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 infinity, of maximum and minimum, of the most perfect, and of allness [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.
Algebra, which we rightly hold in such esteem, is only a part of this general device. Yet algebra accomplished this muchthat we cannot err even if we wish and that truth can be grasped as if pictured on paper 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 usually call a combinatorial characteristic, though it is far different from what may first occur to someone hearing these words. I hope sometime, given health and 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 venture 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 invention of machines will be no more difficult than the construction of geometric problems. And when these studies have been completed though there will always remain to be studied the choicest harmonies of an infinity of theorems, but by observation from day to day rather than by toilmen 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 philosophize in the manner of Boyle, except insofar as nature itself, to the degree to which 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 Letter to Henry Oldenburg (28 December 1675), pp. 165-166; quotation, p. 166.
Meanwhile I admit that no more beautiful example of the art of combinations can be found 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 to 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 developed with its aid which can be learned in a few days and will be adequate to express 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 because the Arabic characters are more convenient, that is, because they better express the genesis of numbers.
No one 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 the 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 adequate 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 the most simple and the primitive concepts, so to speak. These, I admit, it is difficult to give. We must realize, 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 defined. 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 doubt whether the thing defined in it is possible or not.
Besides, anyone who wishes to construct a characteristic or universal analytic can use 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 matters a calculus is necessary. For this is the same as if you had said that characters are necessary, for a calculus is nothing but operation 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 possible, without a prolonged calculation, that is, without paper and pen. For such problems depend as little as possible on external circumstances, being within the power even of a captive who is denied a pen and whose hands are tied. Therefore we ought to practice both in calculating and in meditating, and when we have reached certain results by calculation, we ought to try afterward to demonstrate them by meditation alone, which has in my experience often been successful.
From Letter to Walter von Tschirnhaus [Selection] (May 1678), pp. 192-195; quotations, pp. 193-194.
Finally, 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 truths, 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 among 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 against 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 better source for this than the Congregation for the Propagation of the Faith, which I mentioned above.
From Letter to John Frederick, Duke of Brunswick-Hanover (Fall 1679), pp. 259-262; quotation, pp. 261-262.
I should 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 proposed, for the characters and the words themselves would give directions to reason, and the errorsexcept those of factwould be only mistakes in calculation. It would be very difficult to form or invent this language or characteristic but very easy to learn it without any dictionaries. When we lack sufficient data 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 Letter 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 about 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 anythingexcept the arts of divination, which are nothing but pure cheatingand 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 advancement which much of our knowledge has received from it, however, his philosophy also has its defects, of which you cannot be unaware by this time.
From Letter to Nicolas Raymond, July 1714, pp. 656-658; quotation, p. 657.
[Letters to Nicolas Raymond (1714-15), pp. 654-660.]
SOURCE: Leibniz, Gottfried Wilhelm. Philosophical Papers and Letters, selection translated and 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.
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