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twelve； which results from the synthesis of seven and five。 Such
propositions； then； cannot be termed axioms （for in that case we
should have an infinity of these）； but numerical formulae。
This transcendental principle of the mathematics of phenomena
greatly enlarges our a priori cognition。 For it is by this principle
alone that pure mathematics is rendered applicable in all its
precision to objects of experience； and without it the validity of
this application would not be so self…evident； on the contrary；
contradictions and confusions have often arisen on this very point。
Phenomena are not things in themselves。 Empirical intuition is
possible only through pure intuition （of space and time）；
consequently； what geometry affirms of the latter； is indisputably
valid of the former。 All evasions； such as the statement that
objects of sense do not conform to the rules of construction in
space （for example； to the rule of the infinite divisibility of
lines or angles）； must fall to the ground。 For； if these objections
hold good； we deny to space； and with it to all mathematics； objective
validity； and no longer know wherefore； and how far； mathematics can
be applied to phenomena。 The synthesis of spaces and times as the
essential form of all intuition； is that which renders possible the
apprehension of a phenomenon； and therefore every external experience；
consequently all cognition of the objects of experience； and
whatever mathematics in its pure use proves of the former； must
necessarily hold good of the latter。 All objections are but the
chicaneries of an ill…instructed reason； which erroneously thinks to
liberate the objects of sense from the formal conditions of our
sensibility； and represents these； although mere phenomena； as
things in themselves； presented as such to our understanding。 But in
this case； no a priori synthetical cognition of them could be
possible； consequently not through pure conceptions of space and the
science which determines these conceptions； that is to say；
geometry； would itself be impossible。
2。 ANTICIPATIONS OF PERCEPTION。
The principle of these is： In all phenomena the Real； that
which is an object of sensation； has Intensive Quantity；
that is； has a Degree。
PROOF。
Perception is empirical consciousness； that is to say； a
consciousness which contains an element of sensation。 Phenomena as
objects of perception are not pure； that is； merely formal intuitions；
like space and time； for they cannot be perceived in themselves。
They contain； then； over and above the intuition； the materials for an
object （through which is represented something existing in space or
time）； that is to say； they contain the real of sensation； as a
representation merely subjective； which gives us merely the
consciousness that the subject is affected； and which we refer to some
external object。 Now； a gradual transition from empirical
consciousness to pure consciousness is possible； inasmuch as the
real in this consciousness entirely vanishes； and there remains a
merely formal consciousness （a priori） of the manifold in time and
space； consequently there is possible a synthesis also of the
production of the quantity of a sensation from its commencement；
that is； from the pure intuition = 0 onwards up to a certain
quantity of the sensation。 Now as sensation in itself is not an
objective representation； and in it is to be found neither the
intuition of space nor of time； it cannot possess any extensive
quantity； and yet there does belong to it a quantity （and that by
means of its apprehension； in which empirical consciousness can within
a certain time rise from nothing = 0 up to its given amount）；
consequently an intensive quantity。 And thus we must ascribe intensive
quantity； that is； a degree of influence on sense to all objects of
perception； in so far as this perception contains sensation。
All cognition； by means of which I am enabled to cognize and
determine a priori what belongs to empirical cognition； may be
called an anticipation； and without doubt this is the sense in which
Epicurus employed his expression prholepsis。 But as there is in
phenomena something which is never cognized a priori； which on this
account constitutes the proper difference between pure and empirical
cognition； that is to say； sensation （as the matter of perception）； it
follows； that sensation is just that element in cognition which cannot
be at all anticipated。 On the other hand； we might very well term
the pure determinations in space and time； as well in regard to figure
as to quantity； anticipations of phenomena； because they represent a
priori that which may always be given a posteriori in experience。
But suppose that in every sensation； as sensation in general；
without any particular sensation being thought of； there existed
something which could be cognized a priori； this would deserve to be
called anticipation in a special sense… special； because it may seem
surprising to forestall experience； in that which concerns the
matter of experience； and which we can only derive from itself。 Yet
such really is the case here。
Apprehension； by means of sensation alone； fills only one moment；
that is； if I do not take into consideration a succession of many
sensations。 As that in the phenomenon； the apprehension of which is
not a successive synthesis advancing from parts to an entire
representation； sensation has therefore no extensive quantity； the
want of sensation in a moment of time would represent it as empty；
consequently = O。 That which in the empirical intuition corresponds to
sensation is reality （realitas phaenomenon）； that which corresponds to
the absence of it； negation = O。 Now every sensation is capable of a
diminution； so that it can decrease； and thus gradually disappear。
Therefore； between reality in a phenomenon and negation； there
exists a continuous concatenation of many possible intermediate
sensations； the difference of which from each other is always
smaller than that between the given sensation and zero； or complete
negation。 That is to say； the real in a phenomenon has always a
quantity； which however is not discoverable in apprehension；
inasmuch as apprehension take place by means of mere sensation in
one instant； and not by the successive synthesis of many sensations；
and therefore does not progress from parts to the whole。 Consequently；
it has a quantity； but not an extensive quantity。
Now that quantity which is apprehended only as unity； and in which
plurality can be represented only by approximation to negation = O；
I term intensive quantity。 Consequently； reality in a phenomenon has
intensive quantity； that is； a degree。 if we consider this reality
as cause （be it of sensation or of another reality in the
phenomenon； for example； a change）； we call the degree of reality in
its character of cause a momentum； for example