| Here
under follows the transcription of the chapter Descartes of
Houston Stewart Chamberlain's Immanuel Kant, published by John
Lane,
The Bodley Head, 1914. The note on p. 268 is by me, all
other notes are original.
DESCARTES
From the painting by Mignard,
from the Castle Howard Collection, now in the National Gallery.
|
195
DESCARTES
UNDERSTANDING AND SENSIBILITY
WITH AN EXCURSUS UPON ANALYTICAL GEOMETRY
Water is always like water,
but it has
a quite different
taste when drawn at the
fountain head from what it has
when drunk out of a pitcher.
Descartes.
196
(Blank page)
197
DESCARTES
THERE are days
and days, and I confess
that it is with some hesitation and distrust that I address myself
to-day
to the task of continuing our observations in common. For now I have to
travel with you through regions which it will not be so easy to make
clear
as it was so long as we had the eye of a Goethe and a Leonardo to
lighten
us on our way. The comparison with philosophers who were at the same
time
artists revealed to us much that was of fundamental importance, and
gave
rise to observations which could not but result in a deep insight into
the personality of Kant, in the narrower meaning of the word, but now
we
must face about, we must once more fix the lenses of our eyes upon a
nearer
focus; we must bring into comparison philosophers who in their turn
will
lead us far, but on another road; men, the atmosphere of whose lives
does
not consist in Beauty and Art, but in research and thought. To-day we
will
busy ourselves with Descartes the critically empirical, mathematical
thinker,
and in the next lecture with Bruno the logical schoolman and
enthusiastic
thinker.
You must
not
misunderstand me.
There is no such thing as an absolute artist, no such thing as an
absolute
mathematician, and above all no such thing as an absolute philosopher.
This sort of classification into professions will never succeed even
with
half-important men. Goethe and Leonardo were both of them, as we have
seen,
great investigators of nature, and thinkers:
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DESCARTES
Bruno and Descartes on
their side possess
in a pre-eminent degree the artistic gift of putting into shape: Bruno,
in
his manner of thinking and speaking, is as much a poet as Plato was;
and
Descartes, the masterful thinker, is so penetrated with the value of
perception
and the empirical investigation of nature, that he is the bitter enemy
of genuine professional philosophy. We, however, are dealing to-day
solely
with that which I should like to call the characteristic intellectual
attitude.
In Goethe and in Leonardo it is distinctly directed outwards: the
primacy
of the Eye is dominant in both, and indeed of the eye both as a
receptive
and reproductive machinery of the senses. It is true that we found the
result to be very different in the two men; for behind two equally
powerful
eyes two brains gifted in varying directions take up impressions, and
work
them up each in its own way. In Leonardo the gift of sight is more
precise
and, in the widest sense of the word, more correct in its perspective;
this he owes to the power, which we recognised in the previous lecture,
of referring all that he saw to the inner scheme of perception; before
Goethe's eyes, on the other hand, the outlines are uncertain, his power
of schematising is insufficient, and he mixes up his thought with
everything:
but it is exactly this which bestows on him the gift of illuminating
the
very depths of Nature, depths where without the lamp of creative
thought,
dark night reigns. Leonardo sees the relationship of things to one
another,
Goethe sees their relationship to the human intellect; in Leonardo's
understanding
the masculine element prevails, in Goethe's we find unmistakable
feminine
or receptive constituents; hence Leonardo's thought is keen,
mechanical,
scientific, and easily grasped, whereas Goethe's is deeper, more
iridescent,
baffling conception, because it is pregnant with presentiments too wild
to be tamed into words. We shall go further into this in a future
lecture;
for the moment
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DESCARTES
we must be contented with
recognising
the fact that this precise intellectual habit, the method of looking
outwards,
is the common property of both Leonardo and Goethe. At the same time
this
habit distinguishes both from Kant, even though a closer examination
has
revealed to us so many points of contact in the manner of Seeing
between
him, the artist in thought, and those two artist-sages. But now, for
the
sake of comparison, we will summon into court two men with essentially
different qualifications, — men whose innate intellectual habit points
inwards. I say “inwards“ because these thinkers in the first place
consult
their own thought, and only later on turn to Nature: they do not trust
the impression which comes from without, not, that is to say, until
they
have, as far as may be in any way possible, tested and dissected the
whole
details of the inner diagnosis: this method of procedure is the exact
opposite to that followed by Goethe and Leonardo. This habit I call the
method of looking inwards. René Descartes and Giordano Bruno
will,
as I think, answer our purpose: neither of the two is so nearly akin to
Kant as to prevent dark shadows being thrown upon the picture from them
upon him, and on the other hand, in respect of talent and feeling,
these
two great philosophers are just as fundamentally different from one
another
as Leonardo and Goethe. They have in common only — but this “only“
means
very much — the habit of the specific thinker. Bruno, the Goethe of our
second pair of philosophers, exclaims, Gli beni de la mente non
altronde
che dall' istessa mente rostra riportiamo 1 (it is from the mind itself
and from no other source that we acquire the riches of the mind), — and
Descartes,
the strict empiric, the Leonardo, says deliberately, Il n'est aucune
question
plus importante à résoudre que celle de savoir ce que
c'est
que la connaissance humaine, et jusqu'où elle s'étend,
... Rien ne me semble plus absurde que de discuter audacieusement sur
les
mystères de la nature sans
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DESCARTES
avoir une seule fois
cherché si
l'esprit humain peut atteindre jusque là. 2
These few words
will have sufficed
to show you with what manner of man we have to deal here; at the same
time
the patent relationship to Kant's objects and methods and convictions
is
at once striking. The investigation of the essence and of the limits of
human knowledge describes exactly a great part — the critical part — of
Kant's
Life-work, and that the peculiar riches of the mind must be acquired
from
within and not from without, puts into a few words what Kant looked
upon
as his positive, practical, and edifying achievement. But even the
points
of difference will teach us much. The life-stories of the seigneur du
Perron
(Descartes) and of the man of Nola (Bruno) show conclusively that these
two men as regards their intellectual talents are far removed from
Kant.
In the first lecture we saw how deeply rooted in Kant's method of
perception
and in his adoption of ideas was that peculiar feature which made him
so
painfully avoid even the shortest journey; Bruno and Descartes, on the
contrary, move restlessly from place to place, and from country to
country,
as the spirit moves them. Bruno, with his apostle's nature, needs new
contacts,
new excitements, new disputations; he is bound to strike sparks out of
life, to kindle flames in hearts; wherever he goes he arouses glowing
love
and irreconcilable hatred. Descartes, the reserved man of the world,
travels
in order to be alone, enjoys in cities “the solitude of the remotest
deserts,“
steals away from a place as soon as his presence is noticed, and at the
same time, by a systematic observation of the differently constituted
men
and nations, religions and customs, seeks to free himself from the
prejudices
which are rooted in us all. Je ne
fis autre chose que rouler ça
et là dans le monde, tâchant d'y être spectateur
plutôt
qu'acteur en toutes les comédies qui s'y jouent. 3 Such a funda-
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DESCARTES
mentally different
ordering of life points
to far-reaching differences in the essence of the intellect: we may
premise
without going further that Bruno and Descartes “saw“ otherwise than
Kant
did. This will be especially clear in the case of Bruno, who, in spite
of the purely philosophical tendency of his intellect, is in many
respects
the veriest antipodes of Kant, and as such can render us valuable
service,
whereas in Descartes the close kinship leads us to penetrate the inmost
secrets of Kant's method of perception, while allowing us to leave on
one
side the many points of difference between the two as having no value
for
the object which we have in view.
Among the very
great thinkers of
the world's history perhaps none has been so scurvily treated as
Descartes;
he,— I mean the true Descartes, — is as good as unknown; the shadowy
being
that under this name is represented to our imagination, is a mere
ghost-like
caricature. Here was a man who with desperate energy fought to purge
himself
and us of all philosophical phrases; whose burning endeavour it was to
tear philosophy out of the toils of a logic as arrogant as it was
impotent,
and to open its eyes to the one and only productive authority of pure
perception;
a man who in open and indignant opposition to the schools cried out,
“the
whole sum of human science consists in seeing distinctly“; — and of
this
man the vast majority of cultured people know neither the personality
nor
the life nor the achievements, with the exception of just one single
saying
which has been thrashed out until it has become a mere phrase — cogito,
ergo sum, — a mere jingle of syllables, unless we knew how it
originated
in Descartes, and whither it led him. Just think how it would be if
some
future history should have nothing more to report of Bismarck than that
his was the saying, “We Germans fear God, and nothing else in the
world,“
as if this very disputable
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DESCARTES
phrase represented the
sum-total of the
achievements of his richly active life! Where is the difference, if we
only take count of one ambiguous and much misunderstood saying of the
pioneer
in mathematics, the physicist, the anatomist, the kosmologist, the
philosopher,
of the man who perhaps more than any other has so enriched our treasure
of constructive imagination that to this day philosophy and science are
refreshed by the stimulants of his genius? But as though it were not
enough
that a philosophy resting upon the broadest foundation of an
all-embracing,
manifest consideration of nature, should have been to such an extent
turned
topsy-turvy by degradation into mere logical and psychological
nut-cracking
— beyond all this we are even robbed of the man's personality.
Descartes
was an aristocrat by birth, — by the bent of his intellect an extreme
individualist.
He does not only hold himself aloof from his fellow-men, choosing an
abode
in foreign parts, and leaving a town as soon as he becomes known and
gets
entangled in social relations, — but even intellectually he surrounds
himself
with a high wall lest the doctrines of the contemporary philosophical
guilds
should find their way in, and even for the time being digs a deep moat
to keep the wisdom of the ancients at a respectful distance. To treat
with
scorn the nullities of the professional philosophers — les bagatelles
d'école
— is for him the distinguishing mark of a “princely character,“ and of
himself he confesses, “not the understanding of the arguments of
others,
but personal investigation on my own account is what constitutes for me
the greatest happiness of study.“ It is in a quite different sense from
Schopenhauer that Descartes is a great Eremite; for in him there is
none
of the bitterness or vanity of solitude, it is a proud and peaceful
self-contentment.
It was only after long years that the incessant pressure of so
respected
a friend as Pater Mersenne determined him to publish, and it would have
203
DESCARTES
remained at that
fragmentary beginning,
had not the request of an exalted friend, the Countess Palatine
Elizabeth,
stood in the light of a royal command to so perfect a man of the world.
Je ne recherche point les bonnes
grâces de la populace, he writes
with quiet disdain in a private letter: but with him populace has a
wide
meaning; for when Mersenne communicates to him the criticisms of the
most
learned men in Paris, he answers, “I have long known that there are
asses
in the world, but I set so little store by their judgment, that it
would
vex me to be obliged to spend upon it even a minute of my leisure and
my
peace.“ No more is needed to show that an investigator who so
resolutely
follows his own road, and avoids all contact with the officially
recognised
masters of scholastic thought, will not easily develop a system of
philosophy
fitted to be formulated into a strict scholastic shape. The picture of
the world that Descartes unrolls before us, is no grafted scion such as
we are used to see in philosophy, but a tree grown from the seed. Plato
hangs upon Socrates, and also upon Pythagoras, Anaxagoras, Heraclitus,
and others: Aristotle springs from Plato; Bruno from Plotinus,
Lucretius,
Cusa; Locke, Berkeley, Spinoza, Leibniz from our Descartes; Kant, too,
springs from Descartes, and from Leibniz, Locke, Rousseau, Hume; and so
it is with all of them; Descartes alone stands by himself. And although
he is convinced of the truth of his perceptions, hoping that their
victory
will result in a new birth of the sciences, still he keeps such jealous
watch over his independence, he is so deeply concerned to be left even
after his death inviolate in his proud isolation, that he starts by
declaring
that his method is for himself alone, not for others; mon dessein n'est
pas d'enseigner la méthode que chacun doit suivre pour bien
conduire
sa raison, mais seulement de faire voir en quelle sorte j'ai
tâché
de conduire la mienne; — and so over and over again he does not
shirk
the
paradox
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DESCARTES
that his philosophy is
void of all originality,
which he only admits openly in order that the good people may not fall
into the idea of making his name the centre of a school. The idea was
to
him a scarecrow that there should come men who would imagine that they
could in a day compass that of which he had realised the insight after
twenty years of study and education, and that upon it they should build
up a Philosophy fit to make one's hair stand on end, should delude
themselves
into the notion that this Philosophy was the result of his “Principles,“ and
assure the world
that he, Descartes, was its founder. 4 It is touching to hear how
he
implores
posterity, — “never believe that the things of which people are
assuring
you sum up my teaching, and originate in me: ascribe to me only that
which
you gather from my own mouth“ — and his real wish, that is to say his
wish
in opposition to the founding of a school, he tells us clearly enough
in
the same passage, is ouvrir quelques
fenêtres, not to build up a
system, but to “tear open the windows and let in the light“ for all
those
who have eyes to see. You can now distinguish broadly, what occupied
this
great intellect, and what must needs be his aim when he at last allowed
himself to be talked over into appearing in public. Himself a free
personality,
who at the expense of great labour had torn from his eyes all the
bandages
which education, parentage, the wisdom of the schools, the doctrines of
the Church, had bound round them — his aim is to educate free
personalities,
and with that object not to teach them, — in the sense that is to say
of
the schools, — but to lure them on, and to do for them as he had done
for
himself, namely, to open their eyes, and make them teach themselves by
means of perception. By “philosophy“ he understands literally the
opening
of the eyes, oculos aperire. 5 And since this is the
fundamental
principle
of Descartes' personality and teaching, so he
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DESCARTES
cares nothing for the
fixed establishment
of great, universal, irrefutable principles, but gives himself a free
hand
in the intimate description of his often quaint ideas which only fit in
with his own personality. Only look at his portraits! look at his
innocently
amazed outlook over the world, and his slyly ironical smile at the
wisdom
of mankind! Why! the man is anti-scholastic to his finger-tips. Even
the
famous cogito, ergo sum (“I
think, therefore I am“) is no logical
conclusion,
at any rate for him, but the verbal expression, clothed accordingly in
the rags of logic, for a definite perception: and when the professional
schoolmen want to split hairs with him on the subject he winds up the
argument
by saying, “I do not argue the question of my being by a syllogism, but
I perceive it.“ 6
This
was the man
whose fate it
was to become — beyond the grave — the sacrifice of the populace in a
way
no other thinker did. Hardly was the breath out of his body when the
European
world of learning became divided into two camps, the Cartesians and the
anti-Cartesians. The proud Eye, so wise, so lovable in spite of all its
distrust, was closed; and now it was to be anatomically dissected and
lectured
upon. The teaching of Descartes, “perfected“ — as usual — by all manner
of
insignificant and contradictory minds, was transformed into a system of
scholastic definitions and rigid dogmas. Descartes had said, “as for
the
search after definitions, we can leave that to Messieurs les
Professeurs“;
in very many cases definitions only serve to make dark what is clear;
the
professor with his subtle distinctions clouds the natural light of the
understanding, and ends by making an obscure problem out of what every
peasant knows. Descartes had been indefatigable in confining logic
within
the narrow bounds of its justified effectiveness, since, as he says,
l'art
syllogistique ne sert en rien à la découverte de la
vérité;
whereas the art of logic is a
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DESCARTES
chief instrument of the
schoolmen for
talking of things about which they themselves know nothing. 7 A few
years
after his death there arose a complete logical system, the “Logique de
Port Royal,“ which pretended to be founded on his teaching. A very
short
time elapsed and this so-called Cartesianism was in the very centre of
the conflict over the Eucharist: Calvinists and Jansenists, the deniers
and the champions of the Real Presence of the Body and Blood of Christ
in the bread and wine, both appealed to Descartes: in his grave he was
marked as the founder of the philosophia
eucharistica; his loftily
plain
writings, conspicuous for their frankness, were forced to serve, like
the
arcana disciplinae of the
ancient mysteries, as evidence for and
against
the most abstract cobwebs of the brain, and between whiles the
Physicists
dragged out the over-hurried hypothesis of a genius on the Gyrations of
the Kosmos, fighting for and against it, as if the Personality and
nature-teaching
of Descartes must stand or fall by it; while Freethinkers and Pietists
both took possession of the so-called automatism of beasts, out of
which
they drew opposite conclusions. For more than a century the world was
filled
with the roaring of the Cartesians and the bellowing of the
anti-Cartesians;
of Descartes, the lonely investigator and thinker, there was no longer
any talk. And when at last, in no small measure out of seed which he
had
sown, a new science and a new philosophy had gradually grown up and
waxed
strong, universal contempt washed away the barren Cartesianism and the
equally barren anti-Cartesianism. The great personality of Descartes
had
long since faded away. Only the ill-starred cogito, ergo sum was
bandied
about like sea-wrack on the all-devouring ocean of the world's history.
True,
Descartes
receives honourable
mention in the philosophical histories. Schopenhauer's dictum, “the
Father
of the Modern Philosophy,“ has been universally
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DESCARTES
repeated; but it is
always in the sense
of what is called in stage language un
père noble, an honoured
but
not much noticed person of distinction in the background. I can
unhesitatingly
recommend to you the first volume of Kuno Fischer's comprehensive work
upon the modern philosophy: he gives at any rate a fairly exact
biographical
account of the man: but even here Descartes is so dealt with that he
falls
behind the other philosophers; and although there is much material
given
for a representation of his personality, this very representation, the
portrait of such a wholly individual intellect, the plastic bringing
into
evidence of his special significance, is a failure. In most of the
other
handbooks you will only find one chapter about him, entitled “Descartes
and his school,“ or simply “Cartesianism.“ He who said, “the great
intellects
talk nonsense as soon as it is their disciples who speak for them, for
it is perhaps outside all experience that any pupil should have
equalled
his master,“ that very man hardly exists any longer save in the title
for
a School! Nay, more: when all is said and done, few of our professional
philosophers are so equipped as to be capable of understanding the true
Descartes; for Descartes, as you will already have observed, is far
more
of a contemplator of nature than a philosopher in the scholastic and
still
authoritative meaning of the word:
indeed we might
frankly
call him an
anti-philosopher. For him philosophy, — this is his own literal
definition —
is a tree, “the golden tree of life“; its metaphysical roots strike
into
the dark earth, and as Descartes humorously remarks, it is not upon
roots
that fruit usually grows; the mighty stem is the science of physics,
under
which he comprehends the universal laws of all motion, and this stem
branches
off into the many empirical ramifications of knowledge, at the points
of
which flowers at last bloom, and the blessing of fruit ripens. 8 You
need
only look at Descartes' chief systematic work, the
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DESCARTES
Principles of Philosophy.
In Cousin's
edition the first part, which contains all the psychological and
metaphysical
discussions, needs only 57 pages; the three remaining parts, — Physics,
Kosmology,
and Geognosis, upwards of 400 pages, — while Descartes apologises for
not
yet being able to publish his Zoology, Botany, and Anthropology. He
indeed
was the first to put the problem of perception in the foreground, a
fact
wittily put by Fontenelle in the remark that, avant M. Descartes, on
raisonnait
plus commodément; les siècles passés sont bien
heureux
de ne pas avoir eu cet homme là; 9 and so he was the first man
to
awaken true metaphysical reflection; yet he himself spends but little
time
over it. It was the distinct perception of his own inner being that
served
him as the first step towards distinctness in the perception of visible
Nature. In the same way he made use of metaphysics as an active help to
physics. Anybody who is not competent to follow him in the domain of
natural
science and mathematics will find it difficult to do him justice. He
studies
the functions of his brain as a part of the world which directly
concerns
him, and is therefore of fundamental importance, certainly not in the
sense
of a professed philosopher in the ordinary modern meaning of the word,
whose calling and business it is to think over all matters in the
abstract.
He has no faith in the professional philosophy: he characterises it as
une grande erreur, and says, il est plus facile d'apprendre toutes les
sciences à la fois que d'en détacher une seule. A
man of this
stamp
is far removed from our philosophical professors, not only further than
their own dearly beloved Spinoza, who never once leaves the domain of
the
abstract, but further even than a Francis Bacon, who, it is true,
constructs
a Novum organum for the
dissemination of the knowledge of nature,
without
having ever himself been busied with mathematical and
natural-scientific
work, and whose first principle it is to
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DESCARTES
abandon all philosophy in
favour of a
so-called empiricism; 10
further too than a Locke, or a Berkeley, or a
Hume, or a Leibniz, for the chief element of the philosophy of all
these
men consists in ratiocinatio,
that is to say, the pondering in Reason,
and progress through pure conclusions of Reason. Here, on the contrary,
we see a man whose chief work, unfortunately never finished and only
known
by fragments, was to carry the title of Le Monde, ou Traité de
la
Lumière! So it was the whole great world, the Kosmos as
we
should
call it to-day, and in it first and foremost the medium by which it
becomes
known to us, namely Light, — that it was his aim “to observe, to
investigate,
to grasp,“ and only the man who keeps this aim before his eyes can hope
to gain a correct appreciation of the personality of Descartes, and of
the gifts which it bestowed. If we lay a one-sided stress upon the
intellectual
and theoretical reflections of this man, together with his metaphysical
discussions on mind and matter, and his attempts to set forth
irrefutably
the existence of God and the immortality of the soul, — then we shall
not
only obtain a crooked picture of him, but we shall at the same time not
even be in a position rightly to grasp his peculiar method of looking
upon
these purely speculative questions. The man who does not study
Descartes'
physics and does not penetrate their essence, sees his metaphysics in a
false perspective; that accounts for the inadequateness of all the
representations
of Descartes in philosophical books.
