ma004 - Leibniz and the Spell of the Continuous Hardy Grant...

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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. 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 [
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ma004 - Leibniz and the Spell of the Continuous Hardy Grant...

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