In this book two ethical systems are described in the language of
mathematics. Ordinarily mathematics is thought to be a science of
quantity. Indeed, manipulation of quantities constitutes much of
mathematics. Elementary applied mathematics deals with reckoning and
measurement, where concrete quantities are objects of attention, such as
counting sheep or weighing corno But the operations on these quantities
are performed with the help of symbols, from which concrete referents
have been 'abstracted out': 3 + 5 = 8 regardless of whether the symbols
stand for numbers of sheep or tons of corno Thus, the first principle
that exhibits the power of mathematics is abstraction. It is one ofthe
three pillars on which the edifice of mathematics rests. Another pillar
is precision. Ordinarily, man communicates by words. W ords serve
communication to the extent that they refer to things, events, states of
affairs, feelings of the speaker, and so on. These are the meanings
attributed to words. Communication is successful to the extent that the
meanings coded upon words by the speaker correspond to the meanings
decoded by the hearer. As is weH known, the degree ofthis correspondence
varies enormously in different contexts of discourse and with the back-
grounds or attitudes of the speakers and hearers. Mathematics is a
language in which the meanings ofthe symbols (the 'words' ofthis
language) are absolutely precise. This precision is achieved by
abstraction. Abstract terms are defined by their relations to other
terms and by nothing else.