by Gottfried Wilhelm Leibniz

Preface to the General Science

Since happiness consists in peace of mind, and since durable peace of mind depends on the confidence we have in the future, and since that confidence is based on the science we should have of the nature of God and the soul, it follows that science is necessary for true happiness.

But science depends on demonstration, and the discovery of demonstrations by a certain Method is not known to everybody. For while every man is able to judge a demonstration (it would not deserve this name if all those who consider it attentively were not convinced and persuaded by it), nevertheless not every man is able to discover demonstrations on his own initiative, nor to present them distinctly once they are discovered, if he lacks leisure or method.

The true Method taken in all of its scope is to my mind a thing hitherto quite unknown, and has not been practised except in mathematics. It is even very imperfect in regard to mathematics itself, as I have had the good fortune to reveal by means of surprising proofs to some of those considered to be among the best mathematicians of the century. And I expect to offer some samples of it, which perhaps will not be considered unworthy of posterity.

However, if the Method of Mathematicians has not sufficed to discover everything that might be expected from them, it has remained at least able to save them from mistakes, and if they have not said everything they were supposed to say, they have also not said anything they were not expected to say.

If those who have cultivated the other sciences had imitated the mathematicians at least on this point, we should be quite content, and we should have long since had a secure metaphysics, as well as an ethics depending on metaphysics since the latter includes the sort of knowledge of God and the soul which should rule our life.

In addition, we should have the science of motion which is the key to physics, and consequently, to medicine. True, I believe we are ready now to aspire to it, and some of my first thoughts have been received with such applause by the most learned men of our time on account of the wonderful simplicity introduced, that I believe that all we have to do now is perform certain experiments on a deliberate plan and scale (rather than by the haphazard fumbling which is so common) in order to build thereupon the stronghold of a sure and demonstrative physics.

Now the reason why the art of demonstrating has been until now found only in mathematics has not been well fathomed by the average person, for if the cause of the trouble had been known, the remedy would have long since been found out. The reason is this: Mathematics carries its own test with it. For when I am presented with a false theorem, I do not need to examine or even to know the demonstration, since I shall discover its falsity a posteriori by means of an easy experiment, that is, by a calculation, costing no more than paper and ink, which will show the error no matter how small it is. If it were as easy in other matters to verify reasonings by experiments, there would not be such differing opinions. But the trouble is that experiments in physics are difficult and cost a great deal; and in metaphysics they are impossible, unless God out of love for us perform a miracle in order to acquaint us with remote immaterial things.

This difficulty is not insurmountable though at first it may seem so. But those who will take the trouble to consider what I am going to say about it will soon change their mind. We must then notice that the tests or experiments made in mathematics to guard against mistakes in reasoning (as, for example, the test of casting out nines, the calculation of Ludolph of Cologne concerning the magnitude of circles, tables of sines, etc.), these tests are not made on a thing itself, but on the characters which we have substituted in place of the thing. Take for example a numerical calculation: if 1677 times 365 are 612,105, we should hardly ever have reached this result if it were necessary to make 365 piles of 1677 pebbles each and then finally to count them all in order to know whether the aforementioned number is found. That is why we are satisfied to do it with characters on paper, by means of the test of nines, etc. Similarly, when we propose an approximately exact value of π in the quadrature of a circle, we do not need to make a big material circle and tie a string around it in order to see whether the ratio of the length of this string or the circumference to the diameter has the value proposed; that would be troublesome, for if the error is one-thousandth or less part of the diameter, we should need a large circle constructed with a great deal of accuracy. Yet we still refute the false value of π by experiment and use of the calculus or numerical test. But this test is performed only on paper, and consequently on the characters which represent the thing, and not on the thing itself.

This consideration is fundamental in this matter, and although many persons of great ability, especially in our century, may have claimed to offer us demonstrations in questions of physics, metaphysics, ethics, and even in politics, jurisprudence, and medicine, nevertheless they have either been mistaken (because every step is on slippery ground and it is difficult not to fall unless guided by some tangible directions), or even when they succeed, they have been unable to convince everyone with their reasoning (because there has not yet been a way to examine arguments by means of some easy tests available to everyone).

Whence it is manifest that if we could find characters or signs appropriate for expressing all our thoughts as definitely and as exactly as arithmetic expresses numbers or geometric analysis expresses lines, we could in all subjects in so far as they are amenable to reasoning accomplish what is done in Arithmetic and Geometry.

For all inquiries which depend on reasoning would be performed by the transposition of characters and by a kind of calculus, which would immediately facilitate the discovery of beautiful results. For we should not have to break our heads as much as is necessary today, and yet we should be sure of accomplishing everything the given facts allow.

