Relativity theory of protons and electrons by Sir Arthur. Eddington

By Sir Arthur. Eddington

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Sartre's existentialists, for example, attempted bravely, and perhaps foolishly, to replace conscience with authenticity. The problem, of course, was that Hitler too was authentic. ) Since mathematics is the language of formal relationships, it became increasingly clear that the central formalism was that of mathematics itself, and that if this could not be rendered secure from inconsistency, nothing else could. If, then, formalism is the leitmotif of the twentieth century, and Hilbert's mathematical formalism captures the essence of all other formalisms, then Godel's incompleteness theorem, which dramatically and inescapably refutes Hilbert's program, can well be considered the most significant intellectual accomplishment of the twentieth century.

Hilbert's program consisted in finding a system of primitive formulas called axioms from which, according to fixed rules of proof├│rules of syntax├│one could derive all the theorems of the given mathematical domain. Two features of such a formal system were crucial: consistency and completeness. As a prophylactic against unwelcome surprises, a formal system had to be consistent: two theorems that contradict each other should not be able to be derived from the axioms. And the system should be complete, in the sense that all true statements expressible within the system (under a suitable interpretation) should be derivable from the axioms.

The Leitmotif of the Twentieth Century who pays any attention to the syntax of things will never wholly kiss you. e. cummings (iodel's incompleteness theorem of 1931 began innocently, as an attempt not to refute but to fulfill Hilbert's program. Hilbert's idea was to safeguard mathematics from hidden contradictions by replacing the intuitive mathematics of each mathematical domain with a system of axioms written in a pure formula language that, although having a standard semantic interpretation, could be manipulated according to the mechanical rules of pure syntax (much like a computer program of today).

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