Great Throughts Treasury

This site is dedicated to the memory of Dr. Alan William Smolowe who gave birth to the creation of this database.

Gottfried Leibniz, fully Gottfried Wilhalm von Leibniz, Baron von Leibnitz

German Mathematician, Philosopher, Political Advisor and Logician, Developed Infinitesimal Calculus independently of Isaac Newton

"[Music is] the concealed art of computation for a soul unaware of its counting."

"All things are understood by God a priori, as eternal truths; for he does not need experience, and yet all things are known by him adequately. We, on the other hand, know scarcely anything adequately, and only a few things a priori; most things we know by experience, in the case of which other principles and other criteria must be applied."

"There are never in nature two beings which are exactly alike."

"There is a world of created beings - living things, animals, entelechies, and souls - in the least part of matter... Thus there is nothing waste, nothing sterile, nothing dead in the universe; no chaos, no confusions, save in appearance."

"There is nothing uncultivated, nothing sterile, nothing dead in the universe."

"As there is an infinite number of possible universes in the ideas of God, and as only one can exist, there must be sufficient reason for God’s choice, to determine him to one rather than to another. And this reason can only be found in the fitness, or in the degrees of perfection, which these worlds contain."

"By means of the soul or form there is a true unity which corresponds to what is called the I in us."

"Souls act according to the laws of final causes through appetitions, ends, and means. Bodies act according to the laws of efficient causes or motions. And the two realms, that of efficient causes and that of final causes, are in harmony with one another."

"Senses never give anything but example."

"Knowledge is leagued with the universe, and findeth a friend in all things; but ignorance is everywhere a stranger, unwelcome; ill at ease and out of place."

"The soul is the mirror of an indestructible universe."

"To love is to place our happiness in the happiness of another."

"To realize in its completeness the universal beauty and perfection of the works of God, we must recognize a certain perpetual and very free progress of the whole universe, such that it is always going forward to greater improvement... Although many substances have already attained a great perfection, yet on account of the infinite divisibility of the continuous, there always remain in the abyss of things slumbering parts which have yet to be awakened, to grow in size and worth, and, in a word, to advance to a more perfect state. And hence no end of progress is ever reached."

"A distinction must be made between true and false ideas, and that too much rein must not be given to a man's imagination under pretext of its being a clear and distinct intellection."

"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."

"All nature is a plenum. Everywhere there are simple substances, effectively separated from one another by actions of their own which are continually altering their relations; and each simple substance or distinct monad, which forms the center of a compound substance (e.g.. of an animal) and the principle of its oneness, is surrounded by a mass composed of an infinite number of other monads which constitute the body belonging to this central monad; corresponding to the affections of its body it represents, as in a kind of center, the things that are outside of it. And this body is organic, when it forms a kind of automaton or natural machine, which is a machine not only as a whole but also in its smallest observable parts. And since the world is a plenum everything is connected together, and each body acts on every other body more or less according to the distance, and is affected by it by reaction, it follows that every monad is a mirror that is alive or endowed with inner activity, is representative of the universe from its own point of view, and is as much regulated as the universe itself. The perceptions in the monad spring from one another according to the laws of the or the final causes of good and evil, which consist in the observable perceptions, regulated or unregulated- in the same way as the changes of the bodies and the phenomena outside spring from one another according to the laws of efficient causes, that is to say of motions. Thus there is a perfect harmony between the perceptions of the monad and the motions of the bodies, pre-established at the outset between the system of efficient causes and the system of final causes. Herein consists the concord and the physical union of the soul and the body, which exists without the one being able to change the laws of the other."

"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."

"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 someday 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."

"And although the earth and the air interspersed between the plants in the garden, or the water interspersed between the fish in the pond, are neither plant nor fish, yet they still contain them, though most usually of a subtlety which renders them imperceptible to us."

"An example of a subordinate maxim of the law of nature, in which it is shown that God always regularly conserves the same force, but not the same quantity of motion, contrary to the teaching of the Cartesians and some others."

"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."

"And as to this objection that space and time are quantities, or rather things having quantity, and that situation and order are not such, I reply that order also has its quantity: there is that which precedes and that which follows, there is distance or interval. Relative things have their quantity as well as absolutes: for example, ratios or proportions in mathematics have their quantity and are measured by logarithms, and yet they are relations. Thus although time and space consist in relations, they have their quantity none the less."

"And indeed in actual fact we find that everything in the world takes place in accordance with the laws of the eternal truths, not only geometrical but also metaphysical laws; that is, not only according to material necessities, but also according to formal necessities."

"And the principle of sufficient reason, by virtue of which we consider that no fact can be real or existing and no proposition can be true unless there is a sufficient reason, why it should be thus and not otherwise, even though in most cases these reasons cannot be known to us."

"And there must be simple substances, because there are compounds; for the compound is nothing but a collection or aggregatum of simples."

"As for me, I have more than once stated that I held space to be something purely relative, like time; space being an order of co-existences as time is an order of successions. For space denotes in terms of possibility an order of things which exist at the same time, in so far as they exist together, and is not concerned with their particular ways of existing: and when we see several things together we perceive this order of things among themselves."

"As regards the proposition that three is equal to two and one, which you adduce, Sir, as an example of intuitive knowledge, my comment is that it is simply the definition of the term three; for the simplest definitions of numbers are formed in this manner- two is one and one, three is two and one, four is three and one, and so on."

"As regards the objection that possibles are independent of the decisions of God, I grant that they are so actual decisions (though the Cartesians do not agree with this); but I hold that possible individual notions include a number of possible free decisions."

