Leibniz on the
Universal Characteristic
But =
we
believe that we are thinking of many things (though confusedly) which
nevertheless imply a contradiction; for example,=
the
number of all numbers. We ought strongly to suspect the concepts of infinit=
y,
of maximum and minimum, of the most perfect, and of al=
lness
[omninitas] itself. Nor ought we to believe in =
such
concepts until they have been tested by that criterion I seem to recognize,=
and
which renders truth stable, visible, and irresistible, so to speak, as on a
mechanical basis. Such a criterion nature has granted us as an inexplicable
kindness.
Alge=
bra,
which we rightly hold in such esteem, is only a part of this general device.
Yet algebra accomplished this much—that we cannot err even if we wish and that truth can be grasped as if pictured on p=
aper
with the aid of a machine. I have come to understand that everything of this
kind which algebra proves is only due to a higher science, which I now usua=
lly
call a combinatorial characteristic, though it is far different from what m=
ay
first occur to someone hearing these words. I hope sometime, given health a=
nd
leisure, to explain its remarkable force and power by rules and examples. I
cannot encompass the nature of the method in a few words. Yet I should vent=
ure
to say that nothing more effective can well be conceived for perfecting the
human mind and that if this basis for philosophizing is accepted, there will
come a time, and it will be soon, when we shall have as certain knowledge of
God and the mind as we now have of figures and numbers and when the inventi=
on
of machines will be no more difficult than the construction of geometric
problems. And when these studies have been completed— though there will alw=
ays
remain to be studied the choicest harmonies of an infinity of theorems, but=
by
observation from day to day rather than by toil—men will return to the
investigation of nature alone, which will never be entirely completed. For =
in
experiments good luck is mixed with genius and industry.
Once=
men
carry our method through to the end, therefore, they will always philosophi=
ze
in the manner of Boyle, except insofar as nature itself, to the degree to w=
hich
it is known and can be subjected to this calculus and to the degree that new
qualities are discovered and reduced to this mechanism, will also give to
geometricians new material to which to apply it.
From Le=
tter to
Henry Oldenburg (28 December 1675), pp. 165-166; quotation, p. 166.
Mean=
while
I admit that no more beautiful example of the art of combinations can be fo=
und
anywhere than in algebra and that therefore he who masters algebra will the
more easily establish the general art of combinations, because it is always
easier to arrive at a general science a posteriori from particular
instances than a priori. But there can be no doubt that the general =
art
of combinations or characteristics contains much greater things than algebra
has given, for by its use all our thoughts can be pictured and as it were,
fixed, abridged, and ordered; pictured to others in teaching them, fixed for
ourselves in order to remember them; abridged so that they may be reduced t=
o a
few; ordered so that all of them can be present in our thinking. And though=
I know
you are prejudiced, by reasons which I do not know, to look rather adversely
upon these meditations of mine, I believe that when you examine the matter =
more
seriously, you will agree that this general characteristic will be of
unbelievable value, since a spoken and written language can also be develop=
ed
with its aid which can be learned in a few days and will be adequate to exp=
ress
everything that occurs in everyday practice, and of astonishing value in
criticism and discovery, after the model of the numeral characters. We
certainly calculate much more easily with the characters of arithmetic than=
the
Romans did either with pens or in their heads, and this is undoubtedly beca=
use
the Arabic characters are more convenient, that is, because they better exp=
ress
the genesis of numbers.
No o=
ne
should fear that the contemplation of characters will lead us away from the
things themselves; on the contrary, it leads us into the interior of things.
For we often have confused notions today because the characters we use are
badly arranged; but then, with the aid of characters, we will easily have t=
he
most distinct notions, for we will have at hand a mechanical thread of
meditation, as it were, with whose aid we can very easily resolve any idea
whatever into those of which it is composed. In fact, if the character
expressing any concept is considered attentively, the simpler concepts into
which it is resolvable will at once come to mind. Since the analysis of
concepts thus corresponds exactly to the analysis of a character, we need
merely to see the characters in order to have ad=
equate
notions brought to our mind freely and without effort. We can hope for no
greater aid than this in the perfection of the mind.
I
certainly believe that it is useful to depart from rigorous demonstration in
geometry because errors are easily avoided there, but in metaphysical and
ethical matters I think we should follow the greatest rigor, since error is
very easy here. Yet if we had an established characteristic we might reason=
as
safely in metaphysics as in mathematics.
You =
say
that it is difficult to set up definitions of things; perhaps you mean in t=
he most simple and the primitive concepts, so to speak. T=
hese,
I admit, it is difficult to give. We must realiz=
e,
indeed, that there are several definitions of the same thing, that is,
reciprocal properties which distinguish one thing from all other things and
that from each one we can derive all the other properties of the thing defi=
ned.
You are not unaware of this, but some of these definitions are more perfect
than others, that is, they come nearer to the primary and adequate notions.
Indeed, I hold this to be a certain criterion of a perfect and adequate
definition: that when the definition is once grasped, we cannot further dou=
bt
whether the thing defined in it is possible or not.
