Mathematics: The Loss of Certainty

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.

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.

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.

...in fact that these theorems use the axioms. However, they must be used to derive a large part of classical mathematics. In the second edition 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 Mathematics 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 endowed with an "ultimate furniture" of rather complex structure. Would any realistically-minded man dare say he believes in this transcendental 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.

The upshot of these views is that sound mathematics must be deter- mined not by any one foundation which may some day prove to be right. The "correctness" of mathematics must be judged by its applicability to the physical world. Mathematics is an empirical science much as Newtonian mechanics. It is correct only to the extent that it works and when it does not, it must be modified. It is not a priori knowledge even though it was so regarded for two thousand years. It is not absolute or unchangeable.

Weierstrass endorsed this thought with the words, "The true mathematician is a poet." And Ludwig Wittgenstein (1889-1951), a student of Russell and an authority in his own right, believed that the mathematician is an inventor not a discoverer...

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As Weyl stated, mathematics is an activity of thought, not a body of exact knowledge. It is best viewed historically. The rational constructions and reconstructions of the foundations appear now only as a trav esty of the history.

The most extreme view was expressed by Karl Popper, a notable philosopher of science, in The Logic of Scientific Discovery. Mathematical reasoning is never verifiable but only falsifiable. Mathematical theorems are not guaranteed in any way. One may continue to use the existent theory in the absence of a better one, just as Newton's theory of mechanics was used for two hundred years before relativity, or as Euclidean geometry was before Riemannian geometry. But assurance of correctness is not attainable...

Arthur Stanley Eddington once said, "Proof is an idol before whom the mathematician tortures himself." Why should they continue to do so? We might well ask what mathematicians accomplish with their stress on reasoning if they no longer know that their subject is consistent and if, especially, they no longer agree on what correct proof is. Should they rather become indifferent to rigor, throw up their hands and say that mathematics as a soundly established body of knowledge is an illusion? Should they abandon deductive proof and resort merely to convincing, intuitively sound arguments? After all, the physical sciences use such arguments, and even where they use mathematics they are not too concerned with the mathematician's passion for rigor. Abandonment is not the advisable path. Anyone who has looked into the contributions of mathematics to human thought would not sacrifice the concept of proof.

To these gibes at proof we may add the words of a leading student of the logic of mathematics, Karl Popper: "There are three levels of understanding of a proof. The lowest is the pleasant feeling of having grasped the argument; the second is the ability to repeat it; and the third or top level is that of being able to refute it."