242 
PHILOSOPHY. 
figns, by which ali pofiible numbers may be underflood, 
lie takes his proof from mere conceptions. Now, if we 
only look into the nature of the proof, we perceive that 
he proceeds with thefe figns as with determinate figns, 
that is, as if they reprefented fingle quantities. Yet the 
charaCteriftic of an algebraic proof, by which it is diftin- 
guifhed from an arithmetical one, and even from that 
which is made by means of geometrical conftruCtion, 
confifts in this, that it feleCts that which, in the reprefen- 
tation of the fingle quantity, is neverthelefs univerfally 
valid, by its univerfal figns, (which however are treated 
as determinate figns;) and by this means facilitates our 
underftanding it. In arithmetical and geometrical con¬ 
ftruCtion, the univerfally-valid and the particular run into 
one another; and the view of the univerfally-valid is here 
not fo well prepared. In the preceding example of the 
propofition, that the three angles of a triangle are equal 
to two right, I find, by drawing a triangle, that the 
univerfally-valid and the particular is intermixed, and it 
is here left to our penetration to diftinguifh between the 
two; and in the fame manner the length of the fides, and 
their pofition, as the particular and contiguous; but to 
cor.fider, on the other hand, as the univerfally-valid, both 
drawing a line parallel to its bafe through its point, and 
the equality of the alternate angles which arifes thereby. 
It is cuftomary to extol ancient geometricians for their 
penetration, and to prefer their method to the algebraic, 
by which indeed we arrive in a (liorter way at certain 
refults, but at the fame time lofe many opportunities of 
cxercifing our judgment. I admit this; but I do not 
think that the method of the ancients is preferable to the 
algebraic, on account of its being fynthetical, fince 
algebra is alfo fynthetical; but becaufe it does not fliow 
the univerfal validity by univerfal figns as algebra does, 
and becaufe it gives more fcope to our penetration. 
On the other hand, philofophical knowledge is know¬ 
ledge of reafon by conceptions. When we fpeak of ex¬ 
tending our knowledge, it is clear that it is in no other 
way pofiible, but by a tranfition from the conception to the 
intuition. Now, if it is a pure (formal) intuition, we may 
ifeover in it certain univerfally-valid determinations, 
and. thus extend the conception itfelf which did not 
contain them ; and it is in this manner that mathematical 
knowledge arifes. But, if the intuition is empirical, 
we extend by means of it our knowledge, but not the 
conception, becaufe we cannot transfer the determinations 
of the empirical individual to the whole fpecies. This is 
empirical knowledge, which is never univerfally-valid. 
Now w'e may further think univerfally-valid knowledge 
as pofiible, which does not reft on the reprefentation of 
the pure intuition that correfponds with the conception. 
This, however, will not extend the conceptions, but only 
illuftrate them; this kind of knowledge is termed philo- 
fcphical. It does not go beyond the conception, and 
over to the intuition; but remains with the conceptions, 
and only develops them; it reprefents the particular in 
the univerfal, whereas the mathematics reprefent the uni¬ 
verfal in the particular. Now in philofophical knowledge 
there is indeed alfo a fvnthefis, but it is of a different 
kind from that which is found in mathematical propofi- 
tions. In order to know the latter, we do not remain 
with the conception, but proceed to the intuition. On 
the other hand, in philofophical knowledge I do not for- 
fake the conceptions which I form cl priori. In Analytics, 
it has been fhown how this fynthefis is circumitanced. 
It is nothing more than the & priori determined condition 
of the Jynthetically-objective unity of the empirical intuition. 
The pofition, “ Whatever happens has a caufe,” is fynthe¬ 
tical. Now to thefe conceptions we can give no intuitions. 
