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security； especially when it applies the same method which was
attended with such brilliant results in the science of mathematics。 It
is； therefore； of the highest importance for us to know whether the
method of arriving at demonstrative certainty； which is termed
mathematical； be identical with that by which we endeavour to attain
the same degree of certainty in philosophy； and which is termed in
that science dogmatical。
Philosophical cognition is the cognition of reason by means of
conceptions； mathematical cognition is cognition by means of the
construction of conceptions。 The construction of a conception is the
presentation a priori of the intuition which corresponds to the
conception。 For this purpose a non…empirical intuition is requisite；
which； as an intuition； is an individual object； while； as the
construction of a conception （a general representation）； it must be
seen to be universally valid for all the possible intuitions which
rank under that conception。 Thus I construct a triangle； by the
presentation of the object which corresponds to this conception；
either by mere imagination； in pure intuition； or upon paper； in
empirical intuition； in both cases pletely a priori； without
borrowing the type of that figure from any experience。 The
individual figure drawn upon paper is empirical； but it serves；
notwithstanding； to indicate the conception； even in its universality；
because in this empirical intuition we keep our eye merely on the
act of the construction of the conception； and pay no attention to the
various modes of determining it； for example； its size； the length
of its sides； the size of its angles； these not in the least affecting
the essential character of the conception。
Philosophical cognition； accordingly； regards the particular only in
the general； mathematical the general in the particular； nay； in the
individual。 This is done； however； entirely a priori and by means of
pure reason； so that； as this individual figure is determined under
certain universal conditions of construction； the object of the
conception； to which this individual figure corresponds as its schema；
must be cogitated as universally determined。
The essential difference of these two modes of cognition consists；
therefore； in this formal quality； it does not regard the difference
of the matter or objects of both。 Those thinkers who aim at
distinguishing philosophy from mathematics by asserting that the
former has to do with quality merely； and the latter with quantity；
have mistaken the effect for the cause。 The reason why mathematical
cognition can relate only to quantity is to be found in its form
alone。 For it is the conception of quantities only that is capable
of being constructed； that is； presented a priori in intuition；
while qualities cannot be given in any other than an empirical
intuition。 Hence the cognition of qualities by reason is possible only
through conceptions。 No one can find an intuition which shall
correspond to the conception of reality； except in experience； it
cannot be presented to the mind a priori and antecedently to the
empirical consciousness of a reality。 We can form an intuition； by
means of the mere conception of it； of a cone； without the aid of
experience； but the colour of the cone we cannot know except from
experience。 I cannot present an intuition of a cause； except in an
example which experience offers to me。 Besides； philosophy； as well as
mathematics； treats of quantities； as； for example； of totality；
infinity； and so on。 Mathematics； too； treats of the difference of
lines and surfaces… as spaces of different quality； of the
continuity of extension… as a quality thereof。 But； although in such
cases they have a mon object； the mode in which reason considers
that object is very different in philosophy from what it is in
mathematics。 The former confines itself to the general conceptions；
the latter can do nothing with a mere conception； it hastens to
intuition。 In this intuition it regards the conception in concreto；
not empirically； but in an a priori intuition； which it has
constructed； and in which； all the results which follow from the
general conditions of the construction of the conception are in all
cases valid for the object of the constructed conception。
Suppose that the conception of a triangle is given to a
philosopher and that he is required to discover； by the
philosophical method； what relation the sum of its angles bears to a
right angle。 He has nothing before him but the conception of a
figure enclosed within three right lines； and； consequently； with
the same number of angles。 He may analyse the conception of a right
line； of an angle； or of the number three as long as he pleases； but
he will not discover any properties not contained in these
conceptions。 But； if this question is proposed to a geometrician； he
at once begins by constructing a triangle。 He knows that two right
angles are equal to the sum of all the contiguous angles which proceed
from one point in a straight line； and he goes on to produce one
side of his triangle； thus forming two adjacent angles which are
together equal to two right angles。 He then divides the exterior of
these angles； by drawing a line parallel with the opposite side of the
triangle； and immediately perceives that be has thus got an exterior
adjacent angle which is equal to the interior。 Proceeding in this way；
through a chain of inferences； and always on the ground of
intuition； he arrives at a clear and universally valid solution of the
question。
But mathematics does not confine itself to the construction of
quantities （quanta）； as in the case of geometry； it occupies itself
with pure quantity also （quantitas）； as in the case of algebra；
where plete abstraction is made of the properties of the object
indicated by the conception of quantity。 In algebra； a certain
method of notation by signs is adopted； and these indicate the
different possible constructions of quantities； the extraction of
roots； and so on。 After having thus denoted the general conception
of quantities； according to their different relations； the different
operations by which quantity or number is increased or diminished
are presented in intuition in accordance with general rules。 Thus；
when one quantity is to be divided by another； the signs which
denote both are placed in the form peculiar to the operation of
division； and thus algebra； by means of a symbolical construction of
quantity； just as geometry； with its ostensive or geometrical
construction （a construction of the objects themselves）； arrives at
results which discursive cognition cannot hope to reach by the aid
of mere conceptions。
Now； what is the cause of this difference in the fortune of the
philosopher and the mathematician； the former of whom follows the path
of conceptions； while the latter pursues that of intuitions； which
he represents； a priori； in correspondence with his conceptions？ The
cause is evident from what has been already demonstrated in the
introduction to this Critique。 We do not； in the present case； want to
discover analytical propositions； which may be produced merely by
analy