Leibniz Quotes - Philosophy, Mathematics and Science

Principles of Metaphysical Certainty

Intellectual First Principles of the Essence of Things (a priori)

Every judgment is either true or false.
No judgment is simultaneously true and false.
Either the affirmation or the negation is true.
Either the affirmation or the negation is false.

Nothing can at the same time be and not be, but everything either is or is not.

Intellectual Principles of the Existence of Things (a priori)

Nothing is without a reason.

Of several incompatible possibles, the more perfect exists.

First Principles of Sensation or of First Perceptions (a posteriori)

1. I, who perceive, exist.
2. The things I perceive are various.

For two things above all occur to someone experiencing, that the perceptions are various, and that it is one and the same person who is perceiving. From this it is not only inferred that there is a percipient, but also that the reason that perceptions are so various must be outside the percipient; and therefore that there are other things besides me. From the first I arrive at a knowledge of myself, from the second at a knowledge of the world.

Principles of Opinions

That which is easier is more probable.

By the easier I mean that which has fewer requisites, i.e. that for whose sake fewer suppositions must be made.

Every single thing must be believed to remain in the state it was in, until one sees a reason for believing that it has abandoned this state.

Principle of Moral Certainty

Everything which is confirmed by many indications, which can hardly concur except in the truth, is morally certain, i.e. incomparably more probable than its opposite.

Principles of Physical Certainty

Phenomena which agree with the rest are held to be true, whereby Body, Space, Time, World, Individual are also adumbrated.

Substantial form, or soul, is the principle of unity and of duration, matter is that of multiplicity and change.
There is no place without body, and no time without change.

“All natural phenomena could be explained mechanically if we understood them well enough, but the principles of mechanics themselves cannot be so explained...since they depend on more substantive principles...The final analysis of the laws of nature leads us to the most sublime principles of order and perfection, which indicate that the universe is the effect of a universal intelligent power...It is wrong that laws are entirely indifferent, since they originate in the wisdom of their Author or in the principle of greatest perfection...The most beautiful thing about this view seems to me to be that the principle of perfection is not limited to the general but descends also to the particulars of things and of phenomena and that in this respect it closely resembles the method of optimal forms, that is to say, of forms which provide a maximum or minimum...If in the case of the curve of shortest descent between two given points, we choose any two points on this curve at will, the part of the line intercepted between them is also necessarily the line of shortest descent with regard to them. It is in this way that the smallest parts of the universe are ruled in accordance with the order of greatest perfection; otherwise the whole would not be so ruled.”

“As God calculates, so the world is made.”¹

“Everything happens mechanically in nature, that is, according to certain mathematical laws prescribed by God.²…Unless physical things can be explained by mechanical laws, God cannot, even if he chooses, reveal and explain nature to us.”³

“If there were no best among all possible worlds, God would not have created one.”

“The simplicity of means is balanced against the richness of ends….The simplicity of God’s ways relates to the means he adopts, while their variety, richness or abundance relate to ends or effects. These should be in balance with one another, as the money for putting up a building has to be balanced against its desired size and beauty. Admittedly, whatever God does costs him nothing—even less than it costs a philosopher or scientist to invent theories out of which to build his imaginary world—for God can bring a real world into existence merely by decreeing it. But in the exercise of wisdom by God or a scientist there is something analogous to the cost of a building, namely the number of independent decrees or theories that are involved. For God’s creative activity to be economical is for it to involve very few separate decrees; for a scientific theory to be economical in its means is for it to have very few basic principles or axioms. Reason requires that multiplicity of hypotheses or principles be avoided, rather as the simplest system is always preferred in astronomy.”⁴

“Some hypotheses can satisfy so many phenomena, and so easily, that they can be taken for certain. Among other hypotheses, those are to be chosen which are the simpler; these are to be presented, in the interim, in place of the true causes. The conjectural method a priori proceeds by hypotheses, assuming certain causes, perhaps, without proof, and showing that the things which now happen would follow from these assumptions. A hypothesis of this kind is like the key to a cryptograph, and the simpler it is, and the greater the number of events that can be explained by it; the more probable it is….The same effect can have several causes. Hence no firm demonstration can be made from the success of hypotheses. Yet I shall not deny that the number of phenomena which are happily explained by a given hypothesis may be so great that it must be taken as morally certain. Indeed, hypotheses of these kind are sufficient for everyday use. Yet it is also useful to apply less perfect hypotheses as substitutes for truth until a better one occurs, that is, one which explains the same phenomena more happily or more phenomena with equal felicity. There is no danger in this if we carefully distinguish the certain from the probable.”