But
the same ill
luck pursues
him elsewhere; for he hardly fares better at the hands of the
mathematicians,
mechanicians, physicists, and anatomists than he does at those of the
philosophers.
Inasmuch as we are living under the domination of the extremest
specialisation, every
single branch of
science only
enquires after concrete services rendered within its own especial
kingdom,
and it is upon these that it reports, whereas Descartes' peculiar
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DESCARTES
domain is the
buffer-state. As between
metaphysics and physics, so in all cases Descartes is happiest on the
frontier.
There where union and separation take place, where the coy facts are
forced
in the interests of combination with other series of facts to become
supple
and accommodating, — there where everything arises which we call
“explaining“
and “understanding“ — there it is that Descartes at last feels himself
at home. For that reason, and for that reason only, he devotes himself
passionately to the study of mathematics, the great mediator between
perception
and thought, between things that are visible and thoughts that are
invisible.
But even mathematics, to the furtherance of which he rendered undying
services,
are to him “only the husk, not the essence“; to work at pure
mathematics
for mathematics' sake he looks upon as aimless waste of time, and he
hurries
so that it is difficult to keep up with him through the technicalities
of form and place, in order that he may come at once to Physics and
mechanics;
but here again it is not the detail of the phenomena which interests
him,
but the Essence of Light, the Causes of Gravitation, the relationship
between
the mechanical laws of Matter and the Facts of Life, and so forth. It
is
true that if he dissects a brain he will give an exact anatomical
description
of it, 11 but what
grips him is the hope of discovering a visible
connection
between the morphological figure and the function of memory. This last
example shows you with special clearness how in this peculiar man
theoretical
thought and the desire for concrete perception went hand in hand. It
followed
that Descartes, in the individual sciences, achieved less than might
have
been expected from a man of his genius. His theorising was detrimental
to the freedom of his observation, while at the same time the freedom
of
his theorising was narrowed by the painstaking detail-work of his
observations.
Hence it is that even his undeniable
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services in the domain of
the exact sciences, — his
informing thoughts as well as the discovery of facts, — reached their
goal
for the most part in other hands, not in his own, and therefore are
assigned
to other names. For example, there is documentary proof, though no
notice
is taken of it, that he taught the gravity of air and made experiments
upon it, when Pascal was a boy and Galilei still maintained the horror
vacui as an unassailable dogma,
— as also that the
famous
experiment
of the Puy de Dôme was only undertaken under pressure from the
unbelieving
Pascal; 12 that
Descartes should have discovered the circulation of the
blood independently of Harvey, and the laws of falling bodies
independently
of Galilei, are matters of which the specialists take no heed; but for
the knowledge of his personality they are of the deepest interest;
that
he was the first to expound the mathematical laws of the refraction of
light, was proved by Humboldt as far back as 1847, but I find no
mention
of the fact in any later work; in medical books you will find cursory
mention
of Descartes amongst the leading names under the words “Eye“ and
“Brain“ — as you see mere fragments, mere insignificance, or — Nothing.
That the
perceptible idea of the inertia of matter lies at the bottom of our
whole
mechanical science, is a matter of common knowledge, but few know that
we are indebted to Descartes for it, and there is not one who prefers
to
base his judgment of the nature of such a mind upon an intellectual
feat
like this and others, rather than upon the cogito, ergo sum. 13 Just as
little is it remembered that it was Descartes who paved the way for a
revolution
in Physics similar to that of Copernicus in astronomy, when he
nourished
the inspired conviction — which to his contemporaries was
incomprehensible
and seemed sheer madness; — Light is motion; and that moreover not the
trajectory
motion of a body violently flung, as Newton taught, but the motion of
an
imponderable
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DESCARTES
matter, the aether, by
which our optic
nerve is made to oscillate. Under the passive domination of the clumsy
Newtonian ideas this thought was forgotten, and when, in order to
justify
the facts, it had to be taken up again, men preferred to attach
themselves
to Christian Huyghens — a son and grandson of
two most intimate
friends of Descartes, — who had grown up under the eyes of the great
man,
and who had further developed his inspired thoughts as to Aether and
Light
into the ultimate mathematically and fully developed theory of
undulation.
And so the constructive thoughts of Descartes are not only the basis of
our atomistic physics, but also of our molecular physics. And in spite
of all it is but little that we learn about him in the books on natural
science, and here too his form remains clouded and distorted before our
eyes.
I hope
that I shall
incur no displeasure
for having shown you so circumstantially how far and why Descartes has
seldom been honoured in accordance with his merits, and why his
personality
is perhaps never rightly judged. I had to introduce this negative
method
of dealing with the question, because I had it at heart to upset what
you
might possibly know about him, or rather that is to say, think you
know,
in order to make way for more correct views. In the meantime I hope
that
you will yet have learnt something, and feel yourselves nearer to the
true
Descartes than you did a while ago. And I set great importance upon
your
knowing exactly what were the views of this remarkable man's brain:
for
in my lectures this brain constitutes the turning-point of our
observations
of Kant's personality, just as he himself, in more than one respect,
constitutes
the turning-point of human thought in general. I purposely use the word
Brain, not System, not Metaphysics, not Discoveries: the system of
Descartes,
that is to say, his Kosmology as it is developed in the Principia and
elsewhere,
is distasteful, that is to
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DESCARTES
say, distasteful if we
examine it with
painstaking accuracy like a dogmatic structure, without paying
attention
to the author's warning to read his systematic works as fast as
possible,
comme une fable or ainsi qu'un roman; 14 his metaphysics, in spite
of
the fact that they are the point from which all later thought proceeds,
are at once jejune and extravagant, without ideas and at the same time hyperphantastic; he
never, with
the single exception of the explanation of the rainbow, 15 followed up
and
worked out his discoveries to the end in a satisfactory manner: at one
moment he allows himself to be choked by empirical detail, in the next
he soars into hypotheses which in the plethora of artificially
interlaced
distinctions of detail are but ill calculated to further the strict
beeline
of investigation. We will not dispute with him about that, but far
rather
learn to recognise with Vauvenargues the fact that Descartes has often
seen right and guessed right, even where he was in too great a hurry to
press forward in the combination of hypothetic causes; ordinary
intellects
have nothing to fear from such mistakes, les esprits subalternes n'ont
point d'erreur en leur privé nom, parce qu'ils sont incapables
d'inventer,
même en se trompant. 16 Descartes himself, in his
wisdom, knew
full
well how that matter stood, and often gave expression to this
appreciation
in the words: “it is enough if I clear the road, you must do the rest“ 17 — and therefore I say
once more of him his work is of less importance
than the Man himself, or, as I said before, the Brain. We men are a
right
foolish folk: here is the one philosopher of all others, in whom first
and foremost personality in the very special character of its
intellect,
and only in the second place systematic doctrine, forms the driving
power
and the lasting interest, and yet it is in this very man that we have
allowed
personality to escape us! Still, in the after life of history certain
men
enjoy an inexpressible immortality: this Descartes possesses
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DESCARTES
almost more than any
other man; for the
thoughts which that brain thought, and even more than the thoughts, the
way and manner in which that brain grasped the chief problems of
existence, — what
therefore we must call the Manner of Seeing, the manner of directing
the
Eyes outwards and inwards, — all this has so penetrated, impregnated,
and
informed our philosophy and our natural science, that all of us, no
matter
to what school we belong, are compelled to weave the warp and woof of
our
thoughts in the loom of Descartes. Rightly did Huxley, one of the few
philosophically
trained investigators of Nature of the nineteenth century, remark: “In
all
thoughts which are characteristically modern, whether in the domain of
philosophy or in that of Natural Science, we find, if not always the
form,
still the spirit of the great Frenchman“; an acknowledgment for which
one
of the best authorities upon Descartes, Count Foucher de Careil, coined
the epigram, On se croit nouveau, on
est Cartésien.
It was
first and
foremost the
whole attitude of the intellect, namely the unconditional enquiring,
which
made epoch. Descartes' intellectual attitude is sceptical,
— but in the old
meaning
of the word.
For the verb skeptomai
originally meant to see, to contemplate, to
investigate,
later to ponder, to reflect upon. In the word sceptic in old days the
stress
was laid upon investigation and careful contemplation (Gellius called
the
sceptics quaesitores et
consideratores). The instinctive wisdom of the
language-forming powers united the perception by the senses with the
necessity
of exact careful investigation, but not with the meaning of doubt which
disintegrates everything, which arose in the decadence of Greek
thought,
and impressed a new meaning upon the word skepsis. The barrenness of
philosophical
scepticism is by its narrowed sense confined to logical functions: it
neither reaches outwards to
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DESCARTES
empirical Nature, nor
does it reach inwards
to confident self-consciousness; the outer Nature as well as the inner
essence should have taught the sceptics that that which is a matter of
fact does not necessarily hold its own before the logical Forum. The
ancient
scepticism arose out of shallow thinking, and led to frivolity, whereas
the scepticism of Descartes, on the contrary, means an awakening of
mankind
out of the sleep of dogma to free, enquiring use of the eyes. Descartes
did not doubt for doubting's sake, but, on the contrary, in order to
help
forward the discovery of a possible knowledge. Non que j'imitasse les
sceptiques,
... au contraire tout mon dessein ne tendait qu'à m'assurer,
et à rejeter la terre mouvante et le sable pour trouver le roc
ou
l'argile.
The old sceptics, however superior they might think themselves,
remained
snared in superstition up to their necks; while Descartes was in all
earnest
endeavouring d'entreprendre
d'ôter une bonne fois toutes les
opinions
que j'avais reçues jusques alors en ma créance.
Now if
Descartes'
doubts had contented themselves with leading us back to that perception
which he used to clothe in the words cogito,
ergo sum, or dubito, ergo
sum, or sum, cogito, sum
cogitans, and the rest, that of itself would
have
been something: Kant
calls him on that
account “a benefactor
of the human Reason“: but, in fact, this result of critical
reflection
simply means the solstice of the Cartesian method of thought: it
constitutes
the point where motion reverses its direction to cross over from the
negative
to the positive. The cogito, ergo sum
is a perception on the
boundary-line,
just as with Kant, das ding an sich
(“the thing in itself“) is a
conception
on the boundary line, and it is only fools who find a pleasure in
running
their heads against boundary stones of this sort. Descartes was no such
fool. On this furthest boundary line, upon the “rock“ of his search,
he
raised a church to the God without whom he could not live; to prove the
existence
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DESCARTES
of God is always a thorny
undertaking,
for He stands beyond the boundary of Descartes: yet this God not very
religiously
felt by Descartes, who had been educated in a Jesuit school, is less
pressed
upon us as something proven than made plausible as a necessary
assumption,
and has the one advantage that he is a God of truth. Descartes needs
Him
only in the interests of truth, in order that what is should be true,
and
for no other purpose. 18
And now the bold investigator addresses himself
to constructive intellectual work! He turns his back upon that boundary
stone, — in his church he only kneels now and again for short worship:
on
the other hand he enriches the world with thoughts which are so full of
life and freshness by reason of their visibility, that they have defied
all the storms of time, and he bestows upon it a wealth of perceptions,
which shelter such an inexhaustible symbolical store of truth, that,
while
reminding us of the oldest traditions of our race, they point to times
that are yet to come.
Pray
do not believe
that I am
using the language of hyperbole: my words are to be taken literally. As
examples I will cite a thought introduced by him into philosophy, and
an
idea introduced into natural science. Descartes' analytical reference
of
the united subjective and objective experience of man to the two
conceptions
extension and thought is an idea so simply
perceptible that it never
can
cease working productively: to this day all philosophers fasten on to
it.
They may use different wool and weave different patterns, still they
are
weaving at Descartes' loom — as I said before — all of them. On the
other
hand,
a conception like that of the imponderable matter filling the whole
universe,
the aether, is so rich in symbolical, thoughtful, creative power, that
it
is only now that, in the light of new discoveries, we are at last
beginning
to recognise its great fruitfulness. 19
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DESCARTES
In his work
on the
immortality
of man Herder remarks:
“It is incredible
how few
special forms
in the realm of thought and human activities appear when we put history
to the test. There are far fewer Regents who govern the world of the
sciences
... than Monarchs who rule over countries.“ There you have, expressed
in a short formula, the merit of Descartes. He is one of those
incredibly
few who produce special forms in the realm of thought — and here, since
an
exposition of the philosophy of Descartes would lead us too far, we
must
give up the enumeration of the special forms which he introduced: but
what
we must keep our keenest sight upon is the way in which this man,
receptively
and creatively, looked out upon the world, the way in which he came
upon “the special forms in the realm of thought.“ Let us now apply
ourselves
to this task.
I just
now praised
the great perceptibility
in Descartes' thoughts; at the same time I cited as an example his
theory
of the aether, an imaginary thing, which when we consider it more
nearly
defies all perceptibility. An exact analysis will convince us that, as
a matter of fact, there are two ways of showing this expression of
intellectual
satisfaction which in ordinary life we describe as perceptible
clearness; we are partly dealing with what is seen, partly with what is
thought.
The creative power of the informing faculty of sight, directed upon the
surrounding universe, was in Descartes of such rare might, that a
matter-of-fact
contemporary, the great mathematician Christian Huyghens, on receiving
the news of his death, exclaimed:
Nature!
prends le
deuil, viens plaindre
la première
Le
grand Descartes, et montre ton
désespoir;
Quand
il perdit le jour, tu perdis
la lumière,
Ce n'est qu'à ce flambeau que nous
l'avons pu
voir. 20
As verses these are not
worth much: but
coming from the pen of a Huyghens, they have more significance
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DESCARTES
inasmuch as this
investigator belonged
to the exactest of the exact. And as you hear, he maintains that the
sunlit
world was dark and unseen until Descartes lighted a torch over it, the
torch of thought. We men see nature all blurred, until clear
comprehensions
have reduced the chaos of perceptions to order. Our eye sees but dimly,
until the thinking brain has fixed it sharply, like an optician's
glass,
upon the objects in view. In another stanza of the same poem Huyghens
makes
use of a trope which by the direct opposite completes what he has just
said; for he says of Descartes that he
Faisait
voir aux esprits
ce qui se cache
aux yeux.
This implies that
Descartes gave visibility
to those things which our physical eyes indeed do not see, but which
our
understanding is compelled to think. And so as in the one case he
bestowed
thought upon things, so in the other he conferred upon thoughts the
representations
of the senses: in other words he gave them substance. In the one case
it
was the turning into thought that which had been indistinctly seen, in
the other the turning into something visually perceived an idea which
had
been indistinctly thought.
We
will at once
illustrate these
two sayings of Huyghens by examples. Descartes comes to the help of
perception
when he e.g. explains all the movements of bodies in heaven and on
earth
by the setting up of certain fundamental conceptions such as inertia,
mass,
and others; even these simplest phenomena we never knew how to observe
aright and see aright before the discovery of such ruling conceptions.
To such as these belongs his theory that the Sum of Motion in the
universe
is once for all immutable, a favourite assertion of Descartes which,
for
the first time, brings into the chaotic oscillation backwards and
forwards
and circuitously in the Kosmos, a thought reducing it to order, — a
thought
which, merely
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DESCARTES
amplified by an
additional sentence,
is the foundation of the modern doctrine of the maintenance of energy,
which is at the bottom of our whole science of physics. 21 That will
suffice
for one of Huyghens' affirmations: now for the other. Descartes comes
to
the assistance of thought through perceptibility, when for example he
starts
the theory of the above-named aether. This thought-picture leads us on
to look upon Light as the movement of an endlessly refined,
imponderable,
imperceptible matter, which fills the whole world, a movement which the
optic nerve betrays to us, without showing it, since, of course, aether
is not a thing perceptible and therefore real, but a symbol for
something
which is presupposed in thought, and undefinable. 22 Another example
would
be Descartes' doctrine that it is not the Eye but the Brain that sees;
all impressions of the senses are in the last instance invisible
motions
of imperceptible infinitesimal particles inside the Brain. 23 Here, in
the
case of the hypothetical aether, and in the hypothetical molecular
motions
of the substance of the brain, the visibility which has been acquired
in
what are matters of mere thought serves to a consequential observation
and concatenation of phenomena; true exact science of nature and of
mankind
first became possible by means of this and similar symbols.
Here
you have
obviously two different
intellectual gifts with which our philosopher is accredited, gifts
which
do not necessarily belong to one another, and both of which, if we see
them as purely and absolutely developed as they are here, at once
fascinate
us as something not easy of comprehension. Descartes knew how to give
intelligible
form to that which he saw, and at the same time possessed the power of
transforming that which was only thought into something visible: that
is
the fact to which Huyghens calls our attention. And here in very deed
he
goes straight to the core of the matter,
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DESCARTES
and for that reason his
remark must serve
us as a clue to the further analysis of this unique intellect.
In
order swiftly
and surely to
plumb the depths, I should wish to take the judgment of Huyghens which
I have already traced back to its simplest meaning and reduce it to a
still
more striking, concise, and purposely paradoxical formula. For it is
not
formula but phrases which are a hindrance to vivid insight, whereas a
true
formula serves as a skeleton round which the organs of the living
figure
by degrees arrange themselves. My formula runs thus: — Descartes'
distinguishing
gift was to make the visible invisible, and the invisible visible.
If you
look around
you in the
world of your own contemplative consciousness, you will soon observe
that
the degree of perceptibility of the ideas which fill it is exceedingly
various, and the same holds good of the possibility of conceiving them.
And you will soon be aware that there exists here a very complicated
interchange
of displacements, a mutual give and take. We possess thoughts with
hardly
a shadow of a perception, and we possess perceptions which are attended
only by just such a minimum of thought as is necessary for us to be
conscious
of those perceptions. Our daily life is made up in that way. Without
venturing
further I will only call your attention to one thing, and that is that
a thought that is accompanied by a blurred, hardly realisable
perception,
therefore an “invisible“ thought, can achieve but little, and that on
the contrary pure perception soon grows into something monstrous,
intractable,
inflexible, unless thought takes the pains to seize upon it and convert
it into something unseen. We are in no way embarrassed to find concrete
examples, we need only think of our two first lectures: It was by a
thought
and in the interests of a thought that Goethe brought together the
whole
incalculable mass of animal and vegetable forms into his idea of
metamorphosis: and so he breathed
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DESCARTES
the artist's soul into
what was a mere
brutal observation, furthering the investigation of Nature for all
time;
Helmholtz,
the physicist, rightly taught us that the powers with which
mathematical science deals cannot be “objects of the perception of the
senses,“ but only “objects of the comprehending understanding“; yet
Helmholtz,
in his work on optics, has none the less to take refuge in plain
diagrams,
first the wet thread, then the ray, which like the sailor swarming up a
rope, “produces itself along the particles of aether,“ and so he goes
on
from diagram to diagram because this thought of the “comprehending
understanding“
could not be realised and appreciated without a perceptible
representation.
This is the way in which we human beings, half unconsciously, are for
ever
changing the visible into the invisible — in order to see it better, —
and
the invisible into the visible, — in order to think it better. Kant,
from
his metaphysical eminence, has summed up what I am here only concerned
to show in a concrete and visible shape into the following pithy
sentence:
“thoughts without contents are empty; perceptions without
comprehensions
are blind. Hence it is just as necessary to make our comprehensions
perceptible
to the senses, as it is to make our perception intelligible, that is to
say, to bring it into subjection to comprehensions.“ Kant is here
speaking
of the common, unconsciously proceeding, necessary functions of all
human
reason from the moment that it enters into activity in the new-born
babe:
allow this reason to ripen to such an extent that it desires to build
up
for itself a science and a philosophy, and you will find this reason
standing
as conscious intelligence exactly where at its first awakening it stood
unconscious. Then it begins to take matters easily; it seems so natural
not to follow Kant's warning, but to be busy with empty thoughts and
blind
perceptions, that three-fourths of all philosophy from the earliest
times
to the present day
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DESCARTES
has never busied itself
with other things.
The writings of St. Thomas Aquinas, for instance, are an inexhaustible
arsenal of ideas, which are incapable of exciting the smallest
thought — mere
“blind perceptions“; and if you skip from the thirteenth to the
nineteenth
century, you will find that the most popular of all the more modern
systems,
that of Schopenhauer, takes as its foundation-stone a thought which is,
according to Kant, utterly empty, the one which it calls Will and
which,
according to its definition, is the opposite of an idea and
consequently
contains nothing capable of being in any way perceptibly understood.
All
such thought-structures are extravagance, not knowledge: Kant once
formulated
this very simply. “By mere perception without comprehension the object
is certainly given, but not thought; by comprehension without
corresponding
perception it is thought, but none is given: in neither case,
therefore,
does any recognition take place.“ How, on the other hand, perception
and
thought, the visible and invisible, go hand in hand towards the
building
up of systems of philosophy which explain nature, you may best see from
the histories of our natural sciences, the development of which was
conditioned
by this mutual penetration. Let us here pause for reflection.
