CERTAINTY IN MATHEMATICS
Russell and Whitehead’s attempt to provide logicist foundations for the whole of mathematics in Mathematica Principia was undermined by the discovery of various paradoxes which led to contradictions. A meta-mathematical approach led by Hilbert attempted to sidestep this impasse. He was willing to abandon the Platonist viewpoint―implied in ill-fated logicism―in favor of a Formalist stance which conceded the tautological nature of mathematics. Davis and Hersh cite the following passage from Hilbert:
The goal of my theory is to establish once and for all the certitude of mathematical methods… The present state of affairs where we run up against the paradoxes is intolerable. Just think, the definitions and deductive methods which everyone learns, teaches and uses in mathematics , the paragon of proof and certitude, lead to absurdities! If mathematical thinking is defective, where are we to find truth and certitude?
Hilbert’s formalism was defeated in 1930 by Gödel’s famous Incompleteness Theorem.
Davis, Philip J. and Hersh, Reuben (1981) The Mathematical Experience. Houghton Mifflin, Boston and New York.
Russell, Bertrand (1919) Mysticism and Logic: And Other Essays. Longman, London.
ABSTRACTION
For Davis and Hersh (1981: 113) abstraction is “the life’s blood of mathematics.” Inextricable from the brain’s penchant for discerning pattern, abstraction is “ubiquitous” and “almost characteristic or synonymous with intelligence itself.” Davis and Hersh (1981: 126) pinpoint the truism that “mathematics began when the perception of three apples was freed from apples and became the integer three.” As far as abstraction in geometry is concerned:
Alongside this real, concrete instance of a straight line, there exists the mental idea of the mathematical abstraction of an ideal straight line. In the idealized version, all the accidentals and imperfections of the concrete instance have been miraculously eliminated… The line is conceived of as potentially extending to infinity on both sides.
Bertrand Russell famously wrote that mathematics has “a beauty cold and austere… without appeal to any part of our weaker nature.” In their landmark The Mathematical Experience, Davis and Hersh (1981: 109) take this convention view as a starting point:
the foremost example of a field where reason is supreme and where emotion does not enter; where we know with certainty, and know that we know; where truths of today are truths forever.
Davis and Hersh (1981: 6) initially define mathematics as “the sciences of quantity and of space” which, “in their simpler forms are known as arithmetic and geometry.” During the 20th century the scope of mathematics expanded as new branches of study were born and were proliferated. By now most professional mathematicians would expand this definition, agreeing with Richard Feynman (1999: 75) that, “Mathematics is looking for patterns.”
SOME LANDMARK FACTORS WHICH LED TO THE COLLAPSE OF ABSOLUTE CERTAINTY AND CONSISTENCY IN MATHEMATICS.
Davis and Hersh point to (1999: 341) some landmark developments that undermined mathematical consistency. In the 19th century a loss of certainty in the foundations of Euclidian geometry opened up new mathematical vistas:
The attempt to prove Euclid’s fifth postulate (the postulate of parallels, which was not as “self-evident” as the other four postulates) led to the discovery of non-Euclidean geometry in which the parallel postulate is assumed to be false.