Moreover, we should be able to convince the world what we should have found or concluded, since it would be easy to verify the calculation either by doing it over or by trying tests similar to that of casting out nines in arithmetic. And if someone would doubt my results, I should say to him: "Let us calculate, Sir," and thus by taking to pen and ink, we should soon settle the question.

I still add: in so far as the reasoning allows on the given facts. For although certain experiments are always necessary to serve as a basis for reasoning, nevertheless, once these experiments are given, we should derive from them everything which anyone at all could possibly derive; and we should even discover what experiments remain to be done for the clarification of all further doubts. That would be an admirable help, even in political science and medicine, to steady and perfect reasoning concerning given symptoms and circumstances. For even while there will not be enough given circumstances to form an infallible judgment, we shall always be able to determine what is most probable on the data given. And that is all that reason can do.

Now the characters which express all our thoughts will constitute a new language which can be written and spoken; this language will be very difficult to construct, but very easy to learn. It will be quickly accepted by everybody on account of its great utility and its surprising facility, and it will serve wonderfully in communication among various peoples, which will help get it accepted. Those who will write in this language will not make mistakes provided they avoid the errors of calculation, barbarisms, solecisms, and other errors of grammar and construction. In addition, this language will possess the wonderful property of silencing ignorant people. For people will be unable to speak or write about anything except what they understand, or if they try to do so, one of two things will happen: either the vanity of what they advance will be apparent to everybody, or they will learn by writing or speaking. As indeed those who calculate learn by writing and those who speak sometimes meet with a success they did not imagine, the tongue running ahead of the mind. This will happen especially with our language on account of its exactness. So much so, that there will be no equivocations or amphibolies, and everything which will be said intelligibly in that language will be said with propriety. This language will be the greatest instrument of reason.

I dare say that this is the highest effort of the human mind, and when the project will be accomplished it will simply be up to men to be happy since they will have an instrument which will exalt reason no less than the Telescope perfects our vision. It is one of my ambitions to accomplish this project if God gives me enough time. I owe it to nobody but myself, and I had the first thought about it when I was 18 years old, as I have a little later evidenced in a published treatise (De Arte Combinatoria, 1666). And as I am confident that there is no discovery which approaches this one, I believe there is nothing so capable of immortalizing the name of the inventor. But I have much stronger reasons for thinking so, since the religion I follow closely assures me that the love of God consists in an ardent desire to procure the general welfare, and reason teaches me that there is nothing which contributes more to the general welfare of mankind than the perfection of reason.

Towards a Universal Characteristic

An ancient saying has it that God created everything according to weight, measure, and number. However, there are many things which cannot be weighed, namely, whatever is not affected by force or power; and anything which is not divisible into parts escapes measurement. On the other hand, there is nothing which is not subsumable under number. Number is therefore, so to speak, a fundamental metaphysical form, and arithmetic a sort of statics of the universe, in which the powers of things are revealed.

That the profoundest secrets are hidden in numbers has been a conviction of men ever since the time of Pythagoras himself who, according to a reliable source, transmitted this and many another intuition to Greece from the Orient. However, since the right key to the secret was not possessed, man's curiosity was led to nilities and superstitions of all sorts from which arose a kind of vulgar Cabal, far removed from the true and also -- under the false name of magic -- an abundance of fantasies with which books teem. Meanwhile there are still men who persist in the old belief that wonderful discoveries are imminent with the help of numbers, characters or signs of a new language, which the "adamite" Jacob Böhme calls a nature-language.

Nevertheless, no one perhaps has penetrated to the true principle, namely, that we can assign to every object itsdetermined characteristic number. For the most learned men, whenever I divulged something of the sort to them, led me to believe that they understood nothing of what I meant by it. Indeed for a long time excellent men have brought to light a kind of "universal language" or "characteristic" in which diverse concepts and things were to be brought together in an appropriate order, with its help, it was to become for people of different nations to communicate their thoughts to one and to translate into their own language the written signs of a foreign language. However, nobody, so far, has gotton hold of a language which would embrace both the technique of discovering propositions and their critical examination -- a language whose signs or characters would play the same rôle as the signs of arithmetic for numbers and those of algebra for quantities in general. And yet it is as if God, when he bestowed these two sciences on mankind, wanted us to realize that our understanding conceals a far deeper secret foreshadowed by these two sciences.

Now through some sort of destiny I had as a boy already been led into these reflections, and they have since, as often the case with first inclinations, remained most deeply impressed on my mind. This was wonderfully advantageous to me in two ways -- though often both dubious and injurious to many -- : first, I was thoroughly self-taught; as soon as I entered into the study of any science, I immediately sought out something new, frequently before I even completely understood its known, farniliar contents. Thus I gained in two ways: I did not fill my head with empty assertions (resting on learned authority rather than on actual evidence) which are forgotten sooner or later; furthermore, I did not rest until I had penetrated to the root and fiber of each and every theory and reached the principles themselves from which I might with my own power find out everything I could that was relevant.

