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Home > Dictionary of Science Quotations > Scientist Names Index L > Gottfried Wilhelm Leibniz Quotes

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Gottfried Wilhelm Leibniz
(1 Jul 1646 - 14 Nov 1716)

German philosopher, mathematician and political adviser.


Science Quotes by Gottfried Wilhelm Leibniz (26 quotes)

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Nihil est sine ratione.
There is nothing without a reason.
— Gottfried Wilhelm Leibniz
Attributed.
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According to their [Newton and his followers] doctrine, God Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion. Nay, the machine of God's making, so imperfect, according to these gentlemen; that he is obliged to clean it now and then by an extraordinary concourse, and even to mend it, as clockmaker mends his work.
— Gottfried Wilhelm Leibniz
'Mr. Leibniz's First Paper' (1715). In H. G. Alexander (ed.), The Leibniz-Clarke Correspondence (1956), 11-2.
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All the different classes of beings which taken together make up the universe are, in the ideas of God who knows distinctly their essential gradations, only so many ordinates of a single curve so closely united that it would be impossible to place others between any two of them, since that would imply disorder and imperfection. Thus men are linked with the animals, these with the plants and these with the fossils which in turn merge with those bodies which our senses and our imagination represent to us as absolutely inanimate. And, since the law of continuity requires that when the essential attributes of one being approximate those of another all the properties of the one must likewise gradually approximate those of the other, it is necessary that all the orders of natural beings form but a single chain, in which the various classes, like so many rings, are so closely linked one to another that it is impossible for the senses or the imagination to determine precisely the point at which one ends and the next begins?all the species which, so to say, lie near the borderlands being equivocal, at endowed with characters which might equally well be assigned to either of the neighboring species. Thus there is nothing monstrous in the existence zoophytes, or plant-animals, as Budaeus calls them; on the contrary, it is wholly in keeping with the order of nature that they should exist. And so great is the force of the principle of continuity, to my thinking, that not only should I not be surprised to hear that such beings had been discovered?creatures which in some of their properties, such as nutrition or reproduction, might pass equally well for animals or for plants, and which thus overturn the current laws based upon the supposition of a perfect and absolute separation of the different orders of coexistent beings which fill the universe;?not only, I say, should I not be surprised to hear that they had been discovered, but, in fact, I am convinced that there must be such creatures, and that natural history will perhaps some day become acquainted with them, when it has further studied that infinity of living things whose small size conceals them for ordinary observation and which are hidden in the bowels of the earth and the depth of the sea.
— Gottfried Wilhelm Leibniz
Lettre Prétendue de M. De Leibnitz, à M. Hermann dont M. Koenig a Cité le Fragment (1753), cxi-cxii, trans. in A. O. Lovejoy, Great Chain of Being: A Study of the History of an Idea (1936), 144-5.
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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.
— Gottfried Wilhelm Leibniz
Epigraph, without citation, in J.R. Newman (ed.) The World of Mathematics (1956), 1832.
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He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.
— Gottfried Wilhelm Leibniz
Quoted, without citation, in Max Dehn, 'Mathematics, 300 B.C.-200 B.C.', The American Mathematical Monthly (Jan 1944), 51, No. 1, 31.
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If it were always necessary to reduce everything to intuitive knowledge, demonstration would often be insufferably prolix. This is why mathematicians have had the cleverness to divide the difficulties and to demonstrate separately the intervening propositions. And there is art also in this; for as the mediate truths (which are called lemmas, since they appear to be a digression) may be assigned in many ways, it is well, in order to aid the understanding and memory, to choose of them those which greatly shorten the process, and appear memorable and worthy in themselves of being demonstrated. But there is another obstacle, viz.: that it is not easy to demonstrate all the axioms, and to reduce demonstrations wholly to intuitive knowledge. And if we had chosen to wait for that, perhaps we should not yet have the science of geometry.
— Gottfried Wilhelm Leibniz
In Gottfried Wilhelm Leibnitz and Alfred Gideon Langley (trans.), New Essays Concerning Human Understanding (1896), 413-414.
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Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being. (1702)
[Alternate translation:] The Divine Spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world, that amphibian between being and not-being, which we call the imaginary root of negative unity.
