1 The Birth of Science - MSRI

2.5 Continuous Mathematics 39
The argument given above is a rigorous demonstration of the infiniteness
of a set. Euclid, knowing very well the subtlety of the concept of infinity,
which had been clear at least since Zeno’s time, manages to obtain
a rigorous proof without ever dealing directly with infinity, by reducing
the problem to the study of finite numbers. This is exactly what contemporary
mathematical analysis does. That Euclid does not use the word
“infinite” is of course irrelevant. In any case, the word “infinite” is not a
novelty introduced by modern mathematicians; it is the literal translation
of Greek , which, after a long and complicated history, was eventually
used in mathematics in its current meaning of “infinite” (by Apol- page 68
lonius of Perga, for example). 46 We will return to this question in Section
11.9, where we try to pinpoint the origin of the opinion that Kline and so
many others have held.
2.5 Continuous Mathematics
The use of “magnitudes”, or continuous quantities, in addition to integers,
gave rise to a difficult problem. Consider segments. To operate with
these “magnitudes” one must know how to carry out the basic arithmetic
operations. Addition poses no problem: if a and b are two segments, the
sum a + b is the segment obtained in a natural way by extending the first
segment a length equal to that of the second. Differences are defined analogously.
These rules correspond to what one effectively does in order to
add or subtract noninteger quantities using the geometric method. For
multiplication things were also simple: the product ab was thought of as
a rectangle whose sides were represented by a and b. 47 But what meaning
could be assigned to the ratio a:b? Of course, the operation of addition
between magnitudes induces in an obvious way the operation of multiplication
of a magnitude by a natural number: if k is a natural number,
the product ka can be defined as the sum of k magnitudes, each equal to
a. If two integers, k and h, can be found such that ka = hb, the ratio a:b
can be defined as the ratio between integers h:k. In other words, the ratio
a:b can be defined as a fraction. When there are no two integers h and k
46 For instance, Apollonius of Perga, Conica, II, prop. 44. This use of “infinite” () in the
sense of actual infinity in a mathematical context appears already in Plato’s Theaetetus, 147d.
Theaetetus reports a conversation between the mathematician Theodorus and his students (of
whom he was one), dealing with squares that are multiples of the unit square but whose sides
are not multiples of the unit length (and therefore are incommensurable with it). They remark that
such sides are infinite in number ( ).
47 This way of regarding products returns magnitudes non homogeneous with the factors, so it
makes expressions such as a + ab, where a and b are lengths, nonsensical. This introduces a kind
of automatic “dimension control”.
Revision: 1.12 Date: 2003/01/10 06:11:21