536 19. LOGIC AND SET THEORYarithmeticians and algebraists, for whom inference consists only in
the use of characters, and a mistake in thought and in the calculus
is identical. [Quoted by Bochenski, 1961, p. 275]The ideal enunciated by Leibniz remains largely unfulfilled when it comes to
settling philosophical disagreements. It reflects an oversimplified and optimistic
view of human beings as basically rational creatures. This sort of optimism contin-
ued into the early nineteenth century, as exemplified by the Handbook of Political
Fallacies by the philosopher Jeremy Bentham (1748-1832). But if the complex
questions of the world of nature and society could not be mastered through logic
alone, mathematics proved more amenable to the influences of logic. The influence,
however, was bidirectional. In fact, there is a paradox, if one thinks of logic as
being the rudder that steers mathematical arguments and keeps them from going
astray. As Charles Sanders Peirce wrote in 1896, reviewing a book on logic:
It is a remarkable historical fact that there is a branch of sci-
ence in which there has never been a prolonged dispute concerning
the proper objects of that science. It is mathematics... Hence, we
homely thinkers believe that, considering the immense amount of
disputation there has always been concerning the doctrines of logic,
and especially concerning those which would otherwise be appli-
cable to settle disputes concerning the accuracy of reasonings in
metaphysics, the safest way is to appeal for our logical principles
to the science of mathematics. [Quoted in Bochenski, 1961, pp.
279 280]Peirce seemed to believe that far from sorting out the mathematicians, logi-
cians should turn to them for guidance. But we may dispute his assertion that
there has never been a prolonged dispute about the proper objects of mathematics.
Zeno confronted the Pythagoreans over that very question. In Peirce's own day,
Kronecker and Cantor were at opposite ends of a dispute about what is and is not
proper mathematics, and that discussion continues, politely, down to the present
day. (See, for example, Hersh, 1997.)
Leibniz noted in the passage quoted above that algebra had the advantage of a
precise symbolic language, which he held up as an ideal for clarity of communication.
Algebra, in fact, was one of the mathematical sources of mathematical logic. When
de Morgan translated a French algebra textbook into English in 1828, he defined
algebra as "the part of mathematics in which symbols are employed to abridge and
generalize the reasonings which occur in questions relating to numbers." Thus, for
de Morgan at the time, the symbols represented numbers, but unspecified numbers,
so that reasoning about them applied to any particular numbers. Algebra was a
ship anchored in numbers, but it was about to slip its anchor. In fact, only two years
later (in 1830) George Peacock wrote a treatise on algebra in which he proposed that
algebra be a purely symbolic science independent of any arithmetical interpretation.
This step was a radical innovation at the time, considering that abstract groups,
for example, were not to appear for several more decades. The assertion that
the formula (a - b) (a + b) = a^2 - b^2 holds independently of any numerical values
that replace a and b, for example, almost amounts to an axiomatic approach to
mathematics. De Morgan's ideas on this subject matured during the 1830s, and at
the end of the decade he wrote: