Morris Kline (1908–1992)

He would accept nothing as true that was not so clearly and distinctly so to his own mind as to exclude all doubt. He
would divide difficulties into smaller ones; he would proceed from the simple to the complex; and finally, he would enumerate and review the steps of his reasoning so thoroughly that nothing would be omitted.

Galileo stressed that the way to obtain correct and basic principles is to pay attention to what nature says rather than what the mind prefers.

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Yet the modernist Descartes did not grant the wisdom of Galileo's reliance on experimentation. The facts of the senses, Descartes said, can only lead to delusion. Reason penetrates such delusions. From the innate general principles supplied by the mind we can deduce particular phenomena of nature and understand them. In much of his scientific work Descartes did experiment and require that theory fit facts, but in his philosophy he was still tied to truths of the mind.

...electromagnetic theory is entirely a mathematical theory illustrated by a few crude physical pictures. These pictures are no more than the clothes that dress up the body of mathematics and make it appear presentable in the society of physical sciences. This fact may disturb or elate the mathematical physicist, depending on whether the mathematician or the physicist is dominant.
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We can appreciate now what Alfred North Whitehead meant when he said, "The paradox is now fully established that the utmost abstractions [of mathematics] are the true weapons with which to control our thought of concrete facts."

To account for the motions of the planets in their orderly elliptical paths, Sir Isaac Newton provided a theory of gravitation whose physical nature neither he nor his successors for three hundred years have explained. Sense perception in this case has proved useless.

Purely mathematical considerations led Clerk Maxwell to assert that there are electromagnetic waves that are not at all detectable by any one of the five human senses. Yet the reality of these waves can hardly be questioned; every radio and television set proclaims their reality. Maxwell also concluded that light is one type of electromagnetic wave. In this case, one could say that a mystery was replaced by mathematics.

...physical axioms must be used cautiously. This was emphasized by Bertrand Russell in his The Scientific Outlook (1931). He gave the following example. If one starts from the assumptions that bread is made of stone and that stone is nourishing, one can logically conclude that bread is nourishing. The conclusion can be verified experimentally. However, needless to say, the assumptions are absurd.

On the other hand, mathematics proves to be even sturdier than the physics it purportedly describes. Joseph Fourier (1768-1830) wrote a complete and elaborate mathematical theory of the conduction of heat that seemed to apply to the caloric theory that heat was a fluid, a theory that has long since been discarded. As Edmund Burke put it, "But too often different is rational conjecture from melancholic fact." However, Fourier's mathematics has proved to be essential in the analysis of musical sounds and other phenomena.

There is indeed some reason to question what mathematics tells us about what is real.

Einstein, Planck, and Schrodinger objected to the probabilistic interpretation. Einstein stated his objections in a paper of 1935. His argument was that quantum theory is approximate and incomplete:

I reject the basic idea of contemporary statistical quantum theory, insofar as I do not believe that this fundamental concept will prove a useful basis for all of physics. . . . I am in fact firmly convinced that the essentially statistical character of contemporary quantum theory is solely to be ascribed to the fact that this theory operates with an incomplete description of physical systems.

Although the probabilistic interpretation is now accepted, perhaps with further investigation one might be able to determine the precise location of the electron with certitude. However, according to one of the novel features of quantum theory, some indeterminism is unavoidable. This is the principle of uncertainty enunciated by Werner Heisenberg...

As John Herman Randall says in the The Making of the Modern Mind,
"Science was born of a faith in the mathematical interpretation of Nature, held long before it had been empirically verified."

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The problem has been posed repeatedly, notably by Albert Einstein in his Sidelights on Relativity:

Here arises a puzzle that has disturbed scientists of all periods. How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality? Can human reason without experience discover by pure thinking properties of real things?

Although Einstein understood that the axioms of mathematics and principles of logic derive from experience, he was questioning why the many deductions from these axioms and principles, which are made by human minds, should still fit experience.

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.

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."