I. If one may judge from his publications, Hilbert's conception of the problem of foundations underwent marked developments.

In [1932–5, 3: 145–56; orig. 1918] he still concentrated on the “sound” and rather colourless Independence Problem which may be formulated as follows: given a branch of knowledge which is so well-developed as to be axiomatized, the problem is to get a clear view of the logical relationships (dependence and independence or, derivability and non-derivability) of statements of the axiomatic theory. Hilbert emphasized the consistency problem which is so to speak the weakest non-derivability result, since it is the problem of showing that there exists at least one statement which is not derivable.

But in later writings (though also in 1905, [1899b, 7th ed.: 247–61] the Consistency Problem was associated with the problem of understanding the concept of infinity. He sought such an understanding in understanding the use of transfinite machinery from a finitist point of view. And this he saw in the elimination of transfinite (∈—) symbols from proofs of formulae not containing such symbols. He was convinced from the start that such an elimination was possible, and expressed it by saying that the problems of foundations were to be removed or that doubts were to be eliminated instead of saying that they were to be investigated.

We note at once that there is no evidence in Hilbert's writings of the kind of formalist view suggested by Brouwer when he called Hilbert's approach “formalism.”