ON THE THIRD CLASS OF OBJECTS FOR THE SUBJECT AND THAT FORM OF THE
PRINCIPLE OF SUFFICIENT REASON WHICH PREDOMINATES IN IT.
§ 35. _Explanation of this Class of Objects._
It is the formal part of complete representations--that is to say, the
intuitions given us _à priori_ of the forms of the outer and inner
sense, _i.e._ of Space and of Time--which constitutes the Third Class
of Objects for our representative faculty.
As pure intuitions, these forms are objects for the faculty of
representation by themselves and apart from complete representations
and from the determinations of being empty or filled which these
representations first add to them; since even pure points and pure
lines cannot be brought to sensuous perception, but are only _à
priori_ intuitions, just as the infinite expansion and the infinite
divisibility of Space and of Time are exclusively objects of pure
intuition and foreign to empirical perception. That which distinguishes
the third class of representations, in which Space and Time are _pure
intuitions_, from the first class, in which they are _sensuously_ (and
moreover conjointly) _perceived_, is Matter, which I have therefore
defined, on the one hand, as the perceptibility of Space and Time, on
the other, as objectified Causality.
The form of Causality, on the contrary, which belongs to the
Understanding, is not separately and by itself an object for our
faculty of representation, nor have we consciousness of it, until it is
connected with what is material in our knowledge.
§ 36. _Principle of the Sufficient Reason of Being._
Space and Time are so constituted, that all their parts stand in
mutual relation, so that each of them conditions and is conditioned
by another. We call this relation in Space, _position_; in Time,
_succession_. These relations are peculiar ones, differing entirely
from all other possible relations of our representations; neither the
Understanding nor the Reason are therefore able to grasp them by means
of mere conceptions, and pure intuition _à priori_ alone makes them
intelligible to us; for it is impossible by mere conceptions to explain
clearly what is meant by above and below, right and left, behind and
before, before and after. Kant rightly confirms this by the assertion,
that the distinction between our right and left glove cannot be made
intelligible in any other way than by intuition. Now, the law by which
the divisions of Space and of Time determine one another reciprocally
with reference to these relations (position and succession) is what
I call the _Principle of the Sufficient Reason of Being, principium
rationis sufficientis essendi_. I have already given an example of
this relation in § 15, by which I have shown, through the connection
between the sides and angles of a triangle, that this relation is
not only quite different from that between cause and effect, but
also from that between reason of knowledge and consequent; wherefore
here the condition may be called _Reason of Being_, _ratio essendi_.
The insight into such a _reason of being_ can, of course, become a
reason of knowing: just as the insight into the law of causality
and its application to a particular case is the reason of knowledge
of the effect; but this in no way annuls the complete distinction
between Reason of Being, Reason of Becoming, and Reason of Knowing. It
often happens, that what according to _one_ form of our principle is
_consequence_, is, according to another, _reason_. The rising of the
quicksilver in a thermometer, for instance, is the _consequence_ of
increased heat according to the law of causality, while according to
the principle of the sufficient reason of knowing it is the _reason_,
the ground of knowledge, of the increased heat and also of the judgment
by which this is asserted.
§ 37. _Reason of Being in Space._
The position of each division of Space towards any other, say of any
given line--and this is equally applicable to planes, bodies, and
points--determines also absolutely its totally different position with
reference to any other possible line; so that the latter position
stands to the former in the relation of the consequent to its reason.
As the position of this given line towards any other possible line
likewise determines its position towards all the others, and as
therefore the position of the first two lines is itself determined
by all the others, it is immaterial which we consider as being first
determined and determining the others, _i.e._ which particular one
we regard as _ratio_ and which others as _rationata_. This is so,
because in Space there is no succession; for it is precisely by uniting
Space and Time to form the collective representation of the complex
of experience, that the representation of coexistence arises. Thus an
analogue to so-called reciprocity prevails everywhere in the Reason
of Being in Space, as we shall see in § 48, where I enter more fully
into the reciprocity of reasons. Now, as every line is determined by
all the others just as much as it determines them, it is arbitrary to
consider any line merely as determining and not as being determined,
and the position of each towards any other admits the question as to
its position with reference to some other line, which second position
necessarily determines the first and makes it that which it is. It
is therefore just as impossible to find an end _a parte ante_ in the
series of links in the chain of Reasons of Being as in that of Reasons
of Becoming, nor can we find any _a parte post_ either, because of the
infinity of Space and of the lines possible within Space. All possible
relative spaces are figures, because they are limited; and all these
figures have their Reason of Being in one another, because they are
conterminous. The _series rationum essendi_ in Space therefore, like
the _series rationum fiendi_, proceeds _in infinitum_; and moreover not
only in a single direction, like the latter, but in all directions.
