The History of Mathematics: A Brief Course

1. LOGIC 539

altogether, and consider the word as meaning the state of the mind with respect to
an assertion, a coming event, or any other matter on which absolute knowledge does
not exist." But subjectivity is not the same thing as arbitrariness. De Morgan, like
us, would have labeled insane a person who asserted that the probability of rolling
double sixes with a pair of dice is 50%. In fact, he gave the usual rules for dealing
with probabilites of disjoint and independent events, and even stated Bayes' rule of
inverse probability. He considered two urns, one containing six white balls and one
black ball, the other containing two white balls and nine black ones. Given that
one has drawn a white ball, he asked what the probability is that it came from the
first urn. Noting that the probability of a white ball was f in the first case and ãô
in the second, he concluded that the odds that it came from either of the two urns
must be in the same proportion, | : ãô or 33 : 7. He thus gave |§" as the probability
that the ball came from the first urn. This answer is in numerical agreement with
the answer that would be obtained by Bayes' rule, but de Morgan did not think of
it as revising a preliminary estimate of ^ for the probability that the first urn was
chosen.

1.2. Symbolic calculus. An example of the new freedom in the interpretation

of symbols actually occurred somewhat earlier than the time of de Morgan, in

Lagrange's algebraic approach to analysis. Thinking of Taylor's theorem as

ÄË/(÷) = f(x + ft) - /(*) = hDf(x)h + ^h^2 D^2 f(x) + ^h^3 D^3 f(x) + ... ,

where Df(x) = f'(x), and comparing with the Taylor's series of the exponential

function,

â' = 1+ß+2Àß2 + Ýß3+"'

Lagrange arrived at the formal equation

Ah = ehD-l.

Although the equation is purely formal and should perhaps be thought of only

as a convenient way of remembering Taylor's theorem, it does suggest a converse

relation

D/(x) = i(ln(l + Ä„))/(÷) = \(&hf(x) + \&lf{x) + •••),

and this relation is literally true for polynomials /(x). The formal use of this sym-

bolic calculus may have been merely suggestive, but as Grattan-Guinness remarks

(2000, p. 19), "some people regarded these methods as legitimate in themselves,

not requiring foundations from elsewhere."

1.3. Boole's Mathematical Analysis of Logic. One such person was George

Boole. In a frequently quoted passage from the introduction to his brief 1847

treatise The Mathematical Analysis of Logic, Boole wrote

[T]he validity of the processes of analysis does not depend upon

the interpretation of the symbols which are employed but solely

upon the laws of their combination. Every system of interpretation

which does not affect the truth of the relations supposed is equally

admissible, and it is thus that the same process may under one

scheme of interpretation represent the solution of a question or the