Think
of how, at
the beginning
of the seventeenth century, Copernicus and Kepler are unravelling in
its
main features the course of the planets round the sun; from the leaning
tower of Pisa Galilei makes minute observations of the fall of
bodies, — instead
of merely reasoning logically upon it as all his predecessors had
done, — and
pursues his studies upon inclined planes; Descartes and others with
keen
intellect and patience follow up the mysterious course of the
Light-ray,
its curves, its refraction, its reflection; Gilbert publishes his
observations
on magnetism
... from all sides
there comes in
a stream of additional matter, — that is to say, material of
observation,
and in
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DESCARTES
every single sphere the
empirical investigators
are at work trying to the best of their ability, as Kant demands, to
make
their perceptions intelligible, that is to subject them to
comprehension.
Yet here we discover something over which we need not for the moment
break
our heads, but which we will simply accept as experience; namely that
thought
cannot directly fasten upon the perception of the senses, but must
first
with that intent create its own mental perception, — that which we call
Symbol
when we are wishing rather to bring to the front the perceptible side,
Hypothesis when we are dealing with the mental side. Thought must
create
unity: this is its special function: pure perception only gives a
kaleidoscope
of special cases. Therefore perceptible thought cannot proceed without
Symbol; it cannot, without further help, grasp, comprehend, and absorb
the material of perception: without Symbol it remains empty. I can have
no thoughts about the courses of the constellations, about the fall of
bodies, or about the essence of Light, unless I also possess, besides
the
empirical material, and for its amplification, a symbolical
representation
of what takes place in that connection, — in other words something
intermediate
between perception and thought. And here my intellect makes a further
claim.
Not only must phenomena, within the individual series of phenomena be
joined
together by means of symbols, but all the separate series of phenomena
with which I have become acquainted by means of empirical perception,
must
in addition be capable of being understood as one single comprehensive
unity. For as Kant will teach you later on, that which we call Nature
is
“the unity of the multitude of phenomena,“ as it is set forth as a
matter
of subjective necessity by our thoughts. It is impossible for me to
realise
a number of natures. The grouping of the planets round the sun, the
grouping
of the steel filings round the pole of a magnet on my desk must be
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taken as energies inside
one symmetrical
Whole. And here the great Descartes steps in as a creative power: he
produces
a new “special form in the sphere of thought,“ he changes into
visibility
that invisible something which our understanding insists upon though it
cannot perceive it, — he fills thought with contents: this he is able
to
do in that he sets up the perceptible hypothesis of a medium filling
space,
of a matter absolutely refined, invisible, imponderable, fluidly
moving — the aether, a symbol, the child of his phantasy. 24 At once all the
phenomena
mentioned enter the domain of demonstrability and so become accessible
to the constructive labours of thought: the aether carries and urges
the
stars in their courses, the aether as a driving mass becomes the
foundation
of the phenomena of gravitation, one set of movements of the aether
gives
birth to what we call the warming of bodies, another set to light,
others
to electricity and magnetism, and so forth. I refer you to my former
lecture
and am confident that this one example will show you with extraordinary
clearness what is meant by “making visible the invisible.“ At the same
time you will learn how indispensable perception is to thought, even to
the possibility of thought. Descartes had indeed by his hypothesis
poured
out such a wealth of visibility over the secrets of Nature, while he
Faisait voir aux
esprits
ce qui se cache
aux yeux‚
that the eyes of
men were
dazzled by
it. In those days neither the collected empirical material was
sufficient,
nor was thought adequately trained and refined to be fit for so grandly
simple a symbol for all the physical phenomena of movement of the
Kosmos.
Besides this Descartes in the closer elaboration of the matter had
fallen
into an error for which he was reproved by Goethe; “he attacks the
insoluble
problems with a certain hurry, and for the most part enters the subject
from the side of
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DESCARTES
the most complicated
phenomena.“ 25
There
is much that is artificial and arbitrary in the use which he makes of
the conception of the aether. The startling simplicity
of the general
conception
is marred
by all sorts of hazardous amplifications in detail. But it is just
here,
as is the case with every important man, that we learn how far
greatness
and limitation are set side by side, conditional and conditioned. And
so
it soon came about that Newton with his keen intellect, at once exact
and
barren of all imagination, once more seized upon the scholastic
fictions
of forces working at a distance, and took the old conception of Light
as
a special Matter: Newton's ideas are in the same relation to Descartes'
ideas, as those of a child to those of a man; and yet they corresponded
exactly to the requirements of empirical investigation in those days.
At
the present time, when new matter has been accumulated by the work of
centuries,
we are gradually going back to Descartes and his symbolical method of
thought:
in the case of the understanding of Light this took place about a
hundred
years ago with the introduction of the undulation theory mentioned in
the
last lecture; in the case of the electric magnetic phenomena about half
a century ago; physical experiments to explain gravitation as
conditioned
by the movement of aether, exactly as Descartes postulated, are the
order
of the day, 26 and
the great Hertz, so early torn from the world, was
possessed
in death by the dream of reducing “the putative working of the distant
forces to conditions of motion in a medium filling space.“ 27 Lord
Kelvin — and
following him many modern physicists, go still further and contend that
the various atoms which chemistry admits are only different gyratory
motions
of the one and only aether: that there must therefore be no such thing
as Matter, but Aether only: in this most exact method of investigation
the “Thing“ fades away, the Symbol alone remains. In a symbol so
solidly
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perspicuous is contained
the principle
of robust vital power.
So
much for the
explanation of
the transformation of the invisible into the visible. “Perceptions
without
conceptions are blind,“ says Kant. Even as I could not budge an inch in
the realm of thought unless I possessed a “reasoned“ perception, so I
must remain helplessly stuck in the quagmire of perception, unless I
should
have thoughts to drag me, as horses drag the cart, out of my
difficulties.
So be it. But how am I to obtain conceptions for my perceptions? Here
again
an intermediary something is necessary. Perception cannot directly
become
conception; the intermediary image is the Scheme. We men are incapable
of taking into our inner consciousness anything seen or in any way
perceived
by the senses, unless we have previously in our thoughts reduced it to
a Scheme. This is an aptitude which differs greatly in different
individuals; yet if a man were altogether unable to generalise, that is
to reduce
the many perceptions to few schemes, it would certainly be impossible
for
him to think; for, as Kant hits the point by saying, his perceptions
would
be blind; he would see, but not recognise. In the last lecture we saw
how
the great painters schematise: a purely perceptible scheme is still
sufficient
for their object; only a minimum of conception enters into it. In a
somewhat
different fashion, but in obedience to precisely the same universal law
of human reason, science goes to work. Whereas the painter wishes to
see
yet more clearly that which is already seen, and calls to his aid
conceptions
for that sole purpose, the investigator of nature wishes to conceive
more
clearly that which is seen, and to transform it into something known.
When
in this process of perceptible reasoning it is that which is
perceptible
which is preponderant, we speak of a Scheme; when, on the other hand,
it
is the element of thought which preponderates, we
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speak of a Theory. Theory
and scheme
belong to one another as Hypothesis and Symbol. Now we know exactly
with
what we have to deal; in order to obtain a concrete example, we must
return
once more to the seventeenth century.
This
time we must
work within
narrower limits; we will only take into consideration the works upon
the
visible movements of perceptible Bodies: for we shall busy ourselves
not
with hypotheses but with seen facts. Let us then confine our thoughts
to
the way in which some men in those days busied themselves with the
observation
of the movements of the heavenly bodies, and how others, — the immortal
Galilei
in the forefront, instituted eager experiments on the movements of
bodies
on our earth, that is to say, on the fall, the impetus, the rolling off
upon inclined planes, upon the trajectory of projectiles, upon the
communication
of motion from one body to another, and many other similar matters. The
physical acceptations of the ancients proved themselves to be utterly
false: new, accurately observed facts accumulated. How to order them?
How to
“make the perceptions intelligible“ ? How make what took place on earth
consistent with what took place in Heaven? the fall of the apple from
the tree with the circuit of the moon round the earth? Exactly as man
had
before, by submitting to thought the perceptible idea of the aether,
come
to the assistance of thought, so he had to act now in order to make his
perceptions visible and capable of being surveyed: he had to remove the
cataract from his eye, and that could only be by means of
comprehensions,
by referring all the single conditions of motion to a scheme which
should
be in accordance with rule, artificially thought out, and capable of
being
grasped logically; not given to him by the empirical observation of
Nature,
but set up autocratically between the eye and Nature by the King in
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his Castle of whom I
spoke in the first
lecture. Here again it was Descartes who laid down the principles of
our
modern theory of motion, and at the same time of our whole science of
mechanics.
All
movements of
visible bodies
may, as a matter of common knowledge, be referred to three fundamental
laws, which we usually call after Newton, because he was the first to
crystallise
them in words, and has developed them in all their sequence. 28 But the
third of these, which is not to be found in Descartes, is by universal
consent recognised as a formal amplification of the first, 29 and even
so
very disputable. 30
We have to deal therefore with two, not three,
fundamental
laws, and these two laws were not thought out by Newton but by
Descartes;
Newton took them over almost literally from Descartes, though the
latter
had not worked them up to such perfect refinement. 31 All that the
so-called
“first Law“ of Newton contains — that Rest and Motion are not
opposites,
but only conditions of a body, — that every body left to itself remains
perpetually
in its own condition whether of Rest or of Motion, — that the body
which
is set in motion, unless there be some hindrance, will continue to move
in a straight line with unaltered speed for all time, — all this stands
word
for word in Descartes. And I must call your attention to this, that no
single one of the thoughts uttered in this law is the result of
observation,
or even capable of proof by experiment. 32 The second law of Newton
too,
which treats of the mensuration and direction of the Motion which is
communicated
by one body to another, is contained without a single omission in
Descartes.
It is he then, and no other, who perfected this creative work of
thought.
But here again, as in the case of the aether, Descartes overshot the
mark,
and like Dürer in his doctrine of proportion, introduced
superfluous,
and even in the end false, matter, so that the sure tact of a Newton
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was sadly needed to
purify the core from
the slag. But the only thing that is of interest to us here, is the
fact
that Descartes, by the introduction of a few schematically theoretical
conceptions, contrived to unravel and so make available for mental
elaboration
that which winds itself round our senses from childhood, — that in
connection
with which the whole united antiquity never achieved clear ideas, —
that
which the great calculators and experimenters of the fifteenth and
sixteenth
centuries failed to set free from the entanglements of the whole
material
of perception; I mean the Phenomena of visible motion. Here again as
you
see is a “new form in the realm of thought.“ And here as in the former
case the value of such a creation for science and philosophy is
immeasurable.
For just as the symbolical hypothesis of aether paved the roads for
thought
upon which it was now possible to arrive at a rational appreciation of
the phenomena of light, of electricity, etc., by means of a visible
representation,
so in this case the setting up of a schematic theory of Motion based
upon
metaphysical conceptions allows us to range the over-rich mass of facts
seen into a few schemes of thought, where they can be guarded inclosed
in formulae. For there is the turning-point: since the Visible is as
fully
as possible, — in some lucky cases altogether, — transferred into the
realm
of the Unseen, of that which is as yet only thought, it possesses a
handiness,
a pliability, a movability, which otherwise are foreign to its own
perceptions, — purely
as such — and are dull, inert, awkward: they are, just as Kant taught
us,
blind,
and grope about in the dark; but as soon as the human understanding has
arranged them into comprehensible Schemes then it does with them as
seems
good to it, dissects a Whole into Parts, unites Parts at will, in short
behaves as it chooses: it is Lord in its Castle.
We
have now, as I
believe, made
an important advance in the understanding of the universal relations
between
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thinking and
seeing, — which collaborate
in so peculiar and twofold a combination for the building up of a
system
of philosophy, — as well as in respect of the recognition of Descartes'
special
aptitude for acting as intermediary between them. Our formula that
Descartes'
distinguishing gift was to make the visible invisible and the invisible
visible, is no longer a formula, but an Insight. But I cannot let the
matter
rest there. Kant's thinking is a pinnacle of the human intellect; no
man
can reach him who shirks the trouble of climbing. It is therefore
indispensably
necessary that you should yourselves now enter upon the region which
lies
between perceptive seeing (or the sensitive faculty) and the
understanding,
which binds together comprehensions:
otherwise you will
only
be possessed
of partial, not complete, distinctness.
Let
me, however, in
a parenthesis
introduce a short remark upon the subject of Symbol, Scheme,
Hypothesis,
and Theory. It is not a question of mere terminological clearness, but
of a visible representation, which will also be useful to you
philosophically.
The
Symbol, in
fullest acceptation
of the word, is the perceptive demonstration of that which is thought:
the Scheme, in its widest sense, is the rendering into thought of that
which has been perceived: the Symbol furnishes thought with a thinkable
perception; the Scheme furnishes perception with a visible thought.
Within
the symbol, however, it is possible to distinguish between a more
purely
perceptible and a more mental conception of the demonstration: the
result
of the first is the true Symbol, that of the second is the Hypothesis.
In the same way the Scheme splits up into true Scheme and Theory. From
this I draw the following explanatory diagram.
The
advantage of
this diagram
is that it accurately describes the mutual relationships of these
different
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DESCARTES
conceptions — that is to
say, if I may
so express myself, their mutual position in the Space of Thought. You
see
at a glance that if, on the one hand, Symbol and Hypothesis are
related,
on the other the relationship is between Scheme and Theory, while
Hypothesis
and Theory, Symbol and Scheme in the same way lie close to one another.
A very slight mental impulse suffices to turn a Symbol into a
Hypothesis,
and a Theory into a Scheme; it is a sort of swinging of the pendulum
that
our intellect
is carrying on the
livelong day without
paying attention to it. But even the boundary between Symbol and
Scheme,
as between Hypothesis and Theory, is not insuperable: a small change in
the standpoint suffices to give a colour of Scheme to Symbol, and a
colour
of Symbol to Scheme, and in the sciences Hypotheses have a way of quite
quietly, according to seniority, slipping into Theories. On the other
hand,
as regards the two pairs which stand crosswise to one another, Symbol
and
Theory, Hypothesis and Scheme, it is a matter of impossi-
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bility for them to be
changed into one
another. But what cannot occur directly may sometimes be effected
indirectly,
and so it often happens in the Natural Sciences that a Hypothesis by
degrees
acquires the value of a Symbol, becomes schematised, and at last stands
in all the dignity of a Theory. In the course of time that which is
really
only thought, and as such in a slight degree hypothesised, has managed
to assume the character of perceptibility to such a degree, that it is
conceived as practical perception, and is then converted into thought,
so that it takes the shape of a Scheme, and in the end of a full-grown
Theory. With the aether, for example, it is always the case, until
often
some new discovery suddenly reminds us that this idea only possesses a
symbolically hypothetical value; that is the way in which we men
befool
ourselves without any suspicion that we are doing so. The inverted
process
from Theory over Scheme, and Symbol over Hypothesis, which hardly
occurs
in science, is, on that account, common in everyday life. That which is
seen is converted into thought by Science, but the layman comprehends
scientific
schematic thought as true perception: indeed, we have heard a Helmholtz
talking of particles of aether “along which“ a Ray moves!
This,
however, is
only a side
issue. You must draw from it the one distinction between thinking and
perceiving
which is perpetually being forced to and fro in our brains. Perhaps in
addition to that the small artificial Scheme may render us good
service.
And
now let us go
back to Descartes.
From
the two
examples that we
have taken, aether and the laws of motion, you will perhaps already
have
begun to suspect that thought and perception are not merely
transiently,
but really and permanently divided from one another. A complete fusion
between them never takes place. There is never so much as an attempt at
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such a fusion. The world,
as we perceive
it by our senses, does not satisfy thought, and never has satisfied it:
for the world is incapable of thought, only our brain is that:
and so thought
creates
for itself a
Kosmos of its own, a special perception “converted into thought,“ and
discovers
at one time the Atoms, at another the Aether which the modern
science
of physics designates simply as “unperceivable matter and invisible
motion.“ 33
And yet thought does perceive the unperceivable because it wills to do
so; and thought sees the invisible because in no other way could it
build
a bridge by which to attain perception, or make a road by which to
reach
the dreams and works of Reason. We may grant that this aether, this
atom,
is something perceptible, indeed it is seen with all the special
intensity
of a dream-picture, and it is only thanks to this vision that thought
can
climb aloft. In spite of this the aether, like the Atom, is sicklied
o'er
with the pale cast of thought, and — again like a dream-picture, as we
advance
they retire and ever elude our grasp:
they are indeed
not
perceptions of the
senses, but perception that is thought:
a symbol is not a thing: the
man
who seeks to investigate aether and atom by perception, is tilting
against
something that does not exist. The analogy holds good with our
perception.
The schemes upon which we base our experiences in the matter of the
movement
of bodies have for their aim the transferring of these perceptions into
the domain of the comprehensible: here it is, and nowhere else, that
thought
like a mighty tree must carry and nourish the monstrous rootless liane
of empiricism that is “conscious of no bounds.“ In this case our aim
is
to convert what we have seen into a quantity, that is to say into
something
so far only thought; colour becomes a quantity of oscillation, and a
man
born blind can talk as much wisdom about it as a Titian.
But
should you not
yet be convinced
that it is the
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intimate laws of the
human intellect,
the fundamental facts of metaphysics, that are the informing power that
is at work here, — should you imagine that without calling to your help
metaphysical
discussions you can arrive at clear notions about Time and Space, and
about
Motion in space and time, I will instead of laying before you arguments
for which you are not yet prepared, address one request to you: I would
ask you to refer to the scholion on the eighth definition in Newton's
mathematical
principles of natural science. It is the man of distinctly
anti-metaphysical
principles who is talking to you, and that indeed in a work of
imperishable
importance. In the beginning of the passage in question he declares
with
disconcerting guilelessness — “Time, Space, Place, and Motion, as
matters
of common knowledge, I do not explain.“ 34 If the question were
merely
one
of dealing with the simple perception of these things, then an
explanation
of time and space would be as little necessary for the greatest
intellect
as for the most narrow-minded cow-herd. It seems to me that this
postulate
was altogether insensate: that which is self-evident cannot gain in
value
by explanation: on the contrary, it is out of the life that the word
comes.
Descartes' warning is: il faut
mettre au nombre des principales
erreurs
qui peuvent être commises dans les sciences l'opinion de ceux qui
veulent définir ce qu'on ne peut que concevoir. But there
is no
question of time and space, as they are known to all, — Newton himself
will
presently teach you that this would not lead us one step further in
Science, — but
with that intent it is our business to transfer that which is seen into
that which is thought, and vice versa, and so we arrive at inextricable
confusion until a critique of human Reason has illuminated us. Read a
little
further in Newton's scholion. You will find there things about
“absolute
space“ (spatium absolutum)
which are not less edifying than the
properties
of the absolutum quid of
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DESCARTES
the schoolmen. This “absolute
space is
without relation to any outward object“ (sine relatione ad externum
quodvis);
but there would be little to be made of a thing which stands in no
relation
to anything; therefore, in addition to this absolute space, relative
spaces
are assumed (in quantity), and these relative spaces are movable in
absolute
space of which they constitute the parts! I do not think that the human
intellect has ever attempted to imagine anything so monstrous as this
quantity
of spaces, which move about in confusion. It is true that these
movements
are only a passing idea such as might appeal to the intellect of our
aforesaid
cow-herd, for immediately afterwards Newton gives utterance to this
deep
reflection: “if the parts of space are turned out of their place they
are, so to speak, removed from themselves“; but even that will not do,
and so we receive the amplifying assertion about these relative spaces
—
“the spaces are their own places“ (spatia
sunt sui ipsorum loca). And
when
you are stuck fast in this utterly senseless empirical jumble, you are
taught that this space (of which you were told on the previous page
that
it is such a matter of common knowledge that it needs no
explanation), — is
beyond your ken, and that “you are not able to separate its parts by
means
of your senses“; and therefore, and here comes the gem of the whole,
since you are dealing with something not perceptible to the senses,
something
impossible of distinction, therefore, quoniam,
you must assume
perceptible
mensurations (mensuras sensibiles).
So with perception you are to reach
the invisible, and to measure something the parts of which you are not
able to distinguish! The cause of this confusion which could only be
cleared
up by the highest critical circumspection and the finest analysis, lies
in this, that mankind is not possessed of a clear appreciation of its
own
intellect: we interchange the Scheme which is only capable of being
thought
with the
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true perception of the
senses. There
in the case of aether (just think of the theory of undulation and its
powerlessness
in respect of colour) that which pertains to thought intruded into
perception
with disastrous results; or perhaps it would be more correct to
say, — since
the aether is, as you will remember, a thought converted into
perception — the
human intellect proved incapable of producing out of its own powers a
symbol
which should equal Nature: here, in the fundamental conceptions of
dynamics
as developed by Newton, the same intellect proves incapable of freely
discovering
thoughts in all portions, that is to say, of converting into thoughts
its
perceptions by the senses. In order to bring our perceptions under a
few
fundamental conceptions we invented the law of inertia: but the
thoughts
of absolute space, endless time, the uniformity of a body, which
according
to definition should be alone, and so removed from all comparison, —
all
this is not known to us by perception. From empirical perception we
borrow
that minimum of perceptions of the senses without which our theoretical
thoughts would be empty, that minimum without which the scheme could
not
be fashioned: but true perception never exactly tallies with this
theoretical
schematisation. And so we come to a standstill as soon as we in all too
great simplicity attempt to satisfy the human intellect without a
metaphysical
critique, although in practice all goes well enough, and a Newton
erects
a building worthy of everlasting admiration when once we grant him a
certain
series of premisses as unthinkable as they are imperceptible. 35
You
see from these
considerations
how important it is accurately to investigate the critical domain
between
perception and thought, and also how many difficulties throw us into
confusion
by piling themselves up against our understanding. Happily there is one
function of our intellect, one, only one, mathematics, which allows us
to
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DESCARTES
clear up this matter to
perfect distinctness.
