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I wonder what are mathematical objects - say, the number 1, a circle, etc. - for Kant? Do they have some kind of special status for him compared to ordinary (empirical) objects? Where exactly does he talk about it (references)?

Given his Transcendental Idealism, I am guessing that he wouldn't say that mathematical objects exist in a sort of Fregean/Realist third realm independent of the human mind, but that they have their existence in the human understanding. In some sense, perhaps, mathematical objects could be regarded as the "pure objects of the understanding" - since those objects can be apprehended by the pure intuitions of time and space alone. But I am not an expert in Kant's philosophy.

Is this correct? Could you give me some references about the subject?

Thanks!

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  • It can't be mathematical Platonism, as the possible world that could be a candidate - the noumenal - is defined negatively.
    – Mozibur Ullah
    Mar 1, 2016 at 22:43

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For Kant mathematical objects are not pure objects of the understanding, although this view was later adopted by Marburg neo-Kantians, who rejected his separate faculty of sensibility after non-Euclidean geometries were discovered. They are objects attached to pure intuitions synthesized by productive imagination, which is the constructive aspect of sensibilty, in time for arithmetic, in space for geometry.

Correspondingly, Kant distinguishes symbolic and ostensive constructions. In other words, mathematical objects, while they are a priori, are like empirical objects in that they stand in the same relation to pure intuitions, as empirical objects stand to perceptions. Unlike a pure concept of the understanding, which only enables syntheses of possible intuitions which have to be supplied by sensibility, mathematical one "already contains a pure intuition in itself". This forces Kant to restrict mathematical objects to spatial and temporal magnitudes, because "qualities cannot be exhibited in anything but empirical intuition".

References are scattered throughout the Critique of Pure Reason, e.g. in the Preface to the second edition we find a famous quote:

"...new light flashed upon the mind of the first man (be he Thales or some other) who demonstrated the properties of the isosceles triangle. The true method, so he found, was not to inspect what he discerned either in the figure, or in the bare concept of it, and from this, as it were, read off its properties; but to bring out what was necessarily implied in the concepts that he had himself formed a priori, and had put into the figure in the construction by which he presented it to himself".

Elsewhere in the Critique and in the Prolegomena he describes establishing 7+5=12 by a priori synthesis, see Is number π empirical or a priori? But the central place for it is the section called Discipline of Pure Reason in its Dogmatic Use (SEP has a detailed article on it), where he writes that mathematical

"concepts must immediately be exhibited in concreto in pure intuition, through which anything unfounded and arbitrary instantly becomes obvious... to construct a concept means to exhibit a priori the intuition corresponding to it".

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  • +1 Thanks for your answer! "although this view will later be adopted by Marburg neo-Kantians, who rejected his separate faculty of sensibility after non-Euclidean geometries were discovered" could you develop it a little bit or maybe just give me some references about it? Also, do you know if numbers (and all things we usually call mathematical objects) are objects for Kant? Or they are just concepts?
    – StudentType
    Mar 1, 2016 at 4:11
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    @StudentType "Object" for Kant is an intuitive unity (pure or perceptual) brought under a concept, so yes figures and numbers are "objects" in the same way (except for purity) as empirical "objects". But of course neither are objects in the colloquial sense of "objective reality". I touched on neo-Kantian revisions in this answer philosophy.stackexchange.com/questions/32341/… you can read more in this paper philsci-archive.pitt.edu/9201/1/… and in Friedman's book Parting of the Ways
    – Conifold
    Mar 1, 2016 at 18:45
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    @StudentType It may not help, but Brouwer took the linkage to space and time to indicate that these are products of human creativity, since space and time are forms of intuition and not ideas. If arithmetic procedes from time and geometry from space, then mathematical objects are the consequences of our need to have a framework in which to organize perceptions. They explore the detailed structure of that framework, which is uniquely an aspect of human thought prior to perception. He used this as the basis for mathematical Intuitionism.
    – user9166
    Nov 27, 2018 at 19:07

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