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

anzaurres1@xxxxxxxxxxx wrote:
> 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.

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