Nothing of all this can be proved; for the truth of these principles is
transcendental, they being directly founded upon the intuition of Space
given us _à priori_.
§ 38. _Reason of being in Time. Arithmetic._
Every instant in Time is conditioned by the preceding one. The
Sufficient Reason of Being, as the law of consequence, is so simple
here, because Time has only one dimension, therefore it admits of
no multiplicity of relations. Each instant is conditioned by its
predecessor; we can only reach it through that predecessor: only so far
as this _was_ and has elapsed, does the present one exist. All counting
rests upon this nexus of the divisions of Time, numbers only serving to
mark the single steps in the succession; upon it therefore rests all
arithmetic likewise, which teaches absolutely nothing but methodical
abbreviations of numeration. Each number pre-supposes its predecessors
as the reasons of its being: we can only reach the number _ten_ by
passing through all the preceding numbers, and it is only in virtue
of this insight that I know, that where ten are, there also are eight,
six, four.
§ 39. _Geometry._
The whole science of Geometry likewise rests upon the nexus of the
position of the divisions of Space. It would, accordingly, be an
insight into that nexus; only such an insight being, as we have already
said, impossible by means of mere conceptions, or indeed in any other
way than by intuition, every geometrical proposition would have to be
brought back to sensuous intuition, and the proof would simply consist
in making the particular nexus in question clear; nothing more could
be done. Nevertheless we find Geometry treated quite differently.
Euclid's Twelve Axioms are alone held to be based upon mere intuition,
and even of these only the Ninth, Eleventh, and Twelfth are properly
speaking admitted to be founded upon different, separate intuitions;
while the rest are supposed to be founded upon the knowledge that in
science we do not, as in experience, deal with real things existing
for themselves side by side, and susceptible of endless variety, but
on the contrary with conceptions, and in Mathematics with _normal
intuitions_, i.e. figures and numbers, whose laws are binding for all
experience, and which therefore combine the comprehensiveness of the
conception with the complete definiteness of the single representation.
For although, as intuitive representations, they are throughout
determined with complete precision--no room being left in _this_ way
by anything remaining undetermined--still they are general, because
they are the bare forms of all phenomena, and, as such, applicable to
all real objects to which such forms belong. What Plato says of his
Ideas would therefore, even in Geometry, hold good of these normal
intuitions, just as well as of conceptions, _i.e._ that two cannot be
exactly similar, for then they would be but one.[148] This would, I
say, be applicable also to normal intuitions in Geometry, if it were
not that, as exclusively spacial objects, these differ from one another
in mere juxtaposition, that is, in place. Plato had long ago remarked
this, as we are told by Aristotle:[149] ἔτι δὲ, παρὰ τὰ αἰσθητὰ καὶ τὰ
εἴδη, τὰ μαθηματικὰ τῶν πραγμάτων εἶναί φησι μεταξύ, διαφέροντα τῶν
μὲν αἰσθητῶν τῷ ἀΐδια καὶ ἀκίνητα εἶναι, τῶν δὲ εἰδῶν τῷ τὰ μὲν πόλλ'
ἄττα ὅμοια εἶναι, τὸ δὲ εἶδος αὐτὸ ἓν ἕκαστον μόνον (_item, præter
sensibilia et species, mathematica rerum ait media esse, a sensibilibus
quidem differentia eo, quod perpetua et immobilia sunt, a speciebus
vero eo, quod illorum quidem multa quædam similia sunt, species vero
ipsa unaquæque sola_). Now the mere knowledge that such a difference
of place does not annul the rest of the identity, might surely, it
seems to me, supersede the other nine axioms, and would, I think, be
better suited to the nature of science, whose aim is knowledge of the
particular through the general, than the statement of nine separate
axioms all based upon the same insight. Moreover, what Aristotle says:
ἐν τούτοις ἡ ἰσότης ἑνότης (_in illis æqualitas unitas est_)[150] then
becomes applicable to geometrical figures.
[148] Platonic ideas may, after all, be described as normal
intuitions, which would hold good not only for what is formal, but
also for what is material in complete representations--therefore
as complete representations which, as such, would be determined
throughout, while comprehending many things at once, like
conceptions: that is to say, as representatives of conceptions, but
which are quite adequate to those conceptions, as I have explained
in § 28.
[149] Aristot. "Metaph." i. 6, with which compare x. 1. "Further,
says he, besides things sensible and the ideas, there are things
mathematical coming in between the two, which differ from the
things sensible, inasmuch as they are eternal and immovable, and
from the ideas, inasmuch as many of them are like each other; but
the idea is absolutely and only one." (Tr.'s Add.)
