Mathematicians From Germany

"The lack of early practice in mathematics left its mark on Leibniz's later mathematical style, in which good ideas are sometimes inefficiently developed through lack of technical skill. Often he seemed to lack not only the technique but also the patience to develop the ideas conceived by his wide-ranging imagination."

"Leibniz's work lacked the depth and virtuosity of Newton's, but then Leibniz was a librarian, a philosopher, and a diplomat with only a part-time interest in mathematics."

"Leibniz's apparent corelessness stands for a fundamental philosophical problem, a quandary that reaches to the foundations of his system of philosophy. In the metaphysics he... presented to the world, Leibniz claimed that the one thing of which we can all be certain is the unity, permanence, immateriality, and absolute immunity to outside influence of the individual mind. In identifying the mind as a "monad"—the Greek word for "unity"—he positioned himself in direct opposition to Spinoza, whose allegedly materialist philosophy of mind he adamantly rejected. And yet the philosopher who made the unity of the individual the fundamental principle of the universe was himself incomparably fragmented, multiplicitous, exposed to the influence of others, and impossible to pin down. How could a monad be so multifarious, not to say nefarious?"

"Long before most of these facts were discovered, Leibnitz had conjectured that originally the earth in general, even in the north, enjoyed a much warmer temperature than in the present period of all-ruling and progressive frost; and Buffon and others have established on this idea their hypothesis of a vast central fire in the interior of the earth. The interior parts of the earth and its internal depths are a region totally impervious to the eye of mortal man, and can least of all be approached by those ordinary paths of hypothesis adopted by naturalists and geologists."

"In Leibniz, a vast edifice of deduction is pyramided upon a pin-point of logical principle. In Leibniz, if the principle is completely true and the deductions are entirely valid, all is well; but the structure in unstable, and the slightest flaw anywhere brings it down in ruins."

"In Locke's own day, his chief philosophical opponents were the Cartesians and Leibniz. ...Until the publication of Kant's Critique of Pure Reason in 1781, it might have seemed as if the older philosophical tradition of Descartes, Spinoza, and Leibniz were being definitely overcome by the newer empirical method. The newer method, however, had never prevailed in German universities, and after 1792 it was held responsible for the horrors of the Revolution."

"Plato, in the Theaetetus, had set to work to refute the identification of knowledge with perception, and from his time onwards almost all philosophers, down to and including Descartes and Leibniz, had taught that much of our most valuable knowledge is not derived from experience."

"[We] will discuss Soul and Body, the doctrine of God, and Ethics. ...We shall find that Leibniz no longer shows great originality, but tends, with slight alterations of phraseology, to adopt (without acknowledgment) the views of the decried Spinoza. We shall find also many more minor inconsistencies than in the earlier part of [Leibniz's] system, these being due chiefly to the desire to avoid the impieties of the Jewish Atheist, and the still greater impieties to which Leibniz's own logic should have led him."

"The principle premisses of Leibniz's philosophy appear to me to be five... : I. Every proposition has a subject and a predicate. II. A subject may have predicates which are qualities existing at various times. (Such a subject is called a substance.) III. True propositions not asserting existence at particular times are necessary and analytic, but such as assert existence at particular times are contingent and synthetic. The latter depend upon final causes. IV. The Ego is a substance. V. Perception yields knowledge of an external world, ie. of existents other than myself and my states. The fundamental objection to Leibniz's philosophy will be found to be the inconsistency of the first premiss with the fourth and fifth; and in this inconsistency we shall find a general objection to Monadism."

"As an interpreter of nature... Leibnitz stands in no comparison with Newton. His general views in physics were vague and unsatisfactory; he had no great value for inductive reasoning; it was not the way of arriving at truth which he was accustomed to take; and hence, to the greatest physical discovery of that age, and that which was established by the most ample induction, the existence of gravity as a fact in which all bodies agree, he was always incredulous, because no proof of it, a priori could be given."

"Substances do not interact. Every substance is eternal. Bodies are phenomena, not independently real. Choices are determined but free. This is the best possible world. I first encountered Leibniz in an introduction to Modern Philosophy and the image of him as a philosopher so enthralled with his reasoning as to deny the reality in front of him stuck with me for a long time. It wasn't that his arguments were bad, but that their conclusions seemed obviously false. Wouldn't a swift kick in the shin suffice to prove that substances do interact, that bodies are real, and perhaps even that this is not the best possible world? This image of Leibniz as naive and detached from reality was cemented by Voltaire's satirical character Dr Pangloss, who insists over and over again - in the face of the worst suffering and injustice - that this is the best possible world. There is some irony in this image of Leibniz, as Leibniz was the far opposite of an 'ivory tower' philosopher. He consistently pursued positions that would increase his political influence over positions that would increase his leisure for study and reflection. Leibniz claimed the progress of knowledge as his main goal, but he approached this goal from two sides, on one side through his own research and writing while on the other side promoting institutions that would better support, disseminate, and apply knowledge. Today, Leibniz is best known or at least most widely read for his philosophical writings, but philosophy represents only a small part of his life's work. Although this book will focus on explaining Leibniz's philosophy, that philosophy must be approached from within the broader context of his life and time."

