Mathematics Program, Classical Studies Program, Philosophy Program, and Physics Program Present

The Labyrinth of the Continuum and the Concept of the Infinitely Small

The concept of the continuum is one of the oldest in philosophy and mathematics. A continuum is conceived of as a continuous entity possessing no gaps or interruptions. We commonly suppose that space, time and motion are continua. The continuum concept was first systematically investigated by Aristotle c. 350 B.C. His major conclusion was that a continuum cannot be reduced to a discrete entity such as a collection of points or numbers. In the 17th century Leibniz’s struggle to understand the continuum led him to term it a labyrinth. In modern times mathematicians have formulated a set-theoretic, or “arithmetic” account of the continuum in discrete terms, although certain important thinkers, such as Brentano, Weyl and Brouwer rejected this formulation, upholding to Aristotle’s view that continua cannot be reduced to discreteness.

Closely allied to the continuum concept is that of the infinitely small, or infinitesimal. Traditionally, an infinitesimal has been conceived of, geometrically, as a part of a continuous curve so small that it may be regarded as “straight”, or, numerically, as a “number” so small that, while not coinciding with zero, is smaller than any finite nonzero number. The development of the differential calculus from the 17th century until the 19th century was based on these concepts.

In my talk I shall present a historical survey of these ideas.

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For more information, call 845-758-6822, or e-mail [email protected].

Time: 12:00 pm – 1:00 pm EDT/GMT-4

Location: Hegeman 107

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