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real acquisition。
V。 In all Theoretical Sciences of Reason； Synthetical Judgements
〃a priori〃 are contained as Principles。
1。 Mathematical judgements are always synthetical。 Hitherto this
fact； though incontestably true and very important in its
consequences； seems to have escaped the analysts of the human mind；
nay； to be in plete opposition to all their conjectures。 For as
it was found that mathematical conclusions all proceed according to
the principle of contradiction （which the nature of every apodeictic
certainty requires）； people became persuaded that the fundamental
principles of the science also were recognized and admitted in the
same way。 But the notion is fallacious； for although a synthetical
proposition can certainly be discerned by means of the principle of
contradiction； this is possible only when another synthetical
proposition precedes； from which the latter is deduced； but never of
itself which
Before all； be it observed； that proper mathematical propositions
are always judgements a priori； and not empirical； because they
carry along with them the conception of necessity； which cannot be
given by experience。 If this be demurred to； it matters not； I will
then limit my assertion to pure mathematics； the very conception of
which implies that it consists of knowledge altogether non…empirical
and a priori。
We might； indeed at first suppose that the proposition 7 ＋ 5 = 12 is
a merely analytical proposition； following （according to the principle
of contradiction） from the conception of a sum of seven and five。
But if we regard it more narrowly； we find that our conception of
the sum of seven and five contains nothing more than the uniting of
both sums into one； whereby it cannot at all be cogitated what this
single number is which embraces both。 The conception of twelve is by
no means obtained by merely cogitating the union of seven and five；
and we may analyse our conception of such a possible sum as long as we
will； still we shall never discover in it the notion of twelve。 We
must go beyond these conceptions； and have recourse to an intuition
which corresponds to one of the two… our five fingers； for example； or
like Segner in his Arithmetic five points； and so by degrees； add
the units contained in the five given in the intuition； to the
conception of seven。 For I first take the number 7； and； for the
conception of 5 calling in the aid of the fingers of my hand as
objects of intuition； I add the units； which I before took together to
make up the number 5； gradually now by means of the material image
my hand； to the number 7； and by this process； I at length see the
number 12 arise。 That 7 should be added to 5； I have certainly
cogitated in my conception of a sum = 7 ＋ 5； but not that this sum was
equal to 12。 Arithmetical propositions are therefore always
synthetical； of which we may bee more clearly convinced by trying
large numbers。 For it will thus bee quite evident that； turn and
twist our conceptions as we may； it is impossible； without having
recourse to intuition； to arrive at the sum total or product by
means of the mere analysis of our conceptions。 just as little is any
principle of pure geometry analytical。 〃A straight line between two
points is the shortest；〃 is a synthetical proposition。 For my
conception of straight contains no notion of quantity； but is merely
qualitative。 The conception of the shortest is therefore fore wholly
an addition； and by no analysis can it be extracted from our
conception of a straight line。 Intuition must therefore here lend
its aid； by means of which； and thus only； our synthesis is possible。
Some few principles preposited by geometricians are； indeed；
really analytical； and depend on the principle of contradiction。
They serve； however； like identical propositions； as links in the
chain of method； not as principles… for example； a = a； the whole is
equal to itself； or （a＋b） 》 a； the whole is greater than its part。 And
yet even these principles themselves； though they derive their
validity from pure conceptions； are only admitted in mathematics
because they can be presented in intuition。 What causes us here
monly to believe that the predicate of such apodeictic judgements
is already contained in our conception； and that the judgement is
therefore analytical； is merely the equivocal nature of the
expression。 We must join in thought a certain predicate to a given
conception； and this necessity cleaves already to the conception。
But the question is； not what we must join in thought to the given
conception； but what we really think therein； though only obscurely；
and then it bees manifest that the predicate pertains to these
conceptions； necessarily indeed； yet not as thought in the
conception itself； but by virtue of an intuition； which must be
added to the conception。
2。 The science of natural philosophy （physics） contains in itself
synthetical judgements a priori； as principles。 I shall adduce two
propositions。 For instance； the proposition； 〃In all changes of the
material world； the quantity of matter remains unchanged〃； or； that；
〃In all munication of motion； action and reaction must always be
equal。〃 In both of these； not only is the necessity； and therefore
their origin a priori clear； but also that they are synthetical
propositions。 For in the conception of matter； I do not cogitate its
permanency； but merely its presence in space； which it fills。 I
therefore really go out of and beyond the conception of matter； in
order to think on to it something a priori； which I did not think in
it。 The proposition is therefore not analytical； but synthetical；
and nevertheless conceived a priori； and so it is with regard to the
other propositions of the pure part of natural philosophy。
3。 As to metaphysics； even if we look upon it merely as an attempted
science； yet； from the nature of human reason； an indispensable one；
we find that it must contain synthetical propositions a priori。 It
is not merely the duty of metaphysics to dissect； and thereby
analytically to illustrate the conceptions which we form a priori of
things； but we seek to widen the range of our a priori knowledge。
For this purpose； we must avail ourselves of such principles as add
something to the original conception… something not identical with；
nor contained in it； and by means of synthetical judgements a
priori； leave far behind us the limits of experience； for example；
in the proposition； 〃the world must have a beginning；〃 and such
like。 Thus metaphysics； according to the proper aim of the science；
consists merely of synthetical propositions a priori。
VI。 The Universal Problem of Pure Reason。
It is extremely advantageous to be able to bring a number of
investigations under the formula of a single problem。 For in this
manner； we not only facilitate our own labour； inasmuch as we define
it clearly to ourselves； but also render it more easy for others to
decide whether we have done justice to our undertaking。 The proper
problem of pure reason； then； is contained in the question： 〃How are
synthetical judgements a priori possible？〃
That metaphysical science has hitherto remai