The empirical intuition however would only furnifli 
empirical knowledge ; and in this way could never lead 
to the univerfal validity expreffed by the propofition; not 
to mention that we never fhould even come at the ob¬ 
jective validity of the conception of a Caufe. On the 
other hand, I perceive that, by means of this rule, certain 
things given in the intuition are reprefented as necefiariiy 
belonging to one another, namely, an objeCt and an 
event. This fynthefis is, properly fpeaking, a rule, 
according to wdiich the fynthetical unity in experience 
firft of all becomes poffible. The knowledge of it, how¬ 
ever, is juft as well pofiible from conceptions only, which 
muff be called difeurfive, as it is that knowledge which 
confifts in the mere development of conceptions. 
There are therefore two different kinds of rational 
knowledge. The mathematical is intuitive; it proceeds 
from the univerfal to the particular, and, properly fpeak¬ 
ing, to the individual or lingular; that is, it arifes from 
the conftruCtion of conceptions. It is however able to 
think what is met with in the particular, as univerfally- 
valid, and in this manner to extend the conception fyn- 
thetically. The philofophical, on the other hand, is dif¬ 
eurfive ; it remains with the univerfal, and is of two kinds: 
it either develops the univerfal, by extracting the parti¬ 
cular that is thought in it; or it connefts conceptions 
together, which fynthefis is always tranfcendental, and 
can only be thought as a condition of experience. 
We will now fhow what follows from this; namely, 
that the Definitions, Axioms, and Demonftrations, upon 
which the evidence of the mathematics refts, cannot be 
made nor annihilated in philofophy ; the mathematician 
gives them. 
Pure Mathematics does not concern itfelf with given ob¬ 
jects. The objective reality of its conceptions lies entirely 
out of its fphere, and can only be comprehended after a 
tranfcendental reflection. That the fynthefis of the homo¬ 
geneous variety in apprehenfion is precifely that by which 
a determined fpace is produced in the pure intuition, is 
the ground of the application of the Category of Quantity 
to the external phenomena. It is this fynthefis which 
produces the reprefentation of a necefiary connexion of 
parts ; Quantity, that is, the reprefentation of an object 
in experience. Pure mathematics fets the objective 
validity of its conceptions entirely alide. Inftead of 
enquiring after this, it forms to itfelf conceptions of 
objeCts, which are merely formal and imaginary, which 
fo far are already objeCts, if they can but be exhibited in 
pure intuition. But hence it follows, that, previoufly to 
confidering them, the definition of the conceptions, by 
wdiich they are thought, muft precede. This definition 
rntift give the marks that we mean to unite in one con¬ 
ception completely and precifely. Previous to this defi¬ 
nition we can think nothing, becaufe there is no objeCt 
given, but it arifes firft of all in the definition ; and the 
conception, confidered in itfelf, has no objective validity. 
But, on the other hand, if we turn our attention to con¬ 
ceptions of objeCts which are given, it will be evident that 
they cannot be defined, but only explained. For, as in 
this cafe the objeCt determines the conception, it may 
happen that the conceptions contain fometimes more and 
fometimes fewer marks, according as obfervation and 
experiment have extended our knowledge of objeCts, 
Thus w'e may have introduced into our conception of 
gold, the mark that it does not ruft; and others may 
know nothing of this quality. Nor would definitions of 
thefe empirical conceptions be of any ufe, becaufe in 
experience it does not patter what is thought in the con¬ 
ception, but what properties the objeCts have; for which 
purpofe, however, obfervation is necefiary to render the 
conceptions gradually more complete and more adapted 
to the objeCts. Nor can conceptions, fuch as Subftance, 
Caufe, Right, Equity, See. be defined & priori, in the 
proper fenfe of the word definition. For though, in this 
cafe, the objeCt does not determine the conception, but 
the conception the objeCt, and indeed either tranfeenden- 
tally or practically, ftill I can only become certain in the 
application of the conception, confequently only by ex¬ 
amples of the marks which it contains, by which however 
I never can arrive at the apodiCtical certainty of their 
complete number. If we mean to avail ourfelves of the 
exprefiion definition for the explanation both of empirical 