[150] "In these it is equality that constitutes unity." (Tr.'s Add.)
But with reference to the normal intuitions in Time, _i.e._ to
numbers, even this distinction of juxtaposition no longer exists.
Here, as with conceptions, absolutely nothing but the _identitas
indiscernibilium_ remains: for there is but one five and one seven.
And in this we may perhaps also find a reason why 7 + 5 = 12 is a
synthetical proposition _à priori_, founded upon intuition, as Kant
profoundly discovered, and not an identical one, as it is called by
Herder in his "Metakritik". 12 = 12 is an identical proposition.
In Geometry, it is therefore only in dealing with axioms that we appeal
to intuition. All the other theorems are demonstrated: that is to say,
a reason of knowing is given, the truth of which everyone is bound to
acknowledge. The logical truth of the theorem is thus shown, but not
its transcendental truth (v. §§ 30 and 32), which, as it lies in the
reason of _being_ and not in the reason of _knowing_, never can become
evident excepting by means of intuition. This explains _why_ this sort
of geometrical demonstration, while it no doubt conveys the conviction
that the theorem which has been demonstrated is true, nevertheless
gives no insight as to why that which it asserts is what it is. In
other words, we have not found its Reason of Being; but the desire to
find it is usually then thoroughly roused. For proof by indicating
the reason of knowledge only effects conviction (_convictio_), not
knowledge (_cognitio_): therefore it might perhaps be more correctly
called _elenchus_ than _demonstratio_. This is why, in most cases,
therefore, it leaves behind it that disagreeable feeling which is given
by all want of insight, when perceived; and here, the want of knowledge
_why_ a thing is as it is, makes itself all the more keenly felt,
because of the certainty just attained, _that_ it is as it is. This
impression is very much like the feeling we have, when something has
been conjured into or out of our pocket, and we cannot conceive how.
The reason of knowing which, in such demonstrations as these, is given
without the reason of being, resembles certain physical theories,
which present the phenomenon without being able to indicate its cause:
for instance, Leidenfrost's experiment, inasmuch as it succeeds also
in a platina crucible; whereas the reason of being of a geometrical
proposition which is discovered by intuition, like every knowledge
we acquire, produces satisfaction. When once the reason of being is
found, we base our conviction of the truth of the theorem upon that
reason alone, and no longer upon the reason of knowing given us by the
demonstration. Let us, for instance, take the sixth proposition of the
first Book of Euclid:--
"If two angles of a triangle are equal, the sides also which subtend,
or are opposite to, the equal angles shall be equal to one another."
(See fig. 3.)
[Illustration: _Fig. 3._]
Which Euclid demonstrates as follows:--
"Let _a b c_ be a triangle having the angle _a b c_ equal to the angle
_a c b_, then the side _a c_ must be equal to the side _a b_ also.
"For, if side _a b_ be not equal to side _a c_, one of them is greater
than the other. Let _a b_ be greater than _a c_; and from _b a_ cut off
_b d_ equal to _c a_, and draw _d c_. Then, in the triangles _d b c_,
_a b c_, because _d b_ is equal to _a c_, and _b c_ is common to both
triangles, the two sides _d b_ and _b c_ are equal to the two sides _a
c_, _a b_, each to each; and the angle _d b c_ is equal to the angle
_a c b_, therefore the base _d c_ is equal to the base _a b_, and the
triangle _d b c_ is equal to the triangle _a b c_, the less triangle
equal to the greater,--which is absurd. Therefore _a b_ is not unequal
to _a c_, that is, _a b_ is equal to _a c_."
Now, in this demonstration we have a reason of knowing for the truth
of the proposition. But who bases his conviction of that geometrical
truth upon this proof? Do we not rather base our conviction upon the
reason of being, which we know intuitively, and according to which
(by a necessity which admits of no further demonstration, but only of
evidence through intuition) two lines drawn from both extreme ends of
another line, and inclining equally towards each other, can only meet
at a point which is equally distant from both extremities; since the
two arising angles are properly but one, to which the oppositeness
of position gives the appearance of being two; wherefore there is no
reason why the lines should meet at any point nearer to the one end
than to the other.
It is the knowledge of the reason of being which shows us the necessary
consequence of the conditioned from its condition--in this instance,
the lateral equality from the angular equality--that is, it shows their
connection; whereas the reason of knowing only shows their coexistence.
