CONSENSUS OF THE QUALIFIED
Davis and Hersh insist (1999: 409-410) that we refrain from truncating mathematics “to fit into a philosophy too small to accommodate it.” Rather, they demand that “philosophical categories be enlarged to accept the reality of our mathematical experience.” They characterize mathematics as:
An objective reality that is neither subjective nor physical. It is an ideal (i.e., non-physical) reality that is objective (external to the consciousness of any one person).
They offer a third path which transcends the dichotomy between Platonism and Formalism:
Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather it is the part of human studies which is capable of achieving science-like consensus, capable of establishing reproducible results.
Framed this way, “Mathematics does have a subject matter, and its statements are meaningful” in terms of “the shared understanding of human beings.” As all-too-human, working mathematicians, Davis and Hersh know that:
we invent ideal objects, and then try to discover the facts about them.
Davis, Philip J. and Hersh, Reuben (1981) The Mathematical Experience. Houghton Mifflin, Boston and New York.
Davis and Hersh recognize (1999: 59-60) the partially contingent nature of published and peer-review mathematical work:
There is work, then, which is wrong, is acknowledged to be wrong and which, at some later date may be set to rights. There is work which is dismissed without examination. There is work which is so obscure that it is difficult to interpret and is perforce ignored. Some of it may emerge later. There is work which may be of great importance―such as Cantor’s set theory―which is heterodox, and as a result, is ignored or boycotted. There is work, perhaps the bulk of the mathematical output, which is admittedly correct, but which in the long run is ignored, for lack of interest, or because the main streams of mathematics did not choose to pass that way. In the final analysis, there can be no formalization of what is right and how we know it is right, what is accepted, and what the mechanism for acceptance is.
Davis and Hersh refer to (1999: 354) “real mathematics, with proofs which are established by a “consensus of the qualified.” These “real proofs” are:
not checkable by a machine, or even by any mathematician not privy to the gestalt, the mode of thought of the particular field of mathematics in which the proof is located. Even to the “qualified reader,” there are normally differences of opinion as to whether a real proof (i.e., one that is actually spoken or written down) is complete or correct. These doubts are resolved by communication and explanation, never by transcribing the proof into first-order predicate calculus.
Davis and Hersh remark that a theorem only becomes “regarded as rock bottom” if it:
is widely known and used, its proof frequently studied, if alternative proofs are invented, if it has known applications and generalizations and is analogous to known results in related areas.