One general explanation, and then I propose to start upon a discussion
of Descartes' relationship to mathematics: in this way we shall by
degrees
reach daylight, and we shall have no difficulty in seeing how all this
may be applied to the study of Kant.
I
propose here to
insert a diagram
which will serve as a pause, and give my words a really comprehensible
meaning. If we express the range of the human intellect by a
quadrangle, — a
circle would be better, and a globe of course still better — we can in
general
terms affirm that one half belongs to the senses, that is to say, to
perception,
to that which is perceived, the other to the understanding, thought,
the
formation of comprehensions; those are the “two quite heterogeneous
portions“
of which Kant spoke a while ago. A more minute consideration, however,
such as that which the history of our natural sciences has forced upon
us, will soon convince us that pure perception and pure thought are not
directly in contact, but that there is an intermediate domain which
serves
to help the crossing over of the one to the other. There are certainly
no fixed boundaries; we are not dealing with a machine the wheels of
which
simply lay hold upon one another, but with a living structure in which
every single organ in combination with all the other organs forms a
unity
at once real and ideal. Whereas in a watch the parts come first, and it
is only in the end that the watch as a whole comes into existence by
the
combination of the parts, — in a living body the Being itself is the
first,
and that which we are pleased to distinguish as parts or organs, is
formed
by degrees and has never more than a conditional importance in regard
to
the Being, since the division of the functions does not take place, as
in the watch, according to an immutable stencilled pattern, but one
organ
can even take up the duties of another. Still a Scheme will serve our
present
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purpose, and a Scheme is
only clear when
it is schematic, that is to say, absolutely quadrangular and
rectilinear.
So we will draw our quadrangle and assign one half to the Senses (the
Sinnlichkeit
of Kant) and with them to Perception, — the other half to the
Understanding
(as Kant calls it) with its conceptive Thought. But, towards the
middle,
pure conceptive thought crosses over to perceptible thought, and in the
same way, towards the middle, pure perception of the senses crosses
over
into thoughtfulness. This boundary land I will denote by hatchings.
You
have already
seen how the
understanding strove to annex into its own domain the visually seen
perceptions
in regard to Motion, and how with this intent it drew them over, not
without
violence, by the help of Schemes to its own special boundary land of
perceptible
thought; and before that you had seen how the senses had succeeded in
awakening
to a glorious life scientific thoughts which had up to then remained
unfruitful,
and when well considered generally unthinkable, by the means of the
discovery
of a sensible and perfectly perceptible Symbol, the aether.
The
slightest
reflection will
surely suffice to show you what a travelling backwards and forwards
goes
on within the human intellect. If, for instance, in our laws of Motion
stress should be laid only upon the theoretical and arithmetical, which
was the case with Newton the juggler in figures, then these laws end by
losing all perceptibility, they leave our middle line for the boundary
of the hatched part, they become altogether thoughts: but with
Descartes
in these very same laws of motion it was the conception of the senses
which
prevailed, and more recently with Hertz in the same way the
geometrically
perceptible: by those means the thought shifts towards the middle line,
that is to say, towards the Symbol, and Theory becomes relatively more
schematic
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DESCARTES
than theoretic. The same
thing takes
place with our thoughtful perceptions. They may belong so entirely to
the
senses, that is to say, they may stand so entirely on the edge of this
hatched region, so far therefore from the half assigned to the
understanding,
that comprehension is not in a position to grasp them. Goethe's
metamorphosis
is an example of that. Descartes' aether, on the contrary, belongs in
an
important degree more to the realm of thought, in spite of being still
quite concrete. The symbol of the aether can be drawn into itself from
the conceptive portion of
our being with
such violence that, as you have seen, in the end every concrete
conception
fades away, and aether subtilises itself into a motion as yet only
imagined,
dispensing with every perceptible, material foundation (see page 130).
In this case then not only is the middle line crossed, and the Symbol
turned
into Scheme, but this Scheme itself is as yet little more than Thought.
I commend to your understanding the Physics of Lord Armstrong and the
“Primitive
animal“ (Urtier) of Goethe as
the two most remote and most opposite
ends
of our “buffer state.“ In the one case a conception (the movement of
the
No-Thing in
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DESCARTES
empty space) which wipes
out all conceptibility
down to the uttermost remnant, so that it is impossible to think of it
any more; in the other case a thought (the original creator of all
individuals,
itself without any individuality) has so completely materialised itself
that there remains not even that minimum of conceptibility without
which
no form can be clearly recognised.
From
this
schematisation and this
warning against the misuse of the Scheme, let us now turn to
Mathematics.
The
characteristic
of the science
of Mathematics is that it takes possession of the “buffer state,“ the
hatched part of my diagram, and exactly fills it. Here is a case where
no scheme can be too uncompromising. Both the two forms of Mathematics
(on the one side the perceptible form of the science, — Geometry or the
doctrine
of Forms,
— on the other,
the
comprehensible form, — Arithmetic
or the doctrine of numbers) reach inwardly with exact precision towards
the middle line, that is to say, towards the boundary line between the
two domains of the understanding and Perception by the Senses. But
inasmuch
as mathematical science reaches outwards only exactly so far as the
boundaries
of this intermediate region, and does not cross it, so there arises
between
its two parts a reciprocal independence, an exact Parallelism which is
nowhere else to be found between perception and thought. That which is
thought mathematically contains nothing which might not also be
perceived,
and that which is perceived mathematically embraces no forms which
might
not also be grasped by thought. Here that unconscious shifting to and
fro,
of which we spoke just now, does not take place: every mathematical
conception,
every mathematical representation of ideas, has its appointed and
immovable
place. The two mathematical fields of intellectual operation are not
identical, — the
diagram shows how entirely autonomous they are, — and yet they are a
matched
pair, the one
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being the counterpart of
the other.
On the other hand, the sharp definition of the middle line conditions
such
an uncompromising antithesis of the two mathematical functions as
nowhere
else occurs between perception and the representation of conceptions.
Here
there is no such gradual crossing over as we found between other
Schemes
and Symbols. Geometry is pure Symbolism; the science of numbers is
Schematism
devoid of all perception, it is the prototype of what Kant called
“thoughts
without contents.“ 36
The conversion of the one into the other can only
be effected suddenly, and is, as I shall show presently in detail, the
result of a purely internal and arbitrary deliberation. Even where the
two parts of the middle line are very close to one another — I shall
give
an example immediately, — there are no means of changing form into
numbers
gradually; on the contrary, the concordance between thought and
perception
must be seen directly. If mathematics were not a purely human thought
and
perception, if we had to derive them from experience, as for instance
we
do our perceptions of the movements of bodies, then indeed we should be
in a bad case; for Nature, as outer experience, gives us no handle
whereby
we may bring form and numbers into connection. By good luck, however,
our
empirical shallow pates are at fault, and in geometry we possess our
archetypical
Symbolism, and in algebra our archetypical Schematism, and
therefore, — pray
note this therefore — since
mathematics are a form of thought and
perception
dwelling in us, and since they exactly fill that frontier domain of our
intellect, therefore it is here, and here only, that we are in a
position
to convert Symbol into Scheme and Scheme into Symbol in their absolute
entirety.
37
I
shall make this
conversion clear
to you by an example. When a boy receives his first instruction in
calculation
by letters (Algebra) the poor wretch is in the first place
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compelled to learn by
heart a series
of equations, of which he can make neither head nor tail, not because
there
is no food for thought in them, but because on the contrary they are
matters
exclusively of thought, since they deal with pure and therefore empty
(“void of contents“) conceptions, absolutely without any perception.
The
first of these equations runs thus:
(a+b)2=a2+2ab +b2.
That is to say, a and b
added together
and then multiplied by themselves equal a multiplied by itself, added
to
twice the product of a
multiplied by b, added to b multiplied by
itself.
Is not that a terror to listen to? But if we take heart, and jump out
of schematism into the symbolism of our intellect, we immediately see
the
truth of the proposition, without wasting a single thought on the
matter.
Let me show the thing in a diagram, only begging that you will not
exercise
thought upon it, but just simply open your eyes.
We
take a line a
and add to it
in a straight line the line b.
And now upon this line we
build an equilateral
and right-angled quadrangle.
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What you see here is
(a+b)2. That this
square is equal to the square on a
increased by the square on b,
increased
by twice the right angle which consists of the length a and the breadth
b, you will see directly from
the following construction which I build
into our quadrangle.
In this way Algebra has
been converted
into Geometry, the scheme of numbers into a Form-Symbol. And you need
only
invert this simple example, that is to say, think of the square and the
construction introduced into it as the starting-point, in order to
understand
that it must of necessity be possible to convert every geometrical
construction,
every play of constructive phantasy, into a purely comprehensible,
entirely
perceptible, in other words algebraical, expression of figures.
In the
case which
we have just
been talking about mathematical perception and mathematical thought
were
in close proximity to the dividing middle line: there was therefore no
difficulty in grasping the comprehension as material, the perception as
abstract: generally, however, they are far removed from that line, and
it was Descartes who first taught us how we must set about in order to
succeed in revealing the Scheme as Symbol, the Symbol as Scheme, a
discovery
by means of which he became the founder of the so-called higher
mathematics.
And
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here it is that we must
now follow him
if we wish once for all to ascertain the relationship between thought
and
perception, which is indispensable for any understanding of Kant. 38
The
whole course of
our considerations
up to the present will easily convince you what a special attraction
mathematics
must have exercised on a man like Descartes, on a man whose
distinguishing
gift it was to discover Symbols and Schemes, in other words, to make
the
visible invisible, and the invisible visible. Yet, if we wish to
understand
Descartes' personal method of perception, it is important that we
should
be accurately instructed as to his position in regard to mathematics,
and
that is just where our school-books lead us astray. In order,
therefore,
to be able to speak of Descartes' mathematical achievements, my first
business
must be to dispel the common, and almost without exception ruling,
misunderstanding
about Descartes' conception of mathematics, and about the place which
they
occupy in his whole thought. This is the only way in which we can
extricate
ourselves out of the jungle of meaningless phrases into the free Pamir
of clear insight.
In our
scientific
knowledge of
Nature mathematics play the part of the mechanism which electric
engineer
call a commutator or current reverser. As soon as we succeed in
arriving
at phenomena, — even should it be in so arbitrary and contradictory a
way
as was the case with Newton in his doctrine of gravitation, — the game
is
won; we go on turning the current, i.e. the perceived into the
conceptible,
and the conceptible into perception, exactly as in the (a+b)2.
The
one
helps the other forward, and so we are ever rising higher and higher,
and
that without ever falling into error, for the simple reason that we are
only working within our own intellect, and so make images and thoughts
take their proper places in regard to one another. That was what
Descartes,
after
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DESCARTES
Plato, was the first to
see; he it is
who endowed us with the thought of analytical geometry, with which we
shall
immediately busy ourselves more closely: yet he did not remain caught
in
the meshes of purely mathematical ideas, but his masterful intellect
stretched
out far above the science of mathematics. If it is absurd to follow
Schopenhauer
in representing Descartes as undervaluing mathematics, so it is hardly
less full of misunderstanding and misleading to exaggerate the
significance
of mathematics in his thought and for his philosophy. The image of the
aether and the thought of the law of Inertia
are sufficient proof that
his
development of this mathematical juggling only served him as a
preliminary
exercise, and so he holds that it is to be understood by others, as his
Règles pour la direction de
l'esprit clearly set out. The
doctrine
of numbers and forms does not contain truths; rather is it in one
respect
quite empty, the emptiest thing that one can imagine: for in it neither
is perception nourished by comprehensions from outside, nor do its
conceptive
gymnastics allow of enrichment by special thoughts; mathematics are
simply
a system of formal principles of perception and the concatenation of
conceptions. 39
Descartes is continually laughing at the professional calculateurs and
géomètres, and
says that their business is de
s'occuper
de
bagatelles. Open any work on philosophical history, you will
find
everywhere
that Descartes declared that mathematics are the “origin and source of
all truths.“ Nothing has done so much to turn good brains amongst us
from
Descartes as this reputed saying. For what is one to think of so silly
an assertion — at best a sort of mythical Pythagorean symbol of Nature,
something
which was in truth further from this man than from all others? And yet
no man doubts the authenticity of the position, otherwise it would not
be quoted with the usual inverted commas in one learned German work
after
another, and the whole
246
DESCARTES
thing is just a matter of
mistranslation.
The passage in question occurs in the XIth volume, p. 219, of Cousin's
edition. Descartes has just set out the first principles of his Method,
which he reduces to two principles only:
first and
foremost, and
as indispensable,
the clear perception of the object (l'intuition);
next, as second, the
consistent and unbroken deduction of the propositions (la
déduction).
Here the perception of the senses and understanding appear in their
first
and most elementary relationship. 40 Still their reciprocally
conditioning
interplay cannot but lead us much further. So Descartes points to
Mathematics
as an example, and as the only safe schooling for the application of
this
quite universally adopted Method, — mathematics which he holds to be
incomparable
and indispensable as an exercise of the alliance between the most
manifest
perception and the strictest logic — and then comes the sentence which
has
given rise to the misunderstanding to which I have alluded: je suis
convaincu
qu'elle est supérieure à tout autre moyen humain de
connaître,
parce qu'elle est l'origine et la source de toutes les
vérités.
The pronoun elle refers to
the Method, the great universal Method, the
Method of the reversion of the current, — not to Mathematics! The
Method
of the reciprocal interpenetration between perception and thought is
the
source of all true knowledge — this Method! In no way mathematics by
themselves
and of themselves, of which Descartes on the following page assures us
that there is nothing more empty. Rien
de plus vide. Even as a matter
of
grammar the thing is out of court. Elle
could not refer to mathematics
which are almost always spoken of in the plural, and in this very
passage
are without exception given as les
mathématiques and elles.
How
little Descartes was inclined to look upon mathematics as the “source
of all truths“ is sufficiently manifest from the fact that he reckons
les nombres et les figures
among those ideas qui ne peuvent pas
être
estimées
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DESCARTES
un pur
néant,
quoique peut-être
elles n'aient aucune existence hors de ma
pensée, and that
in another place he says of them, elles
ne peuvent pas étre
considérées
comme des
substances, mais
seulement comme des
termes sous lesquels la substance est contenue. 41 But that is the way
in
which we treat our great men; instead of adopting an infinitely subtle,
vivid, pregnant knowledge, we accredit the genius with any manner of
patent
absurdity at which every commonplace man runs a tilt with solemn
self-satisfaction. 42
Just as little truth is there in the affirmation that Descartes taught
that philosophy was destined to become a “universal system of
mathematics,“
an affirmation which we in the same way meet everywhere. He, on the
contrary,
called attention to the fact, as Plato had already done, that in a
series
of Sciences, — he mentions optics, astronomy, mechanics, acoustics,
everything
must at last come to a question of mensuration and figures, and this
remark
leads him to the affirmation that all these sciences in combination
with
geometry and arithmetic form une
science mathématique en
général,
or une science mathématique
universelle. But this description
holds
good only in contradistinction to the other sciences, and so far from
saying
that the universal science of mathematics is all-embracing, Descartes
asserts
expressly, “I have busied myself so much with it that I think that I
may
henceforth devote myself to higher sciences, without having to fear
being
over-hasty.“ Descartes would have agreed with Kant, “Philosophy makes
use
of mathematics only as an instrument.“ For the rest he himself clenches
the question into a convenient and correct formula when he says, “In my
method the science of mathematics is the husk and not the core.“
It was
indispensable to replace
a conception that is meaningless and false into the bargain by a true
appreciation
of Descartes' conception. So much for that. There is only one more
thing
which ought to be brought out in
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DESCARTES
this connection, and that
is the strong
insistence which he lays upon perception as the source and fountain of
all truths, for that is the true conception of Descartes' teaching. It
would be quite imaginable that a philosopher might have set up this
“mathematical
method,“ and yet have taken the abstract side as his starting-point.
Descartes
did not do that. On the contrary, just as in mathematics he takes his
stand
upon geometry, so he consistently insists that perception (l'intuition
as he calls it) is the one and only indispensable foundation of all
knowledge.
What he prizes above all in mathematics is that “they exercise the
phantasy
in the right conception of forms and motions, and so accustom us to
represent
phenomena to ourselves correctly.“ 43 It is not the least of
the
achievements
of the pioneer that he introduced the principle of perception into
philosophy
in the stead of the method of tyrannical and sterile logic which up to
his time was alone dominant. It you read the writings of Descartes, you
will at once be struck by the frequency with which such expressions as
voir clairement, concevoir fort
clairement et fort distinctement,
imaginer
clairement, la conception évidente d'un esprit sain,
etc.,
occur:
the foundation-stone upon which the whole of this philosophy rests, is
simply clear perception, and so it is that the first power of man which
must be methodically developed, is la
perspicacité en
envisageant
distinctement chaque chose, which means, “the piercing glance
which
shows
itself herein that we should see everything clearly.“ Yes! but
“perceptions
without conceptions are blind“; it is conceptions that first make
them
intelligible. Thus it is that in Descartes the algebra of
déduction
follows upon the geometry of intuition,
and that the sagacité
à
observer rigoureusement l'enchaînement des choses follows
upon
perspicacité. It is
characteristic of geometry that by itself it does not carry us
very
far. It is true that a carefully planned geometrical
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DESCARTES
construction contains all
the connections
which may later be drawn from it, still the eye is clumsy and confused,
and the more we succeed in converting that which is seen into that
which
is thought — in this case connecting forms into symbols of figures, —
the
richer
will be the results. This experience drawn from the practice of
mathematics
was applied by Descartes to all other spheres of thought, exacting that
we should first see clearly, and then dissect with flawless logical
keenness.
Without a brilliantly powerful perception of the material empirical
world,
no true knowledge, — nothing but cobwebs of the brain! Without an
“algebraically“
dexterous analysis of that which has been seen clearly and lightly, no
true science, no philosophy! It is always the same principle: the
interplay
between understanding and the senses, between conception and
perception,
between Scheme and Symbol. And of all importance is the doctrine that
perception
always takes the lead, while logical dissection exclusively comes into
play in the second place. Pure intuitions of reason and pure logical
arguments
have no value for Descartes; they are objectless. In contradistinction
to the schoolmen not only of his own time, but also of the nineteenth
century,
Descartes declares roundly, “logical forms and syllogisms are of
absolutely
no use for the discovery of truth,“ — “Dialectics are rather a
hindrance
than a help.“ They can only play a part secondarily, — only in the
analytical
investigation of that which has been discovered by direct and
experimental
perception. 44
That is what Descartes understands by his
“mathematical
method.“
Fundamentally his attitude towards mathematics is precisely the same as
that of Plato, who had already suspected and preached the intermediary
position
of mathematics, and on that account ascribes to the exercise of
mathematical
methods an incomparable significance for the development of the power
of
knowledge, but nevertheless laughs at the professional mathe-
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DESCARTES
maticians when he says,
“they make themselves
ridiculous with their fussing, as if with their complicated
calculations
and barbarous terminology they were achieving some mighty thing,
whereas
the whole significance of mathematics lies in the fact that they serve
as a medium of philosophical thought and as a road leading to
knowledge.“ 45
Descartes was conscious of this historical connection. According to him
the thinkers of antiquity would have found it impossible to recommend
mathematics
as a philosophical instrument, if by them they had only understood
calculation;
he was more inclined to believe qu'ils
reconnaissaient une certaine
science
mathématique différente de celle de notre âge,
and
it was this other science of mathematics which he once more took up.
I
think we have now
quite intelligibly
shown how there is no inconsistency in Descartes when he at one and the
same time declares that there is nothing “more empty“ than mathematics,
and in spite of that holds that the philosopher is bound to spend much
time over their study. And since you now know that when he busied
himself
with mathematics it was not on account of any formal whim, not on
account
of any Pythagorean cobwebs of the brain, but on the one hand in the
interest
of the precedence of perception over thought in every investigation of
nature and mankind, and on the other hand, in the interest of the
conscious
handling of that method by which perception and thought reciprocally
help
one another. Since you also are in possession of the comforting
assurance
that it is no barren philosophy, but scientific and living perception
of
the world that is at work here, so I hope that you will have the
courage
to climb one last rocky peak with me where the sharp pure air of the
glaciers
will be wafted around us. If Descartes has by others been
misunderstood,
and has remained unrecognised, there is one act of justice rendered to
him
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DESCARTES
by every cyclopaedia.
He
is the first
inventor of analytical geometry, with which he revolutionised our whole
doctrine of geometry and numbers, and gave the impetus to the discovery
of the so-called higher mathematics, upon which again our modern
sciences
of Physics, Mechanics,
and Astronomy are
based.