Nay, we might even maintain that the usual method of proving merely
convinces us of their coexistence in the actual figure given us as
an example, but by no means that they are always coexistent; for, as
the necessary connection is not shown, the conviction we acquire of
this truth rests simply upon induction, and is based upon the fact,
that we find it is so in every figure we make. The reason of being
is certainly not as evident in all cases as it is in simple theorems
like this 6th one of Euclid; still I am persuaded that it might be
brought to evidence in every theorem, however complicated, and that
the proposition can always be reduced to some such simple intuition.
Besides, we are all just as conscious _à priori_ of the necessity of
such a reason of being for each relation of Space, as we are of the
necessity of a cause for each change. In complicated theorems it will,
of course, be very difficult to show that reason of being; and this is
not the place for difficult geometrical researches. Therefore, to make
my meaning somewhat clearer, I will now try to bring back to its reason
of being a moderately complicated proposition, in which nevertheless
that reason is not immediately evident. Passing over the intermediate
theorems, I take the 16th:
"In every triangle in which one side has been produced, the exterior
angle is greater than either of the interior opposite angles."
This Euclid demonstrates in the following manner (see fig. 4):--
[Illustration: _Fig. 4._]
"Let _a b c_ be a triangle; and let the side _b c_ be produced to _d_;
then the exterior angle _a c d_ shall be greater than either of the
interior opposite angles _b a c_ or _c b a_. Bisect the side _a c_ at
_e_, and join _b e_; produce _b e_ to _f_, making _e f_ equal to _e b_,
and join _f c_. Produce _a c_ to _g_. Because _a e_ is equal to _e c_,
and _b e_ to _e f_; the two sides _a e_, _e b_, are equal to the two
sides _c e_, _e f_, each to each; and the angle _a e b_ is equal to the
angle _c e f_, because they are opposite vertical angles; therefore
the base _a b_ is equal to the base _c f_, and the triangle _a e b_ is
equal to the triangle _c e f_, and the remaining angles of one triangle
to the remaining angles of the other, each to each, to which the equal
sides are opposite; therefore the angle _b a e_ is equal to the angle
_e c f_. But the angle _e c d_ is greater than the angle _e c f_.
Therefore the angle _a c d_ is greater than the angle _a b c_."
"In the same manner, if the side _b c_ be bisected, and the side _a c_
be produced to _g_, it may be demonstrated that the angle _b c g_, that
is, the opposite vertical angle _a c d_ is greater than the angle _a b
c_."
My demonstration of the same proposition would be as follows (see fig.
5):--
[Illustration: _Fig. 5._]
For the angle _b a c_ to be even equal to, let alone greater than,
the angle _a c d_, the line _b a_ toward _c a_ would have to lie in
the same direction as _b d_ (for this is precisely what is meant by
equality of the angles), _i.e._, it must be parallel with _b d_; that
is to say, _b a_ and _b d_ must never meet; but in order to form
a triangle they must meet (reason of being), and must thus do the
contrary of that which would be required for the angle _b a c_ to be of
the same size as the angle _a c d_.
For the angle _a b c_ to be even equal to, let alone greater than, the
angle _a c d_, line _b a_ must lie in the same direction towards _b d_
as _a c_ (for this is what is meant by equality of the angles), _i.e._,
it must be parallel with _a c_, that is to say, _b a_ and _a c_ must
never meet; but in order to form a triangle _b a_ and _a c_ must meet
and must thus do the contrary of that which would be required for the
angle _a b c_ to be of the same size as _a c d_.
By all this I do not mean to suggest the introduction of a new method
of mathematical demonstration, nor the substitution of my own proof
for that of Euclid, for which its whole nature unfits it, as well as
the fact that it presupposes the conception of parallel lines, which
in Euclid comes much later. I merely wished to show what the reason of
being is, and wherein lies the difference between it and the reason of
knowing, which latter only effects _convictio_, a thing that differs
entirely from insight into the reason of being. The fact that Geometry
only aims at effecting _convictio_, and that this, as I have said,
leaves behind it a disagreeable impression, but gives no insight into
the reason of being--which insight, like all knowledge, is satisfactory
and pleasing--may perhaps be one of the reasons for the great dislike
which many otherwise eminent heads have for mathematics.
I cannot resist again giving fig. 6, although it has already been
presented elsewhere; because the mere sight of it without words conveys
ten times more persuasion of the truth of the Pythagorean theorem than
Euclid's mouse-trap demonstration.
[Illustration: _Fig. 6._]
Those readers for whom this chapter may have a special interest will
find the subject of it more fully treated in my chief work, "Die Welt
als Wille und Vorstellung," vol. i. § 15; vol. ii. chap. 13.