Science And Math Was: Python's Lisp heritage

[Brian McErlean]

Fernando Pérez <fperez528 at yahoo.com> wrote in message news:<aa1fe9$mn7$1 at peabody.colorado.edu>...
> > Second, math *is* an attempt to describe the physical world - that's what
> > makes it useful. Case in point: calculus, which was invented/discovered
> > specifically to deal with and describe the motion of physical bodies. Had
> > it not been useful in describing such motion, it would have been tossed
> > out, or at least not so widely accepted. Math is useful because of its
> > relevance to the world around us.
> 
> Sorry but no. The fact that certain physical problems may have served as 
> inspiration for some mathematical developments can be considered to be purely 
> accidental. The foundational structure of mathematics is entirely one of 
> logical self consistency, and massive amounts of work was done particularly 
> in the 20th century to build that framework. While it is true that up to the 
> 19th century most (if not all) significant advances in math were prompted by 
> challenges from the physical sciences, the work of the formalists in the 20th 
> century showed that mathematics could be built entirely on an abstract, 
> logical foundation.

But that abstract, formal foundation is built upon axioms that are
direct analogues of real-world properties.  The mathematical concept
of "addition" is directly mappable to the combining groups of discrete
objects.  All of maths is based on a framework which "just happens" to
be isomorphic to counting, measuring and comparing real-world objects.

We build more complex theorems on top of these, and way of combining
rules, (identical to logical rules that seem to hold in the real world
too.)  But all maths is based around a system that makes it applicable
in the real world, and this is no accident.

It would be possible to design a system with rules completely
different from anything "real", but no-one does this because its a
fundamentally pointless task.  Instead, mathematicians use the
existing framework, and while they may work in areas that seem obscure
and impractical, their work is still on some level relevant to a
similar "real" situation.  Since all our concepts are to some degree
shaped by the real world, actually creating such a system may be
harder than it sounds.

> That is not to say that it's not _useful_ to have a reference to the physical 
> world both for inspiration and for applications, but there's a crucial 
> difference between that and saying that 'math *is* an attempt to describe the 
> physical world'.

Real-world relevance may not be the _reason_ for math, but the
relevance is nonetheless there.  At its lower levels, math is an
attempt to descrive the world, and all else is build on top of this.

> What I do find most fascinating is how many seemingly super-abstract results 
> of mathematics turn up (sometimes decades later after they are found) having 
> physical implications which noone ever suspected. That 'mysterious 
> mathematical structure of the physical world' is one of the most beautiful, 
> in my opinion unanswerable questions of the foundation of science.

Beautiful yes, but I don't think unanswerable.  Its not really so
surprising that "real-world" relevance pops up in odd corners of math.
 I'd argue that the relevance was always there - our body of
mathematical knowledge is designed that way.  Perhaps its more
appropriate to speak of the 'mysterious physical structure of the
mathematical world'

> Cheers,
> 
> f.

Brian.