544 19. LOGIC AND SET THEORYagainst the principles of induction and cause in the preceding century. Venn's
reduction risked exposing probability theory to the same demolition.
1.6. Jevons. Both de Morgan and Boole used the syllogism or modus ponens (in-
ferring method) as the basis of logical inference, although de Morgan did warn
against an overemphasis on it. Their successor William Stanley Jevons, formulated
this law algebraically and adjoined to it a principle of indirect inference, which
amounted to inference by exhaustive enumeration of cases. The possibility of doing
the latter by sorting through slips of paper led him to the conclusion that this sort-
ing could be done by machine. Since he had removed much of the mathematical
notation used by Boole, he speculated that the mathematics could be entirely re-
moved from it. He also took the additional step of suggesting, rather hesitantly, that
mathematics was itself a branch of logic. According to Grattan-Guinness (2000, p.
59), this speculation apparently had no influence on the mathematical philosophers
who ultimately developed its implications, Russell and Frege.
Set theory is all-pervasive in modern mathematics. It is the common language used
to express concepts in all areas of mathematics. Because it is the language everyone
writes in, it is difficult to imagine a time when mathematicians did not use the word
set or think of sets of points. Yet that time was not long ago, less than 150 years.
Before that time, mathematicians spoke of geometric figures. Or they spoke of
points and numbers having certain properties, without thinking of those points and
numbers as being assembled in a set. We have seen how concepts similar to those of
set theory arose, in the notion of classes of objects having certain properties, in the
British school of logicians. On the Continent, geometry and analysis provided the
grounds for a development that resulted in a sort of "convergent evolution" with
mathematical logic.
2.1. Technical background. Although the founder of set theory, Georg Cantor,
was motivated by both geometry and analysis, for reasons of space we shall discuss
only the analytic connection, which was the more immediate one. It is necessary to
be slightly technical to explain how a problem in analysis leads to the general notion
of a set and an ordinal number. We begin with the topic that Riemann developed
for his 1854 lecture but did not use because Gauss preferred his geometric lecture.
That topic was uniqueness of trigonometric series, and it was published in 1867,
the year after Riemann's death. Riemann aimed at proving that if a trigonometric
series converged to zero at every point, all of its coefficients were zero. That is,
Riemann assumed that the coefficients an and bn tend to zero, saying that it was
clear to him that without that assumption, the series could converge only at isolated
points.^4 In order to prove this theorem, Riemann integrated twice to form the
continuous function
2. Set theory
-an + / (an cos nx + b„ sin nx) = 0 = 0 = 6, 'n ·
n=lF(x) = Ax + Â + -a 0 x^2 - Ó(an cos nx + bn sin nx)n=l(^4) Kronecker pointed out later that this assumption was dispensible; Cantor showed that it was
deducible from the mere convergence of the series.