Kant, 1787, strives to demonstrate that space and time are neither experience nor concepts, but they are pure intuition.
He calls it as metaphysical demonstrations of space and time; and concludes that: firstly, space is not an empirical concept obtained by abstraction due to any empirical concept obtained from the external senses such as even "next to each other" presupposes the notion of space; and this means that two things are located at two different spaces.
Time is not obtained by abstraction or association from our empirical experience, but is prior to the notion of simultaneous or successive.
Space and time are anticipations of perception and are not the products of our abstraction.
Secondly , the idea of space is necessary due to the fact that we are not able to think of space without everything in it, however we are not able to disregard space itself.
We can think of time without any phenomenon, but it is not possible to think of any phenomenon without time; space and time are a priori as the conditions for the possibility of phenomena.
Thirdly , the idea of space is not a universal concept; it is an individual idea or an intuition. There is only one time and various special times are parts of the whole time and the whole is prior to its parts.
Fourthly, space is infinite and contains in itself infinitely many partial spaces.
Next, Kant, 1787, develops Transcendental Demonstrations to indicate that the possibility of synthetic a priori knowledge is proven only on the basis of Space and Time, as follows: first, if space is a mere concept and not an intuition, a proposition which expands our knowledge about the characters of space beyond the concept cannot be analyzed from that concept.
Therefore, the possibility of synthesis and expansion of Geometric knowledge is thus based on space's being intuited or on the fact that such a proposition may be known true only in intuition.
And thus the truth of a Geometric proposition can be demonstrated only in intuition.
Second , the apodeicticity of Geometric knowledge is explained from the apriority of intuition of space and the apodeicticity of Arithmetics knowledge is explained from the apriority of intuition of time.
If space and time are to be empirical, they do not have necessity; however, both Geometric and Arithmetic propositions are universally valid and necessary true.
Third , mathematical knowledge has the objective reality that based on space and time in which our experiences are possible.
Forth, in regard to time, change and motion are only possible on the basis of time.
ments, by contrast, are non-empirical and non-contingent judgments.
1 Kant, I., 1787, “The Critique of Pure Reason: Preface To The Second Edition”, Translated By J. M. D. Meiklejohn, Retrieved 2003