everything particular under a general, and so on always to what is more
general, that the truth of many propositions is only logically
proved,—that is, through their dependence upon other propositions, through
syllogisms, which at the same time appear as proofs. But we must never
forget that this whole form of science is merely a means of rendering
knowledge more easy, not a means to greater certainty. It is easier to
discover the nature of an animal, by means of the species to which it
belongs, and so on through the genus, family, order, and class, than to
examine on every occasion the animal presented to us: but the truth of all
propositions arrived at syllogistically is always conditioned by and
ultimately dependent upon some truth which rests not upon reasoning but
upon perception. If this perception were always as much within our reach
as a deduction through syllogisms, then it would be in every respect
preferable. For every deduction from concepts is exposed to great danger
of error, on account of the fact we have considered above, that so many
spheres lie partly within each other, and that their content is often
vague or uncertain. This is illustrated by a multitude of demonstrations
of false doctrines and sophisms of every kind. Syllogisms are indeed
perfectly certain as regards form, but they are very uncertain on account
of their matter, the concepts. For, on the one hand, the spheres of these
are not sufficiently sharply defined, and, on the other hand, they
intersect each other in so many ways that one sphere is in part contained
in many others, and we may pass at will from it to one or another of
these, and from this sphere again to others, as we have already shown. Or,
in other words, the minor term and also the middle can always be
subordinated to different concepts, from which we may choose at will the
major and the middle, and the nature of the conclusion depends on this
choice. Consequently immediate evidence is always much to be preferred to
reasoned truth, and the latter is only to be accepted when the former is
too remote, and not when it is as near or indeed nearer than the latter.
Accordingly we saw above that, as a matter of fact, in the case of logic,
in which the immediate knowledge in each individual case lies nearer to
hand than deduced scientific knowledge, we always conduct our thought
according to our immediate knowledge of the laws of thought, and leave
logic unused.(20)
§ 15. If now with our conviction that perception is the primary source of
all evidence, and that only direct or indirect connection with it is
absolute truth; and further, that the shortest way to this is always the
surest, as every interposition of concepts means exposure to many
deceptions; if, I say, we now turn with this conviction to mathematics, as
it was established as a science by Euclid, and has remained as a whole to
our own day, we cannot help regarding the method it adopts, as strange and
indeed perverted. We ask that every logical proof shall be traced back to
an origin in perception; but mathematics, on the contrary, is at great
pains deliberately to throw away the evidence of perception which is
peculiar to it, and always at hand, that it may substitute for it a
logical demonstration. This must seem to us like the action of a man who
cuts off his legs in order to go on crutches, or like that of the prince
in the “_Triumph der Empfindsamkeit_” who flees from the beautiful reality
of nature, to delight in a stage scene that imitates it. I must here refer
to what I have said in the sixth chapter of the essay on the principle of
sufficient reason, and take for granted that it is fresh and present in
the memory of the reader; so that I may link my observations on to it
without explaining again the difference between the mere ground of
knowledge of a mathematical truth, which can be given logically, and the
ground of being, which is the immediate connection of the parts of space
and time, known only in perception. It is only insight into the ground of
being that secures satisfaction and thorough knowledge. The mere ground of
knowledge must always remain superficial; it can afford us indeed rational
knowledge _that_ a thing is as it is, but it cannot tell _why_ it is so.