It is now necessary
that you should
see
Descartes, — who made his
discovery not as a
professed
mathematician, but
as an amateur after a few
months of
self-taught
studìes, — at
work in this direction; the detestation in which we hold all verbosity,
should
steel you not to
rest
before you have
grasped in its solid significance the question which lies at the bottom
of our observations
of
to-day. I admit
that we shall here have to tread the special path of mathematics, and
that
is distasteful to
the
man who is
no mathematician; yet I hope we shall succeed in applying ourselves to
the subject in such a way that even those who are absolutely ignorant
of
mathematics will be able to see exactly what we are driving at. And
with
this we shall in the first place gain the advantage of obtaining a
quite
exact idea of Descartes' individual method of Seeing: in the second
place
we shall gain the knowledge, not merely theoretical but absolutely
concrete
resulting from practical perception, that every transition from thought
to perception and vice versa, — even where (as in mathematics only) it
takes
place with absolute precision — has in itself something artificial and
arbitrary,
from which it results that perception which is thought always remains
more
or less a Scheme, and thought which is perceived always remains more or
less a Symbol; last not least, we shall be driven on a purely
perceptible
and therefore entirely safe road, to the very central point of that
Kantian
perception to which it is otherwise so difficult to gain access, and
which
is so dark and difficult to illuminate. That point is the conception of
the Transcendental. In this way Kant's method of Seeing the world will
no longer
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DESCARTES
be so foreign to us, and
we shall have
gained in addition an advantageous standpoint for a later study of his
philosophy. For these reasons I urge you to follow me for a while in
the
pure domain of mathematics. 46
* * * * * *
In order that you
may make your
way with some pleasure into the subject of analytical geometry, which
touches
the innermost essence of mathematics, I must at the outset tell you
what
was the aim of this discovery of Descartes. It is necessary that you
should
know this, otherwise you would see nothing but a sort of ipse dixit in
the proceeding, and that might mean astonishment, but it could not
mean
understanding.
With
the help of
perceptible mathematics,
namely geometry, simple problems may be solved, but not complex
problems;
human imagination soon gives up the task: a very complicated system of
lines and points and bodies, which assert themselves in various ways,
is
something which we cannot put with perfect clearness before our eyes;
we
are not in a position to compare differently formed bodies directly
with
one another; we are not able to see, to recognise with our eyes, the
outcome
of it all. But in a quite different measure we are able to deal with
the
mathematics of conception, that is to say with numbers or the symbols
of
numbers; for in this case the master law-giver is not perception, but
Logic,
and that implies the opportune succession of a linked chain of insight
into facts, instead of a Present only to be deciphered by a direct and
simultaneous combination. If we deal with numbers logically we need not
trouble ourselves about the perceptible meaning of each single
operation
of calculation; the correctness of the result is the important matter.
That is why men very early came to reduce lines and rectangular figures
to numbers, as, for instance, expounding the relationship
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DESCARTES
of the square on one side
of a rectangular
triangle to the squares on the two other sides, not perceptibly by
drawing
figures, but arithmetically and algebraically. But how arrive at a
universally
valid expression in numbers for complex figures, such, for instance, as
curves? That was the question upon which many men busied themselves,
and
no one found the solution.
Here
it was that
Descartes came
to the front as a creative genius. He perceived that to reduce a curved
line to a symbolical expression in numbers, the first necessity must be
to bring the particular curve (circle, ellipse, volute, etc.), into
relation
with straight lines. The next task to be solved was the discovery of
these
straight lines. Once solve that difficulty and discover the relations
between
the curved line and the straight line, then what was elusive would be
brought
to a standstill, the curve would be bent straight, and the object would
be attained; for as you will see presently, straight lines can always
be
considered as numbers (real or symbolical), and a fixed relationship
between
straight lines is therefore at the same time an arithmetical
relationship.
Thus the curve which is seen, becomes an unseen, logical, arithmetical
expression, and can take its place in every arithmetical series by
means
of various calculations. In this Descartes succeeded. With simple
unconsciousness
of the magnitude of his achievement the first sentence of his
Géométrie
tells us: Tous les problèmes
de Géométrie se
peuvent
facilement réduire à tels termes qu'ils n'est besoin par
après que de connaître la longueur de quelques lignes
droites
pour les construire. As coins and watches disappear in the hands
of a
conjurer,
so in the hands of Descartes the visible became invisible, the
geometrical,
arithmetical. But you will at once remark that with this achievement
the
inversion was of necessity given at the same time. For it was only
necessary
to strike into the opposite direction, and at once we were in
possession
of a form
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DESCARTES
for every futile
arithmetical formula! Here you have the whole Descartes. Goethe
declares that man can wrest
from Nature nothing more valuable than —
Wenn sie
ihm offenbare,
Wie
sie das Feste
lässt zu
Geist verrinnen,
Wie
sie das
Geisterzeugte fest
bewahre —
“When she reveals to him how she lets the substantial lapse
into
the Spirit, how she preserves as substance that which is the child of
the
spirit.“
Since
Descartes has
pervaded the
life of man as teacher, there has been no geometrical form which we
have
not been able to let “lapse into Spirit,“ that is to say, turn into an
arithmetical expression, — into an equation merely thought, — no
arithmetical
picture “child of the Spirit“ which we have not been able to convert to
something seen, something substantial. That is the essence of
analytical
Geometry.
Now we
may proceed
to a closer
exposition.
I hope
that you are
not scared
either by Greek words or by the jargon of mathematics. Both are
accessible
if you only approach them in the right spirit. Greek was once spoken in
a sunny land, — spoken by men who possessed the immeasurable luck not
to
be forced, as we are, to gag their spiritual life into dead idioms, —
men
among whom the sage drew his words from the same living well as the
shepherd,
and so was understood by all:
and as regards
mathematics this discipline
by the application of the right method, was capable of being brought
home
even to the least gifted, — at any rate in a certain measure, — for
mathematical
ideas are common to us all, and in their essence elementary. La
facilité
suprême is what Descartes praises in all true mathematics.
Analysis comes from αναλύείν,
a word which means to unloose and also to set free: it signifies
therefore
the unloosing of a single perception into simpler component parts, —
the
setting free of the elements out of a com-
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DESCARTES
bination. That is why the
resolution
of any body into materials which are not capable of further
disintegration
is called “Analysis.“ In mathematics the word implies in the same way
the
disintegration of a given proposition into its component parts. You
will,
however, at once meet me with the question, How can one disintegrate
figures
into component parts? To represent to myself 70 as 10 times 7, or as 58
plus 12, or as 210 divided by 3, is a purely arbitrary proceeding in my
brain. The number 70 or 7000 or 7,000,000 is just as simple and just as
impossible of disintegration as 7 or 1. Certainly; and yet it is just
as
capable of disintegration, for the number 1 is capable of
disintegration
ad infinitum so soon as it
pleases me to look upon it as a product. The
same holds good of figures; a circle is a circle, a globe a globe, a
pyramid
a pyramid, each positively a symbol of unity: still I am able to
imagine
the globe as actually consisting of segments which have grown together,
as in the case of the orange, and in accordance with that I am also
able
to take it to pieces. I can think of the circle as a line rotating
round
one of its extremities, or as a variety of an ellipse, or as a slice
taken
out of a cylinder or a cone, or as the place of an endless number of
coincident
equilateral triangles with the same vertical point, and in fifty other
ways besides. In this way the structural unity is at my bidding set
free
into multitude. I find myself within the domain of pure human will.
Here
there is no such practical concrete analysis as there is in chemistry,
where by mechanical methods of attack I can resolve a combined body
into
several qualitatively different component parts, nor is there any
operation
analogous to philosophical “analytics,“ in which complicated ideas and
conceptions are reduced to the elements of which they are composed: but
mathematical analysis is the autocratic setting free of a given
magnitude
into several other magnitudes for purely practical reasons, in
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DESCARTES
order, that is to say, in
that way better
to calculate, and this end is attained as soon as the original idea in
space has been reduced to an expression in which there is neither
space,
nor possibility of representation, — an expression which is in accord
with
numbers. In a wider sense the converse process belongs also to
mathematical
analysis, — the construction of a superficial image or of a solid body
out
of a combination of numbers. It had already occurred to the Greeks of
later
times to transfer to the realm of numerical calculation geometrical
problems
which it was difficult or impossible to solve by direct means. But the
next point was one which they did not attain, for it was contrary to
the
genius of that people to convert the visible into the invisible, and
therefore
they made no great progress in that direction. In contradistinction to
the Greeks the Aryan Indian achieved his best work in the logical
calculation
of conceptions (arithmetic and algebra): but he lacked that
geometrical
eye which is dominant in matters of form. It was the Teuton who was the
first to possess the right intellectual aptitude for this twofold work,
and Descartes was the one and only man who stood so exactly upon the
boundary
line that, without being a mathematician, and after a short period of
study,
he by pure instinct forced the door through which hundreds and
thousands
dashed after him. Car en
mathématiques — he says in the last
sentence
of his Géométrie —
lorsqu'on a les deux ou trois premiers
termes,
il n'est pas malaisé de trouver les autres. Et j'espère
que
nos neveux me sauront gré, non seulement des choses que j'ai ici
expliquées, mais aussi de celles que j'ai omises volontairement,
afin de leur laisser le plaisir de les inventer.
Descartes, as I
have said, set
to work with the utmost simplicity. He was in the twenties, and an
officer;
in order to fill the leisure of winter quarters, and because he had
remarked
that the study of the mathematical sciences is of incomparable
methodical
value (elles accoutument
257
DESCARTES
l'esprit à se
repaître de
vérités), 47
he undertook to take a bird's-eye view of
this
discipline. But he had always abominated numbers, the wading about in a
sea of endless
calculations; pour
ce qui est
des nombres je n'ai jamais prétendu d'y rien savoir, he
writes
to
his mathematical confessor Père Mersenne; he belongs to the
open-eyed division
of
mankind; mathematics
are for him the science of forms and motions: his repugnance to
arithmetic
is so strong that it is only geometrically that he establishes all its
operations, addition, subtraction, division, multiplication, even the
extraction
of Roots. Toutes ces
opérations doivent être
ramenées
à l'examen de l'imagination, et il faut les figurer aux yeux,
pour
ensuite en expliquer l'usage et la pratique. 48 But Descartes soon
remarked
that mathematics as taught by the professors are a prosy, dull affair,
a compound of many parts: he despaired of learning them in this way,
and
of making them into a living knowledge. And yet, he said to himself,
all
these branches of mathematics deal with the same thing, the relation of
magnitudes to one another: a plague upon all their Geometry and
arithmetic!
I shall henceforth only fix my eyes directly upon these relative
magnitudes:
je pensai qu'il valait mieux que
j'examinasse seulement ces proportions
en général.
Of
course you
understand what
he means by the word proportions.
It may be a matter of comparison of
absolute
magnitude between similarly formed bodies; that is the simplest case,
and
always without more ado to be referred to the difference of numbers, in
other words to arithmetic; but the comparison may also be the relation
to one another of different forms, and this is what is so actively
present
to Descartes. In this case it is not a question of whether a thing is
great
or small, but of the forms which are possible to our phantasy, seeing
that
each is different from the other, and without the absolute magnitude
coming
under observation. The circle is a form differing
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DESCARTES
from the square, it is
also a form differing
from the ellipse or the volute. The same holds good of a globe, a cube,
a pyramid. They are different forms. Humanly speaking each one of these
forms is subject to a special law, or if you prefer it a special
thought,
and this thought is a fixed relation to the extension in space towards
the various directions. Goethe says of the perception of natural
objects,
“There is in the Object something of an unknown law which corresponds
to the unknown law in the Subject.“ 49 In geometrical forms we
supply
both
object and subject, and thence the idea of law is at once a matter of
compelling
strictness and of boundless elasticity. A ball of the size of the
planet
Jupiter is, as a matter of thought, as like a billiard ball as two
hairs
are to each other, for the reason that the relative proportions are the
same: the comparison of the two is exhaustively expressed by a simple
arithmetical
proportional equation
a : b = 1 : x
a, the billiard ball, is
in relation
to b (Jupiter) as 1 to x (the requisite number). But, on
the contrary,
we are dealing with a quite different sort of comparison if I hold the
billiard ball and the cue together, and do not wish simply to establish
the relative bulk of the two, for which a pair of scales would suffice,
but to bring into comparative relation the law of form of the one, and
the law of form of the other, that is to say, reduce them to an
equation
of mutual relation. All this, the interrelation of magnitudes and the
interrelation
of forms, is what Descartes speaks of as proportions. What he asserts
is
this, that the different branches of mathematics ultimately ne
considèrent
autre chose que les diverses proportions qui s'y trouvent. And
inasmuch
as the monotony of mere arithmetic is repugnant to him, and he is
wearied
by the calculations incident to the geometry of solids, he just asks
himself
one question: What will be the simplest
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DESCARTES
way for me to compare
different forms
with one another? The answer is, by reducing the problem to one of the
relations of various straight lines to one another, à cause que
je ne trouvais rien de plus simple ni que je pusse plus distinctement
représenter
à mon imagination et à mes sens. You see then the
first
and
constant thing postulated is something which can be represented,
clearly
represented, obvious to the senses. Yes. But supposing that I should
analyse
many forms in this way, I shall obtain a whole forest of lines. How can
I carry them in my memory? And how can I continue my investigations of
their reciprocal interrelations? For this purpose the lines must be
reduced
to an expression in numbers: il
fallait que je les expliquasse par
quelques
chiffres, les plus courts qu'il serait possible. And he ends by
saying,
je pensais que par ce moyen
j'emprunterais tout le meilleur de la
géométrie
et de l'algèbre, et corrigerais tous les défauts de l'une
par l'autre.
You
now know how it
was that the
idea of analytical geometry arose in the intellect of Descartes, and in
what form it floated before him. It contains absolutely nothing
abstruse
or learned which need scare us laymen. On the contrary, it was in
direct
opposition to the professorial men of science that Descartes invented
his
method, and in answer to a friend who communicates to him attacks from
all sides on his geometry, he writes, J'aurais
mauvaise opinion de mes
pensées, si je voyais que les doctes les approuvassent. I
imagine
that you will already have remarked what is the turning-point of the
whole
method. It is the establishment of the line as intermediate between
form
and Numbers. And this means exactly: it is the discovery of that point
wherein the doctrine of magnitudes, sense and understanding, perception
and thought merge into one another, where the visible becomes
invisible,
and vice versa. In a late work, which unfortunately remained
unfinished,
the Règles
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DESCARTES
pour la direction de
l'esprit, Descartes
very clearly laid down this mediatory principle of his: par les lignes
il faut représenter tantôt des grandeurs continues
(i.e.
forms)
tantôt la pluralité et
le nombre; l'industrie humaine ne
peut
rien trouver de plus simple pour exposer toutes les différences
des rapports; so the relation between the straight lines stands
exactly
in the middle, pointing on one side to the visible form, on the other
to
the essence of abstract numbers.
We are
now
sufficiently equipped
to start upon the concrete observation of analytical geometry. But I
have
to insist that what follows must be treated as a series of ideas
without
your ever for a moment being contented with thought alone, as apart
from
ocular demonstration.
Surely
it is plain
to the eyes
that I can, if I so choose, conceive every straight line as a number?
For
example, if three straight lines stand in relation to one another as 5,
4, and 3 (it is immaterial whether we are speaking of yards, or feet,
or
metres, or miles), I can call them simply 5, 4, and 3, and so calculate
with them: every builder does that daily, and that is really
geometrical
analysis, for it is the conversion of a conception in space into a
number
which has nothing to do with space. But your builder now goes a step
further.
Supposing that the line 5 represents the one side of the house which is
to be built quadrilaterally and that the builder wishes to know the
size
of the area for his work, there is no need for him to measure it with
his
measuring tape, nor to set it out on paper: the required area results
from
the sum 5 times 5. This sum 5 times 5 is what the science of
arithmetic
calls the square of 5, or 52, or 5 in the second power. And
if the
house
were to be of the same height as the breadth, the builder need only
write
53 that is to say, 5 times 5 multiplied by 5, and he would
know exactly
what would be the space included in the cube-shaped
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DESCARTES
house. If the sides of
the house are
unequal, he has to multiply the one by the other as a x b, then
by
the height, c, and with this abc he has at once at his command
all
the
conjuring tricks known to him by the study of arithmetical logic,
without
reference to the concrete house which has to be
built.
For as soon as I have
written down the
numbers instead of the line or the area or the body, the visible form
fades
away — a2 is
simply the number a
multiplied by itself, b3
is b multiplied
by itself and then again by the product, abc the multiplication of
the
numbers a, b, and c with one another. 50 The form absolutely
disappears,
and only the
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DESCARTES
measurements remain. Up
to this point
the matter is simplicity itself. But if you examine your power of
imagination,
it will soon answer that of all forms it is only those that are
rectangular
that allow themselves to be reduced in this simple way to lines, and
consequently
again resolved into numbers, whether these rectangles be plain or
solid.
All these rectangular forms can be imagined as originating in the
rectilinear
movement of limited straight lines, and this rectilinear movement may,
like the straight line itself, be expressed as number without further
trouble.
If I say a I name the line:
if I say a2 I name
the superficies which
comes
into existence when a moves
lengthwise along its length; if I say a3
I
am naming the cube which arises when this superficies moves upwards to
the length of a. The same
thing takes place in the case of rectangles
with
unequal sides; we can represent all of these to ourselves as proceeding
from the movement of two or, as the case may be, three lines of
different
lengths. Thus ab is the
movement of a along b, and abc is the
movement
of the superficies ab along c. The line is therefore
comprehensible as
a number (explete or symbolical), and what is perceived as the movement
of this line is to be understood as the multiplication of the number:
a2
is the multiplication of the number by itself, ab the multiplication
of
the number by another number. Therefore, inasmuch as rectangular
figures
can without more ado be reduced to single straight lines and to single
rectilinear movements, — it is easy to reduce them to an expression in
numbers.
The numbers 5, 4, 3, or the letters a,
b, c, correspond to the length
of
the component lines, and what we call exponents,
that is to say, those
smaller cyphers which Descartes taught us to write above and to the
right
of the larger figures, — for that was his invention, — denote the
movement
of the lines. As soon as you conceive of the matter as visible, these
algebraic
figures are shorn of all their abstract
263
DESCARTES
terrors. The small 2, as
in a2, points
to a simple movement out of which only a superficies arises, therefore
a space of two dimensions, hence the 2; the small 3 points to a double
movement, and consequently to a solid body, that is to say, a form of
three
dimensions, hence the 3. When therefore I reduce rectangular figures to
the measurement of length and indications of movement, denoting the
measurement
by ordinary cyphers, and add the movements by small cyphers written in
above, as exponents, I have obtained a very simple expression which I
can,
at will, look upon as a visible form, or as an arithmetical conception.
But how am I to deal with forms which are not rectangular? Question
your
own sound natural power of conception. Unless a man be a second
Descartes
he will have difficulty in finding the answer.
Not to
extend this
mathematical
excursus too far, we will only take into consideration one single case,
that of the curves in a plane, that is to say, of such curves as you
may
draw upon a sheet of paper, and which correspond proportionally to the
rectangular superficies. How can these curved lines be made capable of
a similar solution into arithmetical magnitudes? Without the help of
straight
lines the transition from curves to numbers is unthinkable. Numbers
have
no analogy in any shape with visible things, beyond on the one hand the
circumstantial analogy with objects exhibited side by side, and on the
other hand with straight lines. This second analogy is not, as you
might
think, drawn from the first, but arises out of the essence of numbers
which
are to be thought of as a rectilinear continuation. The numbers 5, 6,
7,
are essentially identical in their nature, only 6 is longer than 5 and
shorter than 7. 51
The curve, on the other hand, is an idea which
arithmetical
conception can never reach: it lacks the necessary pliability. The
essence
of the curve is form, the essence of arithmetic is indifference to
form.
It is therefore only the straight
264
DESCARTES
line that can be of any
assistance in
the task of converting form into numbers. For we may define the
straight
line as follows: it is the only line which even if it be produced to
infinity
creates no form. It is pure magnitude and pure numbers. How then can I
bend into pure formless magnitude and pure comprehensible numbers
magnitude
which is possessed of form and numbers conditioned by form, and at the
same time locked in form? Here, too, I can only succeed if I reduce
form
to movement, but even so the movement must be rectilinear.
Take a
ruler: let a
slider with
a pencil slide to and fro from one end to the other on this ruler, and
let this
pencil be placed
perpendicularly to the
ruler in a capsule which may be drawn out and compressed at pleasure:
if
you hold the ruler immovably on a sheet of paper you are able to draw
the
most complicated curves with the point of the pencil, as you, on the
one
hand, push along the ruler the sliding capsule which carries the
pencil,
and on the other hand, by lengthening and shortening the distance
between
the pencil's point and the ruler. You must now consider the resulting
visible
curve as being produced by the length of the ruler and the pencil; and
as a matter of fact that is what it is. So this curve expresses the
varying
relation between three straight lines, of which the one, the ruler, has
retained its length unaltered, while the two others which express the
length
265
DESCARTES
of the pencil and the
position of the
slider on the ruler, have been changeable. Looked upon purely from the
mechanical point
of view,
that is the
proceeding of Descartes in the analytical dissection of a curve. It was
with the help of such considerations and instruments that he arrived at
his thought. You see how this man is always and everywhere wandering on
the boundary line. The problem as a whole deals with the conversion of
the visible into the invisible and vice versa. Its solution he arrives
at by a perpetual shifting to and fro of the ideas of rest and
movement.