"At first, when I had freed myself from the yoke of Aristotle, I had believed in the void and atoms, for it is this which best satisfies the imagination. But returning to this view after much meditation, I perceived that it is impossible to find the principles of a true unity in matter alone, or in what is merely passive, since everything in it is but a collection or accumulation of parts ad infinitum. Now a multiplicity can be real only if it is made up of true unities which come from elsewhere and are altogether different from mathematical points, which are nothing but extremities of the extended and modifications out of which it is certain that nothing continuous could be compounded. Therefore, to find these real unities, I was constrained to have recourse to what might be called a real and animated point or to an atom of substance which must embrace some element of form or of activity in order to make a complete being."

"As I objected that space, taken as something real and absolute without bodies, would be a thing eternal, impassive, and independent of God, our author has tried to elude this difficulty by saying that space is a property of God."

"But at the present, men barely touch what is difficult and has not yet been attempted; but all run in crowds to what others have already done, where they cease not from copying and even from striving with one another. What one has built is first overthrown by another, who claims to found his reputation on the ruin of someone else's; but his own reign is no better established nor of longer duration. The fact is that they seek glory much more than truth, and seek rather to dazzle others than to enlighten themselves. To escape from this unhappy position, we must abandon the spirit of sect, and the affectation of novelty. We must imitate the geometers, who are not Euclideans or Archimedeans. They are all for Euclid and all for Archimedes, because they are all for their common master, that is, divine truth."

"But atoms of matter are contrary to reason, besides the fact that they also are composed of parts, since the invincible attachment of one part of another (granted that this could be reasonably conceived or supposed) would not destroy their diversity. It is only atoms of substance, that is to say unities which are real and absolutely without parts, which can be the sources of actions, and the absolute first principles of the composition of things, and as it were the ultimate elements into which substantial things, and as it were the ultimate elements into which substantial things can be analyzed. They might be called metaphysical points; there is about them something vital and a kind of perception, and mathematical points are their points of view for expressing the universe."

"But in simple substances the influence of one monad over another is ideal only; it can have its effect only through the intervention of God, inasmuch as in the ideas of God a monad rightly demands that God, in regulating the rest from the beginning of things, should have regard to itself. For since it is impossible for a created monad to have a physical influence on the inner nature of another, this is the only way in which one can be dependent on another."

"But I grant that there is a difference between a genuine absolute movement of a body and a simple relative change of its situation with respect to another body."

"But it is nowise follows that matter is eternal and necessary, unless we suppose that space is eternal and necessary: an altogether ill-founded supposition."

"But it is the knowledge of necessary and eternal truths that distinguishes us from the mere animals and gives us Reason and the sciences, raising us to the knowledge of ourselves and of God. And it is this in us that is called the rational soul or mind. It is also through the knowledge of necessary truths, and through their abstract expression, that we rise to acts of reflection, which make us think of what is called I, and observe that this or that is within us: and thus, thinking of ourselves, we think of being, of substance, of the simple and the compound, of the immaterial, and of God Himself, conceiving that what is limited in us is in Him without limits."

"But the greatest genius of the world is unable to force matters, and we must of necessity enter by the gates provided by nature if we are not to go astray. Moreover, one man alone cannot do everything at the outset; and for my part when I consider all the fine things M. Descartes has said, and said by himself, I marvel rather at what he has done than that there is something he has failed to do."

"But, besides the principle of the change, there must be a particular series of changes [un detail de ce qui change], which constitutes, so to speak, the specific nature and variety of the simple substances."

"But though the existence of necessities comes first of all in itself and in the order of nature, I agree none the less that it is not the first in the order of our knowledge. For you see that in order to prove its existence I have taken for granted that we think and that we have sensations. Here then are two absolute general truths, truths that is to say which treat of the actual existence of things: the one that we think, the other that there is a great variety in our thoughts. From the first it follows that we are, from the other it follows that there is something other than us; something other, that is to say, than that which thinks, which is the cause of the variety of our appearances."

"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."

"Everything that is possible demands to exist."

"Finally there are simple ideas of which no definition can be given; there are also axioms or postulates, or in a word primary principles, which cannot be proved and have no need of proof."

"Even in the games of children there are things to interest the greatest mathematician."

"Deeds make people."

"Concerning the important question of the author of sin, it is commonly believed that one may avoid the difficulty by claiming that sin in its essence is nothing but a pure privation without any reality, and that God is not the author of privations."

"Every man who acts wisely considers all the circumstances and connections of the decision he is taking, and the more so in proportion to his capacity."

"Every opinion has two causes: the temperament of the one with the opinion and the disposition of the object of the opinion."

"Every substance is as a world apart, independent of everything else except God."

"For as the whole is a plenum, which means that the whole of matter is connected, and as in a plenum every movement has some effect on distant bodies in proportion to their distance, so that each body not only is affected by those which touch it, and is in some way sensitive to whatever happens to them, but also by means of them is sensitive to those to those which touch the first bodies by which it is itself directly touched; it follows that this communication stretches out indefinitely. Consequently every body is sensitive to everything which is happening in the universe, so much so that one who saw everything could read in each body what is happening everywhere, and even what has happened or what will happen, by observing in the present the things that are distant in time as well as space."

"Fate is the decree of God or the necessity of events. Fatal things are those that will necessarily happen. God either does not decree concerning everything or, if he does decree concerning everything, then he is the author of absolutely everything."