— Gottfried Wilhelm Leibniz
Quoted in Félix Klein, Elementary Mathematics From an Advanced Standpoint: Arithmetic, Algebra, Analysis (1924), 56. Alternate translation as quoted in Tobias Dantzig, Number, the Language of Science: a Critical Survey Written for the Cultured Non-Mathematician (1930), 204
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In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.
— Gottfried Wilhelm Leibniz
In letter to Tschirnhaus. As quoted in George F. Simmons Calculus Gems (1992), 156, citing Dirk Jan Struik, 281-282.
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It follows from the supreme perfection of God, that in creating the universe has chosen the best possible plan, in which there is the greatest variety together with the greatest order; the best arranged ground, place, time; the most results produced in the most simple ways; the most of power, knowledge, happiness and goodness the creatures that the universe could permit. For since all the possibles in I understanding of God laid claim to existence in proportion to their perfections, the actual world, as the resultant of all these claims, must be the most perfect possible. And without this it would not be possible to give a reason why things have turned out so rather than otherwise.
— Gottfried Wilhelm Leibniz
The Principles of Nature and Grace (1714), The Philosophical Works of Leibnitz (1890), ed. G. M. Duncan, 213-4.
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It is God who is the ultimate reason things, and the Knowledge of God is no less the beginning of science than his essence and will are the beginning of things.
— Gottfried Wilhelm Leibniz
Letter on a General Principle Useful in Explaining the Laws of Nature (1687).
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It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used.
Describing, in 1685, the value to astronomers of the hand-cranked calculating machine he had invented in 1673.
— Gottfried Wilhelm Leibniz
From 'Machina Arithmetica in qua non Aditio tantum Subtractio', as translated by Mark Kormes in David Eugene Smith, A Source Book in Mathematics (1929), 181.
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It is worth noting that the notation facilitates discovery. This, in a most wonderful way, reduces the mind's labour.
— Gottfried Wilhelm Leibniz
In Eberhard Zeidler, Applied Functional Analysis: main principles and their applications (1995), 225.
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Music is the pleasure the human soul experiences from counting without being aware that it is counting.
— Gottfried Wilhelm Leibniz
As quoted, without citation, in William L. Schaaf, 'The Highest Rung', National Mathematics Magazine (May 1942), 16, 8, 395.
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Nothing is accomplished all at once, and it is one of my great maxims, and one of the most completely verified, that Nature makes no leaps: a maxim which I have called the law of continuity.
[Referring to the gradual nature of all change from an initial state, through a continuous series of intermediate stages, to a final state.]
— Gottfried Wilhelm Leibniz
Letter to de Volder, in Nouveaux Essais, New Essays (1703) translated by A. G. Langley (3rd ed. 1949). In Frederick Copleston, A History of Philosophy (1999), 292.
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Nothing is more important than to see the sources of invention which are, in my opinion more interesting than the inventions themselves.
— Gottfried Wilhelm Leibniz
Epigraph, without citation, in R.P. Watkins, Computer Problem Solving (1980). Webmaster has not yet been able to find a primary source. Can you help?
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Now this supreme wisdom, united to goodness that is no less infinite, cannot but have chosen the best. For as a lesser evil is a kind of good, even so a lesser good is a kind of evil if it stands in the way of a greater good; and the would be something to correct in the actions of God if it were possible to the better. As in mathematics, when there is no maximum nor minimum, in short nothing distinguished, everything is done equally, or when that is not nothing at all is done: so it may be said likewise in respect of perfect wisdom, which is no less orderly than mathematics, that if there were not the best (optimum) among all possible worlds, God would not have produced any.
— Gottfried Wilhelm Leibniz
Theodicy: Essays on the Goodness of God and Freedom of Man and the Origin of Evil (1710), 128.
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One cannot explain words without making incursions into the sciences themselves, as is evident from dictionaries; and, conversely, one cannot present a science without at the same time defining its terms.
— Gottfried Wilhelm Leibniz
'Of the Division of the Sciences' (1765), Book 4, Chap. 21, in New Essays on Human Understanding, trans. and ed. Peter Remnal (1981), 522.
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Our reasonings are grounded upon two great principles, that of contradiction, in virtue of which we judge false that which involves a contradiction, and true that which is opposed or contradictory to the false.
