Russell’s discovery of a hidden contradiction in Frege’s attempt to formalize set theory, with the help of his simple comprehension scheme, caused some mathematicians to wonder how one could make sure that no other contradictions existed. Hilbert’s program, called formalism, was to concentrate on the formal language of mathematics and to study its syntax. In particular, the consistency of mathematics, which may be taken, for instance, to be the metamathematical assertion that the mathematical statement 0 = 1 is not provable, ought to be a metatheorem—that is, provable within the syntax of mathematics. This formalization project made sense only if the syntax of mathematics was consistent, for otherwise every syntactical statement would be provable, including that which asserts the consistency of mathematics.
Unfortunately, a consequence of Gödel’s incompleteness theorem is that the consistency of mathematics can be proved only in a language which is stronger than the language of mathematics itself. Yet, formalism is not dead—in fact, most pure mathematicians are tacit formalists—but the naive attempt to prove the consistency of mathematics in a weaker system had to be abandoned.
While no one, except an extremist intuitionist, will deny the importance of the language of mathematics, most mathematicians are also philosophical realists who believe that the words of this language denote entities in the real world. Following the Swiss mathematician Paul Bernays (1888–1977), this position is also called Platonism, since Plato believed that mathematical entities really exist.
Zenos-paradox-illustrated-by-Achilles-racing-a-tortoiseFigure 1: Zeno’s paradox, illustrated by Achilles racing a tortoise.[Credits : Encyclopædia Britannica, Inc.]
Contrasting-triangles-in-Euclidean-elliptic-and-hyperbolic-spacesFigure 2: Contrasting triangles in Euclidean, elliptic, and hyperbolic spaces.[Credits : Encyclopædia Britannica, Inc.]
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