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arguments to put forth the coherence of his theory of transfinite numbers. 106 As a consequence,
Cantor devoted many pages of his writings to opinions on infinity held by past mathematicians
and philosophers, including Aristotle, Descartes, Spinoza, Hobbes, Berkeley, Locke, Leibniz and
Bolzano, as well as theologians and Fathers of the Church. 107
The notion of infinity lies at the heart of mathematics and is most clearly manifested in
the attempts to apply arithmetic to geometry. It is a classical conclusion of ancient mathematics
that the length of the diagonal of a square is not commensurable with the side; i.e., the length of
the diagonal cannot be expressed as a ratio of two whole numbers. The non-commensurability of
the diagonal of a square with its side resulted in a sharp philosophical, if not practical, division
between ancient arithmetic and geometry. However, by the 19 th century, mathematics practically
identified geometry with arithmetic and identified numbers with points of a line. 108 The length of
the diagonal of the unit square is a well-defined length, and hence it defines a point on the
number line. That point, however, does not correspond to any points defined by the rational
numbers; therefore, there must exist another kind of number, an irrational number, specifically
√2, which is the measure of the length of the diagonal of the unit square. The value of √2 can be
approximated by rational numbers to any desired accuracy, but in order to measure √2 exactly an
infinite sequence of rational numbers is needed. In the nineteenth century, due to developments
in calculus and functional analysis, limiting process and irrational numbers were a common place
in mathematics. Very few mathematicians rejected the meaningfulness or existence of the
irrational numbers. Cantor’s notion of transfinite numbers arises from a similar limiting process.
106
Ibid.
Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, tr. Phillip E. B. Jourdain (New
York: Dover Publications, 1915), 55.
108
Ibid, 15.
107
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