Gottfried Leibniz



Gottfried Wilhelm von Leibniz (1 July 1646 {21 June O.S.} – 14 November 1716) was a German philosopher and mathematician.

Quotes



 * quando orientur controversiae, non magis disputatione opus erit inter duos philosophos, quam inter duos computistas. Sufficiet enim calamos in manus sumere sedereque ad abacos, et sibi mutuo (accito si placet amico) dicere: calculemus
 * De arte characteristica ad perficiendas scientias ratione nitentes in C. I. Gerhardt (ed.), Die philosophischen Schriften von Gottfried Wilhelm Leibniz (7 vols. 1875–1890) VII 200.
 * "[...] if controversies were to arise, there would be no more need of disputation between two philosophers than between two calculators. For it would suffice for them to take their pencils in their hands and to sit down at the abacus, and say to each other (and if they so wish also to a friend called to help): Let us calculate."
 * The famous calculemus of Leibniz appears in several places of his writing; this is the most frequently quoted; variants are found in the Preface to his New Essays on Human Understanding, and in Dissertatio de Arte Combinatoria (1666). See R. Chrisley, Artificial Intelligence (2000), p. 14; H. Busche, Leibniz' Weg ins perspektivische Universum (1997), p. 134.


 * Theologus: Amare autem? Philosophus: Felicitate alterius delectari.
 * Theologian: But what is to love? Philosopher: To be delighted by the happiness of another.
 * Confessio philosophi (1673)


 * 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.
 * Only geometry can hand us the thread [which will lead us through] the labyrinth of the continuum’s composition, the maximum and the minimum, the infinitesimal and the infinite; and no one will arrive at a truly solid metaphysic except he who has passed through this [labyrinth].
 * Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae (Spring 1676) 


 * 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.
 * The Elements of True Piety (c. 1677), The Shorter Leibniz Texts (2006) edited by Lloyd H. Strickland, p. 189


 * Omne possibile exigit existere.
 * Everything that is possible demands to exist.
 * De veritatibus primis (1686)


 * Chaque substance est comme un monde à part, indépendant de toute autre chose, hors de Dieu...
 * Every substance is as a world apart, independent of everything else except God.
 * Discours de métaphysique (1686)


 * 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.
 * Gottfried Leibniz (May, 1686) as quoted in George R. Montgomery, Tr., "Correspondence between Leibniz and Arnauld," Leibniz: Discourse on metaphysics; correspondence with Arnauld, and Monadology (1916) VIII, p. 108


 * 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.
 * "A Dialogue" (after 1695), as quoted in The Shorter Leibniz Texts (2006) edited by Lloyd H. Strickland, p. 170


 * 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.
 * I am convinced that the unwritten knowledge scattered among men of different callings surpasses in quantity and in importance anything we find in books, and that the greater part of our wealth has yet to be recorded.
 * Discours touchant la méthode de la certitude et de l'art d'inventer pour finir les disputes et pour faire en peu de temps de grands progrès (1688–1690)


 * Il y a jusque dans les exercices des enfants ce qui pourrait arrêter le plus grand Mathématicien.
 * Even in the games of children there are things to interest the greatest mathematician.
 * Discours touchant la méthode de la certitude et de l'art d'inventer pour finir les disputes et pour faire en peu de temps de grands progrès (1688–1690)


 * 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.
 * Anecdote (1701) from John Conduitt's manuscript, as quoted by Sir David Brewster, Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton (1855) Vol.2


 * La nature ne fait jamais des sauts.
 * Nature does not make leaps.
 * Avant-propos to Nouveaux essais sur l'entendement humain (1704).
 * A later, more famous Latin version — "Natura non facit saltus" — is from the Philosophia Botanica (1751) by Linnaeus.
 * A variant translation is "natura non saltum facit" (literally, "Nature does not make a jump") ( Extract of page 289).


 * [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 future generations before fortune and even before life.
 * Nouveaux essais sur l'entendement humain (1704)


 * 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.
 * Letter 11 to Grimarest: Passages Concerning the Abbe de St. Pierre's 'Project for Perpetual Peace' (June 1712). Taken from Leibniz: Political Writings (2nd Edition, 1988), Edited by Patrick Riley.


