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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； constitute the
object…matter of the other division of pure mathematics； this is the notion itself in the form of
existence。 The principle of quantity； of difference which is not determined by the notion； and the
principle of equality； of abstract； lifeless unity； are incapable of dealing with that sheer restlessness
of life and its absolute and inherent process of differentiation。 It is therefore only in an arrested；
paralysed form； only in the form of the quantitative unit； that this essentially negative activity
becomes the second object…matter of this way of knowing； which； itself an external operation；
degrades what is self…moving to the level of mere matter； in order thus to get an indifferent；
external； lifeless content。
13。 The nature of philosophical truth and its method
Philosophy； on the contrary； does not deal with a determination that is non…essential； but with a
determination so far as it is an essential factor。 The abstract or unreal is not its element and
content； but the real； what is self…establishing； has life within itself； existence in its very notion。 It is
the process that creates its own moments in its course； and goes through them all； and the whole
of this movement constitutes its positive content and its truth。 This movement includes； therefore；
within it the negative factor as well； the element which would be named falsity if it could be
considered one from which we had to abstract。 The element that disappears has rather to be
looked at as itself essential； not in the sense of being something fixed； that has to be cut off from
truth and allowed to lie outside it； heaven knows where； just as similarly the truth is not to be held
to stand on the other side as an immovable lifeless positive element。 Appearance is the process of
arising into being and passing away again； a process that itself does not arise and does not pass
away； but is per se； and constitutes reality and the life…movement of truth。 The truth is thus the
bacchanalian revel； where not a member is sober； and because every member no sooner becomes
detached than it eo ipso collapses straightway； the revel is just as much a state of transparent
unbroken calm。 Judged by that movement； the particular shapes which mind assumes do not
indeed subsist any more than do determinate thoughts or ideas； but they are； all the same； as much
positive and necessary moments； as negative and transitory。 In the entirety of the movement； taken
as an unbroken quiescent whole； that which obtains distinctness in the course of its process and
secures specific existence； is preserved in the form of a self …recollection； in which existence is
self…knowledge； and self…knowledge； again； is immediate existence。
It might well seem necessary to state at the outset the chief points in connexion with the method of
this process； the way in which science operates。 Its nature； however； is to be found in what has
already been said； while the proper systematic exposition of it is the special business of Logic； or
rather is Logic itself。 For the method is nothing else than the structure of the whole in its pure and
essential form。 In regard； however； to what has been hitherto currently held on this point； we must
be sensible that the system of ideas bearing on the question of philosophical method； belongs also
to a stage of mental culture that has now passed away。 This may perhaps seem somewhat boastful
or revolutionary； and I am far from adopting an attitude of that sort； but it is significant that the
scientific régime bequeathed by mathematics — a régime of explanations； divisions； axioms； an
array of theorems； with proofs； principles； and the consequences and conclusions drawn from
them — all this has already come to be generally considered as at any rate out of date。 Even
though there is no clear idea why it is unsuitable； yet little or no use is made of it any longer； and
even though it is not condemned outright； it is all the same not in favour。 And we must be so far
prejudiced in favour of what is excellent to bel