MATHEMATICAL
AND FORMAL LOGIC
What is mathematical logic?
Mathematical logic is not the logic of
mathematics, but is really the mathemat-
ics of logic composed of those parts of
logic that can be modeled mathematical-
ly. Overall, it was invented to understand
and present the work of Austrian-Ameri-
can mathematician and logician Kurt
Gödel (1906–1978) and his interpretation
of the foundations of mathematics in the
early 20th century. Although mathemati-
cians use mathematical logic to have
rational and reasonable discussions of the
many issues in the foundations of mathe-
matics, not everything is agreed upon.
What is intuitionism?
There are some people within philosophy and mathematics who reject the formalism
of mathematics and believe in intuitionism,which says that words and formulas have
significance only as a reflection of the mind’s activity. Intuitionists believe that a theo-
rem is meaningful only if it represents a mental construction of a mathematical or
logical entity. This is different from the classical approach that states that the exis-
tence of an entity can be proven by refuting its non-existence. For example, if you said
“A or B” to an intuitionist, he or she believes that either A or B can be proved; but if
you said, “A or not A,” this is not allowed, since you cannot assume that it is always
possible to either prove or disprove statement A.
What is a propositionin mathematical logic?
A propositionin mathematical logic is a statement that can be proven to be either true
or false. For example, if you say, “The bear is black,” that is a proposition; but the
statement “the bear is x,” can’t be true or false until a particular value for xis chosen;
therefore, it is not a proposition.
What is symbolic logic?
Symbolic logic (also called formal logic) is mainly concerned with the structure of rea-
soning. It determines the meaning and relationship of statements used to represent 107
FOUNDATIONS OF MATHEMATICS
The human brain is more than just a remarkable
calculating device; it also possesses the capacity for
intuition. Some mathematicians have created the
concept of “intuitionism” to reflect the idea that
concepts of language and math are really just all in
your head. Visuals Unlimited/Getty Images.