I had early in my youth shown a preference for historical books and rhetorical exercises, and shown such facility in prose and poetry that my teachers feared I might remain suspended in these delights. Consequently, I was led to logic and philosophy. Scarcely hefore I had understood anything at all of these subjects, I set down on paper an abundance of fanciful thoughts which had risen to the surface of my brain, and when I presented them to my teachers they were amazed. One of the things I explored was the problem of the categories. What I intended especially to show was that just as we have categories predicating classes of simple concepts, so there must be a new sort of category which embraces propositions themselves or complex terms in their natural order. I had no inkling at the time of methods of proof, and did not know that what I was advancing was already being done by geometers when they arrange their propositions in a consecutive order so that in a proof one proposition proceeds from others in an orderly way. Thus my reflection was absolutely superfluous, but since my teachers did not satisfy my doubts, I had to take on the task all by myself to establish the aforementioned categories of complex terms or theorems.

As a result of my assiduous concern with this problem I arrived by a kind of internal necessity at a reflection of astounding import: there must be invented, I reflected, a kind of alphabet of human thoughts, and through the connection of its letters and the analysis of words which are composed out of them, everything else can be discovered and judged. This inspiration gave me then a very rare joy which was, of course, quite premature, for I did not yet then grasp the true significance of the matter. Later, however, the conclusion forced itself on me, with every step in the growth of my knowledge, that an object of such significance had to be pursued further. Chance had it then that as a young man of twenty I had to compose an academic dissertation. So I wrote the dissertation on "ars combinatoria" (art of combination) which in 1666 was published in book form, and thus my astounding discovery was made public. Of course, people observe that this treatise is the work of a youngster just out of school, not yet familiar with the sciences; for I lived in a place where mathematics was not cultivated, and had I, like Pascal, lived my early life in Paris, I should have succeeded earlier in advancing the sciences. Still I do not regret having written this dissertation, for two reasons: first, because it met with the approbation of many men of the highest intellect; secondly, because it already gave the world an intimation of my discovery so that the suspicion that it was discovered only recently cannot be supported.

I have often wondered why nobody until now, so far as any written evidence indicates, had ever put his hands on such an important subject. For if one had only followed step by step a strict method of procedure from the start there should have immediately been forced on one's mind considerations of this sort -- which I still as a boy missed in the study of Logic, without any acquaintance with mathematics, natural and moral sciences -- considerations which came home to me simply because I always sought first, original principles. The main reason, however, why people fail to go so far lies in the fact that abstract principles are usually dry and not very exciting, and after a momentary brush with them people let them alone. But there were three men, especially, who left me wondering why they had not entered into a problem of this significance: Aristotle, Joachim Jungius, and René Descartes. For Aristotle in the Organon and Metaphysics investigated with the greatest acuteness of mind the innermost nature of concepts. Joachim Jungius of Lijbeck, however -- who, of course, was himself scarcely known in Germany -- is a man of such penetrating judgment and of so comprehensive a mind that one should have expected from him as from no other person -- not excepting even Descartes -- a fundamental renovation of the sciences, had he only been known and supported. He was already an old man when Descartes' work took effect, and it is regrettable that they both did not get to know each other. This is not the place to indicate what is to be extolled in Descartes whose mind stands far above any praise. He surely set foot on the true and right path in the country of ideas, the path which might have led to our goal -- but for the fact that later, as it appears, in the course of his essay [Discourse on Method] he dropped the burden of the problem of method, and contented himself with metaphysical meditations and applications of his analytic geometry with which he attracted so much attention. In addition, he decided to investigate the nature of bodies for the purposes of medicine, which he was surely right in doing, had he only first solved the other problem, namely, the ordering of judgments and ideas. For from the latter there might have emerged a greater intellectual illumination than one was to believe possible, which would have shed light on experimental subjects also. That he did not direct his efforts to this end can be explained only by the fact that he did not grasp the deeper significance of the problem. Had he seen a method for establishing a rational philosophy with the same incomparable clarity as that of arithmetic, he would have chosen no other path than this one in order to establish a school which he so ambitiously strove to do. For a school which followed such a method in philosophy would naturally attract from among its tyros the same leadership in the kingdom of reason as geometry has, and would not totter or collapse if as a result of invasion, in a new barbaric era, the sciences themselves went under with mankind.

