Re: [Phys-l] Mathematics and science are different (was "the role of equations")

[ludwik kowalski <kowalskil@mail.montclair.edu>]

Than you John D.

Yes, an axiom is an accepted (in a specific context) hypothesis. It  
does not have to be justified in a debate; it has to be accepted (as  
if it were self-evident). A "hypothesis" is a better term.

Here is my message again; the term (self-evident truth) has been  
replaced by (***)



1) What are our "basic principles"? Are they the same things as axioms  
(***) or do they also include "already derived" theorems?

2) In mathematics (not only in Euclidean geometry), I used to tell  
students, everything is derived. Mathematical claims are validated by  
logical derivations.

3) Scientists and engineers do use mathematics but their methods of  
validation of claims are different. Their claims are finally validated  
by results from reproducible experiments and observations.

4) I know that many scientific discoveries were predicted by  
mathematicians. Being a mathematician and a scientist can be very  
useful.

Ludwik
====================


On Feb 1, 2011, at 11:48 AM, John Denker wrote:

> On 02/01/2011 07:18 AM, ludwik kowalski wrote in part:
>
> > axioms (self-evident truth)
>
> Axioms are not truth, let alone self-evident truth.
>
> Axioms are essentially hypotheses, put forth *without regard* to  
> whether
> they are "true" or not.  The Euclidean geometry book is one long  
> "if...then"
> statement.  For instance, it says *if* the Euclidean axioms apply  
> *then*
> the sum of the interior angles in a triangle is 180 degrees.
>
> You can perfectly well hypothesize Euclidean geometry on one side of  
> the
> page and hypothesize non-Euclidean geometry on the other side.  You  
> can
> do that with hypotheses ... but you cannot do that with truth.   
> Truth is
> not so flexible.
>
> The theorems of geometry are not "true" in any absolute sense,  
> either.  I
> suppose you could say that they are conditionally true, conditioned  
> on the
> axioms.
>
> > 2) In mathematics (not only in Euclidean geometry), I used to tell
> > students, everything is derived. Mathematical claims are validated by
> > logical derivations.
>
> No, not "everything".
>
> You could say that everything you have derived is derived ... but
> that is not particularly informative.
>
> In contrast, *not* everything that is a valid consequence of the  
> axioms
> is derivable.  Gödel proved this.  Indeed he gave a _constructive_
> proof, exhibiting a statement that must be true (given almost any
> reasonable set of axioms) but is not derivable.
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Ludwik

http://csam.montclair.edu/~kowalski/life/intro.html