"In letters which went between me and that most excellent geometer. G.G. Leibniz, ten years ago, when I signified that I was in the knowledge of a method of determining maxima and minima, of drawing tangents, and the like, and when I concealed it in transposed letters involving this sentence (Data æquatione, etc., above cited) that most distinguished man wrote back that he had also fallen upon a method of the same kind, and communicated his method, which hardly differed from mine, except in his forms of words and symbols."

"Plus un, moins un, plus un, moins un, etc. En ajoutant les deux premiers termes, les deux suivans, et ainsi du reste, on transforme la suite dans une autre dont chaque terme est zéro. Grandi, jésuite italien, en avait conclu la possibilité de la création; parce que la suite étant toujours égale à ½, il voyait cette fraction naìtre d'une infinité de zéros, ou du néant. Ce fut ainsi que Leibnitz crut voir l'image de la création, dans son arithmétique binaire ou il n'employait que les deux caractères zéro et l'unité. Il imagina que l'unité pouvait représenter Dieu, et zéro, lé néant; et que l'Être Suprême avait tiré du néant, tous les êtres; comme l'unité avec le zéro, exprime tous les nombres dans ce système. Cette idée plut tellement à Leibnitz, qu'il en fit part au jésuite Grimaldi, président du tribunal des mathématiques à la Chine, dans l'espérance que cet emblème de la création convertirait au christianisme, l'empereur d'alors qui aimait particulièrement le sciences. Je ne rapporte ce trait, que pour montrer jusqu'à quel point les préjugés de l'enfance peuvent égarer les plus grands hommes."

"When the Eleatic School denied the possibility of motion, Diogenes, as everybody knows, stepped forth as an opponent. He stepped forth literally, for he said not a word, but merely walked several times back and forth, thinking that thereby he had sufficiently refuted those philosophers. Inasmuch as for a long time I have been engaged, at least occasionally, with the problem whether a repetition is possible and what significance it has, whether a thing gains or loses by being repeated, it suddenly occurred to me, "Thou canst take a trip to Berlin, where thou hast been before, and convince thyself now whether a repetition is possible and what significance it may have." At home I had almost been brought to a standstill by the problem. Say what one will, it is sure to play a very important role in modem philosophy; for repetition is a decisive expression for what "recollection" was for the Greeks. Just as they taught that all knowledge is a recollection, so will modem philosophy teach that the whole of life is a repetition. The only modem philosopher who had an inkling of this was Leibnitz ."

"Of all the works of Leibnitz, the "Theodicee" is the one most spoken of in Germany. Yet it is his feeblest production. This book, like several other writings in which Leibnitz expresses his religious sentiments, has obtained for its author an evil reputation, and has caused him to be cruelly misunderstood. His enemies have accused him of maudlin sentimentality and weakness of intellect; his friends, in defending, have proved him an accomplished hypocrite. The character of Leibnitz was for long a subject of controversy amongst us : the most partial critics could not absolve him from the accusation of duplicity; his most eager detractors were the freethinkers and the men of enlightenment. How could they pardon in a philosopher defence of the Trinity, eternal punishment, and the divinity of Christ! Their tolerance did not extend so far as that. But Leibnitz was neither fool nor knave, and by the lofty harmony of his intellect he was well able to defend Christianity in its integrity. I say, in its integrity, for he defended it against semi-Christianity. He established the consistency of the orthodox as opposed to the inconsistency of their adversaries. More than this he never attempted. He thus stood at that point of indifference where diverse systems appear as merely different sides of the same truth."

"As to Leibnitz, he is certainly a good philosopher, but in his Theodicée he goes too far and would have all actions necessary. His foreordained harmony is not the least credible nor feasible. If you can get a book entitled: An Essay on the Origin of Evil, by Dr W. King, you will find a much better solution of the question: 'whence comes evil?' Leibnitz does indeed reconcile it all with the goodness of God, but not so reasonably as Dr King."

"As Leibniz was fond of saying, it is one and the same to be a thing and to be a thing. In other words the “really real” is free from otherness, because what we could ascribe to it as other than what it is would actually be “another being.” For the same reason, being as such is free from change. In a doctrine where to be is to be the same, otherness is the very negation of being. Thus in virtue of its self-identity, which forbids it to change unless indeed it ceased to be, true being is immutable in its own right. This permanency in self-identity is the chief mark of the “really real,” that is, of being."