 * 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.
 * Letter to Thomas Burnet (1697), as quoted in Platonism, Aristotelianism and Cabalism in the Philosophy of Leibniz (1938) by Joseph Politella, p. 18


 * 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.
 * There are two famous labyrinths where our reason very often goes astray. One concerns the great question of the free and the necessary, above all in the production and the origin of Evil. The other consists in the discussion of continuity, and of the indivisibles which appear to be the elements thereof, and where the consideration of the infinite must enter in.
 * Théodicée (1710)ː Préface


 * Musica est exercitium arithmeticae occultum nescientis se numerare animi.
 * Music is a hidden arithmetic exercise of the soul, which does not know that it is counting.
 * Letter to Christian Goldbach, April 17, 1712.
 * Arthur Schopenhauer paraphrased this quotation in the first book of Die Welt als Wille und Vorstellung: Musica est exercitium metaphysices occultum nescientis se philosophari animi. (Music is a hidden metaphysical exercise of the soul, which does not know that it is philosophizing.)


 * 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.
 * I have said more than once, that I hold space to be something purely relative, as time; an order of coexistences, as time is an order of successions.
 * Third letter to Samuel Clarke, February 25, 1716


 * cur aliquid potius extiterit quam nihil
 * Why is there anything at all rather than nothing whatsoever?
 * De rerum originatione radicali (1697); reprinted in God. Guil. Leibnitii Opera philosophica quae exstant latina, gallica, germanica omniaː 1 (1840), p. 148
 * Cf. Martin Heidegger, What is Metaphysics? (1929)ː "Warum ist überhaupt Seiendes und nicht vielmehr Nichts? Das ist die Frage."


 * 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.
 * As quoted in The World of Mathematics (1956) by J. R. Newman, p. 1832


 * 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...
 * In whatever manner God created the world, it would always have been regular and in a certain general order. God, however, has chosen the most perfect, that is to say, the one which is at the same time the simplest in hypothesis and the richest in phenomena.
 * Discours de métaphysique (1686); Leibniz famously tried to show that ours is the best of all possible worlds (see also Monadologie (53 & 54) below and compare Maimonides from Guide for the Perplexed (c. 1190), "Whatever is formed of matter receives the most perfect form possible in that species of matter.") These attempts were mercilessly parodied in Voltaire's Candide. Quotations from Voltaire's novel are often mistakenly attributed to Leibniz. Other statements by Leibniz upon the subject include these:
 * S'il n'y avait pas le meilleur (optimum) parmi tous les mondes possibles, Dieu n'en aurait produit aucun.
 * If there were no best among all possible worlds, God would not have created one.
 * Théodicée (1710)ː I. 8
 * I do not believe that a world without evil, preferable in order to ours, is possible; otherwise it would have been preferred. It is necessary to believe that the mixture of evil has produced the greatest possible good: otherwise the evil would not have been permitted. The combination of all the tendencies to the good has produced the best; but as there are goods that are incompatible together, this combination and this result can introduce the destruction of some good, and as a result some evil.
 * Letter to Bourguet (late 1712)], as translated in The Shorter Leibniz Texts (2006) edited by Lloyd H. Strickland, p. 208


 * 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.
 * This miracle of analysis, this marvel of the world of ideas, an almost amphibian object between Being and Non-being that we call the imaginary number.
 * Quoted in Singularités : individus et relations dans le système de Leibniz (2003) by Christiane Frémont


 * 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.
 * The Shorter Leibniz Texts (2006) edited by Lloyd H. Strickland, p. 111


 * 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.
 * Closing sentence of the Preface to the general science (1677) (in P. Wiener (ed.), Leibniz Selections, Macmilland Press Ltd, 1951).

The Monadology (1714)



 * 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.
 * Moreover, it must be confessed that perception and that which depends upon it are inexplicable on mechanical grounds, that is to say, by means of figures and motions. And supposing there were a machine, so constructed as to think, feel, and have perception, it might be conceived as increased in size, while keeping the same proportions, so that one might go into it as into a mill. That being so, we should, on examining its interior, find only parts which work one upon another, and never anything by which to explain a perception. Thus it is in a simple substance, and not in a compound or in a machine, that perception must be sought for.
 * La monadologie (17).


