Review: Mathematics: The Loss of Certainty

Mathematics: The Loss of Certainty by Morris Kline
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From the standpoint of the search for truths, it is noteworthy that Ptolemy, like Eudoxus, fully realized that his theory was just a convenient mathematical description which fit the observations and was not necessarily the true design of nature. For some planets he had a choice of alternative schemes and he chose the mathematically simpler one. Ptolemy says in Book XIII of his Almagest that in astronomy one ought to seek as simple a mathematical model as possible. But Ptolemy's mathematical model was received as the truth by the Christian world.

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There are mathematicians who believe that the differing views on what can be accepted as sound mathematics will some day be reconciled. Prominent among these is a group of leading French mathematicians who write under the pseudonym of Nicholas Bourbaki:

Since the earliest times, all critical revisions of the principles of mathematics as a whole, or of any branch of it, have almost invariably followed periods of uncertainty, where contradictions did appear and had to be resolved.. There are now twenty-five centuries during which the mathematicians have had the practice of correcting their errors and thereby seeing their science enriched, not impoverished; this gives them the right to view the future with serenity.

However, many more mathematicians are pessimistic. Hermann Weyl, one of the greatest mathematicians of this century, said in 1944:

The question of the foundations and the ultimate meaning of mathematics remains open; we do not know in what direction it will find its final solution or even whether a final objective answer can be expected at all. "Mathematizing" may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization.

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Faith


fact that these theorems use the axioms. However, they must be used to derive a large part of classical mathematics. In the second edi. tion of his Principles (1937), Russell backtracked still more. He said that "The whole question of what are logical principles becomes to a very considerable extent arbitrary." The axioms of infinity and choice "can only be proved or disproved by empirical evidence." Nevertheless, he insisted that logic and mathematics are a unity.

However, the critics could not be stilled. In his Philosophy of Mathemat- ics and Natural Science (1949), Hermann Weyl said the Principia based mathematics

not on logic alone, but on a sort of logician's paradise, a universe en- dowed with an "ultimate furniture" of rather complex structure. Would any realistically-minded man dare say he believes in this tran- scendental world?... This complex structure taxes the strength of our faith hardly less than the doctrines of the early Fathers of the Church or of the Scholastic philosophers of the Middle Ages.

Still another criticism has been directed against logicism. Though ge- ometry was not developed in the three volumes of the Principia, it seemed clear, as previously noted, that by using analytic geometry, one

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