I didn't mean to imply that you were in any way promoting your own view
and justifying it with a definition that fit your needs. However,
reading the definition, I couldn't help but viewing it as promoting an
oddly narrow view of mathematics.
> "Circular"? I don't think so. "Circular" would imply that two fields of
> study were bouncing a single set of ideas back and forth. I was implying
> a two-way interchange of NEW concepts and ideas.
Circular, because it depends on itself to determine what belongs in
mathematics and what does not. We could say, for instance, that logic is
not a part of mathematics, but the content of some logical arguments may
well be. This means that doing logic for its own sake, without directly
affecting other branches of mathematics, would not be mathematics, since
the results thus obtained would not be mathematical. However, if we
consider any of these results to be mathematics, by your definition,
suddenly, the entire field becomes a part of mathematics and ALL the
results fall into mathematics with it.
Look at your definition:
> It seems to me that in order for a scientific study to be considered
> part of mathematics, it must meet two criteria:
> 1) its practicioners make heavy demands on their math
> 2) its practitioners in turn make original mathematical
> contributions which find their homes in other branches of math, or
> at least that the study presents problems which cause other people
> to develop new mathematical techniques to solve.
If we consider some areas of study to be sufficient to qualify as test
items for (1), then all a field of study needs to do to be a part of
mathematics is claim that it contributes to any of these under (2). It
would make sense to me to consider either (1) or (2) by themselves as
inclusive criteria, but not demand that they be applied in conjunction.
Even then, I suspect, the definition would be too narrow.
> > If we only define mathematics as geometry, than we would have to
> > dismiss most of the XIXth and XXth century mathematicians as
> > non-practicioners of mathematics.
>
> Let us take as our INITIAL definition of mathematics "the set consisting
> only of geometry".
>
> Now, what about number theory?
>
> The recent proof of Fermat's Last Theorem required considerable input
> from geometry. In the other direction, in the field of Public-Key
> Cryptosystems (which is a branch of applied number theory), there is
> considerable work being done on elliptic curves, and I'm sure some of
> this work will prove useful to geometrers.
>
> So we have one example of a two-way transfer of concepts and techniques
> between geometry and number theory. Anybody care to come up with some
> more?
However, even twenty years ago, neither of these would have qualified
number theory for inclusion in this hypothetically defined mathematics.
This sets up an odd scenario that I have already described above--at
some point an entire domain may become a part of mathematics, along with
all its methods and results that previously had not been considered
mathematics. So indeed we have a recursive definition. However, in
normal life, history of science is not rewritten so easily (even though
we sometimes try). You cannot retroactively relabel an entire field of
study, just because you happen to like it more now than earlier. The
reverse should also be true--statistics was undoubtedly a part of
mathematics for over a century. It is only in the mid-century that
statisticians proclaimed themselves to be independent. The unintended
consequence was a partial loss of rigor, logical accuracy and a few
other changes that would make professional mathematicians cringe. But
can we simply retroactively define statistics out of mathematics? Or
should we differentiate between "data processing" and "mathematical
statistics" (although this term already has a somewhat different
technical meaning)?
> (If I were doing this seriously, I would have gone from geometry to
> calculus, then from calculus to algebra, and then have demonstrated a
> full-duplex transfer of ideas between algebra and number theory).
>
> I don't know if you realize it, but you have come up with a recursive
Not only do I realize it, but that was my intent.
> technique (one that could even be expressed in Backus Normal Form) to
> define the term "mathematics":
>
> Mathematics is the set containing the element "Geometry" plus any field
> of study which meets the two criteria in my original message for
> interchanging information with an element or elements already in the set
> Mathematics.
Returning to the original question of whether physics is a part of
mathematics, etc. In general, the answer should justifiably be "No!"
However, this should not prevent us from claiming large parts of
physics, especially the older theoretical apparatus, for mathematics. It
is simply an area of overlap, and overlaps are good for development of
the sciences. Is all theoretical physics a part of mathematics? This is
a more complicated question and since there is not one person who can
even answer the question of "what is mathematical physics?", the entire
enterprise seems more rhetorical than meaningful. A proper response may
well be "Who cares?", but not an unequivocal "No!"
VS-)