[FOM] Real numbers
John Pais
pais at kinetigram.com
Sat May 10 12:58:58 EDT 2003
Hartley Slater wrote:
> ................................ One categorical difference here is that the
> equivalence class has members while the rational number does not.
> There is nothing on the geometric line which corresponds to the
> inside of an equivalence class, even if the appropriate equivalence
> classes map onto all the points on that line. Cauchy, for instance,
> took infinitesimals to be members of [<0,0,0,...>], but they have no
> decimal representation. The rational number zero has no inside.
Mathematically, these observations seem peculiar and naive. They seize on
unimportant peripheral features of some specific constructions, while ignoring
(missing) the mathematical content and purpose of these constructions.
Unfortunately, they seem to involve some sort of mirage that attempts to locate
rational (and real) numbers in one particular construction or another.
Mathematicians use these constructions to address questions like: "how
algebraically and order-theoretically different can two ordered fields
(satisfying a certain set of axioms) be?" Their first concern is existence, and
their second is uniqueness. The constructions are constructed in order to deal
with the task at hand--not as an end (or ultimate characterization) in
themselves. So, after the proof of the unique characterization up to
isomorphism, these constructions are discarded, and one just works in a generic
structure R (possibly a term or Herbrand model) satisfying the appropriate
first-order axioms, and the process starts all over again, e.g. for complex
numbers, vector spaces, real functions, etc.
Trying hard to first understand the significance of the conceptually deep
mathematical results, regarding the real number system and its subsystems (e.g.
as set out in Rudin's book), can pay great dividends in terms of mathematical
insight. At this point one is in a position to better understand what it means
logically and mathematically for a lub-complete ordered field to exist and be
unique up to isomorphism, and then try to align, compare, and contrast this
abstract conception with other (possibly more naive) conceptions of real number.
Best wishes,
John Pais
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