— Gottfried Wilhelm Leibniz
The Monadology and Other Philosophical Writings (1714), trans. Robert Latta (1898) 235.
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Taking mathematics from the beginning of the world to the time when Newton lived, what he had done was much the better half.
— Gottfried Wilhelm Leibniz
As quoted in Edmund Fillingham King, A Biographical Sketch of Sir Isaac Newton (1858), 97, stating this was Leibniz’s reply “when asked at the royal table in Berlin his opinion of Newton.” No source citation was given, although all the next quotes that followed had footnotes. The lack of citation leaves the accuracy of the quote unverified. If you know a primary source, please contact the Webmaster.
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The art of discovering the causes of phenomena, or true hypothesis, is like the art of deciphering, in which an ingenious conjecture greatly shortens the road.
— Gottfried Wilhelm Leibniz
In Gottfried Leibniz and Alfred Fideon Langley (trans.), 'Leibniz’s Critique of Locke', New Essays Concerning Human Understanding (1896), Book 4, Chap. 12, 526.
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The knowledge we have aquired ought not to resemble a great shop without order, and without inventory; we ought to know what we possess, and be able to make it serve us in our need.
— Gottfried Wilhelm Leibniz
In Hialmer Day Gould, New Practical Spelling (1905), 27
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The order, the symmetry, the harmony enchant us ... God is pure order. He is the originator of universal harmony
— Gottfried Wilhelm Leibniz
…...
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The soul [is a] mirror of an indestructible universe.
— Gottfried Wilhelm Leibniz
In Gottfried von Leibniz and ‎Robert Latta (trans.), The Monadology and Other Philosophical Writings (1898), 262.
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There are also two kinds of truths, those of reasoning and those of fact. Truths of reasoning are necessary and their opposite is impossible: truths of fact are contingent and their opposite is possible. When a truth is necessary, reason can be found by analysis, resolving it into more simple ideas and truths, until we come to those which are primary.
— Gottfried Wilhelm Leibniz
The Monadology and Other Philosophical Writings (1714), trans. Robert Latta (1898), 235-6.
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These principles have given me a way of explaining naturally the union or rather the mutual agreement [conformité] of the soul and the organic body. The soul follows its own laws, and the body likewise follows its own laws; and they agree with each other in virtue of the pre-established harmony between all substances, since they are all representations of one and the same universe.
— Gottfried Wilhelm Leibniz
The Monadology and Other Philosophical Writings (1714), trans. Robert Latta (1898), 262.
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We should like Nature to go no further; we should like it to be finite, like our mind; but this is to ignore the greatness and majesty of the Author of things.
— Gottfried Wilhelm Leibniz
Letter to S. Clarke, 1715. Trans. M. Morris and G. H. R. Parkinson, Leibniz: PhilosophicalWritings (1973), 220.
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Quotes by others about Gottfried Wilhelm Leibniz (23)

I see no good reason why the views given this volume [The Origin of Species] should shock the religious feelings of any one. It is satisfactory, as showing how transient such impressions are, to remember that the greatest discovery ever made by man, namely, the law of attraction of gravity, was also attacked by Leibnitz, “as subversive of natural, and inferentially of revealed, religion.”
The Origin of Species (1909), 520.
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The stone that Dr. Johnson once kicked to demonstrate the reality of matter has become dissipated in a diffuse distribution of mathematical probabilities. The ladder that Descartes, Galileo, Newton, and Leibniz erected in order to scale the heavens rests upon a continually shifting, unstable foundation.
Mathematics in Western Culture (1953), 382.
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Laplace would have found it child's-play to fix a ratio of progression in mathematical science between Descartes, Leibnitz, Newton and himself
The Education of Henry Adams: An Autobiography? (1918), 491.
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The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.
In 'What Knowledge is of Most Worth?', Presidential address to the National Education Association, Denver, Colorado (9 Jul 1895). In Educational Review (Sep 1895), 10, 109.
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We hold these truths to be self-evident.
