Logical Dilemmas Summary

Summary (Literary Masterpieces, Critical Compilation)

In 1930, shortly after receiving his Ph.D. from the University of Vienna, logician Kurt Gödel announced to friends the discovery of a formally undecidable proposition. He proved that there is a statement in the language of elementary number theory which is true but which can neither be proved true nor false. Thus, he concluded, any formal mathematical system strong enough to include elementary number theory cannot prove its own consistency.

These unexpected results, known as Gödel’s incompleteness theorems, not only ran counter to most scientists’ implicit assumptions about the ultimate decidability of truth but also seemed to challenge the widespread view that all problems may be solved through scientific inquiry. Paradoxically, Gödel retained that view for the rest of his life and championed scientific inquiry as the best tool for potentially solving any problem. Rather than having discovered limitations on the scope of intelligence in problem solving as Alan Turing, the artificial intelligence pioneer, famously interpreted Gödel’s incompleteness theorems, Gödel thought his incompleteness discovery actually affirmed the unique role of the human mind and its dynamism in problem solving. If there were inherent limitations in every fixed logical scheme, he reasoned, then the evolution of intelligence as embodied by computers and machines could never completely surpass the human mind and its capacity to change schemes. Trying to reconcile the contradictions posed by his own epochal discoveries and seemingly anachronistic philosophy, Gödel suffered sporadically from paranoia and eventually, overcome by a fear of being poisoned, died of starvation.

Gödel’s biographer, John Dawson, Jr., set himself a formidable task in weaving his subject’s intricate ideas in mathematics and logic together with details of his life. As a partial indication of the difficulty of Gödel’s work, Dawson reports that there are still many professional mathematicians who have little awareness or understanding of what Gödel accomplished—this, in spite of Gödel’s greater posthumous visibility afforded by his prominent appearance in Douglas Hofstadter’s best-seller Gödel, Escher, Bach (1979). Even stating correctly the essence of a Gödel theorem is a delicate matter requiring some advanced terminology and knowledge of logic or set theory. Dawson presupposes such knowledge in his brief descriptions of Gödel’s mathematical ideas and novel methods of proof. Yet Gödel’s personal story, with its cast of important supporting characters such as Albert Einstein, likely will compel even a reader with little mathematical background to eagerly continue turning the pages of Dawson’s important chronicle.

After establishing himself internationally among mathematicians and logicians with the incompleteness theorems and some groundbreaking work in set theory, Gödel emigrated to the United States in 1940 where his research turned toward philosophy. By presenting Gödel as one who did not believe in chance events, one of Dawson’s main themes concerns Gödel’s search for hidden causes for all things. Instead of the dour “nothing but the facts” empiricist one might initially expect Gödel to be, this logician’s interest in supernatural phenomena (including extrasensory perception, or ESP), his published work asserting that Einstein’s field equations imply the possibility of time travel into the past, and his proof of the existence of God reveal a more reflective character. As opposed to maintaining a state of alertness so as to observe the world with maximum accuracy, Gödel suggested that introspection was the right road to true knowledge of concepts and ideas whose existence, he claimed, was every bit as real as the material realm.

Dawson contextualizes the later half of Gödel’s research career, with its frequent changes of direction, by relating these shifts in focus to specific distractions in the logician’s personal life. Although Dawson’s claims in this area are as cautious as those pronounced by his subject (whose desire to avoid confrontation was manifested as his legendary reclusiveness and hesitancy to publicly announce research results), one wonders if Dawson, too, is excessive in singling out hidden causes to flatteringly account for Gödel’s seemingly erratic research activity after emigrating to the United States. As evidence to the contrary, Dawson cites the continuing admiration for Gödel by esteemed friends such as Einstein, mathematician John von Neumann, and economist Oskar Morgenstern, as recorded in their personal letters and diaries, which attest to Gödel’s continuing brilliance throughout this period. Dawson also compares Gödel’s post-emigration research, whose...

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