Leibniz and the Spell of the Continuous
Hardy Grant
The College Mathematics Journal
, September 1994, Volume 25, Number 4, pp. 291-294.
Hardy Grant
was born in Ottawa, Canada, and holds degrees in mathematics from
Queen’s University, Kingston, Ontario, and McGill University, Montreal. He has taught
at York University, Toronto, since 1965. He takes a special interest in the role of
mathematics in cultural history, and he has given an undergraduate humanities course on
this subject for many years. His other enthusiasms include travel, birding, computer
programming and pre-1950s popular music.
T
he famous fictional detective Philo Vance once dabbled in the history of
mathematics. One of the keys to a particularly baffling murder, he told a bemused
policeman, was the fact that the mathematicians of the seventeenth century, unlike
their modern descendants, dealt only with well-behaved functions. “Neither Newton nor
Leibniz nor Bernoulli,” said the great sleuth, “ever dreamed of a continuous function
without a tangent” [
4
]. Vance’s legendary erudition was usually sound, and this case was
no exception. In the seventeenth century such mathematical bizarreries as continuous but
nowhere differentiable functions were indeed still far in the future.
The Law of Continuity
For no thinker of that age was the seeming regularity of the mathematical universe more
significant than for Leibniz. This pioneer contributor to the infinitesimal calculus was
also (of course) a great philosopher, whose metaphysical views were profoundly shaped
by his mathematical knowledge and experience. Mathematics was for him a body of
eternal truths describing an objectively known reality; moreover he felt, like many
before and after him, that the clarity of its ideas and the rigor of its arguments made
mathematics the paradigm of an exact and certain science. Hence he came to see it as a
model for inquiry in other fields, and as a source of potential insight into God’s creation
and governance of the world. In particular the continuity so conspicuous in the curves
and functions of contemporary mathematics underwrote for Leibniz one of the cardinal
principles of all his thought. In what follows I sketch the impact of a mathematically
conceived
Law of Continuity
on several diverse aspects of this protean thinker’s mature
philosophy.
He expressed this fundamental
lex continui
in various ways. In one informal statement
he identified it with the old saying that “nature makes no leaps,” adding by way of
elaboration that “we pass always from the small to the great, and the reverse, through
the medium” [
1
]. But his attempts to describe the Law of Continuity more rigorously
have a decidedly mathematical air. He wrote in 1687:
When the difference between two instances in a given series or that which is
presupposed [
deux cas.
...in datis ou dans ce qui est pos
é] can be diminished until
it becomes smaller than any given quantity whatever, the corresponding difference
in what is sought or in their results [