Euclid chose the latter way to the obvious detriment of the science. For
just at the beginning, for example, when he ought to show once for all how
in a triangle the angles and sides reciprocally determine each other, and
stand to each other in the relation of reason and consequent, in
accordance with the form which the principle of sufficient reason has in
pure space, and which there, as in every other sphere, always affords the
necessity that a thing is as it is, because something quite different from
it, is as it is; instead of in this way giving a thorough insight into the
nature of the triangle, he sets up certain disconnected arbitrarily chosen
propositions concerning the triangle, and gives a logical ground of
knowledge of them, through a laborious logical demonstration, based upon
the principle of contradiction. Instead of an exhaustive knowledge of
these space-relations we therefore receive merely certain results of them,
imparted to us at pleasure, and in fact we are very much in the position
of a man to whom the different effects of an ingenious machine are shown,
but from whom its inner connection and construction are withheld. We are
compelled by the principle of contradiction to admit that what Euclid
demonstrates is true, but we do not comprehend _why_ it is so. We have
therefore almost the same uncomfortable feeling that we experience after a
juggling trick, and, in fact, most of Euclid’s demonstrations are
remarkably like such feats. The truth almost always enters by the back
door, for it manifests itself _per accidens_ through some contingent
circumstance. Often a _reductio ad absurdum_ shuts all the doors one after
another, until only one is left through which we are therefore compelled
to enter. Often, as in the proposition of Pythagoras, lines are drawn, we
don’t know why, and it afterwards appears that they were traps which close
unexpectedly and take prisoner the assent of the astonished learner, who
must now admit what remains wholly inconceivable in its inner connection,
so much so, that he may study the whole of Euclid through and through
without gaining a real insight into the laws of space-relations, but
instead of them he only learns by heart certain results which follow from
them. This specially empirical and unscientific knowledge is like that of
the doctor who knows both the disease and the cure for it, but does not
know the connection between them. But all this is the necessary
consequence if we capriciously reject the special kind of proof and
evidence of one species of knowledge, and forcibly introduce in its stead
a kind which is quite foreign to its nature. However, in other respects
the manner in which this has been accomplished by Euclid deserves all the
praise which has been bestowed on him through so many centuries, and which
has been carried so far that his method of treating mathematics has been
set up as the pattern of all scientific exposition. Men tried indeed to
model all the sciences after it, but later they gave up the attempt
without quite knowing why. Yet in our eyes this method of Euclid in
mathematics can appear only as a very brilliant piece of perversity. But
when a great error in life or in science has been intentionally and
methodically carried out with universal applause, it is always possible to
discover its source in the philosophy which prevailed at the time. The
Eleatics first brought out the difference, and indeed often the conflict,
that exists between what is perceived, φαινομενον,(21) and what is
thought, νουμενον, and used it in many ways in their philosophical
epigrams, and also in sophisms. They were followed later by the Megarics,
the Dialecticians, the Sophists, the New-Academy, and the Sceptics; these
drew attention to the illusion, that is to say, to the deception of the
senses, or rather of the understanding which transforms the data of the
senses into perception, and which often causes us to see things to which
the reason unhesitatingly denies reality; for example, a stick broken in
water, and such like. It came to be known that sense-perception was not to
be trusted unconditionally, and it was therefore hastily concluded that
only rational, logical thought could establish truth; although Plato (in
the Parmenides), the Megarics, Pyrrho, and the New-Academy, showed by
examples (in the manner which was afterwards adopted by Sextus Empiricus)
how syllogisms and concepts were also sometimes misleading, and indeed
produced paralogisms and sophisms which arise much more easily and are far
harder to explain than the illusion of sense-perception. However, this
rationalism, which arose in opposition to empiricism, kept the upper hand,
and Euclid constructed the science of mathematics in accordance with it.
He was compelled by necessity to found the axioms upon evidence of
perception (φαινομενον), but all the rest he based upon reasoning
(νουμενον). His method reigned supreme through all the succeeding
centuries, and it could not but do so as long as pure intuition or
perception, _a priori_, was not distinguished from empirical perception.
Certain passages from the works of Proclus, the commentator of Euclid,
which Kepler translated into Latin in his book, “De Harmonia Mundi,” seem
to show that he fully recognised this distinction. But Proclus did not
attach enough importance to the matter; he merely mentioned it by the way,
so that he remained unnoticed and accomplished nothing. Therefore, not
till two thousand years later will the doctrine of Kant, which is destined
to make such great changes in all the knowledge, thought, and action of
European nations, produce this change in mathematics also. For it is only
after we have learned from this great man that the intuitions or
perceptions of space and time are quite different from empirical
perceptions, entirely independent of any impression of the senses,
conditioning it, not conditioned by it, _i.e._, are _a priori_, and
therefore are not exposed to the illusions of sense; only after we have
learned this, I say, can we comprehend that Euclid’s logical method of
treating mathematics is a useless precaution, a crutch for sound legs,
that it is like a wanderer who during the night mistakes a bright, firm
road for water, and carefully avoiding it, toils over the broken ground
beside it, content to keep from point to point along the edge of the
supposed water. Only now can we affirm with certainty that what presents
itself to us as necessary in the perception of a figure, does not come
from the figure on the paper, which is perhaps very defectively drawn, nor
from the abstract concept under which we think it, but immediately from
the form of all knowledge of which we are conscious _a priori_. This is
