Re: Help in answering news story on refutation of fermat's last theorem

Torkel Franzen wrote:
> anzaurres1@xxxxxxxxxxx writes:
>
> > When we, mathematicians, say that a statement is true in a given
> > axiomatic system, we mean that one can logically derive this statement
> > from the axioms.

> People do indeed often speak of a statement being "true in a given
> axiomatic system" when they mean that it is provable in that system.
> While mostly harmless, this terminology promotes needless confusion.
> For example, it sometimes prompts them to contradict the simple
> observation that there are theories with false axioms.

Whom "them"? Non-logician mathematicians? Name one active mathematical
non-logician theory, which contains "false axioms", whatever that
means. There are none. Only idiots would work on thories that contain
false axioms.

In fact, there are very few axioms in mathematics. Pretty much
everything is just definitions:

"A group is a set of elements with a two-to-one mapping called
"multiplication" such that ....."

"A metric space is a set of elements with a mapping into reals such
that ...."

"Housdorff space is a topological space such that ...."

etc.

In productive areas of math, all you need to know are a set of most
common axioms for integers and an understanding of what is meant by
terms like "set", "element", "mapping", etc. Everything else, including
rationals and reals, are just definitions... No axioms.

.

Relevant Pages