.
>
> I hope by "came up with" you mean "realized", as that definition is
at
> least a century old.
>

You sure? Got a source to cite?

> > 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".
>
> The problem is not everyone agrees on what is logical or even what is
> true. I've always found the law of the excluded middle particularly
> troubling myself. Of course, as others have said, mathematicians
> reason about axioms, not truths. If the axioms happen to be true, in
> some sense, that may or may not be useful.
>

An axiom is considered to be true, but it's generally accepted that it
can't be proven to be true.

The best way to look at my definition is in a Platonic way, as what a
mathematical proof is.

Then you can consider that what some person gives as a "proof" is just
an attempt at describing.

So, let's say that truth is independent of human thought, as in, there
are truths that exist.

Then a mathematical proof is an example of such a truth that exists.

We may find mathematical proofs and attempt to describe them, but
whether we succeed or not, the proof still exists.

Put another way, truth doesn't care if you believe, if you know, or
even if you exist.

Your beliefs, your opinions, your reality is irrelevant, no matter how
much you may believe otherwise to whether or not something is
mathematically true.

For instance, 2+2 = 4, can be reduced to various axioms, and one can
debate whether or not they are true, but if you die today, will that
change?

Will things go up in the air?

Of course not.

It's a truth whether you choose to believe it or not.

>
> Have you read "The Mathematical Experience"? It's slightly dated but
> quite good at expaining the human process of doing mathematics. For
> example, which is better, the theory of limits or the theory of
> inifinitesmals?

Yes I read it years ago. Excellent book and I highly recommend it.

As for your other question, short answer is, Newton was right.

James Harris

Relevant Pages

  • Re: JSH: Authority from Google search results?
    ... "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." ... you'll probably find english translations in a couple of minutes) §2.7 to 2.10 seems pretty similar to your definition. ... (reasonning which deductively establishes the truth of a proposition, that is proceeding by necessarily steps to other propositions admitted as true or previously prooved). ...
    (comp.theory)
  • Re: Mathematical proofs
    ... > A mathematical proof begins with a truth and proceeds by logical ... You never start with a "truth" unless it's supporting argument. ... > truth and proceeds by logical steps to a conclusion which then must ... The manner of science doesn't change [that ...
    (sci.math)
  • Re: Mathematical proofs
    ... > A mathematical proof begins with a truth and proceeds by logical ... You never start with a "truth" unless it's supporting argument. ... > truth and proceeds by logical steps to a conclusion which then must ... The manner of science doesn't change [that ...
    (sci.crypt)
  • JSH: But what is truth?
    ... mathematical proof was, and after some years, I had something. ... A mathematical proof begins with a truth and proceeds by logical steps ... years of human history, you kind of wonder a bit, but it's like, hey, ... Ok, so, I got a functional definition. ...
    (sci.math)
  • Re: JSH: Simple math but bad math habits
    ... define mathematical proof ... To quote from ... What is the truth that one begins ... axioms of Euclidean plane geometry, and the rules of mathematical logic. ...
    (sci.math)