On this necessity of an a priori representation of space rests the apodictic certainty of all geometrical principles, and the possibility of their construction a priori.
On this a priori necessity depends also the possibility of apodictic principles of the relations of time, or of axioms of time in general.
Such principles cannot be derived from experience, because experience could not impart to them absolute universality nor apodictic certainty.
The apodictic quality of mathematical thought, the certainty and correctness of its conclusions, are due, not to a special mode of ratiocination, but to the character of the concepts with which it deals.
Kant speaks of the problematic, the assertoric, and the apodictic forms of judgment as representing the stages through which knowledge passes in the process of its development.
From the already demonstrated a priori character of space, Kant deduces the apodictic certainty of all geometrical principles.
The objection to (b) is that it cannot account for the apodictic certainty of geometry, nor guarantee its application to experience.
The synthetic character of mathematical judgments has hitherto escaped observation owing to their being proved (as is required of all apodictic certainty) according to the principle of contradiction.
Apodictic propositions, he declares, are either dogmata or mathemata; and the former are beyond the competence of the human mind.
Then from the apodictic character of geometry he infers that space exists in us as pure and a priori;[457] no experience can ever reveal necessity.
There is in regard to both [space and time] a large number of a priori apodictic and synthetic propositions.
An a priori intuition will alone account for such apodictic knowledge.
The origin of metaphysics in empirical sources of knowledge, which is here set forth, and which cannot fairly be denied, deprives it certainly of that kind of apodictic certainty which is only possible through knowledge a priori.
Certainly only its a priori origin can give apodictic certainty to any knowledge; but this limits it to the mere form of experience in general, for it shows that it is conditioned by the subjective nature of the intellect.
I may mention, by the way, that I am well aware Sigwart and others influenced by him have recently questioned the peculiar nature of apodictic as opposed to assertorical judgments.
I regard, for example (of course in opposition to Windelband), the distinction between assertorical and apodictic judgments (cf.
That Sigwart, in opposing the view which regards the apodictic judgment as a special class, also occasionally bears witness against himself is clear from what has been already said (4).
All judgments without exception are, on account of this peculiarity, called by Sigwart apodictic: nor will he admit the validity of any distinction between the assertorical and apodictic forms of judgment (cf.
Sigwart denies (31) the distinction between assertorical and apodictic judgments, since in every judgment the sense of necessity in respect of its function is essential.
That is the meaning of Fichte's apodictic statement: "That which does not meet with the approval of one's own conscience is necessarily sin.
The first observation of Kant refers to the word impossible, which he considers unnecessarily added, since the apodictic certainty, which we wish to express, should be contained in the proposition itself.
