Hilbert’s Program a proposal in the foundations of mathematics, named for its developer, the German mathematician-philosopher David Hilbert, who first formulated it fully in the 1920s. Its aim was to justify classical mathematics (in particular, classical analysis and set theory), though only as a Kantian regulative device and not as descriptive science. The justification thus presupposed a division of classical mathematics into two parts: the part (termed real mathematics by Hilbert) to be regulated, and the part (termed ideal mathematics by Hilbert) serving as regulator.
Real mathematics was taken to consist of the meaningful, true propositions of mathematics and their justifying proofs. These proofs – commonly known as finitary proofs – were taken to be of an especially elementary epistemic character, reducing, ultimately, to quasi-perceptual intuitions concerning finite assemblages of perceptually intuitable signs regarded from the point of view of their shapes and sequential arrangement. Ideal mathematics, on the other hand, was taken to consist of sentences that do not express genuine propositions and derivations that do not constitute genuine proofs or justifications. The epistemic utility of ideal sentences (typically referred to as ideal propositions, though, as noted above, they do not express genuine propositions at all) and proofs was taken to derive not from their meaning and/or evidentness, but rather from the role they play in some formal algebraic or calculary scheme intended to identify or locate the real truths. It is thus a metatheoretic function of the formal or algebraic properties induced on those propositions and proofs by their positions in a larger derivational scheme. Hilbert’s ideal mathematics was thus intended to bear the same relation to his real mathematics as Kant’s faculty of pure reason was intended to bear to his faculty of understanding. It was to be a regulative device whose proper function is to guide and facilitate the development of our system of real judgments. Indeed, in his 1925 essay ‘Über das Unendliche,’ Hilbert made just this point, noting that ideal elements do not correspond to anything in reality but serve only as ideas ‘if, following Kant’s terminology, one understands as an idea a concept of reason which transcends all experience and by means of which the concrete is to be completed into a totality.’ The structure of Hilbert’s scheme, however, involves more than just the division of classical mathematics into real and ideal propositions and proofs. It uses, in addition, a subdivision of the real propositions into the problematic and the unproblematic. Indeed, it is this subdivision of the reals that is at bottom responsible for the introduction of the ideals. Unproblematic real propositions, described by Hilbert as the basic equalities and inequalities of arithmetic (e.g., ‘3 ( 2’, ‘2 ‹ 3’, ‘2 ! 3 % 3 ! 2’) together with their sentential (and certain of their bounded quantificational) compounds, are the evidentially most basic judgments of mathematics. They are immediately intelligible and decidable by finitary intuition. More importantly, they can be logically manipulated in all the ways that classical logic allows without leading outside the class of real propositions. The characteristic feature of the problematic reals, on the other hand, is that they cannot be so manipulated. Hilbert gave two kinds of examples of problematic real propositions. One consisted of universal generalizations like ‘for any non-negative integer a, a ! 1 % 1 ! a’, which Hilbert termed hypothetical judgments. Such propositions are problematic because their denials do not bound the search for counterexamples. Hence, the instance of the (classical) law of excluded middle that is obtained by disjoining it with its denial is not itself a real proposition. Consequently, it cannot be manipulated in all the ways permitted by classical logic without going outside the class of real propositions. Similarly for the other kind of problematic real discussed by Hilbert, which was a bounded existential quantification. Every such sentence has as one of its classical consequents an unbounded existential quantification of the same matrix. Hence, since the latter is not a real proposition, the former is not a real proposition that can be fully manipulated by classical logical means without going outside the class of real propositions. It is therefore ‘problematic.’
The question why full classical logical manipulability should be given such weight points up an important element in Hilbert’s thinking: namely, that classical logic is regarded as the preferred logic of human thinking – the logic of the optimally functioning human epistemic engine, the logic according to which the human mind most naturally and efficiently conducts its inferential affairs. It therefore has a special psychological status and it is because of this that the right to its continued use must be preserved. As just indicated, however, preservation of this right requires addition of ideal propositions and proofs to their real counterparts, since applying classical logic to the truths of real mathematics leads to a system that contains ideal as well as real elements.
Hilbert believed that to justify such an addition, all that was necessary was to show it to be consistent with real mathematics (i.e., to show that it proves no real proposition that is itself refutable by real means). Moreover, Hilbert believed that this must be done by finitary means. The proof of Gödel’s second incompleteness theorem in 1931 brought considerable pressure to bear on this part of Hilbert’s Program even though it may not have demonstrated its unattainability.
See also BROUWER , GÖDEL’S INCOMPLETE- NESS THEOREMS , HILBERT , PHILOSOPHY OF MATHEMATIC. M.D.