 * 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.
 * And as every present state of a simple substance is naturally a consequence of its preceding state, so its present is pregnant with its future.
 * La monadologie (22).


 * 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.
 * There are two kinds of truths: those of reasoning and those of fact. The truths of reasoning are necessary and their opposite is impossible; the truths of fact are contingent and their opposites are possible.
 * La monadologie (33).


 * 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.
 * Now, as there is an infinity of possible universes in the Ideas of God, and as only one of them can exist, there must be a sufficient reason for God's choice, which determines him toward one rather than another. And this reason can be found only in the fitness, or the degrees of perfection, that these worlds contain, since each possible thing has the right to claim existence in proportion to the perfection it involves.
 * La monadologie (53 & 54).


 * 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 it may be said that not only the soul, the mirror of an indestructible universe, is indestructible, but also the animal itself, though its mechanism may often perish in part and take off or put on an organic slough.
 * La monadologie (77).
 * Sometimes paraphrased as: The soul is the mirror of an indestructible universe.


 * 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.


 * Monadology, $$ 69-70.

Quotes about Leibniz

 * 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.
 * Michael J. Beeson, "The Mechanization of Mathematics," in Alan Turing: Life and Legacy of a Great Thinker (2004)


 * [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.
 * Jacob Bronowski, The Origins of Knowledge and Imagination (1978)


 * In the interval of 200 years between Kepler and Gauss there arose no great mathematician in Germany excepting Leibniz.
 * Florian Cajori, A History of Mathematics (1893).


 * Leibniz was certainly not alone among great men in presenting in his early work almost all the important mathematical ideas contained in his mature work.
 * J. M. Child, Preface, The Early Mathematical Manuscripts of Leibniz (1920)


 * The main ideas of his philosophy are to be attributed to his mathematical work, and not vice versa.
 * J. M. Child, Preface, The Early Mathematical Manuscripts of Leibniz (1920)


 * The manuscripts of Leibniz... show, perhaps more clearly than his published work, the great importance which Leibniz attached to suitable notation in mathematics and... in logic generally. He was perhaps the earliest to realize fully and correctly the important influence of a calculus [some mindless method of calculation] on discovery. ...There is a frivolous objection... to the effect that such economy of thought is an attempt to substitute unthinking mechanism for living thought. This contention fails... through the simple fact that this economy is only used in certain circumstances. In no science do we try to make subject to a mechanical calculus any trains of reasoning except such that have not been the object of careful thought many times previously. ...this reasoning has been universally recognized as valid, and we do not wish to waste energy of thought in repeating it when so much remains to be discovered by means of this energy. Since the time of Leibniz, this truth has been recognized, explicitly or implicitly, by all the greatest mathematical analysts.
 * J. M. Child, Preface, The Early Mathematical Manuscripts of Leibniz (1920)


 * When one compares the talents one has with those of a Leibniz, one is tempted to throw away one's books and go die quietly in the dark of some forgotten corner.
 * Denis Diderot, Oeuvres complètes, vol. 7, p. 678, quoted in "Gottfried Wilhelm Leibniz" (2013) by Brandon C. Look


 * Perhaps never has a man read as much, studied as much, meditated more, and written more than Leibniz… What he has composed on the world, God, nature, and the soul is of the most sublime eloquence. If his ideas had been expressed with the flair of Plato, the philosopher of Leipzig would cede nothing to the philosopher of Athens.
 * Denis Diderot, Oeuvres complètes, vol. 7, p. 709, quoted in "Gottfried Wilhelm Leibniz" (2013) by Brandon C. Look


 * 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.
 * Alan Ebenstein, Hayek's Journey: The Mind of Friedrich Hayek (2003), Ch. 2. German and Viennese Intellectual Thought


 * 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.
 * C. I. Gerhardt, "Leibniz and Pascal," (1891) Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin as quoted in The Early Mathematical Manuscripts of Leibniz (1920) Tr. J. M. Child


 * 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.
 * Being and Some Philosophers 1949, 1952 Étienne Gilson 1884-1978


