Immanuel Kant's Critique of Pure Reason — trans. F. Max Müller

the use of these numbers becomes general. If I say, a triangle can be constructed with three lines, two of which together are greater than the third, I have before me the mere function of productive imagination, which may draw the lines greater or smaller, and bring them together at various angles. The number 7, on the contrary, [p. 165] is possible in one way only, and so likewise the number 12, which is produced by the synthesis of the former with 5. Such propositions therefore must not be called axioms (for their number would be endless) but numerical formulas.

This transcendental principle of phenomenal mathematics adds considerably to our knowledge a priori. Through it alone it becomes possible to make pure mathematics in their full precision applicable to objects of experience, which without that principle would by no means be selfevident, nay, has actually provoked much contradiction. Phenomena are not things in themselves. Empirical intuition is possible only through pure intuition (of space and time), and whatever geometry says of the latter is valid without contradiction of the former. All evasions, as if objects of the senses should not conform to the rules of construction in space (for instance, to the rule of the infinite divisibility of lines or angles) must cease, for one would thus deny all objective validity to space and with it to all mathematics, and would no longer know why and how far mathematics can be applied to phenomena. The synthesis of spaces and times, as the synthesis of the essential form of all intuition, is that which renders possible at the same time the apprehension of phenomena, that is, every external [p. 166] experience, and therefore also all knowledge of its ob136 Transcendental Analytic 2jects, and whatever mathematics, in their pure use prove of that synthesis is valid necessarily also of this knowledge. All objections to this are only the chicaneries of a falsely guided reason, which wrongly imagines that it can separate the objects of the senses from the formal conditions of our sensibility, and represents them, though they are phenomena only, as objects by themselves, given to the understanding. In this case, however, nothing could be known of them a prio^d, nothing could be known synthetically through pure concepts of space, and the science which determines those concepts, namely, geometry, would itself become impossible.

[Anticipations of Perception

The principle which anticipates all perceptions as such, is this : In all phenomena sensation, and the Real which corresponds to it in the object (rea/ifas phaenomenon), has an intensive quantity, that is, a degree ^ }

All knowledge by means of which I may know and determine a priori whatever belongs to empirical knowledge, may be called an anticipation, and it is no doubt in this sense that Epicurus used the expression [p. 167] 7r/3dX7;'i/rt9. But as there is always in phenomena something which can never be known a priori^ and constitutes the real difference between empirical and a priori knowledge, namely, sensation (as matter of perception), it follows that this can never be anticipated. The pure determinations, on the contrary, in space and time, as

1 Here follows in the Second Edition, Supplement XVI b.