I, on the contrary, no matter how busy or diverted I might be, have steadfastly persisted in this line of reflection; I was alone in this matter, because I had intuited its whole significance and had perceived a marvelous and easy way to reach the goal. It took strenuous reflection on my part, but I finally discovered the way. In order to establish the Characteristic which I was after -- at least in what pertains to the grammar of this wonderful universal language and to a dictionary which would be adequate for the most numerous and most recurrent cases -- in order to establish, in other words, the characteristic numbers for all ideas, nothing less is required than the founding of a mathematical-philosophical course of study according to a new method, which I can offer, and which involves no greater difficulties than any other procedure not too far removed from familiar concepts and the method of writing. Also it would not require more work than is now already expended on lectures or encyclopedias. I believe that a few selected persons might be able to do the whole thing in five years, and that they will in any case after only two years arrive at a mastery of the doctrines most needed in practical life, namely, the propositions of morals and metaphysics, according to an infallible method of calculation. Once the characteristic numbers are established for most concepts, mankind will then possess a new instrument which will enhance the capabilities of the mind to a far greater extent than optical instruments strengthen the eyes, and will supersede the microscope and telescope to the same extent that reason is superior to eyesight. Great as is the benefit which the magnetic needle has brought to sailors, far greater will be the benefits which this constellation will bring to all those who ply the seas of investigation and experiment. What further will come out of it, lies within the lap of destiny, but it can only be results of significance and excellence. For all other gifts may corrupt man, but genuine reason alone is unconditionally wholesome for him. Its authority, however, will not be open any longer to doubt when it becomes possible to reveal the reason in all things with the clarity and certainty which was hitherto possible only in arithmetic. It will put an end to that sort of tedious objecting with which people plague each other, and which takes away for many the pleasure of reasoning and arguing in general. For instead of testing an argument, an adversary usually makes the following objection: "How do you know that your reason is better than mine? What criterion of truth do you have?" If the first party then again refers to his reasons, his interlocutor lacks the patience to test them; since for the most part there must still be a great many Other questions to settle, which would take a week's labor if he observed the traditionally valid procedures and rules of reasoning. Instead, after long pro and con discussions, most of the time it is emotion rather than reason that claims the victory, and the struggle ends there with the Gordian knot cut rather than untied. This is especially pertinent to the deliberations of practical life in which some decision must be finally made. Here it is only rarely the case that advantages and disadvantages which are so often distributed in many different ways on both sides, are weighed as on a balance. The stronger the representation or rather misrepresentation one party makes of this or that point according to his variable disposition, persuading others against his adversary by rhetorical effects of sharp relief and contrasting colors, the more dogmatically does he make up his own mind or indoctrinate others especially when he skillfully appeals to their prejudices. On the other hand, there is hardly anyone who is ever able to weigh and figure out the whole table of pros and cons on both sides, i.e., not only to count the advantages and disadvantages, but also to weigh them accurately against one another. Hence, I regard the two disputants as though they were two merchants who owe each other various moneys and have never drawn up a financial statement of their balance, but instead, always cross out the various postings of their outstanding debts and insist on inserting their own claims with respect to the legitimacy and magnitude of their debts. In this way, of course, conflicts could never end. We need not be surprised then that most disputes arise from the lack of clarity in things, that is, from the failure to reduce them to numbers.

Our Characteristic, however, will reduce all questions to numbers, and thus present a sort of statics by virtue of which rational evidence may be weighed. Besides, since probabilities lie at the basis of estimation and proof, we can consequently always estimate which event under given circumstances can be expected with the highest probability. Whoever is firmly convinced of the truth of religion and its implications and at the same time in his love of mankind longs for its conversion, will surely have to understand, as soon as he grasps our method, that (besides the miracles and acts of the saints or the conquests of a great ruler) there can be no more effective means conceived for the spread of the faith than the discovery under discussion here. For once missionaries are able to introduce this universal language, then also will the true religion, which stands in intimate harmony with reason, be established, and there will be as little reason to fear any apostasy in the future as to fear a renunciation of arithmetic and geometry once they have been learnt. I repeat, therefore, what I have frequently said, that nobody, whether a prophet or prince, can set himself a task of greater significance for human welfare as well as for the glory of God. We should nevertheless not remain content with words. Since, however, the wonderful interrelatedness of all things makes it extremely difficult to formulate explicitly the characteristic numbers of individual things, I have invented an elegant artifice by virtue of which certain relations may be represented and fixed numerically and which may thus then be further determined in numerical calculation. I make the arbitrary assumption, namely, that some special characteristic numbers are already given, and that some peculiar general property is observable in them. Thus, in the meantime, I take numbers which are correlated with the peculiar property, and then am able with their help immediately to demonstrate with astonishing facility all the rules of logic numerically, and can offer a criterion to ascertain whether a given argument is formally conclusive. Whether, however, a demonstration is materially conclusive may for the first time be judged without any trouble and without the danger of error, once we are put in possession of the true characteristic numbers of things themselves.

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