specific mathematical concepts and provides a means to compose proofs of statements.
Symbolic logic draws most notably on set theory. It uses variables combined by opera-
tions such as notor and,and assigns symbols to them (“~” and “&”, respectively).What are truth valuesand truth functions?
As seen above, when discussing propositional calculus, a proposition is any declarative
sentence that is either true (T) or false (F). Mathematicians refer to T or F as the truth
valueof the statement.
The combinations of such statements are known as truth functions,with their
true values determined from the overall true values of their contents. Truth-function-
al analysis includes the following logical operators:
Negation—The negation of a statement is false if the original statement is true,
and true if the original statement is false; it refers to “it is not the case that” or simply
“not” in natural language.
Conjunction—The conjunction of two statements is true only if both are true and
false in all other instances; it refers to “and” in natural language.
Alteration—Alteration (or disjunction) of two statements is false only if both are false
and true in all other instances; it refers to “or” (and “either ... or”) in natural language.
Conditional—Conditional (or implication) is false only if the antecedent is true
and the consequent is false, and is true in all other instances; it refers to “if ... then”
108 or “implies” in natural language.
What contributions did David Hilbert make to mathematics?
G
erman mathematician David Hilbert (1862–1943) contributed a great deal to
mathematical logic, as well as mathematics in general. In 1890 his proof of
the theorem of invariants replaced earlier work on the subject and paved the way
for modern algebraic geometry; by 1897, his algebraic number theory led to
many developments in that field. His contributions also included discoveries in
number theory, mathematical logic, differential equations, multivariable calcu-
lus, Euclidean geometry, and even mathematical (theoretical) physics.
Hilbert is most well known for presenting “Hilbert’s problems,” which origi-
nally were a set of 23 unsolved mathematical problems that he hoped would
eventually lead to many more disciplines within the field of mathematics. His
idea worked: As mathematicians attempted to solve the problems their efforts
led to mathematical discoveries in the 20th century, although a number of the
problems have yet to be solved. (For more information about Hilbert, see “Histo-
ry of Mathematics.”)