Subject: [HM] Mathematics and Time
From: Clark Kimberling (ck6@evansville.edu)
Date: Wed Mar 08 2000 - 09:40:33 EST
One difference between mathematics and other sciences is the dependence of
the latter on time. Every object of the physical and natural sciences
changes with time, as each object consists of atomic constituents whose
locations and interrelationships vary with time.
Mathematics, however, inasmuch as it consists of basic assumptions,
definitions, and implications, rests on time-independent pillars, such as
Peano's Axioms, one of which is, essentially, the Principle of
Mathematical Induction. This particular pillar is curiously like time,
especially when it takes the form of an inductive definition. Consider,
for example, the sequence a(1), a(2), a(3), ... defined as follows:
Beginning with k = 1, let p(k) be the least natural number not already an
a(i), let q(k) = p(k) + k, a(p(k)) = q(k), and a(q(k)) = p(k).
The point of this example is illustrated by the use of the words
"beginning" and "already". The act of writing out terms
2, 1, 5, 7, 3, 10, 4, ...
depends on "beginning at some point in time" and "already done in time".
Nevertheless, the definition itself (which could be recast more wordily so
as to avoid "beginning" and "already") is invitingly free of time and
completely determines the object defined. For contrast, trying defining,
with comparable time-independence and completeness, a quark, atom, star,
virus, or DNA. (Here, one could become distracted with questions about
"reality" and "truth", but these are notions which, for many people, are
time-dependent and/or matter-and-energy-dependent, whereas our focus for
the moment is on definitions and implications based only on irreducibly
basic assumptions.)
These musings lead to a historical question: how far back in time can we
date recorded recognitions of the time-independence of mathematics? Or,
more largely: in what early writings is it argued that mathematics is a
separate discipline, strikingly unlike its various realms of application,
especially in its independence of time?
Clark Kimberling
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