Erin,
erin: "I was thinking about whether mathematics can be considered
completely divorced from language.
Are math terms concepts? one, infinity, negative, square root etc."
The key thing about mathematical terms is that, though they are taken
from natural language, they become self-contained in the mathematical
system. For example, perhaps the most basic term of all mathematics is
the "set", but the term "set" is never defined, in the form:
Definition: a set is ...
for the simple reason that anything to put in there (e.g. "collection",
grouping") is equally in need of definition.
Instead, what is done is to put down some axioms that include the terms
"set" and "element of a set", and one then knows how to use the terms to
define other things. For example, from the axioms one can prove that
there is an empty set (a set with no elements). One can then define
"zero" as "the empty set", "one" as "the set containing zero", "two" as
"the set containing zero and one", and hence get the natural number
system. From the natural numbers one can define negative numbers, then
rational numbers, then real numbers, and eventually get to calculus and
other goodies.
One does of course use images (like boxes containing things) to help one
understand what the set theorist is doing, but these images are
irrelevant to the mathematics itself.
As to whether one wants to call these terms "concepts", I don't see why
not. One gets into trouble when one insists that a concept must refer to
some non-concept, but that is a SOM failing.
- Scott
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