“It must be admitted that a hypothesis becomes the more probable as it is simpler to understand and wider in force and power, that is, the greater the number of phenomena that can be explained by it, and the fewer the further assumptions. It may even turn out that a certain hypothesis can be accepted as physically certain if, namely, it completely satisfies all the phenomena which occur, as does the key to a cryptography. Those hypotheses deserve the highest praise (next to truth), however, by whose aid predictions can be made, even about phenomena or observations which have not been tested before; for a hypothesis of this kind can be applied, in practice, in place of truth.”

“God does nothing disorderly, and it isn’t possible even to feign events that are not regular….Not only does nothing absolutely irregular ever happen in the world, but we cannot even feign such a thing. Suppose that someone haphazardly draws points on a page, like people who practice the ridiculous art of fortune-telling through geometrical figures. I say that it is possible to find a single formula that generates a geometrical line passing through all those points in the order in which they were drawn. And if someone drew a continuous line which was now straight, now circular, now of some other kind, it would be possible to find a notion or rule or equation that would generate it. The contours of anyone’s face could be traced by a single geometrical curve governed by a formula. But when a rule is extremely complex, that which conforms to it passes for random. 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 hypotheses and the richest in phenomena, as might be a geometric line whose construction would be easy but whose properties and effects would be extremely remarkable and of great significance.”

“If arbitrarily complex laws are permitted, then the concept of law becomes vacuous, because there is always a law!”

“It follows from the supreme perfection of God, that in creating the universe He has chosen the best possible plan, in which there is

• the greatest variety along with the greatest order;

• the best arranged situation, space and time;

• the maximum effect produced by the simplest means;

• the highest levels of power, knowledge, happiness and goodness in the creatures that the universe could allow.

For since all the possibles in the 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 any reason why things have turned out so rather than otherwise.”⁵

“Nature in general always pursues its own end.”

“The Universe is only the collection of a certain kind of compossibles; and the actual Universe is the collection of all existent possibles, i.e. of those which form the richest compound. And as there are different combinations of possibles, some better than others, there are many possible Universes, each collection of compossibles making one of them.”

“Everything possible demands existence, and hence will exist unless something else prevents it, which also demands existence and is incompatible with the former.”

“The existent is the being which is compatible with the most things.”

“The possible demands existence by its very nature, in proportion to its possibility, that is to say, its degree of essence.”

“Even if we should imagine the world to be eternal, therefore, the reason for it would clearly have to be sought elsewhere, since we would still be assuming nothing but a succession of states, in anyone of which We can find no sufficient reason, nor can we advance the slightest toward establishing a reason, no matter how many of these states we assume. For even though there be no cause for eternal things, there must yet be understood to be a reason for them. For permanent things this reason is their necessity or essence itself; but in a series of changing things (if this is taken a priori to be eternal) it is a prevailing of inclinations, as we shall see presently, for here reasons do not necessitate (in the sense of an absolute or metaphysical necessity, whose contrary implies a contradiction) but incline. These considerations show clearly that we cannot escape an ultimate extramundane reason for things, or God, even by assuming the eternity of the world….

Furthermore, in order to explain a bit more distinctly how temporal, contingent, or physical truths arise from eternal, essential or metaphysical truths, we must first acknowledge that since something rather than nothing exists, there is a certain urge for existence or (so to speak) a straining toward existence in possible things or in possibility or essence itself; in a word, essence in and of itself strives for existence. Furthermore, it follows from this that all possibles, that is, everything that expresses essence or possible reality, strive with equal right for existence in proportion to the amount of essence or reality or the degree of perfection they contain, for perfection is nothing but the amount of essence.

Hence it is very clearly understood that out of the infinite combinations and series of possible things, one exists through which the greatest amount of essence or possibility is brought into existence. There is always a principle of determination in nature which must be sought by maxima and minima; namely, that a maximum effect should be achieved with a minimum outlay, so to speak….

