G. Cantor (1845-1918):
Founder of the Set Theory, Inquiry into the Infinity. "There are more real numbers than natural numbers";"Is there any infinity which is larger than the denumerable and smaller than the continuum?"
G. Frege (1848-1925):
Inventor of the Mathematical Logic, Logicist Foundation of Arithmetic. "A proposition is made of a function and arguments"; "Mathematics can be reduced to logic."
B. Russell (1872-1970):
Russell Paradox, Logicist Philosopher of Mathematics. "Is the present King of France bald?"; "Consider the set of all sets which do not belong to themselves; does this set belong to itself?"
D. Hilbert (1862-1943):
Formalist Philosopher of Mathematics, Hilbert's program. "We should be able to say that axioms may be satisfied by a table, a chair, or a mug"; "If we show consistency within Finitism, classical mathematics can be saved."
L.E.J. Brower (1881-1966):
Intuitionist Philosopher of Mathematics. "What is not constructive is not mathematics"; Proof is more fundamental than truth.
J. von Neumann (1903-1957):
Automata Theory, Design of Computer. "Five tubes for logic, five tubes for electronics"; his calculation was faster than computers!
K. Goedel (1906-1978):
Highlights of mathematical logic and axiomatic set theory. "Any consistent axiomatization of arithmetic is incomplete; i.e., there is an arithmetical truth such that neither itself nor its negation can be proved in it."
A.M. Turing (1912-1954):
"Computable means what can be done by a Turing machine"; "If a machine passes certain tests, it may be regarded as intelligent and capable of thinking."