works were based on ordinary language—
making them a matter of interpretation
and subject to ambiguities.
It was not until the development of
calculus that most of mathematics was
put on a logical foundation. By the 17th
century, people such as German mathe-
matician Gottfried Wilhelm Leibniz
(1646–1716) began to demand a more
regular and symbolic way to express
logic. Logic truly became a part of mathe-
matics around the mid-19th century,
especially with the 1847 publication of
English mathematician George Boole’s
(1815–1864) The Mathematical Analysis
of Logic and English mathematician
Augustus De Morgan’s (1806–1871) For-
mal Logic. Thus, mathematics began to
encompass symbolic logic with precise
rules to manipulate those symbols (see
below for more about symbolic logic).
Of course, nothing is perfect, although
mathematicians in the late 19th and early 20th centuries hoped it would be. They
believed that all of mathematics could be described using symbolic logic and made
purely formal. But in the 1930s, Austrian-American mathematician and logician Kurt
Gödel (1906–1978) put a damper on such an idea by showing that not all truths could
be derived by a formal logic system.
What were Aristotle’s syllogisms?
Syllogisms, which are often attributed to Aristotle, are the verbal versions of the formal
deductive rules for logic. Aristotle believed that any logical argument could be
explained in these standard forms. He divided them into a major premise (“all squirrels
eat nuts”), a minor premise (“Fred is a squirrel”), and a conclusion derived by a rule of
logic (“Fred eats nuts”). The classical syllogism is, “All men are mortal. Socrates is a
man. Therefore Socrates is mortal.” This form of logic—called syllogistic logic—would
dominate Western cultural thought for more than 2,000 years.
What are subjects and predicatesin Aristotelian logic?
In Aristotelian logic there are grammatical distinctions between a subject and a predi-
cate. The subjectis usually an individual entity (an object, house, city, man, animal); 105
FOUNDATIONS OF MATHEMATICS
Syllogisms, a simple exercise developed by Greek
philosopher Aristotle, use basic concepts of logic
also seen in mathematics. Aristotle also contributed
to mathematics by originating the concept of proofs.
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