Thus we now have a physical necessity derived from a metaphysical necessity, for although the world is not metaphysically necessary, where its contrary would imply a contradiction or logical absurdity, nevertheless it is physically necessary, that is, determined so that its contrary would imply imperfection or moral absurdity. And just as possibility is the principle of essence, so perfection or degree of essence is the principle of existence (since the degree of perfection determines the greatest number of things that are compossible). This shows at once how there may be freedom in the Author of the world, even though he does all things determinately because he acts on the principle of wisdom or perfection. Indifference arises from ignorance, and the wiser a man is, the more determined he is toward the most perfect….

We therefore have the ultimate reason for the reality of essences as well as existences in one being, which must necessarily be greater, higher, and prior to the world itself, since not only the existing things which compose the world but also all possibilities have their reality through it. But because of the interconnection of all these things, this ultimate reason can be found only in a single source. It is evident, however, that existing things are continuously issuing from this source and are being produced and have been produced by it, since no reason appears why one state of the world should issue from it rather than another, that of yesterday rather than today’s. It is clear, too, how God acts not merely physically but freely as well, and how there is in him not only the efficient but the final cause of the world. Thus we have in him the reason not merely for the greatness and power in the world mechanism as already established. but also for the goodness and wisdom exerted in establishing it….

And in general, I hold that there is nothing truer than happiness and nothing happier and sweeter than truth.”⁶

“There were an infinity of possible ways of creating the world, according to the different designs which God might form, and each possible world depends upon certain principal designs or ends of God proper to itself, i.e. certain free primitive decrees (conceived subratione possibilitatis), or laws of the general order of this possible universe, with which they are in accord, and whose notion they determine, as well as the notions of all the individual substances which must belong to this same Universe.”

“It is as yet unknown to men whence arises the incompossibility of different things, or of how different essences can conflict with each other, since all purely positive terms seem to be compatible with each other.”

“As there are an infinity of possible worlds, there are also an infinity of laws, some proper to one, others to another, and each possible individual of any world contains in its notion the laws of its world.”

“God can do everything that is possible, but he will do only what is best.”

“In God there is

• power, which is the source of everything, then

• knowledge, which contains every single idea, and then finally

• will, which produces changes in accordance with the principle of what is best.

And these are what correspond, respectively, to what in created monads constitute

• the subject, or base, or basic nature of the monad itself,

• the faculty of perception, and

• the appetitive faculty.

But in God these attributes are absolutely infinite or perfect, whereas in created monads …they are only imitations of the divine attributes, imitations that are more or less close depending on how much perfection they possess.”

“This simple, primal substance must include eminently the perfections contained in the derivative substances which are its effects. Thus it will have perfect power, knowledge, and will; that is to say, it will have omnipotence, omniscience, and omnibenevolence/sovereign goodness. And since justice, taken in its most general sense, is nothing but goodness in conformity with wisdom, there is also necessarily a sovereign justice in God. The primal Reason which has made things come to exist through himself has also made them depend on him for their staying in existence and for their operations. Whatever perfections they possess they continually receive from him; But whatever imperfection remains with them comes from the essential and original limitation of the created beings.”

“There is a reason in Nature why something should exist rather than nothing. This is consequence of the great principle that nothing happens without a reason, and also that there must be a reason why this thing exists rather than another.”

“It can be said that everything possible demands existence, inasmuch as it is founded on a necessary being which actually exists, and without which there is no way by which something possible may arrive at actuality.”

“From the conflict of all possibles demanding existence this at any rate follows, that there exists that series of things through which the greatest amount exists, or, the greatest of all possible series.”

“There exists, therefore, that which is the most perfect, since perfection is simply quantify of reality.”

“Distinct cogitability gives order to a thing and beauty to a thinker. For order is simply a distinctive relation of several things; confusion is when several things are present, but there is no way of distinguishing one from another.”

“The world is a cosmos, full of ornament; that is, that it is made in such a way that it gives the greatest satisfaction to an intelligent being.”

“An intelligent being’s pleasure is simply the perception of beauty, order and perfection. All pain contains something disordered, though only relative to the percipient; for in the absolute sense all things are ordered.”

