In this paper by Cassou-Nogues which is on an aspect of the mathematical philosophy of Cavailles he quotes the mathematician Hilbert (a colloborator of Einstein in Gottingen)

...We find ourselves in agreement with the philosophers, especially with Kant. Kant already taught - and indeed it is part and parcel of his doctrine - that mathematics has at its disposal a content secured independently of all logic and hence can never be provided by a foundation by means of logic alone; that is why the efforts of Frege and Dedekind were bound to fail


Rather as a condition for the use of logical inferences and the performance of logical operations, something must be given to our faculty of representation, certain extra logical concrete objects that are intuitively present as immediate experience prior to all thought.

(In Kantian terms, his immediate experience is pure intuition; and thought is that thought after the apperception of unity; and after the synthesis of the intuition - and this carried out in the Imagination)

He's claiming contra the Fregean programme which insisted that mathematical propositions were always reduced to logic, that is in Kants language, are analytic and in character are a priori; that they are in fact synthetic.

This Kantian influence on Hilbert I find surprising as I have associated the name of Hilbert with the formalist programme; and Kant with Brouwer's Intuitionism - which in some ways were in opposition.

Is there a more pronounced influence of Kant on the Formalist programme? Apart from the one here where he uses Kantian thought to demonstrate why the Fregean programme - in strict terms, and in this sense - was bound to fail; and it seems post-hoc.

(This is not to say that the Fregean programme wasn't important in other ways).

  • I agree with your view. My understanding of Brouwer's foundational philosophy is that mathematics comes before language and logic. Indeed, it is the mathematics that determines the logic - viz. intuitionist logic. – Nick R Sep 15 '15 at 23:30
  • Liebniz, I think associated the nature of time with ordering events, and this informs our understanding of what it means to count, ie succession; in fact the Peano axioms for the integers take exactly this tack - the successor function. – Mozibur Ullah Sep 15 '15 at 23:43

I think the confusion comes from schematic identification of logicism with realism, intuitionism with conceptualism, and formalism with nominalism, referencing positions in the old debate on the nature of universals. This is mostly right, but not quite: Hilbert is a nominalist about mathematical objects, but he is a conceptualist (Kantian) about mathematical symbols and their manipulation. "The subject matter of mathematics is... the concrete symbols themselves, whose structure is immediately clear and recognizable". The difference with intuitionists like Brouwer is that they were conceptualists about mathematical objects, not just symbols.

In fact, this was Hilbert's original innovation. He considered (idealized) mathematical symbols as objects of a priori perception in a way similar to Kant's view of arithmetic as a priori synthesis in time (hence their agreement against Frege, to whom arithmetic was analytic), and geometry as a priori synthesis in space. But Hilbert extends this to formulas of algebra, formal logic, etc., by merging both space and time into a joint medium of syntheses. These are the "logical concrete objects that are intuitively present as immediate experience prior to all thought", "a condition for the use of logical inferences and the performance of logical operations". A condition of the possibility of certain knowledge, also very Kantian. But Hilbert's extension of Kant gives much more: we can have synthetic a priori knowledge of logical consequences of all our axiomatic theories. Indeed, their proofs are analogous to Euclidean constructions in geometry, they are a priori syntheses of imagination, but based on symbols rather than figures.

This Kantian, in spirit, view of symbolic manipulation explains the key goal of Hilbert's programme: establishing completeness and consistency of mathematics by finitary means. While there is no restriction on the nature of objects that we choose our symbols to represent, there is a restriction on what we can do with those symbols. Only constructions of finite length, although potentially unbounded, are accessible to our Kantian faculties. But should we manage to reduce all our proofs to such constructions we will get the holy Grail: a synthetic a priori certitude for all of mathematics. Alas, this optimistic hope was proved unattainable by Gödel.

See Brown's Philosophy of Mathematics for more details.

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