"In the History of Mathematics it is generally stated that the higher analysis took its rise in the method of indivisibles of Cavalieri (1635). This assertion... is erroneous. ...Leibniz was led to his invention of the algorithm of the higher analysis by a study of the writings of Pascal, more than by anything else."

"The German idealist philosophical tradition from which Hayek emerged is usually held to begin with Gottfried Leibniz, who wrote mostly during the second half of the seventeenth century. Leibniz put forward the idea of “monads,” a starkly idealist conception. Essentially, “each monad is a soul,” in the words of Bertrand Russell. Leibniz reversed the traditional conception of mind and matter by applying attributes of matter (in terms of sensory experience) to mind. Mind is what it experiences. Every mind or soul becomes an independent attribute of the universe, divinely ordered or arranged. Leibniz’s focus truly was mind. […] Leibniz was born at the end of the Thirty Years’ War. Religious struggles, such as the Thirty Years’ War, are often protracted and intense because they concern fundamental individual beliefs and values to which compromise is not always applicable. Chaos and disorder reigned in the larger society from which Leibniz emerged. It is not surprising that his philosophy moved in the direction of mind from a strictly sociological perspective, for the world was too hard to bear."

"[T]he program which Immanuel Kant proposed back in the 1760s... was this: our knowledge of the outside world depends on our modes of perception... Unfortunately, a great revolution took place in or about the year 1768, when he read a paper by Euler which intended to show that space was indeed absolute as Newton had suggested and not relative as Leibnitz suggested. (...in the eighteenth century the question of whether Newton's... or Leibnitz's view of the world was right profoundly affected all philosophy.) After reading Euler's argument... Kant... for the first time proposed that... we must be conscious of [absolute space] a priori. ...Kant died in 1804, long before new ideas about space... had been published... And since one of the things that happened in [our] lifetime has been the substitution of... a Leibnitz universe, the universe of relativity, for Newton's universe... we should think that out again."

"Gottfried Leibniz is famous... for his slogan Calculemus, which means "Let us calculate." He envisioned a formal language to reduce reasoning to calculation, and he said that reasonable men, faced with a difficult question of philosophy or policy, would express the question in a precise language and use rules of calculation to carry out precise reasoning. This is the first reduction of reasoning to calculation ever envisioned. ...he actually designed and built a working calculating machine, the Stepped Reckoner ...inspired by the somewhat earlier work of Pascal, who built a machine that could add and subtract. Leibniz's machine could add, subtract, divide, and multiply, and was apparently the first machine with all four arithmetic capabilities."

"Ainsi on peut dire que non seulement l'âme, miroir d'un univers indestructible, est indestructible, mais encore l'animal même, quoique sa machine périsse souvent en partie, et quitte ou prenne des dépouilles organiques."

"Thus there is nothing waste, nothing dead in the universe; no chaos, no confusions, save in appearence. We might compare this to the appearence of a pond in the distance, where we can see the confused movement and swarming of the fish, without distinguishing the fish themselves. Thus we are that each living body has a dominante entelechy, which in case of an animal is the soul, but the members of this living body are full of other living things, plants and animals, of which each has in turn ita dominant entelechy or soul."

"Or, comme il y a une infinité d'univers possibles dans les idées de Dieu, et qu'il n'en peut exister qu'un seul, il faut qu'il y ait une raison suffisante du choix de Dieu qui le détermine à l'un plutôt qu'à l'autre. Et cette raison ne peut se trouver que dans la convenance, dans les degrés de perfection que ces mondes contiennent, chaque possible ayant droit de prétendre à l'existence à mesure de la perfection qu'il enveloppe."

"Il y a aussi deux sortes de vérités, celles de Raisonnement et celle de Fait. Les vérités de Raisonnement sont nécessaires et leur opposé est impossible, et celles de Fait sont contingentes et leur opposé est possible."

"Et comme tout présent état d'une substance simple est naturellement une suite de son état précédent, tellement, que le présent y est gros de l'avenir."

"On est obligé d’ailleurs de confesser que la Perception et ce qui en dépend, est inexplicable par des raisons mécaniques, c’est-à-dire par les figures et par les mouvements. Et feignant qu'il y ait une Machine, dont la structure fasse penser, sentir, avoir perception ; on pourra la concevoir agrandie en conservant les mêmes proportions, en sorte qu’on y puisse entrer, comme dans un moulin. Et cela posé, on ne trouvera en la visitant au dedans, que des pièces, qui poussent les unes les autres, et jamais de quoi expliquer une perception. Ainsi c'est dans la substance simple, et non dans le composé, ou dans la machine qu’il la faut chercher."

"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."