“Further, the first cause is of the highest goodness, for whist it produces as much perfection as possible in things, at the same time it bestows on minds as much pleasure as possible, since pleasure consists in the perception of perfection.”⁷

“Knowledge is either

dim or vivid;
vivid knowledge is either

confused or clear;
clear knowledge is either

inadequate or adequate;
and adequate knowledge is either

symbolic or intuitive.

Knowledge that was at the same time both adequate and intuitive would be absolutely perfect.”

\[\text{knowledge}\begin{cases} \text{dim}\\ \text{vivid}\begin{cases} \text{confused}\\ \text{clear}\begin{cases} \text{inadequate}\\ \text{adequate}\begin{cases} \text{symbolic}\\ \text{intuitive} \end{cases} \end{cases} \end{cases} \end{cases}\]

“Our reasoning is based upon two great principles: first, the principle of contradiction, by means of which we decide that to be false which involves contradiction and that to be true which contradicts or is opposed to the false.

And second, the principle of sufficient reason, in virtue of which we believe that no fact can be real or existing and no statement true unless it has a sufficient reason why it should be thus and not otherwise. Most frequently, however, these reasons cannot be known by us.”⁸

“First truths are the ones that assert something of itself or deny something of its opposite.”

“As in Geometry and numbers, through the principle of the equality of the whole to all its parts, Geometry is subjected to an analytical Calculus, so in Mechanics, through the equality of the effect to all its causes, or of the cause to all its effects, we obtain certain equations, as it were, and a kind of mechanical Algebra by the use of this axiom.”

“The great foundation of mathematics is the principle of contradiction or identity, that is, that a proposition cannot be true and false at the same time, and that therefore $A$ is $A$ and cannot be not-$A$. This single principle is sufficient to demonstrate every part of arithmetic and geometry, that is, all mathematical principles. But in order to proceed from mathematics to natural philosophy, another principle is required …the principle of sufficient reason, namely, that nothing happens without a reason why it should be so rather than otherwise.”

“No one fittingly understands the works of God unless he sufficiently recognizes in them this fact: that the effect is the trace of its cause.”

“Thus far we have spoken only of what goes on in the natural world; now we must rise to metaphysics, by making use of the great principle, little employed in general, which affirms that nothing happens without a sufficient reason; i.e. that

for any true proposition $P$, it is possible for someone who understands things well enough to give a sufficient reason why it the case that $P$ rather than not-$P$.

This principle being laid down, the first question we are entitled to put will be, why is there something rather than nothing? After all, nothing is simpler and easier than something. Also, given that things have to exist, we must be able to give a reason why they have to exist as they are and not otherwise.”⁹

“Because all truths follow from first truths with the help of definitions, it follows that in any true proposition the predicate or consequent is always in the subject or antecedent. It is just this — as Aristotle observes — that constitutes the nature of truth in general, or the true-making connection between the terms of a statement. In identities the connection of the predicate with the subject (its inclusion in the subject) is explicit; in all other true propositions it is implicit, and has to be shown through the analysis of notions; a priori demonstration rests on this.”

“Finally I gave a decisive argument, which in my view amounts to a proof. This is that in all true affirmative propositions, necessary or contingent, universal or singular, the notion of the predicate is always in some way included in that of the subject—the predicate is present in the subject—or I do not know what truth is.

Now, I want nothing more in the way of connectedness here than what is found objectively between the terms of a true proposition, and it is only in this sense that I say that the notion of an individual substance involves all its events and all its denominations, even those that are commonly called extrinsic (that is to say, which belong to it only in virtue of the general interconnectedness of things, and of the fact that it expresses the whole universe in its way) because there must always be some foundation for the connection between the terms of a proposition, and it must be found in their notions. That is my great principle, with which I believe all philosophers should agree, and of which one of the corollaries is the common axiom that nothing happens without a reason, and that one can always explain why things have gone as they have rather than otherwise, even though that reason often inclines without necessitating, since perfect indifference is a chimerical and incomplete supposition.”

“Each individual substance expresses the whole universe in its own way, and that all its events, together with all their circumstances and the whole sequence of external things, are included in its notion.”

“Each substance is like a world apart, independent of all other things, except for God.”