always the principle of sufficient reason; here as the form of perception,
_i.e._, space, it is the principle of the ground of being, the evidence
and validity of which is, however, just as great and as immediate as that
of the principle of the ground of knowing, _i.e._, logical certainty. Thus
we need not and ought not to leave the peculiar province of mathematics in
order to put our trust only in logical proof, and seek to authenticate
mathematics in a sphere which is quite foreign to it, that of concepts. If
we confine ourselves to the ground peculiar to mathematics, we gain the
great advantage that in it the rational knowledge _that_ something is, is
one with the knowledge _why_ it is so, whereas the method of Euclid
entirely separates these two, and lets us know only the first, not the
second. Aristotle says admirably in the Analyt., post. i. 27: “Ακριβεστερα
δ᾽ επιστημη επιστημης και προτερα, ἡτε του ὁτι και του διοτι ἡ αυτη, αλλα
μη χωρις του ὁτι, της του διοτι” (_Subtilior autem et praestantior ea est
scientia, quâ_ QUOD _aliquid sit, et_ CUR _sit una simulque intelligimus
non separatim_ QUOD, _et_ CUR _sit_). In physics we are only satisfied
when the knowledge that a thing is as it is is combined with the knowledge
why it is so. To know that the mercury in the Torricellian tube stands
thirty inches high is not really rational knowledge if we do not know that
it is sustained at this height by the counterbalancing weight of the
atmosphere. Shall we then be satisfied in mathematics with the _qualitas
occulta_ of the circle that the segments of any two intersecting chords
always contain equal rectangles? That it is so Euclid certainly
demonstrates in the 35th Prop. of the Third Book; _why_ it is so remains
doubtful. In the same way the proposition of Pythagoras teaches us a
_qualitas occulta_ of the right-angled triangle; the stilted and indeed
fallacious demonstration of Euclid forsakes us at the _why_, and a simple
figure, which we already know, and which is present to us, gives at a
glance far more insight into the matter, and firm inner conviction of that
necessity, and of the dependence of that quality upon the right angle:—
[Illustration]
In the case of unequal catheti also, and indeed generally in the case of
every possible geometrical truth, it is quite possible to obtain such a
conviction based on perception, because these truths were always
discovered by such an empirically known necessity, and their demonstration
was only thought out afterwards in addition. Thus we only require an
analysis of the process of thought in the first discovery of a geometrical
truth in order to know its necessity empirically. It is the analytical
method in general that I wish for the exposition of mathematics, instead
of the synthetical method which Euclid made use of. Yet this would have
very great, though not insuperable, difficulties in the case of
complicated mathematical truths. Here and there in Germany men are
beginning to alter the exposition of mathematics, and to proceed more in
this analytical way. The greatest effort in this direction has been made
by Herr Kosack, teacher of mathematics and physics in the Gymnasium at
Nordhausen, who added a thorough attempt to teach geometry according to my
principles to the programme of the school examination on the 6th of April
1852.
In order to improve the method of mathematics, it is especially necessary
to overcome the prejudice that demonstrated truth has any superiority over
what is known through perception, or that logical truth founded upon the
principle of contradiction has any superiority over metaphysical truth,
which is immediately evident, and to which belongs the pure intuition or
perception of space.
That which is most certain, and yet always inexplicable, is what is
involved in the principle of sufficient reason, for this principle, in its
different aspects, expresses the universal form of all our ideas and
knowledge. All explanation consists of reduction to it, exemplification in
the particular case of the connection of ideas expressed generally through
it. It is thus the principle of all explanation, and therefore it is
neither susceptible of an explanation itself, nor does it stand in need of
it; for every explanation presupposes it, and only obtains meaning through
it. Now, none of its forms are superior to the rest; it is equally certain
and incapable of demonstration as the principle of the ground of being, or
of change, or of action, or of knowing. The relation of reason and
consequent is a necessity in all its forms, and indeed it is, in general,
the source of the concept of necessity, for necessity has no other
meaning. If the reason is given there is no other necessity than that of
the consequent, and there is no reason that does not involve the necessity
of the consequent. Just as surely then as the consequent expressed in the
conclusion follows from the ground of knowledge given in the premises,
does the ground of being in space determine its consequent in space: if I
know through perception the relation of these two, this certainty is just
as great as any logical certainty. But every geometrical proposition is
just as good an expression of such a relation as one of the twelve axioms;
it is a metaphysical truth, and as such, just as certain as the principle
of contradiction itself, which is a metalogical truth, and the common
foundation of all logical demonstration. Whoever denies the necessity,
exhibited for intuition or perception, of the space-relations expressed in
any proposition, may just as well deny the axioms, or that the conclusion
follows from the premises, or, indeed, he may as well deny the principle
of contradiction itself, for all these relations are equally
undemonstrable, immediately evident and known _a priori_. For any one to
wish to derive the necessity of space-relations, known in intuition or
perception, from the principle of contradiction by means of a logical
demonstration is just the same as for the feudal superior of an estate to
wish to hold it as the vassal of another. Yet this is what Euclid has
done. His axioms only, he is compelled to leave resting upon immediate
evidence; all the geometrical truths which follow are demonstrated
logically, that is to say, from the agreement of the assumptions made in
the proposition with the axioms which are presupposed, or with some
earlier proposition; or from the contradiction between the opposite of the
proposition and the assumptions made in it, or the axioms, or earlier
propositions, or even itself. But the axioms themselves have no more
immediate evidence than any other geometrical problem, but only more
simplicity on account of their smaller content.