Franklin's edit to the assertion of religion in Thomas Jefferson's original wording, “We hold these truths to be sacred and undeniable” in a draft of the Declaration of Independence changes it instead into an assertion of rationality. The scientific mind of Franklin drew on the scientific determinism of Isaac Newton and the analytic empiricism of David Hume and Gottfried Leibniz. In what became known as “Hume's Fork” the latters' theory distinguished between synthetic truths that describe matters of fact, and analytic truths that are self-evident by virtue of reason and definition.
As explained by Walter Isaacson in Benjamin Franklin: An American Life (2004), 312.
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It is not always the most brilliant speculations nor the choice of the most exotic materials that is most profitable. I prefer Monsieur de Reaumur busy exterminating moths by means of an oily fleece; or increasing fowl production by making them hatch without the help of their mothers, than Monsieur Bemouilli absorbed in algebra, or Monsieur Leibniz calculating the various advantages and disadvantages of the possible worlds.
Spectacle, 1, 475. Quoted in Camille Limoges, 'Noel-Antoine Pluche', in C. C. Gillispie (ed.), Dictionary of Scientific Biography (1974 ), Vol. 11, 43.
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It appears that the solution of the problem of time and space is reserved to philosophers who, like Leibniz, are mathematicians, or to mathematicians who, like Einstein, are philosophers.
Collected in Paul Arthur Schilpp (ed.), Albert Einstein: Philosopher-Scientist (1959), Vol. 1, 307. Also, in James Louis Jarrett and Sterling M. McMurrin (eds.), Contemporary Philosophy: A Book of Readings (1954), 71.
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Was it not the great philosopher and mathematician Leibnitz who said that the more knowledge advances the more it becomes possible to condense it into little books?
Opening remark in 'Introductory Note', Outline of Science: A Plain Story Simply Told (1922), Vol. 1, iii. Webmaster has not yet identified the quote in Leibnitz's original words (translated). If you know the primary source, please contact Webmaster.
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Libraries will in the end become cities, said Leibniz.
Aphorism 29 in Notebook C (1772-1773), as translated by R.J. Hollingdale in Aphorisms (1990). Reprinted as The Waste Books (2000), 37.
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[Isaac Newton] regarded the Universe as a cryptogram set by the Almighty—just as he himself wrapt the discovery of the calculus in a cryptogram when he communicated with Leibniz. By pure thought, by concentration of mind, the riddle, he believed, would be revealed to the initiate.
In 'Newton, the Man' (1946). In Geoffrey Keynes (ed.), Essays in Biography, 2nd edition (1951), 314.
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I am a Christian which means that I believe in the deity of Christ, like Tycho de Brahe, Copernicus, Descartes, Newton, Leibnitz, Pascal ... like all great astronomers mathematicians of the past.
…...
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Notable enough, however, are the controversies over the series 1 – 1 + 1 – 1 + 1 – … whose sum was given by Leibniz as 1/2, although others disagree. … Understanding of this question is to be sought in the word “sum”; this idea, if thus conceived—namely, the sum of a series is said to be that quantity to which it is brought closer as more terms of the series are taken—has relevance only for convergent series, and we should in general give up the idea of sum for divergent series.
…...
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Leibniz never married; he had considered it at the age of fifty; but the person he had in mind asked for time to reflect. This gave Leibniz time to reflect, too, and so he never married.
From the original French, “Leibnitz ne s'était point marié ; il y avait pensé à l'âge de cinquante ans; mais la personne qu’il avait en vue voulut avoir le temps de faire ses réflexions. Cela donna à Leibnitz le loisir de faire aussi les siennes, et il ne se maria point.” In 'Éloge de Leibniz' (1768), in Éloges de Fontenelle (1883), 132.
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Marly 30 July 1705. From all I hear of Leibniz he must be very intelligent, and pleasant company in consequence. It is rare to find learned men who are clean, do not stink and have a sense of humour.
From Letter (30 Jul 1705), to Sophie of Hanover, in Maria Kroll (trans.), Letters from Liselotte: Elisabeth-Charlotte, Princess Palatine and Duchess of Orléans, "Madame" 1652-1722 (1970), 131. Also seen in Alan L. MacKay, A Dictionary of Scientific Quotations (1991), 175. Also seen attributed to Charles-Louis de Secondat Montesquieu, in Alan L. MacKay, Dictionary of Scientific Quotations (1977), 175, citing Letter LXVI (30 Jul 1705) in Les Lettres Persanes. However, Webmaster does not find this quote when looking at Letter LXVI in Les Lettres Persanes (1721), text prepared by André Lefèvre (1873). (Can you help).