For the curve which he wishes to “analyse,“ the circle, the ellipse,
the
spiral, the volute, etc., is in the first place something granted, a
symbol
of that which is perfected, eternal, immovable. But next he considers
how
he may regard it as arising out of the movement of straight lines, and
thus rest becomes movement. Then there is the return of movement into
rest.
For these lines in motion serve to attain an immovable arithmetical
expression.
A
concrete example
will at once
show how Descartes obtains the straight lines for a given curve, and
out
of the lines an arithmetical expression. I choose for the purpose the
simplest
curve in a plane, the circle.
With
the help of a
piece of packthread
and a piece of blue chalk I draw a circle on the wall. Our circle is
obtained
by turning a straight line round one of its ends: that, however,
gives us nothing
available for analysis, which needs the relation to one another of
several
lines. This
causes Descartes to
refer the generative
law of this fixed figure to the relation between one immutable straight
line of an ascertained length and two other movable lines (cf. our
immovable
ruler with the movable pencil attached to it). Only you must not for a
moment imagine as regards the construction, which is the result, that
it
possesses any thinkable significance in nature, outside the human
brain,
or that in practice a circle can
266
DESCARTES
come into existence in
that manner; Descartes
only delineates it, because it is his pleasure to do so, because the
thing
can be thought of in that way, and because all sorts of amusing results
arise out of it. Well then, how does Descartes set to work? He accepts
the circle as given, within it he draws two straight lines
perpendicular
to one another, and the feat is done. The one line — the most important
one — he
draws from the circumference to the centre. For the sake of greater
clearness
I draw the line horizontally, but I might if I chose draw it in any
other
position. This line is always called R,
from the initial letter of the
Latin word Radius which signifies the spoke of a wheel, and later was
adopted
into scientific language as a description of the half of the diameter
of
the circle. Descartes had all the more reason for retaining the sign R
in that the French word for the half-diameter is Rayon, and the Germans
have only to think of their own original word Radspeiche (the spoke)
for
the R to lose all the evil
taste of the dust of the schools. This line
R is a fixed, immutable,
ascertained mathematical magnitude. If the
circle
is a concrete and present figure I can measure it with a yard measure:
if we are only dealing with the form of the circle in general, I cannot
represent any length in cyphers, but the line R is none the less a
recognised,
immutable magnitude, that is to say, in relation to whatever may be the
circular line of which it denotes the half-diameter. Well then, upon
this
immutable line I set up at right angles to it a second line which,
inasmuch
as it is an unknown magnitude, I call y,
and which I represent to
myself
as movable upon R, that is to
say, which I can move to and fro upon R,
from one end to the other, exactly as we did just now with our pencil.
But this second line is not only movable, but also of variable length.
In every single place, that is to say along the whole length of R, its
length differs; and indeed the organic relation between its length and
its
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DESCARTES
place is settled by the
curve in question,
in this case therefore by the line of the circle, inasmuch as we always
produce this line y to the
periphery of the circle by which we allow it
to be cut off. If, therefore, this movable line is raised at the inmost
point of R, that is to say,
at the centre of the circle, it becomes
itself
a half-diameter and its length equals that of the line R; that is the
maximum
length which it can attain; if on the contrary it is set up at the
outermost
point of R, it is immediately
cut off by the circumference line, and
its
length becomes zero. Between zero and a length equal to R the line y
can
have every conceivable measurement. And, as a mere glance at the
diagram
will show, its length will everywhere be determined by its place, and
its
place by its length. One word more and then we shall have gathered all
that we want. The line R is,
as we know, immutable: but it now contains
a movable element, namely the point at which the movable line y is
erected.
You need only think of the slider in our mechanical example. I will now
make use of the centre of the circle as a starting-point, and from that
measure the lines to the point where the line y meets the line R, and
this
line I will call x. Since y moves along R this line x is manifestly
variable
and its value, as a single glance at the diagram will show, will always
diminish or increase in an inverse ratio to that of
y. If y is at the centre,
x dwindles
away at once to zero. If y is
at the outer end of the Radius, x
becomes
equal to R. x is, as you see, in the same case
as y; its length value
can take every step between zero and the length of R; but, in addition,
its value must always of necessity be conditioned by y. As a result of
this construction we have now three values, of which one, R, is
immutable
and the two others, x and y, are mutable. What unites these
three
values
into one organic relation to one another, is first their fixed
reciprocal
position in space, secondly their fixed relation to the centre-point
and
to the peri-
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DESCARTES
phery of the circle by
which they are
bounded. Now as these values stand in a relation to the circle, so too
does the circle stand in a relation to them, and in this way they will
serve us to gain an expression for the curve in lines, and that means
in
numbers. This relation we can describe in the following very simple
fashion:
R2 = x2 + y2
That is to say, expressed
in words, the
square on R, — no matter
whether the circle be great or small, and no
matter
what may be the position of y,
is always equal to the square on y
multiplied by *) the square of x.
But you must not be led astray if this equation
talks
of squares:
for R, x, and y,
are
lines, and the
exponent 2 points, as you will remember, to the movement of a line
along
its own length; every one of these three squares is therefore
————
*) Should
be: “added to“, Redesdale made a translation mistake here. Chamberlain
wrote “vermehrt um“.
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DESCARTES
resolved by this formula
into a line
and a movement:
but the line as
well as
the movement
can be comprehended without more ado as a number; consequently that
representation
of a visible geometrical relation is at the same time, if we choose, an
algebraical equation, and that means a purely arithmetical expression.
As such it belongs to the protean domain of abstract mathematics,
“thought
without contents“; it is conception without perception, and so gains in
pliability and logical multiplicity of significations what it loses in
visibility. This algebraical equation (R2 = x2 + y2
) is the analysis
of
that
flat curve which we call a circle.
I have
no intention
of stopping
to furnish a proof of the correctness of this affirmation that R2 = x2 + y2.
It is very easily demonstrated geometrically, and is to be found in
Euclid
(as the Pythagorean proposition). Its interest for us only lies in the
fundamental idea of Descartes, the idea of the resolution of the
visible
relations of measurement and form into invisible, abstract arithmetical
relations. Any one who is interested in the matter can construct the
proof
empirically with the help of a circle and a millimetre measure. Nor
shall
I wait to show that there are other analytical equations which can be
made
up in behoof of the circle, and that for many other curves, as also for
the analysis of bodies of three dimensions, a far more circumstantial
process
is needed; the principle remains the same. Moreover, my conscience
pricks
me in that I have not led you precisely on the same way as that which
Descartes
followed, and because I have been so bold as to exhibit analytical
geometry
in a manner completely different from that ordinarily adopted. You can
read Descartes, if you are able, for his Géométrie is not
easy, inasmuch as he wrote it with purposeful obscurity in order to
avoid
plagiarisms; 52 or
you can take up that beautiful monument of German
industry,
Cantor's Vorlesungen über
Geschichte der mathe-
270
DESCARTES
matik, if you wish to
make acquaintance
with pure mathematics in their historical development. In neither place
will you meet with my exposition; if I have failed, this hint may be
taken
as an apology. 53
If I have taken my own road it has been because I had
a goal of my own in view. My peculiar way of looking at the subject
grew
out of our precedent course of thought, to which it now carries us
back.
Let me therefore, in closing this mathematical excursus, only say
briefly
that this algebraical analysis of geometrical perceptions is the
foundation
of almost the whole immense development of modern mathematics, and with
them of all physics. The expert mathematician, it may be said, sees in
his mind's eye, in such a scheme as R2 = x2 + y2, things which otherwise he
would never have seen in the mere visible symbol of the circle. True,
the
seen curve has faded away, but in its place that creative law of form,
as we have called it, appears perhaps even more distinct, — at any rate
as a stimulus to new thoughts. The analytical equation is to the
mathematician
what the ground plan is to the architect; unintelligible to the layman,
such a manner of schematising reveals to the expert things which he
would
never have been able to see in the concrete: that is to say, it leads
him to the discovery of relations between the different forms which no
power of perception would have been able to reach — and he has only to
discover
another, cleverly chosen, algebraical formula for his curve, in order
to
obtain an elevation in addition to his ground plan. He is now also in a
position to investigate the properties of forms which, on account of
their
great complications, would be beyond the power of the eye to unravel,
and
perhaps impossible to represent mechanically. Thanks to this method, he
has reached a point where he can investigate the properties of figures
of four dimensions, as well as of others that are beyond the power of
imagination.
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DESCARTES
You
see that this
road leads to
the deepest depths of metaphysics, but at the same time and in the same
measure to the
contemplation of the
unseen. For now all equations can be converted into form, and with
the help of two
lines
which exactly
correspond to the
R and y of our
example,
dreary, dull
rows of cyphers, such,
for
instance, as
statistics,
are conjured into curves
which at once
furnish
every layman with
intelligible ideas, and allow the mathematician to penetrate the
mysterious
laws to which Phenomena are obedient. 54
We
need go no
further now. Let
me add a general survey.
What
Descartes'
intervention has
signified for mathematics in general may, I think, be summed up in a
remark
which at the same time points directly to that which we have had in
view
in this excursus. We might indeed, unless I am mistaken, show that the
peculiar duplex character of the infinitesimal calculus, called into
life
by Descartes and followed up under his instigation by Ferrat, Pascal,
Barrow,
Newton, Leibniz, the brothers Bernouilli, and others, rests upon the
fact
that it stands with one foot in perception, with the other in
abstraction.
To grasp the fundamental conception of the infinitesimal calculus (that
is to say, remember, as “thought“), is so infinitely difficult, not
to
say impossible, that Carnot, one of the most competent of specialists,
assures us that very many professional mathematicians have not
understood
the significance of their own calculations; yet as a consolation he
adds:
il est certaines idées
primitives qui laissent toujours quelque
nuage dans l'esprit, mais dont les premières conséquences
une fois tirées, ouvrent un champ vaste et facile à
parcourir. 55
Historically the infinitesimal calculus grew out of the observation of
geometrical problems, and out of the lucky inspiration to consider
these
as phenomena of motion: far from being an abstraction, this mode of
calculation
is unthinkable
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DESCARTES
unless we take the
perception of the
senses as a starting point. Infinitely small magnitudes are magnitudes
that the eye no longer can see, but only the conception can still
imagine:
the transition from sensibility to understanding takes place here
materially:
the calculation by letters penetrates like a microscope where the
object
fades before the naked eye, and communicates “imaginary images“ to the
brain (see page 74 seq.).
Perception and abstraction are both of them
moving
in a region near the central line of demarcation. Up to Descartes'
time,
then, mathematics had, so to say, always been hopping upon one leg,
either
in perception or in abstraction. He taught them to stride forward
vigorously
on both feet; the start in mathematics could not long be delayed. We
too
to-day, within our modest limits shall gain a similar advantage.
* * * * * *
We
have now come to
an end of the
constructive part of this lecture. It would be delightful to follow
Descartes
still further; the proud, angular, domineering, and at the same time
aristocratically
reserved and sensitive nature of this thinker, fills us with respect
and
sympathy, and there would still be much to bring forward about him and
his life in amplification and correction of the known descriptions of
him;
something of this will perhaps, now that we are quite accurately
informed
as to the principles of his method of perception, weave itself in
automatically
in the further course of our studies. But at this moment another duty
lies
before us, that is to say, to turn to account the sum of our labours of
to-day for the recognition of Kant's intellectual aptitudes. 56
It is
the
fashion, — wrong as a
matter of method, — to start from the simplest point, from that which
analysis
shows as the simplest component parts. Far rather should that which is
best known serve as starting-point
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DESCARTES
in expositions and
explanations, whether
it be complicated or simple. This is the only way in which direct
perceptions with
that
power of persuasion
of which it has the monopoly, can maintain its rights. That is why
I chose as the
main theme
of my first
lecture the conflict between Idea and Experience. In that way we
certainly
gripped the
problem of
perception by
the most complex and difficult phenomenon that it perhaps ever
exhibits.
But the advantage was just this, that we at once faced the whole, that
is to say, that which is living, true, and sure, as it is common and
well
known to us all. We all have our experiences and our ideas, and even if
we are not accustomed to analyse them, one word is enough, and every
one
knows what we are talking about; and even though Goethe's perceptions
and
Goethe's ideas were of an august nature, they none the less spoke directly to our
understanding,
and that which was perceptible might almost have filled the whole
lecture. Next,
however, we
followed Goethe's
advice to “work our way out of the whole into the parts“; in the second
lecture we grasped the problem more closely on both sides, when we made
the conflict between the pure form of all perception and the empirical
material of perception our chief subject of study. Simple, and
apparently
easy to survey, was the relation between the two in the plastic artist, who kept
before his
eyes the scheme of his understanding, half
pure and half
perceptible, in order that he might see more
exactly, that
is to say, in order to “think“ more clearly that
which was seen,
to comprehend it more exactly: far more
complicated
did it become, — harder, that is to say, to
expound, and
so also harder rightly to grasp, — as soon as
the understanding
drew the phenomenon
over to itself, so
that the pure
scheme of the senses became the main point, whilst the empirical phenomenon itself, or at
any rate its
foundation in the
perception of the
senses, — paled almost
to fading away.
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DESCARTES
Calling to our support
the conflict between
physical optics and Goethe's doctrine of colours, we tried to gain as
clear
an explanation as possible of these relations. To-day a new conflict
has
arisen before us, no longer the one between pure perception and
empirical
perception, but that between perception as a function of the senses,
and
conceptible thought as a function of the understanding. Carefully
considered,
this conflict is far simpler than that between idea and experience, and
even in its essence easier to grasp than that between pure and
empirical; what makes it difficult to unravel is the complex
interlocking of the
parts: what I have had before me to-day as my chief aim has been to
arrive
at a clear conception upon this point; no man in the whole history of
the
world could render us such conspicuous service in this as Descartes.
You
must know that
with the help
of Descartes we have become acquainted with a way of seeing, a
recognition,
a conviction, a view, a method, — call it what you will — which is
absolutely
fundamental for Kant's philosophy. And this point is just the one of
all
others which is looked upon as the darkest in his philosophy; it is the
pons asinorum before which the
great majority of the flighty searchers
after knowledge turn tail — and not they alone! I could name a worthy
modern
student and editor of Kant, who only so far masters the difficulties
which
are to him insuperable, that he declares roundly that this fundamental
thought of Kant's “has no scientific value,“ and therefore that it is
not worth while to break one's head over what Kant may have meant:
indeed
that the whole difficulty was only an “invention“ of Kant's — Guter Mut,
halbe Arbeit, says the proverb, and so apparently thought the
learned
Professor.
Still we look at the matter from another point; happily it is not my
business
to explain the famous and dreaded chapter of the Kritik der reinen
Vernunft — Von
dem Schematismus der reinen
275
DESCARTES
Verstandesbegriffe — I
have
only to show in the commonest outline those foundations of
perception
which later, in the artistic connection of the system, go to the
greatest
depths, and are therefore the subjects of the most secret exposition.
“Unhappy is the speaking
man,“ cries Emerson. “If I speak I define, I confine, and am less.“
This
“unhappiness of the speaking man“ Kant had to experience: still he
would
not consent to make his thoughts less, the crabbed genius of truth
forbade
it; and so they became dark, dark as the powers which rise in the
growing
life of the golden-cocooned chrysalis; he who does not call eyes and
heart
to his help will never understand this thinker
and will never,
freed from the darkness of the pupa, fly aloft with him on the wings of
a new knowledge.
But
to-day, as I
said before, our office is far more modest, and I am glad to be able to
give the surprising assertion that we have achieved our task, and we
now
need only recapitulate it briefly, systematically, and with peculiar
reference
to Kant.
Descartes is of
special value
for the understanding of Kant because, with a striking resemblance in
his
intellectual aptitude in general, he has little capability and still
less
inclination to busy himself with the nice analysis of abstract
comprehensions;
that is why with him everything remains so concretely visible. That he
insisted upon the critique of the human intellect as an indispensable
foundation
for all science is proved by a quotation at the very outset of this
lecture;
moreover the expression “pure reason“ occurs often in his works; 57 yet
whatever there is of pure metaphysics in his philosophy is rather
symbolical
than critical. Masterfully and
forcibly he
simplifies,
and then he
places his rough-hewn blocks as landmarks to show that he too has
travelled
through this domain, and then hurries on further to those scientific
investigations
which take complete hold of him. Still these somewhat rough-hewn
symbols
of
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metaphysical knowledge
have exercised
an incomparable power upon later thought, as for example the
distinction
of every substance into thought and expansion. Like an Alexander among
philosophers he thus cut a Gordian Knot which all the desperate
attempts
of the spiritualists and materialists have never been able to join
together
again.
“Reason,“ says Kant,
“proves its
loftiest duties when it distinguishes between the world of the senses
and
the world of the understanding.“ Its loftiest duties! So great weight
does
he attach to this first and elementary direction of critical
reflection! The senses and the understanding are in his view “the two
extreme
ends“
of human knowledge. The shortest formula is as follows: “the business of the
senses is
to perceive, that of the understanding to think.“ More closely thought
out, and more accurately analysed, it runs thus: our knowledge springs
from two intellectual sources, of which the first is the reception of
notions
(receptivity of impressions), the second the power of appreciating a
thing
through the agency of the notion so received (spontaneity of
conceptions); the first gives us the object: by the second the object
is conceived
in relation to the notion (as a mere diagnosis of the mind), and all
this
is rather an accessory, a preamble, a preparation, an exercise of the
understanding
in the intellectual nursery; the true depth of the Kantian method of
perception
is first attained when the philosopher reaches the certainty that the
one “end“ of knowledge (the senses) is incapable of the smallest result
without
the other “end“ (the understanding). Unless the senses afford notions
no thought can arise; and unless thought furnishes its directing power,
no perception of an object can take place. Experience, — and in that
word
we express all that we are, — is therefore always “a product.“ If
experience
is always a product, then it would be simpler not to think of it as
arising
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and combined out of two
different and
separate origins, but rather as an original unity, which is only split into two component parts
by analysis.
Yet this objection
is in reality
very superficial,
and all that it effects is the reopening of the door to insipid
empiricism,
according to which
the
understanding
arises out of the senses, and to objectless mysticism, which makes the
world of perception arise out of the reason or the will; against which
Kant at once admits that “the two stems arise perhaps out of a common,
but to us unknown, root,“ while he declines to waste his strength upon
this unknown and unknowable thing (unknowable inasmuch as it lies
outside
experience), 58
but at once declares that we possess no organ or power
by
which we can ever go beyond experience, and that in all experience the
two stems are there, always capable of being proved to be distinct, and
always postulated as united. 59
Kant
is the only
philosopher of
experience — I wish to lay earnest stress upon that — he is the only
strict
philosopher of experience known to the history of human thought. That
makes
his greatness, and it is that which makes him so unapproachable to most
people. To philosophise with Schopenhauer is a delight, not to say a
luxury;
Kant, on the contrary, warns us with inexorable earnestness, “That the
understanding of which the first duty is to think, should instead of
that
fall into extravagance, is something not to be forgiven.“ To
philosophise
with Büchner,
Haeckel, and their like seems to comfort many brains
that we may presume to be atavistically retrograde; but Kant finds only
one predicate, impertinent,
adequate to the affirmations of materialism
and naturalism, and he exclaims with loathing, “Whoso has once tasted
Criticism,
is for ever disgusted with all dogmatic nonsense.“ You must not be
misled
by that much-abused word Idealism. In my further lectures I shall have
to offer a few remarks upon Kant's nomen-
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clature; here one word
will suffice:
when a critic of the Reine Vernunft
described Kant's teaching as a
“system
of the higher idealism,“ the sage at once answered wittily and with
fine
solemnity, “For goodness' sake not higher!
High towers and the
metaphysically
great men who resemble them, both surrounded by much wind, are not for
me. My place is the fruitful depth of experience.“ 60
If you
wish to know
Kant's method
of Seeing, if you wish to overlook no fundamental feature of his
intellectual
personality, then you must never lose sight of this famous saying, “my
place is the fruitful depth of experience.“ This limitation brings into
play at the same time a second fundamental characteristic, — that of
unconditional
truthfulness. The very same truthfulness which finds such crabbedly
lofty
expression in his moral writings rules here in the philosophic critique
of human reason. And even this removes Kant far away from us, makes him
inaccessible to most of us. It is not the truth that we long for, but
lies: and lies are on the watch for us everywhere; the lie invisible
and
unnoticed,
like the bacteria and microbes, worms its way into our brain in the
character
of “suggestion,“ nests there and multiplies, until even if we were
able
to get rid of the intruder and its brood, we should still be unable to
destroy its network without ruining our own power of thought. It needs
not only extraordinary keenness of thought, but also extraordinary
honesty
of thought, and incorruptible love of truth, even a whole life of
self-discipline,
to fit oneself to the fact that our whole thought and being is
surrounded
by a brazen wall, and that we must resign ourselves to our fate, since
we have neither wings to fly over the ramparts, nor the power of
reaching
the other side by burrowing under the earth.
From
this strict
limitation to
experience we arrive not only at Kant's peculiar method of perception,
but also at the special difficulties which many of his perceptions
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present to our
understanding. One
word more in this or in that direction, — the least little strain
exercised
upon our
thought, — what
Kant calls
experience, — and
all difficulty would disappear. But Kant never makes allowances. “My
place
is the fruitful depth of experience!“ I believe that I know that Kant
may be called the greatest of all thinkers: yet I know with absolute
certainty
that he is the honestest of all men, and that this loftiness of
character
means the Sun under whose rays his mental work ripens.