Besi=
des,
anyone who wishes to construct a characteristic or universal analytic can u=
se
any definitions whatever in the beginning, since all will eventually lead to
the same result when the analysis is continued. You are entirely of my opinion when you say that in very composite matt=
ers a
calculus is necessary. For this is the same as if you had said that charact=
ers
are necessary, for a calculus is nothing but ope=
ration
through characters, and this has its place not only in matters of quantity =
but
in all other reasoning as well. Meanwhile I have a very high regard for such
problems as can be solved by mental powers alone insofar as this is possibl=
e,
without a prolonged calculation, that is, without paper and pen. For such
problems depend as little as possible on external circumstances, being with=
in
the power even of a captive who is denied a pen and whose hands are tied. <=
span
class=3DGramE>Therefore we ought to practice both in calculating and=
in
meditating, and when we have reached certain results by calculation, we oug=
ht
to try afterward to demonstrate them by meditation alone, which has in my
experience often been successful.
From Le=
tter to
Walter von Tschirnhaus [Selection] (May 1678), =
pp.
192-195; quotations, pp. 193-194.
Fina=
lly,
to render my demonstrations absolutely incontestable,
and as certain as anything that can be proved by arithmetical calculation, I
shall offer an essay on my new writing or characteristic or, if you prefer,
language. This is undoubtedly one of the greatest projects to which men have
ever set themselves. It will be an instrument even more useful to the mind =
than
telescopes and microscopes are to the eyes. Every line of this writing will=
be
equivalent to a demonstration. The only fallacies will be easily detected
errors in calculation. This will become the great method of discovering tru=
ths,
establishing them, and teaching them irresistibly when they are established.
Nothing could be proposed that would be more important for the Congregation=
for
the Propagation of the Faith. For when this language is once established am=
ong
missionaries, it will spread at once around the world. It can be learned in
several days by using it and will be of the greatest convenience in general
intercourse. And wherever it is received, there will be no difficulty in
establishing the true religion which is always the most reasonable and in a
word everything which I shall develop in my work on Catholic Demonstrations=
. It
will be as impossible to resist its sound reasoning as it is to argue again=
st
arithmetic. You can judge what advantageous changes will follow everywhere =
in
piety and morals and in short, in increasing the perfection of mankind. But=
to
achieve this end, I shall certainly need great assistance, and I see no bet=
ter
source for this than the Congregation for the Propagation of the Faith, whi=
ch I
mentioned above.
From Le=
tter to
John Frederick, Duke of Brunswick-Hanover (Fall 1679), pp. 259-262; quotati=
on,
pp. 261-262.
I sh=
ould
venture to add that if I had been less distracted, or if I were younger or =
had
talented young men to help me, I should still hope to create a kind of
universal symbolistic [spécieuse générale]
in which all truths of reason would be reduced to a kind of calculus. At the
same time this could be a kind of universal language or writing, though
infinitely different from all such languages which have thus far been propo=
sed,
for the characters and the words themselves would give directions to reason,
and the errors—except those of fact—would be only mistakes in calculation. =
It
would be very difficult to form or invent this language or characteristic b=
ut
very easy to learn it without any dictionaries. When we lack sufficient dat=
a to
arrive at certainty in our truths, it would also serve to estimate degrees =
of
probability and to see what is needed to provide this certainty. Such an
estimate would be most important for the problems of life and for practical
considerations, where our errors in estimating probabilities often amount to
more than a half. . . .
From Le=
tter to
Nicolas Raymond, 10 January 1714, pp. 654-655; quotation, p. 654.
When=
I
was young, I found some pleasure in the Lullian=
art,
yet I thought also that I found some defects in it, and I said something ab=
out
these in a little schoolboyish essay called On =
the
Art of Combinations, published in 1666, and later reprinted without my
permission. But I do not readily disdain anything—except the arts of
divination, which are nothing but pure cheating—and I have found something
valuable, too, in the art of Lully and in the Digestum=
sapientiae of the Capuchin, Father Ives, which =
pleased
me greatly because he found a way to apply Lully’s generalities to useful
particular problems. But it seems to me that Descartes had a profundity of =
an
entirely different level. In spite of the advanc=
ement
which much of our knowledge has received from it, however, his philosophy a=
lso
has its defects, of which you cannot be unaware by this time.
From Le=
tter to
Nicolas Raymond, July 1714, pp. 656-658; quotation, p. 657.
[Letter=
s to
Nicolas Raymond (1714-15), pp. 654-660.]
SOURCE:=
Leibniz,
Gottfried Wilhelm. Philosophical Papers and Letters, selection translated a=
nd
edited with an introduction by Leroy E. Loemker=
, 2nd
ed. Dordrecht, Holland; Boston: D. Reidel Pub. Co., 1976 [1969, 1st
ed. 1956]). (Synthese Historical Library; =
v. 2)
Footnotes omitted.
<= o:p>
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