When a criminal is examined, a _procès-verbal_ is made of his statement in
order that we may judge of its truth from its consistency. But this is
only a makeshift, and we are not satisfied with it if it is possible to
investigate the truth of each of his answers for itself; especially as he
might lie consistently from the beginning. But Euclid investigated space
according to this first method. He set about it, indeed, under the correct
assumption that nature must everywhere be consistent, and that therefore
it must also be so in space, its fundamental form. Since then the parts of
space stand to each other in a relation of reason and consequent, no
single property of space can be different from what it is without being in
contradiction with all the others. But this is a very troublesome,
unsatisfactory, and roundabout way to follow. It prefers indirect
knowledge to direct, which is just as certain, and it separates the
knowledge that a thing is from the knowledge why it is, to the great
disadvantage of the science; and lastly, it entirely withholds from the
beginner insight into the laws of space, and indeed renders him
unaccustomed to the special investigation of the ground and inner
connection of things, inclining him to be satisfied with a mere historical
knowledge that a thing is as it is. The exercise of acuteness which this
method is unceasingly extolled as affording consists merely in this, that
the pupil practises drawing conclusions, _i.e._, he practises applying the
principle of contradiction, but specially he exerts his memory to retain
all those data whose agreement is to be tested. Moreover, it is worth
noticing that this method of proof was applied only to geometry and not to
arithmetic. In arithmetic the truth is really allowed to come home to us
through perception alone, which in it consists simply in counting. As the
perception of numbers is in _time alone_, and therefore cannot be
represented by a sensuous schema like the geometrical figure, the
suspicion that perception is merely empirical, and possibly illusive,
disappeared in arithmetic, and the introduction of the logical method of
proof into geometry was entirely due to this suspicion. As time has only
one dimension, counting is the only arithmetical operation, to which all
others may be reduced; and yet counting is just intuition or perception _a
priori_, to which there is no hesitation in appealing here, and through
which alone everything else, every sum and every equation, is ultimately
proved. We prove, for example, not that (7 + 9 × 8 - 2)/3 = 42; but we
refer to the pure perception in time, counting thus makes each individual
problem an axiom. Instead of the demonstrations that fill geometry, the
whole content of arithmetic and algebra is thus simply a method of
abbreviating counting. We mentioned above that our immediate perception of
numbers in time extends only to about ten. Beyond this an abstract concept
of the numbers, fixed by a word, must take the place of the perception;
which does not therefore actually occur any longer, but is only indicated
in a thoroughly definite manner. Yet even so, by the important assistance
of the system of figures which enables us to represent all larger numbers
by the same small ones, intuitive or perceptive evidence of every sum is
made possible, even where we make such use of abstraction that not only
the numbers, but indefinite quantities and whole operations are thought
only in the abstract and indicated as so thought, as [sqrt](r^b) so that
we do not perform them, but merely symbolise them.
We might establish truth in geometry also, through pure _a priori_
perception, with the same right and certainty as in arithmetic. It is in
fact always this necessity, known through perception in accordance with
the principle of sufficient reason of being, which gives to geometry its
principal evidence, and upon which in the consciousness of every one, the
certainty of its propositions rests. The stilted logical demonstration is
always foreign to the matter, and is generally soon forgotten, without
weakening our conviction. It might indeed be dispensed with altogether
without diminishing the evidence of geometry, for this is always quite
independent of such demonstration, which never proves anything we are not
convinced of already, through another kind of knowledge. So far then it is
like a cowardly soldier, who adds a wound to an enemy slain by another,
and then boasts that he slew him himself.(22)