Mathematics Provides Certain Knowledge

In the same 1945 essay cited previously, Hempel stated: 

"The most distinctive characteristic which differentiates mathematics from the various branches of empirical science... is no doubt the peculiar certainty and necessity of its results." And, he added: "a mathematical theorem, once proved, is established once and for all.... "

In noting the certainty of mathematics, Hempel was merely reasserting a view proclaimed for centuries by dozens of authors who frequently cited Euclid's Elements as the prime exemplification of that certainty. Writing in 1843, Philip Kelland remarked: "It is certain that from its completeness, uniformity and faultlessness,... and from the universal adoption of the completest and best line of argument, Euclid's 'Elements' stand preeminently at the head of all human productions." A careful reading of Hempel's essay reveals a striking feature; immediately after noting the certainty of mathematics, he devoted a section to "The Inadequacy of Euclid's Postulates." Here in Hilbertian fashion, Hempel showed that Euclid's geometry is marred by the fact that it does not contain a number of postulates necessary for proving many of its propositions. Hempel was of course correct; as early as 1892, C. S. Peirce had dramatically summarized a conclusion reached by most late-nineteenth-century mathematicians: "The truth is, that elementary [Euclidean] geometry, instead of being the perfection of human reasoning, is riddled with fallacies... ."

What is striking in Hempel's essay is that he seems not to have realized the tension between his claim for the certainty of mathematics and his demonstration that perhaps the most famous exemplar of that certainty contains numerous faulty arguments. Hempel's claim may be construed as containing the implicit assertion that a mathematical system embodies certainty only after all defects have been removed from it. What is problematic is whether we can ever be certain that this has been done. Surely the fact that the inadequacy of some of Euclid's arguments escaped detection for over two millennia suggests that certainty is more elusive than usually assumed. Moreover, in opposition to the belief that certainty can be secured for formalized mathematical systems, Reuben Hersh has stated: 

"It is just not the case that a doubtful proof would become certain by being formalized. On the contrary, the doubtfulness of the proof would then be replaced by the doubtfulness of the coding and programming."

Morris Kline has recently presented a powerful demonstration that the certainty purportedly present throughout the development of mathematics is an illusion; I refer to his Mathematics: The Loss of Certainty, in which he states: "The hope of finding objective, infallible laws and standards has faded. The Age of Reason is gone." Much in what follows sheds further light on the purported certainty of mathematics, but let us now proceed to two related claims.