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The union of philosophical and mathematical productivity, which besides in Plato we find only in Pythagoras, Descartes and Leibnitz, has always yielded the choicest fruits to mathematics; To the first we owe scientific mathematics in general, Plato discovered the analytic method, by means of which mathematics was elevated above the view-point of the elements, Descartes created the analytical geometry, our own illustrious countryman discovered the infinitesimal calculus—and just these are the four greatest steps in the development of mathematics.
In Geschichte der Mathematik im Altertum und im Mittelalter (1874), 149-150. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 210. From the original German, “Die Verbindung philosophischer und mathematischer Productivität, wie wir sie ausser in Platon wohl nur noch in Pythagoras, Descartes, Leibnitz vorfinden, hat der Mathematik immer die schönsten Früchte gebracht: Ersterem verdanken wir die wissenschaftliche Mathematik überhaupt, Platon erfand die analytische Methode, durch welche sich die Mathematik über den Standpunct der Elemente erhob, Descartes schuf die analytische Geometrie, unser berühmter Landsmann den Infinitesimalcalcül—und eben daß sind die vier grössten Stufen in der Entwickelung der Mathematik.”
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It would be difficult and perhaps foolhardy to analyze the chances of further progress in almost every part of mathematics one is stopped by unsurmountable difficulties, improvements in the details seem to be the only possibilities which are left… All these difficulties seem to announce that the power of our analysis is almost exhausted, even as the power of ordinary algebra with regard to transcendental geometry in the time of Leibniz and Newton, and that there is a need of combinations opening a new field to the calculation of transcendental quantities and to the solution of the equations including them.
From Rapport historique sur les progrès des sciences mathématiques depuis 1789, et sur leur état actuel (1810), 131. As translated in George Sarton, The Study of the History of Mathematics (1936), 13. In the original French: “Il seroit difficile et peut-être téméraire d’analyser les chances que l’avenir offre à l’avancement des mathématiques: dans presque toutes les parties, on est arrêté par des difficultés insurmontables; des perfectionnements de détail semblent la seule chose qui reste à faire… Toutes ces difficultés semblent annoncer que la puissance de notre analyse est à-peu-près épuisée, comme celle de l’algèbre ordinaire l’étoit par rapport à la géométrie transcendante au temps de Leibnitz et de Newton, et qu’il faut des combinaisons qui ouvrent un nouveau champ au calcul des transcendantes et à la résolution des équations qui les contiennent.” Sarton states this comes from “the report on mathematical progress prepared for the French Academy of Sciences at Napoleon’s request”.
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There is something sublime in the secrecy in which the really great deeds of the mathematician are done. No popular applause follows the act; neither contemporary nor succeeding generations of the people understand it. The geometer must be tried by his peers, and those who truly deserve the title of geometer or analyst have usually been unable to find so many as twelve living peers to form a jury. Archimedes so far outstripped his competitors in the race, that more than a thousand years elapsed before any man appeared, able to sit in judgment on his work, and to say how far he had really gone. And in judging of those men whose names are worthy of being mentioned in connection with his,—Galileo, Descartes, Leibnitz, Newton, and the mathematicians created by Leibnitz and Newton’s calculus,—we are forced to depend upon their testimony of one another. They are too far above our reach for us to judge of them.
In 'Imagination in Mathematics', North American Review, 86, 223.
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It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heros of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that flourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment—judgment, that is, in matters not admitting of certainty—balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a tune, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on a priori grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic Applications de l’analyse à la géométrie; Lazare Carnot, author of the celebrated works, Géométrie de position, and Réflections sur la Métaphysique du Calcul infinitesimal; Fourier, immortal creator of the Théorie analytique de la chaleur; Arago, rightful inheritor of Monge’s chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service.
In Lectures on Science, Philosophy and Art (1908), 32-33.
