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



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.