Mathematical and philosophical thought about continuity has changed
considerably over the ages. Aristotle insisted that continuous
substances are not composed of points, and that they can only be
divided into parts potentially. There is something viscous about
the continuous. It is a unified whole. This is in stark contrast
with the prevailing contemporary account, which takes a continuum
to be composed of an uncountably infinite set of points. This vlume
presents a collective study of key ideas and debates within this
history. The opening chapters focus on the ancient world, covering
the pre-Socratics, Plato, Aristotle, and Alexander. The treatment
of the medieval period focuses on a (relatively) recently
discovered manuscript, by Bradwardine, and its relation to medieval
views before, during, and after Bradwardine's time. In the
so-called early modern period, mathematicians developed the
calculus and, with that, the rise of infinitesimal techniques, thus
transforming the notion of continuity. The main figures treated
here include Galileo, Cavalieri, Leibniz, and Kant. In the early
party of the nineteenth century, Bolzano was one of the first
important mathematicians and philosophers to insist that continua
are composed of points, and he made a heroic attempt to come to
grips with the underlying issues concerning the infinite. The two
figures most responsible for the contemporary orthodoxy regarding
continuity are Cantor and Dedekind. Each is treated in an article,
investigating their precursors and influences in both mathematics
and philosophy. A new chapter then provides a lucid analysis of the
work of the mathematician Paul Du Bois-Reymond, to argue for a
constructive account of continuity, in opposition to the dominant
Dedekind-Cantor account. This leads to consideration of the
contributions of Weyl, Brouwer, and Peirce, who once dubbed the
notion of continuity "the master-key which . . . unlocks the arcana
of philosophy". And we see that later in the twentieth century
Whitehead presented a point-free, or gunky, account of continuity,
showing how to recover points as a kind of "extensive abstraction".
The final four chapters each focus on a more or less contemporary
take on continuity that is outside the Dedekind-Cantor hegemony: a
predicative approach, accounts that do not take continua to be
composed of points, constructive approaches, and non-Archimedean
accounts that make essential use of infinitesimals.
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