Hi folks,
Glenn, thanks for the pointer to the Peano site.
Peter
And, in fact, one of the most powerful features of Mathematics is the
ability to change it around at will. Since axioms are so arbitrary, one can
make up as many mathematical systems as one likes.
Frozzbozz
Mathematics, from my understanding is used to describe a problem we already
understand so that we can solve problems of that type in the future. But,
if we want to make something new, mathematics can't really describe it yet,
and thus, we are forced to use our own experience.
David
That is an interesting point. There are however, other ways to stretch
Mathematics to make it fit. One way is to remove it from any binding to
reality, and make it its own reality. When you do this, you can make
mathematics do whatever you wish. ... The possibilities are as endless as
the universes and axioms you can create.
I agree in general. Mathematics is dynamic, and the bases can be chosen to
suit a particular need or interest.
I don't see, however, the axioms as being as arbitrary as is suggested. Of
course, one can select any set of axioms and call it a system. But what
worth would a haphazard system have? Rather, I think we are nudged somehow
to discover the systems that are of benefit in some way. These are the ones
that gain acceptance and longevity.
Riemann geometry, for example, is better than Euclidean geometry when you
are traveling upon a sphere, like the earth. I read an account describing
how a boat taking a "curved" path to an island travelled a shorter distance
than a boat going "straight" to the island. Riemann geometry can explain the
reason for this more readily than Euclidean geometry.
By removing the "binding reality" of Euclid's postulate that states,
"Through any point outside a line one and only one parallel can be drawn,"
and replacing it with one that says no parallel lines can be drawn, you
obtain the set of axioms for Riemann geometry. As this example illustrates,
mathematics is dynamic and can extend past itself; it can evolve.
Yet the choice of the new set of axioms does not appear to be arbitrary, and
hence you can't quite make mathematics do whatever you wish. I expect the
selection of axioms is similar to how Pirsig discusses how hypotheses come
to be. He quotes Einstein p115 ZAM "Evolution has shown that at any given
moment out of all conceivable constructions a single one has always proved
itself absolutely superior to the rest."
Ed
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