Mathematical Statements Are Invariably Correct

The most challenging aspect of the question of the cumulative character of mathematics concerns whether mathematical assertions are ever refuted.  The previously cited quotations from Hankel and Duhem typify the

widespread belief that Joseph Fourier expressed in 1822 by stating that mathematics "is formed slowly, but it preserves every principle it has once acquired.... " Although mathematicians may lose interest in a particular

principle, proof, or problem solution, although more elegant ways of formulating them may be found, nonetheless they purportedly remain. Influenced by this belief, I stated in a 1975 paper that "Revolutions never occur in mathematics." In making this claim, I added two important qualifications: the first of these was the "minimal stipulation that a necessary characteristic of a revolution is that some previously existing entity (be it king, constitution, or theory) must be overthrown and irrevocably discarded"; second, I stressed the significance of the phrase "in mathematics," urging that although "revolutions may occur in mathematical

nomenclature, symbolism, metamathematics, [and] methodology...," they do not occur within mathematics itself.  In making that claim concerning revolutions, I was influenced by the widespread belief that mathematical statements and proofs have invariably been correct. I was first led to question this belief by reading Imre Lakatos's brilliant Proofs and Refutations, which contains a history of Euler's claim that for polyhedra V-E + F = 2, where Kis the number of vertices, E the number of edges, and F the number of faces. Lakatos showed not only that Euler's claim was repeatedly falsified, but also that published proofs for it were on many occasions found to be flawed. Lakatos's history also displayed the rich repertoire of techniques mathematicians possess for rescuing theorems from refutations.

Whereas Lakatos had focused on a single area, Philip J. Davis took a broader view when in 1972 he listed an array of errors in mathematics that he had encountered. Philip Kitcher, in his recent Nature of Mathematical Knowledge, has also discussed this issue, noting numerous errors, especially from the history of analysis. Morris Kline called attention to many faulty mathematical claims and proofs in his Mathematics: The Loss of Certainty. For example, he noted that Ampere in 1806 proved that every function is differentiable at every point where it is continuous, and that Lacroix, Bertrand, and others also provided proofs until Weierstrass dramatically demonstrated the existence of functions that are everywhere continuous but nowhere differentiable.20 In studying the history of complex numbers, Ernest Nagel found that such mathematicians as Cardan, Simson, Playfair, and Frend denied their existence.

 Moreover, Maurice Lecat in a 1935 book listed nearly 500 errors published by over 300 mathematicians.22 On the other hand, Rene Thorn has asserted: "There is no case in the history of mathematics where the mistake of one man has thrown the entire field on the wrong track.... Never has a significant error slipped into a conclusion without almost immediately being discovered."23 Even if Thorn's claim is correct, the quotations from Duhem and Fourier seem difficult to reconcile with the information cited above concerning cases in which concepts and conjectures, principles and proofs within mathematics have been rejected.