|
|
gsax wrote:
> I read in Penrose's "Emperor's new Mind ", that Euclidean geometry is
> actually a Physical theory derived from our sense-experiences of the
> world around us.
Yes and no. It was inspired by the world around us, that is true. But
internally, it is a mathematical model or system, which stands apart
from whether or not it happens to model the world.
This kind of interplay between mathematics and physics is common. The
calculus is also a "physical theory" in this sense, but the language is
misleading: mathematical systems are mathematical systems, regardless
of what led us to discover them. Sometimes they are inspired by the
physics, sometimes it is later discovered that mathematics which was
developed without any obvious connection to the physical worlds has
physical application. I think that group theory falls under this
heading.
> That is why its axioms seemed so obviously true..
>
> Hence Non-Euclidean geometries took time to be accepted as vaild.
>
> Is it possible that even our concept of numbers is similarly inspired.
>
> And so Number theory is as much a Physical theory as Euclidean
> geometry.
With the above qualifications.
> If this is true , then what would be the analog of Non-euclidiean
> geometries, to out 'conventional' number theory..
Obviously, some system which looked something like the integers, but in
which one or more of the axioms applicable to the integers were
modified or dropped. I don't know if such systems have been explored
or named, but given the hunger of mathematicians to explore such
things, it would seem unlikely that they have not.
|
|