“Generally, every true proposition (which is not identical or true per se) can be proved a priori by the help of axioms, or propositions true per se, and by the help of definitions or ideas. For as often as a predicate is truly affirmed of a subject, some real connection is always judged to hold between the predicate and the subject, and thus in any proposition: $A$ is $B$ (or, $B$ is truly predicated of $A$), $B$ is always in $A$ itself, or its notion is in some way contained in the notion of $A$ itself; and this either with absolute necessity, in propositions of eternal truth, or with a kind of certainty, depending upon a supposed decree of a free substance, in contingent things; and this decree is never wholly arbitrary and destitute of foundation, but always some reason for it (which however inclines, and does not necessitate), can be given, which could itself be deduced from analysis of the notions (if this were always within human power), and certainly does not escape the omniscient substance, which sees everything a priori by means of ideas themselves and its own decrees. It is certain, therefore, that all truths, even the most contingent, have an a priori proof, or some reason why they are rather than are not. And this is itself what people commonly say, that nothing happens without a cause, or that nothing is without a reason.”

“This demonstration of this predicate of Caesar [that he resolved to cross the Rubicon] is not as absolute as those of numbers or of Geometry, but presupposes the series of things which God has chosen freely, and which is founded on the first free decree of God, namely, to do always what is most perfect, and on the decree which God has made (in consequence of the first), in regard to human nature, which is that man will always do (though freely) what appears best. Now every truth which is founded on decrees of this kind is contingent, although it is certain….All contingent propositions have reasons for being as they are rather than otherwise, or (what is the same thing) they have a priori proofs of their truth, which render them certain, and show that the connection of subject and predicate in these propositions has its foundation in the nature of the one and the other; but they do not have demonstrations of necessity, since these reasons are only founded on the principle of contingency, or of the existence of things, i.e. on what is or appears the best among several equally possible things.”

“The nature of an individual substance or of a complete being is to have a notion so complete that it is sufficient to contain and to allow us to deduce from it all the predicates of the subject to which this notion is attributed.”

“Every extrinsic denomination has an intrinsic one for its foundation.”

“There are no purely extrinsic denominations.”

“Every individual substance contains in its complete notion the entire universe and everything that exists in it—past, present, and future.”

”[The complete notion] can be approached more and more nearly, so that the difference shall be less than any given difference.”

“There are also two kinds of truths, truths of reasoning and truths 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, we can find the reason for it by analysis, breaking it down into simple ideas and truths until we reach the primitive….But there must also be a sufficient reason for contingent truths or truths of fact, i.e. for the sequence of things which are dispersed throughout the universe of created beings, in which the resolution into particular reasons might go on into endless detail.”

“It is essential to discriminate between necessary or eternal truths, and contingent truths or truths of fact; and these differ from each other almost as rational numbers and surds. For necessary truths can be resolved into such as are identical, as commensurable quantities can be brought to a common measure; but in contingent truths, as in surd numbers, the resolution proceeds to infinity without ever terminating. And thus the certainty and the perfect reason of contingent truths is known to God only, who embraces the infinite in one intuition.”

“Paradoxical as it appears, it is impossible for us to have knowledge of individuals, and to find the means of determining exactly the individuality of any thing, unless we keep it [the thing?] itself; for all the circumstances may recur; the smallest differences are insensible to us; the place and the time, far from determining [things] of themselves, need to be themselves determined by the things they contain. What is most noteworthy in this is, that individuality involves infinity, and only he who is capable of understanding it [infinity] can have knowledge of the principle of individuation of such or such a thing; which comes from the influence (rightly understood) of all the things in the universe on one another. It is true that the matter would be otherwise if there were atoms of Democritus; but also there would then be no difference between two different individuals of the same shape and size.”

“All individual things are successions or are subject to succession….For me nothing is permanent in things except the law itself which involves a continuous succession and which corresponds, in individual things, to that law which determines the whole world.”

“By the persistence of the same law of the series, or of continuous simple transition, which leads us to the opinion that one and the same subject or monad is changing. That there should be a persistent law, involving the future states of that which we conceive as the same, is just what I assert to constitute it the same substance.”