With
all this you
must be familiar
if you wish to understand Kant's position in regard to the “twofold
stem“
of all human knowledge, and actually to know why he looked upon the
world
in this way and in no other. His iron law binds him to experience
alone;
he will neither dig for that “common root,“ — always destined to remain
hypothetical
and perfectly incomprehensible, since we can only understand that which
belongs to experience, — nor will he have aught to say to dreaming and
Dogma.
That is why he considers the twofold sense and understanding as double,
and that is why the organism of the practical union of the two inside
all
experience can only be disentangled by the most painfully exact
observation
and critique of the
facts of
experience in
the mental life.
Kant is not concerned with being easily understood; what he is
concerned
with is spotless truth, above all with never overstepping the boundary
of experience. Great is the reward! Kant is right:
the bathos
of experience is fruitful. What we learn here is inexhaustible, and it
is not only true but useful. Kant's philosophy distinguishes itself in
toto from all other methods of philosophy in this, that it
watches over
itself practically step by step; it is always directing itself towards
two goals, natural science and moral doctrine. What can I know? What am
I to do? Those are the two great questions which exercised
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the sage of
Königsberg. And what
characterises him alone among all others is that in the answer to both
questions he forbids any overstepping of the boundary of experience.
Thence
we see not only a Goethe but also a Johannes Müller leaning upon
Kant,
and thence to-day not only our most important and freest professors of
philosophy, but in the same way many of our leading investigators of
Nature,
go back to Kant. Few have been adequately schooled to grasp Kant purely
and fully; but merely to touch the banner-bearer of crabbed, and at the
same time energetic and richly active truth, suffices to ennoble all
thought.
For
to-day we must
content ourselves
with the standpoint of a Descartes who looked upon the last questions
of
philosophy rather from a psychological than from a purely metaphysical
point of view. That the understanding and the senses are two is a
matter
known to us clearly and in detail by practical examples from the
history
of the Sciences. By seeing Descartes at work, first in the domain of
the
physical sciences, where on the one side the aether and on the other
the
laws of motion served us as main examples, and secondly within the
narrower
field of mathematics, we became aware of a tolerably complicated
relation,
which might otherwise easily have remained unknown to us. We discovered
that between those “two extreme ends of human knowledge,“ as Kant
called
them, there lies a uniting middle land: — outwards the boundaries of
this
buffer-country are rather indistinct, while, on the contrary, the
dividing
line which runs through the middle, and separates the two halves of our
intellect from one another, remains clear and sharply defined, even to
a hair's breadth. We are taught that we must make our comprehensions
evident
to the senses, otherwise they remain empty: true:
but what the
understanding sees in making
its comprehensions evident to the senses, is not that other “extreme
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end,“ not the
unadulterated perception
given by the senses, but only a schematism of the senses, schemes which
at their best reach the divisional middle line. We have to submit our
perceptions
to comprehensions, otherwise they are blind: to
be sure, however,
these conceptions must for this purpose be very essentially
materialised,
and the result is not pure thoughts, but a symbolism. It
is then certainly
no
simple occurrence
when we make our conceptions perceptible to the senses, and bring our
perceptions
under the category of conceptions.
Here is our
ultraschematic diagram,
ready to render us further service. If we were to direct our eyes
simply
towards the general division into understanding and perception by the
senses,
we should not reach far beyond Aristotle, who also in close connection
with Plato distinguished the nature of thought (νοήτικον) from the
nature
of perception by the senses (ἀισθἡτικον), and who consequently was
like
Kant and Descartes, antimaterialist and antispiritualist. The matter
first
gains a living interest, as well psychologically as metaphysically,
through
the discovery of the intermediary domain, and of the complicated
phenomena
which take place there. “Never
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is a comprehension
directly in relation
to an object, but only to some different idea of that object.“ That was
the one great discovery. It is amplified by the next: even if all
perception
of the senses brings to us something manifold, “we receive in the first
place no objects of empirical knowledge, and therefore no experience“;
the rather does experience first come into being by the co-operation of
a function of the understanding, since it is comprehensible imagination
which brings into combination, “the many-sidedness of perception“ —
with
that unity which is indispensable to all experience. There you have
Scheme
and Symbol. And I am convinced that you now accurately understand the
whole
matter at issue, since you have seen by Descartes' great thoughts of
Inertia
on the one side and the Aether on the other, what a Scheme is and what
a Symbol is, — how each arises, what it means, and what are its limits.
I
should like to
remind you once
more of Goethe's precious saying, “all thinking is useless for
thought.“
True thoughts always come as it were of themselves, their birthplace
lies
just in that middle land where perception and conception join hands.
And
it is the same in the case of the understanding of the thoughts of
another,
where the chief matter of importance is far more the subjection, than
any
exertion, of the intellect. Kant himself warns us that “Insight cannot
be forced and hurried by exertion.“ The man who wrinkles his forehead
and draws his eyebrows together, will never make any progress. The
expression
of the true desire to understand is the widely open eye which shows how
inwardly as well the mind greedily sucks up every ray of light in the
one
endeavour — to See. If you have yourself already seen what the other
man
saw and how he saw it, then his thoughts will automatically reach you.
Hence, now that we are about to take a very decisive step, I repeat the
petition which I have already made to you, to think
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as little as possible,
and
see as much
as possible. “In order to be comprehensible one must talk for the eye,“
says Herder in his Reisejournal.
You
remember the
first steps which
we had to take in order to arrive at a comprehension of the
boundary-land
between the senses and the understanding. Here again Descartes helped
us:
we only needed to see him at work in the province of pure mathematics.
Without the assistance of mathematics we should never have arrived at
complete
distinctness. But since without perception there can be no thought, and
since almost all perceptions are of empirical origin, that is to say,
arise
through impressions from without taken up by our senses, so in the
majority
of cases the problem of experience is from the outset very
complicated, — as
we saw in the case of metamorphosis. You have only to examine our
Scheme
in order to convince yourselves how difficult it must be to ascertain
the
precise mental topography, that is to say, the exact place of an idea,
which is forced upon us, given by perception, thought by
conceptions, — but
only thought when it is given, only given when it is thought. Such an
idea
has generally speaking no fixed place; it is shifted to and fro; the
commutator
of the middle land suddenly converts the one into the other, and e.g.
what
was
in Goethe a pure intensive Symbol of the senses, in Darwin is converted
into a perfectly artificial, abstract, logical Scheme. The advantage of
mathematics was that we found there pure schematic thought and pure
symbolical
perception within our own mind and without any adulteration from
outside.
Hence the topography was perfectly fixed, and hence with mathematical
precision,
— as we may well say, — scheme and symbol corresponded to one another.
The most important point which we gained from analytical geometry, was
this, that the dualism of our intellect, expounded by Plato, Aristotle,
and Descartes, but first accurately analysed by
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Kant, is no matter of
theoretical acceptation,
but a mathematically assured fact. All Monism is a lie, not of course a
subjective lie, nor a lie for those who are, or think they are, capable
of soaring above all experience, still an objective lie, a lie so soon
as Monism is to have value inside of experience. A gradual transition
from
perception to conceptions, or vice versa, from conceptions to
perception, — an
absorption of any sort, — is something which never takes place. In
empirical
experience there is still room for doubt; the invisibly complicated
relations
lead to many a deception: Mathematics, however, teach us something
better.
We certainly shall never deny that
and R2=x2+y2
are two exactly
corresponding expressions;
but no unprejudiced man will be able to avoid feeling the artificiality
and arbitrariness, I might almost say the tyranny, of such a
proposition.
Logic is powerless against it, for such propositions are outside the
pale
of logic: perception loses its rights in face of it, for perception is
suppressed. The proposition possesses no trace of a meaning beyond the
connection which I, as man, assign to it. That I have the courage of
such
a proposition does not prove that it has any objective sense outside of
my own intellect, but only that there is a subject which is capable of
uniting into symmetrical relation the two dissimilar parts of its
intellectual
organism. And it is just this construction of relations, not drawn from
experience, but by means of which, uniting the two parts of our mind,
we
first make experience possible, — just as by the relations between
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mental calculation
and perception
we make the higher mathematics possible, — it is just this that Kant
describes
by the dreaded, often used and seldom understood word
“transcendental.“ 61
Mathematical analysis has
here served
us as an example; but I must ask you to make a careful distinction. It
was
the single man of
genius
who succeeded
in setting up the equation R2=x2+y2, and in endowing it with
meaning;
another man may
introduce
another equation
with the same object. What is no achievement of ours, but is simply a
fundamental
law of the human intellect, is the fact that it is only the straight
line
that has the power to convert form into numbers and vice versa. Thus in
our mathematical undertaking we were in reality bound to a
transcendental
principle, though we were hardly conscious of it. Now we must go a step
further and enter upon the field where the arbitrary will of man has no
voice, but where inexorable laws of our mind are the informing power
— the
transcendental laws
of our reason.
In our
schematic
diagram we have
left white the level spaces on either side of the hatching: these were
supposed to represent pure sensibility and pure understanding. Now what
Kant detects is as follows: the Symbol on the one side and the Scheme
on
the other, do not originate in the middle region, in transition and in
combination, but all perception is at its very outset symbolical, and
all
thought is at its birth schematic. Although the commutation, although
the
switches which are to alter our direction may only exist in the middle
domain, that is merely a matter of psychological insight:
metaphysically,
on the other hand, the knowledge that our reason is as a general
proposition
confined within Scheme and Symbol, is of fundamental importance. That
is
the transcendental fixed boundary of all that of which we are conscious
as experience; Experience is never a pure apperception of what is and
takes
place outside our
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human mind, but it is
always a question
of an experience which is schematised and symbolised. Almost all men,
including
our so-called empirical investigators of Nature, maintain that the
human
understanding possesses capabilities which are independent of scheme;
that is to say, that it is at any rate partially set free from the
bondage
of fixed methods of thought; and they hold that human perception can
equally
see things as they are, and not as the tyranny of our one-eyed
cyclopean
form-sense of space transforms them for the benefit of mankind: but the
man who maintains this doctrine is defending conceptions which are far
beyond all experience, and indeed beyond all possible experience: that
man is a dogmatist. Kant refuses to take this aeronautic flight: he
remains
prosaically, heroically, and recusantly on the terra firma of facts,
and
says: all human perception happens through the intermediary of a fixed
Symbol — this Symbol of all pure perception is Space; all human thought
only
moves within a perfectly fixed, limited, inevitable Scheme — this
Scheme
of all pure thought is the table of the Primary Conceptions of pure
understanding,
also called “Categories.“ 62
The fact that the conceptions of the
understanding
do not permit of being referred to any single conception (like
Perception
to Space), is one which, as you will see presently, is founded upon the
essence of our intellectual mechanism; but those conceptions do form a
simple, strictly united scheme, acting on all sides as condition. We
have,
therefore, on the one side the one idea Space as an indispensable
fundamental
form of the senses, and on the other side the single group of the few
pure
conceptions of the understanding which make up an organic unity.
I
should like briefly
to limit and more closely define a saying of which I made use just now
in order to serve as a support to your ideas, but which might possibly
lead to misunderstandings later. I said: all perception
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is at its very outset
symbolical, and
all thought is at its birth schematic: I should like you to grasp that
not strictly, but only as analogy. You have seen how Symbol and Scheme
arose out of the reciprocal interpenetration of senses and
understanding;
briefly, therefore, Symbol and Scheme are not original, but derived;
but
what space is according to Kant's conviction, that you can first and
best
imagine by the analogy of a Symbol, — what is the table of the
comprehensions
of pure understanding you can first and best imagine by the analogy
with
a Scheme — and now that the above reservation has been made you can
fearlessly
facilitate your entrance into Kant's world of ideas by the following
formula:
in the last resort all different symbols may be referred to one symbol,
all different schemes of thought may be referred to a many-branched but
yet single and united scheme. And these are Space and the Table of the
Primary Conceptions.
This
is a point which
we may say we have reached with the help of the scope (étendue)
and thought (pensée) of
Descartes. But a mere lively
perception
is not enough; it must also be correct. And in order that your view of
the world may be the same as that of Kant I will cite two short
passages.
First as regards Space. “Space is nothing more than the mere form of
all
phenomena of the outward senses, that is to say, the subjective
condition
of the power of the senses, by which alone outer perception becomes
possible.
Now since the liability of the subject to be affected by circumstances
necessarily precedes all perception of these objects, we can understand
how the form of all phenomena can be given in the mind before all true
perceptions, and how they, as a pure perception in which all objects
must
be fixed, are able to contain principles of the relations of the
objects
to one another before all experience.“ It is a little more difficult to
find words for the pure primary conceptions
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of the understanding, —
words which,
without a previous exegesis of the Kantian system, should be directly
intelligible
and yet express a great deal, — perhaps the following might serve the
purpose.
“Just as space implies the condition of perception in a possible
experience, so are the Categories nothing more than the Conditions of
thought
in a possible experience; they are forms of thought which imply the
power
of uniting into one consciousness the Manifold which is given in
perception;
and since experience is knowledge by perceptions linked together, so
the
Categories are conditions of the possibility of experience.“ 63
If you
have paid
attention to
my request not to cramp your thought in a narrow gangway, but rather to
yield yourselves openly and without reserve to a new method of
perception,
you surely will have succeeded in following me so far. Of the utmost
importance
are two notions which are easy to retain. Space is the necessary form,
the Symbol, of all phenomena: the uniting of that which is manifold in
experience into a single consciousness takes place by the intermediary
of an immovable Scheme of thought. The little which remains to be said
will offer no difficulties if only you never for a moment turn away
from
the principle of the perceptible incorporation of thought.
As you
will have
gathered from
his words, Kant believes in a condition of the power of the senses and
in a condition of thought: it is the interplay between these two
conditions
that gives birth to “experience.“ And we may be sure that this
conviction
of Kant's does not rest upon logical system-mongering, but, quite on
the
contrary, on precise analytical observation of the functions of the
mind;
his method is, as he says himself, “imitated from that of the
investigator
of nature.“ It is here important in the first place to remark that,
even
if all the power of the senses is subjected to one condition —
extension
in space
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—
it is still the one
thing which brings
about the manifold in
nature, whereas, on
the other
hand, it is the prime function of the many-branched understanding to
bind this multiplicity into unity. Beyond the power of the senses, — it
is impossible not to premise this — lies the objective world, a real
chaos
of multiplicity: on the hither side of the understanding lies —
nothing!
nothing but a unity, that unity which Kant calls Reason, and which is
familiar
to us as the true ego.
According to Kant, moreover, there is, as you
see,
a progressive simplification and unification. Here again a schematic
diagram
will render us preliminary services. Think of the limitless objective
world
outside a circle.
It is
from that
world that the
senses take their impressions, and that too under the strictly
simplified
law that they force it by the compulsion of a single form, that of
extension
in space, to what, if I might so call it, is a “simple multiplicity.“
Then
the understanding reduces this multiplicity to a few primary
conceptions
standing in
relation to one
another, and
these primary conceptions coalesce in a consciousness of their unity
which
might be called Reason. In this way the circles are packed the one
within
the other.
Here
there are two things
deserving
of special attention: first the peculiar intermediary position and
function of the
understanding, and next
the special relation of the inmost circle (Reason) to the outermost
circle, that is to
the
surrounding world.
Further in the way
of
simplification
perception cannot
go, inasmuch as it
brings
all the impressions
under the
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one form of extension; on
the other hand,
the understanding, if it were to proceed in the same fashion, would
bring
up no further simplification, but only a reflected image. For simply
thinking
as a merely formal conception, while it no doubt leads to the idea of
oneness, can yet embrace all multiplicity: for example, to the
mathematician
the number 1 means the “infinitely great,“ and the single “type“ of the
zoologist can embrace a boundless wealth of forms. The conception
“unity“
gains a living meaning only when used to distinguish the idea of
organic
combination as opposed to mere formal fusion; as soon as the one
conditions
the other, by which in its turn it is conditioned, the two together
form
a true unity. True organic unity can never arise out of singularity,
but
only out of plurality. That is why the essence of thought is systematic
organisation, dissection, conjunction. You cannot think without passing
judgment, and you cannot pass the simplest judgment, — for instance
“the
room is big“ — unless you are in possession of three several
conceptions,
the subject, the predicate, the copula. Each of these three is derived
from a special primary conception, substantia,
existentia,
multiplicitas.
The leading simplification, carried out by the senses, is violent and
coarse
like the first preparation of some material, like the dressing of yarn;
it is only later that the threads are woven into an organic unity.
There
must, therefore, be numerous primary conceptions, otherwise it would be
impossible to introduce order, connection, unity (and that means sense)
into the mass of impressions which are afforded by perception. Kant
writes,
“Combination does not exist in the things, and cannot be in any way
borrowed
from them, and so, in the first instance, be taken up through
perception
into the understanding, — but is a function of the understanding which
is
of itself no more than the power to combine and to bring into unity
that
which our senses give us as
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manifold.“ That is a
memorable saying.
Our understanding is a power to combine and to reduce that which Is
manifold
into unity. 64 It
will be understood moreover, I hope, that the
multiplicity
of conceptions in the face
of the unity of
the form
of perception
means a progressive unification, and that without this “wholesale
conjunction“
there would be no such thing as knowledge,
nothing but chaos,
or as
Kant puts it,
a “rhapsody of perceptions.“ (R.V. 495.)
The
second
condition which here
deserves your attention is the intimate connection between the
innermost
circle and the outermost. The ego and the world stand in reciprocal
interchange:
each is necessary to the other:
neither can be
grasped,
seen and dissected
except so far as it is reflected in the other. The powers of the senses
and understanding hover between two Unknowns: the one immeasurably
great,
the other without any magnitude, without space; the one imaginably
rich
in an inexhaustible multitude of forms, the other completely devoid of
form, and for that very reason unthinkable. If we consider the relation
from the standpoint of perceptibility, then we must say with Kant, “the
world is the sum total of all phenomena“ (R.V. 391), the ego, on the
contrary,
“the poorest of all ideas“ (R.V. 408, 404), indeed “an idea empty of
all
contents.“ Still, if we pursue the matter conceptively we discover that
the “world“ is really only an idea, an image in the focus imaginarius
such
as we made acquaintance with in the first lecture, an image projected
out
of the ego into the Inscrutable. 65 And so the two stand over
against
one
another as correlatives: without the world no ego, without the ego no
world.
It is,
of course,
impossible for
me to engage in a more searching discussion of this subject: but it
will
be worth your while to follow up the lead which I have given. Later on
you will find in Kant the most fascinatingly
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DESCARTES
deep amplifications of
the matter. My
object for the moment has been above all to show accurately the twofold
boundary of experience, because that is so essential to Kant's method
of
Seeing. I spoke a while ago of a wall, and said that Kant's principle
was
to confine himself within this boundary of experience. But there are
indeed
two walls. One wall inwards, and a second wall outwards, and it is the
intervening space to which Kant confines himself as the only space of
experience.
And here again is something of which you must possess an exact and
comprehensible
idea, otherwise the next thing will be that you will once more fall
into
the clutches of the all-wise dogmatists, and will lose the moral and
intellectual
greatness as well as the scientific certainty of Kant's renunciation.
It
is manifest how empty is the purpose to try and solve the great riddles
of existence, out of a nature which is of our own creation, of which
the
necessary laws are the laws of our own understanding:
but Kant will tell
you
that the opposite
proceeding is exactly as deceptive. There is no ego of experience which
might serve as a foundation, upon which to raise a dogmatic erection
either
of the comprehension with Fichte and Hegel, or of the senses with
Schopenhauer;
the ego lies beyond, or, if you prefer it, on the hither side of,
experience.
You will see more clearly from the juxtaposition of the two following
short
formulae than from long arguments, the yawning gulf that separates the
schools
of philosophy; Schopenhauer teaches us that
The world IS MY idea.
Kant says —
MY
WORLD is
idea.
The difference is
immense. For the one
is a monstrous, indeed, if you look closely into it, a mad Dogma which
presupposes Nature and the ego as peculiar existences, and then sets up
categorical conclusions as to the relation
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DESCARTES
between them: the
other is the simple
affirmation of the result of critical reflection within the boundaries
of experience — a
reflection
which teaches
that whether inwardly or outwardly we are concerned with nothing except
Symbols and Schemes, so that we can make no pronouncement about
a “world“
beyond the fact that we human beings are compelled to imagine one.
Besides
this there results
from
Kant's doctrine
the significant inversion, I am an idea of the world, a fact upon which
Kant never wearies to descant, since this is really what is contained
in
the otherwise empty ego. Whereas from Schopenhauer's fundamental
doctrine
there is no result beyond the necessity of inventing a second dogma in
addition to the first, which is what happens with the dictum “I am
Will.“ 66
But
now and in
conclusion there
still remains a question for us to examine, which equally moves
entirely
within the frame of Descartes' philosophy and of the material for ideas
which we have gained in the course of this lecture. With this object
let
us return to the region of experience and to our old Scheme, which I
shall
furnish with new terms, for now we shall look upon the matter
“objectively“ instead of “subjectively.“ Understanding and the power of
the senses
have
been considered as functions of the human mind; instead of that we will
now take into consideration that which corresponds with those functions, so
to speak,
as object. Where before we wrote “the senses,“ we now write “space“;
where
we wrote
“understanding“
we will now
write “the primary conceptions of the understanding.“ But what are we
to
write in the
hatched
middle space? We
learnt from the history of sciences, and specially from analytical
geometry, that no
transition
takes place
from the one side of the middle line of separation to the other except
violently, suddenly, and through transcendental encroachment. That will clearly be the
case here,
for if in mathematics we were
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DESCARTES
dealing with quite pure,
that is to say,
quite human, ideas and conceptions, we have now risen a step higher: we
have here space as the primitive form of all possible perception and
the
pure conceptions of the understanding, or categories, as the
all-embracing
primitive forms of all thought. What is to take the part here which
mathematics
played in empiricism? Where shall we find a transcendental commutator,
a transcendental straight line?