 * 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.
 * William Herschel, June 12, 1761 Letter, "William Herschel to Jacob Hershel" in Constance Ann Lubbock's The Herschel Chronicle: The Life-story of William Herschel and His Sister, Caroline Herschel p. 27


 * 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.
 * Heinrich Heine Religion and Philosophy in Germany, A fragment, pp. 66-67


 * 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.
 * Repetition by Soren Kierkegaard 1843 Lowrie translation 1941 p. 33

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 &frac12;, 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.'' By adding the first two terms, the next two, and so forth, the result is converted into another for which each term is zero. Grandi, the Italian Jesuit, had concluded the possibility of creation from this series; because the result is always equal to &frac12;, he saw the unborn fraction of infinitely many zeros, or nothingness. '''It was thus that Leibnitz saw in his binary arithmetic the image of creation. He imagined that Unity represented God, and Zero the void; and that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in this system of numeration'''. This conception was so pleasing to Leibnitz that he communicated it to the Jesuit, [Claudio Filippo] Grimaldi, president of the Chinese tribunal for mathematics, in the hope that this emblem of creation would convert the Emperor of China, since he was very fond of the sciences, to Christianity. I mention this merely to show how childhood prejudices may lead astray even the greatest men.
 * ''Plus un, moins un, plus un, moins un, etc.
 * Plus one, minus one, plus one, minus one, etc.
 * Pierre-Simon Laplace, Essai Philosophique sur les Probabilitésas (1814) p. 82, partially quoted in Tobias Dantzig,  (1930) p. 15 and in Richard Courant, Herbert Robbins What is Mathematics? (1941) revised by Ian Stewart (1996)


 * 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.
 * Isaac Newton, Principia (1687) 1st edition, Book II. Prop.7, scholium reference the co-discovery of the Calculus, as quoted by Florian Cajori, A History of Mathematics (1893)


 * 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.
 * Franklin Perkins, Leibniz: A Guide for the Perplexed (2007), p. 1


 * 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.
 * John Playfair, Dissertations on the History of Metaphysical and Ethical: And of Mathematical and Physical Science, Vol. 1. Dissertation Third, p. 572.

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.
 * The principle premisses of Leibniz's philosophy appear to me to be five... :
 * Bertrand Russell, A Critical Exposition of the Philosophy of Leibniz (1900) Ch. 1, Leibniz's Premisses, p, 4.


 * [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.
 * Bertrand Russell, A Critical Exposition of the Philosophy of Leibniz (1900) Ch. 1, Leibniz's Premisses, p, 5.


 * 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.
 * Bertrand Russell, A History of Western Philosophy (1945) Book Three, Part I, Chapter XIII, Locke's Theory of Knowledge, p. 610


 * 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.
 * Bertrand Russell, A History of Western Philosophy (1945) Book Three, Part I, Chapter XV, Locke's Influence, p. 641-642


 * 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.
 * Bertrand Russell, A History of Western Philosophy (1945) Book Three, Part I, Chapter XV, Locke's Influence, p. 644


 * 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.
 * The Philosophy of History, Vol. 1 of 2 by Friedrich von Schlegel


 * 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?
 * Matthew Stewart, The Courtier and the Heretic (2006)


 * 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.
 * , Mathematics and Its History (1989)


 * 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.
 * , Mathematics and Its History (1989)


 * Leibnitz, though not propounding any full doctrine on evolution, gave it an impulse by suggesting a view contrary to the sacrosanct belief in the immutability of species... His punishment at the hands of the Church came a few years later, when in 1712, the Jesuits defeated his attempt to found an Academy of Science at Vienna.
 * Andrew Dickson White, A History of the Warfare of Science with Theology in Christendom Ch.1, p. 58 (1896)


 * Leibnitz, dominated by ideas of communication, is, in more than one way, the intellectual ancestor of the ideas of this book, for he was also interested in machine computation and in automata. My views in this book are very far from being Leibnitzian, but the problems with which I am concerned are most certainly Leibnitzian. Leibnitz's computing machines were only an offshoot of his interest in a computing language, a reasoning calculus which again was in his mind, merely an extention of his idea of a complete artificial language. Thus, even in his computing machine, Leibnitz's preoccupations were mostly linguistic and communicational.
 * Norbert Wiener, The Human Use of Human Beings (1950)