Claire Ortiz Hill on Hilbert's philosophy (excepts)

Upon several occasions, using almost exactly the same words, Hilbert described what he called the basic philosophical position that he considered necessary for all scientific thinking, understanding, and communicating, and without which no intellectual activity is possible. He stressed that this basic philosophical position was the very least thing that had to be assumed, that it was something that no scientific thinker could dispense with, and that everyone had to adopt, whether consciously or not. According to this approach, as a precondition for the use of logical inferences and the performance of logical operations, certain extra-logical, concrete objects had to be already given in our faculty of presentation and be intuitively present as immediate experience prior to all thought. For logical inferences to be reliable, those objects had to be completely surveyable in all their parts, and the fact that they occurred, differed from one another, followed one another, or were concatenated also had to be immediately given intuitively with them as something neither reducible to anything else, nor requiring reduction. Hilbert labelled this concern for concrete content the finite approach.

 In recognizing that such conditions necessary for the use of “contentual” (inhaltlich) logical inference existed and had to be respected, Hilbert saw himself as being in agreement with philosophers. Specifically, he considered this basic philosophical position to be part and parcel of the teachings of Immanuel Kant who had maintained that extra-logical concrete objects intuitively present as immediate experience prior to all thought had to be given, that, in particular, mathematics could never be provided with a foundation by means of logic alone, that it has at its disposal a content secured independently of all logic. In contrast, he repeatedly stressed that Frege’s and Dedekind’s attempts to provide arithmetic with foundations independent of all intuition and experience and to derive arithmetic by means of logic alone were bound to fail, because logic alone could not suffice, and certain intuitive conceptions and insights were indispensable for scientific knowledge to be possible.

 In the case of mathematics, Hilbert explained, the extra-logical, concrete objects intuitively present prior to all thought were the concrete signs themselves, whose shape was immediately clear and recognizable. As perceptually recognizable, objective and displayable numerals, the numbers of concrete-intuitive number theory met Hilbert’s requirements. They and the proofs of theorems about numbers fell into the domain of the thinkable. Formalized proofs were concrete, surveyable objects communicable from beginning to end. He defined proofs as arrays, that is, objects composed of primitive signs given as such to perceptual intuition and consisting of inferences where each premise is an axiom, directly results from an axiom by substitution, or coincides with the end formula of an inference occurring earlier in the proof or results from it by substitution. The axioms and provable propositions resulting from this procedure were, Hilbert contended, “copies of the thoughts constituting customary mathematics as it has developed till now”.

In his proof theory, only real propositions can be directly verified. “The formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner and develop it in such a way that at the same time the interconnections between the individual propositions and facts become clear. To make it a universal requirement that each individual formula be interpretable by itself is by no means reasonable; on the contrary, a theory by its very nature is such that we do not need to fall back upon intuition or meaning in the midst of some argument”.