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Leibnitz believed he saw the image of creation in his binary arithmetic in which he employed only two characters, unity and zero. Since God may be represented by unity, and nothing by zero, he imagined that the Supreme Being might have drawn all things from nothing, just as in the binary arithmetic all numbers are expressed by unity with zero. This idea was so pleasing to Leibnitz, that he communicated it to the Jesuit Grimaldi, President of the Mathematical Board of China, with the hope that this emblem of the creation might convert to Christianity the reigning emperor who was particularly attached to the sciences.
In 'Essai Philosophique sur les Probabiliés', Oeuvres (1896), t. 7, 119.
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It would be difficult to name a man more remarkable for the greatness and the universality of his intellectual powers than Leibnitz.
In System of Logic, Bk. 2, chap. 5, sect. 6.
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This [the fact that the pursuit of mathematics brings into harmonious action all the faculties of the human mind] accounts for the extraordinary longevity of all the greatest masters of the Analytic art, the Dii Majores of the mathematical Pantheon. Leibnitz lived to the age of 70; Euler to 76; Lagrange to 77; Laplace to 78; Gauss to 78; Plato, the supposed inventor of the conic sections, who made mathematics his study and delight, who called them the handles or aids to philosophy, the medicine of the soul, and is said never to have let a day go by without inventing some new theorems, lived to 82; Newton, the crown and glory of his race, to 85; Archimedes, the nearest akin, probably, to Newton in genius, was 75, and might have lived on to be 100, for aught we can guess to the contrary, when he was slain by the impatient and ill mannered sergeant, sent to bring him before the Roman general, in the full vigour of his faculties, and in the very act of working out a problem; Pythagoras, in whose school, I believe, the word mathematician (used, however, in a somewhat wider than its present sense) originated, the second founder of geometry, the inventor of the matchless theorem which goes by his name, the pre-cognizer of the undoubtedly mis-called Copernican theory, the discoverer of the regular solids and the musical canon who stands at the very apex of this pyramid of fame, (if we may credit the tradition) after spending 22 years studying in Egypt, and 12 in Babylon, opened school when 56 or 57 years old in Magna Græcia, married a young wife when past 60, and died, carrying on his work with energy unspent to the last, at the age of 99. The mathematician lives long and lives young; the wings of his soul do not early drop off, nor do its pores become clogged with the earthy particles blown from the dusty highways of vulgar life.
In Presidential Address to the British Association, Collected Mathematical Papers, Vol. 2 (1908), 658.
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Leibnitz’s discoveries lay in the direction in which all modern progress in science lies, in establishing order, symmetry, and harmony, i.e., comprehensiveness and perspicuity,—rather than in dealing with single problems, in the solution of which followers soon attained greater dexterity than himself.
In Leibnitz (1884), 112.
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It was his [Leibnitz’s] love of method and order, and the conviction that such order and harmony existed in the real world, and that our success in understanding it depended upon the degree and order which we could attain in our own thoughts, that originally was probably nothing more than a habit which by degrees grew into a formal rule. This habit was acquired by early occupation with legal and mathematical questions. We have seen how the theory of combinations and arrangements of elements had a special interest for him. We also saw how mathematical calculations served him as a type and model of clear and orderly reasoning, and how he tried to introduce method and system into logical discussions, by reducing to a small number of terms the multitude of compound notions he had to deal with. This tendency increased in strength, and even in those early years he elaborated the idea of a general arithmetic, with a universal language of symbols, or a characteristic which would be applicable to all reasoning processes, and reduce philosophical investigations to that simplicity and certainty which the use of algebraic symbols had introduced into mathematics.
A mental attitude such as this is always highly favorable for mathematical as well as for philosophical investigations. Wherever progress depends upon precision and clearness of thought, and wherever such can be gained by reducing a variety of investigations to a general method, by bringing a multitude of notions under a common term or symbol, it proves inestimable. It necessarily imports the special qualities of number—viz., their continuity, infinity and infinite divisibility—like mathematical quantities—and destroys the notion that irreconcilable contrasts exist in nature, or gaps which cannot be bridged over. Thus, in his letter to Arnaud, Leibnitz expresses it as his opinion that geometry, or the philosophy of space, forms a step to the philosophy of motion—i.e., of corporeal things—and the philosophy of motion a step to the philosophy of mind.
In Leibnitz (1884), 44-45. [The first sentence is reworded to better introduce the quotation. —Webmaster]
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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