"We never have a full demonstration, although there is always an underlying reason for the truth, even if it is only perfectly understood by God, who alone penetrated the infinite series in one stroke of the mind."

"Ce miracle de l'Analyse, prodige du monde des idées, objet presque amphibie entre l'Être et le Non-être, que nous appelons racine imaginaire."

"De quelque manière que Dieu aurait créé le monde, il aurait toujours été régulier et dans un certain ordre général. Mais Dieu a choisi celui qui est le plus parfait, c’est-à-dire celui qui est en même temps le plus simple en hypothèses et le plus riche en phénomènes..."

"Although the whole of this life were said to be nothing but a dream and the physical world nothing but a phantasm, I should call this dream or phantasm real enough, if, using reason well, we were never deceived by it."

"cur aliquid potius extiterit quam nihil"

"J'ay marqué plus d'une fois, que je tenois l'espace pour quelque chose de purement relatif, comme le temps; pour un ordre des coëxistences, comme le temps est un ordre des successions."

"It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it."

"Musica est exercitium arithmeticae occultum nescientis se numerare animi."

"Il y a deux labyrinthes fameux où notre raison s’égare bien souvent : l'un regarde la grande question du libre et du nécessaire, surtout dans la production et dans l'origine du mal ; l'autre consiste dans la discussion de la continuité et des indivisibles qui en paraissent les éléments, et où doit entrer la considération de l'infini."

"My philosophical views approach somewhat closely those of the late Countess of Conway, and hold a middle position between Plato and Democritus, because I hold that all things take place mechanically as Democritus and Descartes contend against the views of Henry More and his followers, and hold too, nevertheless, that everything takes place according to a living principle and according to final causes — all things are full of life and consciousness, contrary to the views of the Atomists."

"I have seen something of the project of M. de St. Pierre, for maintaining a perpetual peace in Europe. I am reminded of a device in a cemetery, with the words: Pax perpetua; for the dead do not fight any longer: but the living are of another humor; and the most powerful do not respect tribunals at all."

"[The consequences of] beliefs that go against the providence of a perfectly good, wise, and just God, or against that immortality of souls which lays them open to the operations of justice.... I even find that somewhat similar opinions, by stealing gradually into the minds of men of high station who rule the rest and on whom affairs depend, and by slithering into fashionable books, are inclining everything toward the universal revolution with which Europe is threatened, and are completing the destruction of what still remains in the world of the generous Greeks and Romans who placed love of country and of the public good, and the welfare of before fortune and even before life."

"La nature ne fait jamais des sauts."

"When Sir A. Fountaine was at Berlin with Leibnitz in 1701, and at supper with the Queen of Prussia, she asked Leibnitz his opinion of Sir Isaac Newton. Leibnitz said that taking mathematicians from the beginning of the world to the time when Sir Isaac lived, what he had done was much the better half; and added that he had consulted all the learned in Europe upon some difficult points without having any satisfaction, and that when he applied to Sir Isaac, he wrote him in answer by the first post, to do so and so, and then he would find it."

"Il y a jusque dans les exercices des enfants ce qui pourrait arrêter le plus grand Mathématicien."

"Pour ce qui est des connaissances non-écrites qui se trouvent dispersées parmi les hommes de différents professions, je suis persuadé qu’ils passent de beaucoup tant à l'égard de la multitude que de l'importance, tout ce qui se trouve marqué dans les livres, et que la meilleure partie de notre trésor n'est pas encore enregistrée."

"TO LOVE is to find pleasure in the happiness of others. Thus the habit of loving someone is nothing other than BENEVOLENCE by which we want the good of others, not for the profit that we gain from it, but because it is agreeable to us in itself. CHARITY is a general benevolence. And JUSTICE is charity in accordance with wisdom. … so that one does not do harm to someone without necessity, and that one does as much good as one can, but especially where it is best employed."

"As regards the objection that possibles are independent of the decrees of God I grant it of actual decrees (although the Cartesians do not at all agree to this), but I maintain that the possible individual concepts involve certain possible free decrees; for example, if this world was only possible, the individual concept of a particular body in this world would involve certain movements as possible, it would also involve the laws of motion, which are the free decrees of God; but these, also, only as possibilities. Because, as there are an infinity of possible worlds, there are also an infinity of laws, certain ones appropriate to one; others, to another, and each possible individual of any world involves in its concept the laws of its world."

"Chaque substance est comme un monde à part, indépendant de toute autre chose, hors de Dieu..."

"Omne possibile exigit existere."

"To love is to be delighted by the happiness of someone, or to experience pleasure upon the happiness of another. I define this as true love."

"Nam filum labyrintho de compositione continui deque maximo et minimo ac indesignabili at que infinito non nisi geometria praebere potest, ad metaphysicam vero solidam nemo veniet, nisi qui illac transiverit."