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Thus； the nature of a right…angled triangle does not break itself up into factors in the manner set
forth in the mathematical construction which is required to prove the proposition expressing the
relation of its parts。 The entire process of producing the result is an affair of knowledge which
takes its own way of going about it。 In philosophical knowledge； too； the way existence； qua
existence； comes about （Werden） is different from that whereby the essence or inner nature of the
fact comes into being。 But philosophical knowledge； for one thing； contains both； while
mathematical knowledge sets forth merely the way an existence comes about； i。e。 the way the
nature of the fact gets to be in the sphere of knowledge as such。 For another thing； too；
philosophical knowledge unites both these particular movements。 The inward rising into being； the
process of substance； is an unbroken transition into outwardness； into existence or being for
another； and conversely； the coming of existence into being is withdrawal into the inner essence。
The movement is the twofold process in which the whole comes to be； and is such that each at the
same time posits the other； and each on that account has in it both as its two aspects。 Together
they make the whole； through their resolving each other； and making themselves into moments of
the whole。
In mathematical knowledge the insight required is an external function so far as the subject…matter
dealt with is concerned。 It follows that the actual fact is thereby altered。 The means taken；
construction and proof； contain； no doubt； true propositions； but all the same we are bound to say
that the content is false。 The triangle in the above example is taken to pieces； and its parts made
into other figures to which the construction in the triangle gives rise。 It is only at the end that we
find again reinstated the triangle we are really concerned with； it was lost sight of in the course of
the construction； and was present merely in fragments； that belonged to other wholes。 Thus we
find negativity of content coming in here too； a negativity which would have to be called falsity； just
as much as in the case of the movement of the notion where thoughts that are taken to be fixed
pass away and disappear。
The real defect of this kind of knowledge； however； affects its process of knowing as much as its
material。 As to that process； in the first place we do not see any necessity in the construction。 The
necessity does not arise from the nature of the theorem： it is imposed； and the injunction to draw
just these lines； an infinite number of others being equally possible； is blindly acquiesced in； without
our knowing anything further； except that； as we fondly believe； this will serve our purpose in
producing the proof。 Later on this design then comes out too； and is therefore merely external in
character； just because it is only after the proof is found that it comes to be known。 In the same
way； again； the proof takes a direction that begins anywhere we like； without our knowing as yet
what relation this beginning has to the result to be brought out。 In its course； it takes up certain
specific elements and relations and lets others alone； without its being directly obvious what
necessity there is in the matter。 An external purpose controls this process。
The evidence peculiar to this defective way of knowing — an evidence on the strength of which
mathematics plumes itself and proudly struts before philosophy — rests solely on the poverty of its
purpose and the defectiveness of its material； and is on that account of a kind that philosophy must
scorn to have anything to do with。 Its purpose or principle is quantity。 This is precisely the
relationship that is non…essential； alien to the character of the notion。 The process of knowledge
goes on； therefore； on the surface； does not affect the concrete fact itself； does not touch its inner
nature or lotion； and is hence not a conceptual way of comprehending。 The material which
provides mathematics with these welcome treasures of truth consists of space and numerical units
（das Eins）。 Space is that kind of existence wherein the concrete notion inscribes the diversity it
contains； as in an empty； lifeless element in which its differences likewise subsist in passive； lifeless
form。 What is concretely actual is not something spatial； such as is treated of in mathematics。 With
unrealities like the things mathematics takes account of； neither concrete sensuous perception nor
philosophy has anything to do。 In an unreal element of that sort we find； then； only unreal truth；
fixed lifeless propositions。 We can call a halt at any of them； the next begins of itself de novo；
without the first having led up to the one that follows； and without any necessary connexion having
in this way arisen from the nature of the subject…matter itself。 So； too — and herein consists the
formal character of mathematical evidence because of that principle and the element where it
applies； knowledge advances along the lines of bare equality； of abstract identity。 For what is
lifeless； not being self…moved； does not bring about distinction within its essential nature； does not
attain to essential opposition or unlikeness； and hence involves no transition of one opposite
element into its other； no qualitative； immanent movement； no self…movement； It is quantity； a form
of difference that does not touch the essential nature； which alone mathematics deals with。 It
abstracts from the fact that it is the notion which separates space into its dimensions； and
determines the connexions between them and in them。 It does not consider； for example； the
relation of line to surface； and when it compares the diameter of a circle with its circumference； it
runs up against their incommensurability； i。e。 a relation in terms of the notion； an infinite element；
that escapes mathematical determination。
Immanent or so…called pure mathematics； again； does not oppose time qua time to space； as a
second subject…matter for consideration。 Applied mathematics； no doubt； treats of time； as also of
motion； and other concrete things as well； but it picks up from experience synthetic propositions
— i。e。 statements of their relations； which are determined by their conceptual nature — and merely
applies its formulae to those propositions assumed to start with。 That the so…called proofs of
propositions like that concerning the equilibrium of the lever； the relation of space and time in
gravitation； etc。； which applied mathematics frequently gives； should be taken and given as proofs；
is itself merely a proof of how great the need is for knowledge to have a process of proof； seeing
that； even where proof is not to be had； knowledge yet puts a value on the mere semblance of it；
and gets thereby a certain sense of satisfaction。 A criticism of those proofs would be as instructive
as it would be significant； if the criticism could strip mathematics of this artificial finery； and bring
out its limitations； and thence show the necessity for another type of knowledge。
As to time； which； it is to be presumed； would； by way of the counterpart to space； co