Before
answering
the question
I must call attention to another trifling matter, because it will help
towards the clear setting out of the problem. Everybody without
exception
finds it much easier to understand Kant's doctrine of space than his
doctrine
of the pure primitive conceptions of the understanding. Of the readers
of the Reine Vernunft,
perhaps ninety per cent do not get beyond the
first
part, which treats of the form of perception (space). And that does not
happen, as one might imagine, because it is easier to understand that
which
is perceptible than that which is abstract. On the contrary, it arises
from the fact that it is easier for a man to follow a logical
demonstration
than to accustom himself to a new and strange method of perception. It
is from out of the understanding that space is contemplated, and
therefore
it is, as our first Scheme shows us, a “perception of thought,“ or if
you please a symbolical thought. For that reason the argument can be
presented
in almost pure logical form with firstly, secondly, thirdly: nothing
remains
hazy. But the categories, on the contrary must, if we wish to
understand
them rightly, be approached from the side of the senses, that is to
say,
we must contemplate them as Schemes, and yet see them. Pure conceptions
of the understanding cannot be further analysed and explained in terms
of logic, for they themselves are the simplest elements of thought: a
subject which can only be grasped by perception, defies all definition.
You may define space, but you cannot
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DESCARTES
define the single pure
primary conceptions
of the understanding, at any rate not logically. The name “space“
expresses
something
fixed; whereas all
names for primary conceptions are mere helps in need, only, as it
stands
in the Dissertation, — cognitio symbolica. (D. § 10.)
The name substance
(or
stability), for
instance, is nothing but hocus-pocus until you learn
to understand
that we are here face to face with “perceptible thought,“ a pre-logical
thought,
a thought lying on
that
middle territory
where conceptions are first born of the union with the senses. The same
observation applies to causality, reality, etc. Here, standing as we do
upon the two topmost rungs of the two ladders, I can only see the one
when
I take my place on the other. That is the one great, perhaps the
greatest,
difficulty against which Kant has to fight, and a chief cause of the
much
complained of “darkness“ of his philosophy. If the primary conceptions
of our human thought were abstract, self-reliant thoughts, we should in
any case be able to talk about them; but they are nothing outside their
relation to the power of the senses:
that at any rate
is
Kant's view, and
that is the reason why nobody by mere thought and without joining with
it active ideas, can grasp Kant's real meaning upon the subject of the
primary conceptions. He says, “the categories afford us no knowledge of
things excepting by their possible application to empirical
perception,“
and hence “we cannot really define any single category without having
recourse
to conditions of the senses, therefore to the form of phenomena to
which
they must consequently be confined as their only objects, because if this condition
is removed all
significance, that is to say, relation to the object, falls away, and
it
is then impossible for
us by any
example to make
ourselves grasp what sort of thing is meant by such-like conceptions.“
(R.V. 147,
300.)
Here
again I am
forced to content
myself with a mere
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DESCARTES
suggestion: but you will
very soon realise
what is the purport of this reminder.
Supposing then that
the primary
conceptions of our understanding existed without any relation to the
senses,
it would be impossible to see how the requisite “application to
perception“
could take place, and our knowledge could in no case be a “knowledge of
things“: it would be at best a “cloud-cuckoo-town“ * knowledge or, as
Kant
puts it, a too ambitious knowledge. It would be just the same as if we
imagined a mathematical proposition, purely logical, incapable of being
turned into anything perceptible, a calculation with an unknown x, y,
z,
in which everything would naturally fit in correctly without conveying
the slightest meaning. And the same holds good, though of course in an
inverted sense, in the case of space, as to which we have already seen
in Newton, that thought cannot easily grasp it, and yet loves to busy
itself
with it. And you will understand me when I maintain: that that
“transcendental
straight line“ of our research, which is to serve us as a commutator
between
the primary conceptions of thought and the primary form of perception,
must not only, like mathematics, be at the same time perception and
thought,
but must turn its perceptive side to thought, and its thinking side to
perception.
These
are
conditions which Time
alone can fulfil.
Time
is at once
conception and
perception; Kant introduces it sometimes as the one, sometimes as the
other. He calls it “Inner perception,“ or “Form of inner perception,“
but then again “Form of the inner sense.“ 67 However you look at it,
it
always remains something “inner,“ because like its empirical
embodiment,
mathematics, it fills the inner or middle domain. And yet it is in so
far
something “outer“ as it serves to transport each of the two fundamental
functions of our
* See the
“Birds“ of Aristophanes.
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DESCARTES
knowledge out of itself
and into the
other form. “The conception of change, and with it the conception of
motion
as change of place, is only intelligible through and in the idea of
Time;
unless this idea were perception
(inner perception) no
conception, whatever
it might be, would make the possibility of a change comprehensible.“ 68
It is only in Time therefore that a world can exist for us. That is one
side of the question. On the other side it is precisely Time that gives
us the idea of stability, and as such the form of the perception of
ourselves
and of our subjective condition. For, as Kant says, “Time can be no
definition
of outer phenomena: it belongs neither to form nor position, etc.; on
the
other hand, it does define the relation of ideas in our subjective
condition.“ 69
Without Time, then, no Ego! and not only no Ego, but generally no
conception
of substance, that is to say, no idea of any object which remains
stable
in the midst of all change. We have to thank Time for motion and
stability,
for development and being, for World and Ego.
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The analogy
with
mathematics must
strike you at once: and you will perceive that in our new Scheme we
have
placed our indications on their proper sides; for Time as stability
manifestly
corresponds with Geometry which is mathematical form, while Time as
fluent
motion is the foundation of the conception of numbers. 70 It is true
that time as stability is not form, but it furnishes the commutator,
the
salient point at which the idea of stability arises through the
transcendental
union of understanding and the power of the senses, that is to say, of
space and conception. In the same way the logically unthinkable
conception
of a change of place in space is impossible without a similar
intermediary
position of Time.
Let me
go more
closely into what
I have hinted at. For the function of Time must be made clear to you
without
any reservation, and it can be made so if you will only take Descartes'
analytical geometry as your counsellor step by step. Descartes would
have
been unable to carry out the critique of the human intellect as Kant
did,
and yet in the practice of his method of thinking he has shown a manner
of Seeing which is in complete harmony with that of Kant. The part
played
in Descartes' mathematics by the straight line, is played in Kant's
analysis
of the human intellect by Time. 71
As you will remember, the straight
line
is not form, inasmuch as it is the only visible thing which gives birth
to no form; and it is not a number, inasmuch as every straight line can
represent every possible number, and therefore remains entirely
indifferent
to the conception of numbers:
and in spite of
that it
alone furnishes
a footing for the conversion of form into numbers and vice versa. It is
just the same with Time. It lies altogether outside of the true
conceptions
of the understanding; it also lies outside of all perception. Time, as
we said before, is at once conception and perception; now we will speak
with
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DESCARTES
greater precision
and say: Time
is neither perception nor conception, but can never be far removed from
either; that is to say, without Time we can neither perceive nor
conceive.
That is why Kant calls Time “the constant correlatum of all
existence.“
(R.V. 226.) Space too is “no real object which can be outwardly
perceived,“
but is “the very form of phenomena“ (R.V. 459), and indeed in the
famous
dispute as to whether it was possible or not that there should exist an
empty space, — in which Descartes with unerring genius took the part of
the
impossibility, Newton with the childish simplicity which was peculiar
to
him asserted the certainty of this monstrous idea, — our abstract
jingle
of words is based upon a perception which is at least imagined. Time,
on
the other hand, offers no handle to perception, for where we conceive
it
as extension and quasi-visible, that happens by means of an allegory
inasmuch
as we draw a line in our thoughts and analogically use it to represent
Time, which “outward figurative idea of Time“ Kant has discussed in
detail.
(R.V. 154, seq.) Neither has
thought any more power of
comprehending Time;
it always contrives to elude thought. St. Augustine with a sigh says, et
confiteor tibi, Domine, adhuc ignorare me quid sit tempus. 72 The old
Greeks
had already discovered that Time could only exist for the thinker who
measures
and counts its hours, and Descartes, who has no objection to predicate
“lasting,“ that is “stability,“ of things, gives it as his opinion
that
le temps n'est rien qu'une
façon de penser. 73
And yet the
measures
of time deduced from various motions, such as those of heavenly bodies,
for instance, do not correspond to the actual lived or living life of a
man: for him a minute may contain years, years may glide by unobserved
like a short autumn morning: he measures Time not by length, but by
gradation,
that is to say, by the sensations which it contains. Here there is no
possibility
of bridging over the gulf, our double nature asserts itself too
abruptly.
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If ever you study
Kant's Reine
Vernunft you must not linger too long over the preliminary
doctrine of
the senses; for here unfortunately Time is at first dealt with as if it
belonged to space, and indeed to space alone, and that means,
therefore,
to the power of the senses. That has given birth to a misunderstanding
widely spread and disseminated by Schopenhauer into the remotest strata
of the imperfectly cultured, by which Space and Time are represented as
two forms of perception of equal value and parallel to one another. To
talk of the “ideality of Space and Time“ has become a commonplace
platitude;
and yet we have to deal with two completely different things: Space is
the only form of all pure perception; Time is an intermediary between
perception
and understanding, which in itself and by itself can neither be
perceived
nor imagined. That is why I recommend you to hurry on further through
Kant's
work, till you reach the place where it will be shown to you how the
conjunction
between understanding and the power of the senses (i.e. between the
primary
conceptions and the idea of space), takes place. Here you will once
again
meet Time introduced as a twin-sister of space, but as precisely in the
same relationship to understanding, and so “as an intermediary idea, on
the one side a matter of the intellect, on the other of the senses“;
on the one side “similar to the category, on the other side similar to
phenomenon in so far as Time is contained in every empirical idea of
the
Manifold.“ Then will Kant's perception for the first time really become
clear to you. Time is an intermediary idea, on the one side belonging
to
the intellect, on the other to the senses: what that means you now
understand
thoroughly and in detail. You need only think of the analogously
intermediate
part which the straight line plays in mathematics; and you will see
clearly
how important such an intermediary is for our whole intellectual life,
if you remember from the
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history of our sciences
which we touched
upon at the beginning of the lecture, that nothing which has been
observed
can be thought without the intermediary of a Scheme, and no thought can
gain any constructive value without the intermediary of a symbol: Time
is the foundation of all these intermediary processes. And now if you
look
at the title of this chapter of the Reine
Vernunft, you will discover
with
amazement that this chapter so impatiently awaited because it is the
most
indispensable of all, the one by which at last the goal is reached,
this
brilliant solution of riddles, the famous, dreaded chapter decried as
impenetrably
dark, of which I spoke at the beginning of this passage, is the chapter
on “the schematisation of the pure conceptions of the understanding.“
And
I think you will ask with me, what sort of perception can there be in
the
brains of a professional, state-paid commentator on Kant, who singles
out
this chapter to condemn it as valueless? 74
That,
however, is a
matter of
indifference. If Kant himself in the chapter in question asks, as he
does
literally, “How is it possible that pure conceptions of the
understanding
can be applied to phenomena?“ we cannot but think him fully justified.
As he says, “It is clear that there must be a third.“ We too see that
clearly.
And when after proving that this “third“ must be Time — a demonstration
of
the all-conquering power of conviction — he goes on to show that the
combination
of the “first“ and the “second“ which takes place inside this “third“
is no fusion, but nothing more and nothing less than a “placing of the
two together in relation to one another“ by intermediation; and when he
calls this intermediation a Scheme, and consequently makes the
relations
of the primary conceptions of the understanding to the nature of our
impressions
of the senses, take place through the intermediary of a schematisation
of pure understanding; — then all this is perfectly clear and
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DESCARTES
natural and according to
expectation;
we knew it already from empiricism and mathematics: what is new to us
is at most the fact that this relation is fundamental in all human
knowledge
without exception, since it is that which gives birth to what may be
called
Experience. In my humble opinion there is only one thing lacking to
make
Kant perfectly clear: that is to say, a chapter upon “the symbolism of
the pure power of the senses.“ Not that I should wish to obliterate
Kant's
distinction between spontaneity and receptivity, between function and
inclination,
between activity and passivity, 75
as characteristic in the
understanding
and in the power of the senses, — for mathematics and Time itself teach
us,
as Kant himself has done, that we not only comprehend our perceptions,
but also perceive our thoughts, and that both these processes, not one
alone, take place through the intermediary of that “third.“ Because
Kant
at the beginning of his critique lays stress upon a one-sided view of
the
relation of Time to Space, the reader is taken by surprise when he
finds
it brought into an equally close relation to the comprehensions of the
understanding, and described as “a third“; and next the first
one-sidedness
is amplified by a second, since now he only lays stress upon the
schematic
intermediation of Time, therefore, to use Kant's expression, upon the
way
in which the power of the senses realises understanding, not on the way
in which the understanding realises the power of the senses, that is to
say, the symbolising activity of Time. 76 Yet there can be no doubt
that
we have correctly represented Kant's perception: the whole Kritik der
Reinen
Vernunft and that of the Urteilskraft
(power of judgment) bears witness
to that; Kant may have had good metaphysical reasons for his
inconsistent
exposition
— indeed, he
certainly did
have them,
though they have not come under our observation here. We have only been
dealing with his method of Seeing, and of the many
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proofs which have
been laid before
you, I need only remind
you of one,
which furnished
the clue to the whole examination of Descartes: “Thoughts without
contents
are empty,
perceptions
without conceptions
are blind. Therefore it is just as necessary to make our comprehensions
obvious to the senses, — i.e. to add to them the object of perception,
—
as
to make our perception comprehensible to ourselves, — i.e. to subject
it
to comprehensions.“ To bring one's perceptions under conceptions is
called
Schematising: to make conceptions obvious to the senses is
called Symbolising.
Neither can take place
except by one and
the same intermediation,
that is to say, through a single element essentially unified, even
though it should
appear
iridised in
two colours, — otherwise no unity would exist in the understanding —
that
transcendental
unity, which arises in consciousness by the combination of Scheme and
Symbol.
Time is that commutator. Precisely where the middle line separates,
there
the two-sided “commutation“ takes place, the conversion of the one into
the other. We saw it in mathematics, we saw it in the empirical
sciences,
we shall be aware of it in every single one of our thoughts, as soon as
we have been attentive and have learnt to appreciate the indispensable
intermediation of the Proteus Time.
One
thing must be
fixedly borne
in mind, that for Kant, Time, like mathematics, is a purely formal
principle.
For that reason and because its special function is combination,
therefore
it is present everywhere, in every thought and in every perception. In
order to communicate my comprehensions to the senses, I need Time: in
order
to make my perceptions comprehensible, again I need Time. Two examples:
the Aether is little more than the vanishing thought of
stability
hardly felt by the senses: the observations on motion teach us that
the
same point may be in two places, which would have
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DESCARTES
no sense for pure
understanding, unless
the drawing of a line of Time made it thinkable. Even so Time can
nowhere,
neither in thought nor in perception, be grasped otherwise than as
something
existing for itself. Thought and perception cannot exist apart from
Time,
yet Time is nothing apart from its relation to thought and perception;
its essence is to be the fundamental relation of all relations, that
through
which relations as a general principle arise and have their being.
Everywhere
the strict analogy with mathematics! and therefore for us the
relatively
easy mastery of an otherwise so difficult, so incomprehensible subject.
If you
wish briefly
to sum up
what twofold Time has achieved on behalf of our knowledge you may say:
since Time as
Stability
is the means
of subjecting the manifold power of the senses under the yoke of the
conceptions
of the understanding, it bestows upon that power Unity: inasmuch as
Time
as Motion combines the unity of the inner sense (i.e. the unity of
Reason)
with the power of the senses, here it bestows manifoldness. 77 Two
examples:
Kant has shown how every one of our primary conceptions, magnitude,
gradation,
causality, reciprocal action, reality, necessity, etc., grasps the
matter
of perception by the intermediation of a Scheme of Time, and draws it
together
into unity; 78 on
the other hand, Kant has also shown that every
affection
of the senses can only be perceived as motion, from which he draws the
definition, “The fundamental principle of a Something which is an
object
of the outer senses must be Motion,“ and motion equally demands Time as
a correlative, and can only by the intermediation of Time give
manifoldness
to thought. And now at last the knot is tied fast, since understanding
assimilates motion, and produces perfect Scheme, while perception takes
causality, reciprocal action, necessity, and other pure primary
thoughts,
and amalgamates them so completely with
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that which has been
observed, that it
almost fancies that it can see them with its eyes.
We too
must tie a
famous knot
to end this long and laborious lecture. You remember how as a condition
for the understanding of Descartes' special talent we set up the
formula
that he knew how to make the invisible visible, and the visible
invisible.
How far this formula is applicable to Kant you now know too accurately
for any further explanation to be necessary. The saying, put so simply
and abruptly, has but little significance if it is used in connection
with
Kant's rich world of thought. Still, even so it can render certain
services
in this direction. That the invisible comprehension is powerless till
the
actual visible object has been offered to it by perception, and that,
on
the other hand, this same perception remains blind, unless
comprehensions transfer
this visibility into the invisibility of the world of thought — this,
combined
with the doctrine that it is Time which schematically and symbolically
cares for the hither and thither of the transformations: this it is
which
taken together makes up the essence of Kant's perception in regard to
human
knowledge. Now the limitation to experience, forms, as you will
remember,
an indispensable part of this perception of Kant's. What we see are
only
appearances due to the twofold conditions of Form of the Senses and
Schematisation
of the understanding: and if we add “the third,“ Time, the conditioned
phenomena become threefold. As for the things themselves, and what may
be their essence, we have neither the disposition nor the possibility
to
form an opinion, and it is just such a riddle that the individual
remains
to himself. Yet we cannot prevent two powerful ideas from growing out
of
this experience of ours, however strictly we may imprison them within
their
double rampart: — the World and the Ego. We have already spoken of
this,
and I will only add that World and Ego are as it were the two ends of
the
knot that I have in
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view: the Visible and the
Invisible κατ' έξοχήν. The
World, the visible end, is nothing else but that
threefold conditioned phenomenon in its highest, all-embracing
potentiality,
the symbol of all symbols; if I remove the Ego to which it appears,
nothing
remains, nothing, that is to say, that would have any possible meaning
for us men; for it is only the Ego that can bring forward the idea of
the
World. Yet the converse holds equally good. The Ego can be neither
thought
nor perceived unless it be mirrored by the World; if I remove the
World,
the Ego, the Inner thing, fades away. What remains is an empty Scheme
of
all Schemes, that is to say, a Nothing. Here too there is an
interchange,
and we can tie a knot as we bend the Inner outwards, the Outer inwards.
Nothing hinders us from conceiving the Ego as the invisible World, the
World as the visible Ego. That we are accustomed to look upon the
senses
as outer, the understanding as inner, is after all nothing more than a
convention, than a superficial analogical deduction from the organs of
sense and the brain in the bodies of the vertebrate animals. The
diagram
on p. 289 might just as well be reversed; reason or the Ego the
all-embracing
circle, the World in the inmost circle.
In
this way do
relations complicate
themselves as soon as we cease to limit ourselves to the domain of
experience.
This limitation, however, is not always possible. We cannot simply go
to
the order of the day about Ego and World, about Soul and God. And so
many
an idle chatterer, and also many a noble man, and among the latter none
bolder than Giordano Bruno, has soared aloft upon the wings of
fictitious
knowledge, in order to solve the riddle of the world and the riddle of
his own being outside the boundaries of experience. In what a different
spirit Kant set to work upon such questions, you will have suspected
from
our work to-day, and you will guess that he must also have reached
different
results:
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DESCARTES
I hope that this
will
show itself fully
as a result of a comparison with Bruno. In his case, as in that of
Descartes,
we have to deal with a specific thinker, and yet their attitude towards
the material at issue is almost directly opposite: for whereas
Descartes
only exercises the critical function in the domain of knowledge as a
discipline
of limitation in order to be able to devote himself in safety and
freedom
to empirical observation, and to the hypothetical and theoretical
significance
of concrete Nature, Bruno lives only in the empyrean of abstraction and
speculation, and accredits human reason and its logical inferences with
all knowledge and all power. That is why he has to take up an
essentially
different relation to Kant. In order to promote the interests of our
investigation
we gave precedence, whilst dealing with Descartes, to the similarity
with
Kant, leaving unnoticed the points in which they differed; in Giordano
Bruno we shall, on the contrary, gain our brightest illumination from
the
points of difference. And so we shall let our day of empiricism and the
critique of experience be followed by a morrow of dialectics and
dreams.
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Last update
September 25th, 2004