“Each substance expresses the whole universe, but some more distinctly than others, especially each in regard to certain things, and according to its point of view. The union of soul and body, and even the operation of one substance on another, consists only in this perfect mutual agreement, purposely established by the order of the first creation, in virtue of which each substance, following its own laws, falls in with what the others demand, and the operations of the one thus follow or accompany the operation or change of the other.”¹⁰

“Terms are the same if one can be substituted in place of the other without destroying truth. If there are $A$ and $B$, and $A$ is an ingredient of some true proposition, and on substituting $B$ for $A$ in some place a new proposition is formed which is also true, and this always holds good in the case of any such proposition, then $A$ and $B$ are said to be the same; and conversely if $A$ and $B$ are the same, the substitutions I have mentioned will hold for good. That $A$ is the same as $B$ signifies that the one can be substituted for the other, salva veritate, in any proposition whatever.”

“I infer from [the Principle of Sufficient Reason]…that there are not in nature two real, absolute beings, indiscernible from each other; because if there were, God and nature would act without reason, in treating the one otherwise than the other; and that therefore God does not produce two pieces of matter perfectly equal and alike.”

“In nature there can’t be two individual things that differ in number alone.”

“Indistinguishable things are identical.”

“Things which are different must differ in some way, or have in themselves some assignable diversity.”

“What is not truly one being is not one being either. ‘One’ and ‘being’ are reciprocal.”

“It is always necessary that, besides the difference of time and place, there should be an internal principle of distinction, and though there be several things of the same species, it is none the less true that there are none perfectly similar: thus, though time and place (i.e. relation to the external) help us to distinguish things which by themselves we do not well distinguish, things are none the less distinguishable in themselves. Thus the essence (le précis) of identity and diversity consists not in time and place, though it is true that the diversity of things is accompanied by that of time and place, because they bring with them different impressions on the thing.”

“It is never true that two substances are entirely alike, differing only in being two rather than one. It also follows that a substance cannot begin except by creation, nor come to an end except by annihilation; and because one substance can’t be destroyed by being split up, or brought into existence by the assembling of parts, in the natural course of events the number of substances remains the same, although substances are often transformed. Moreover, each substance is like a whole world, and like a mirror of God, or indeed of the whole universe, which each substance expresses in its own fashion—rather as the same town looks different according to the position from which it is viewed. In a way, then, the universe is multiplied as many times as there are substances, and in the same way the glory of God is magnified by so many quite different representations of his work. It can even be said that each substance carries within it, in a certain way, the imprint of God’s infinite wisdom and omnipotence, and imitates him as far as it can. For it expresses (though confusedly) everything that happens in the universe—past, present, and future—and this is a little like infinite perception or knowledge. And as all the other substances express this one in their turn, and adapt themselves to it—that is, they are as they are because it is as it is—it can be said to have power over all the others, imitating the creator’s omnipotence.”

“Every individual created substance exercises physical action and passion on all the others. Any change made in one substance leads to corresponding changes in all the others, because the change in the one makes a difference to the relational properties of the others.”

“No created substance exerts a metaphysical action or influence on another, for to say nothing of the fact that it cannot be explained how anything can pass over from one thing into the substance of another, it has already been shown that all the futures of each thing follow from its own concept. What we call causes are in metaphysical rigour only concomitant requisites. What we call ‘causes’ are, speaking with metaphysical strictness, only concurrent requirements. This too is illustrated by our experience of nature. For bodies really rebound from others through the force of their own elasticity, and not through the force of other things, even if a body other than $x$ is required in order for $x$’s elasticity to be able to act.”

“This variety of thoughts cannot come from what thinks, since a single thing cannot be the cause of the changes in itself. For everything remains in the state in which it is, if there is nothing to change it; and not having been determined of itself to have certain changes rather than others, we could not begin attributing any variety to it, without saying something for which there is confessedly no reason, which is absurd.”

“There is no vacuum. For if there were empty space, two different parts of it could be perfectly similar and congruent and indistinguishable from one another. Thus, they would differ in number alone—differ in being two, but not in any other way—which is absurd. One can also prove that time is not a thing, …if time were a thing there could be stretches of empty time, i.e. time when nothing happens; and two parts of such empty time would be exactly alike, differing only in number, which is absurd.”

“Things which are uniform and contain no variety are never anything but abstractions, like time, space, and the other entities of pure mathematics.”

“Extension and motion, are not substances, but true phenomena (like rainbows and reflections). The same holds for bodies, to the extent that there is nothing to them but extension and motion.”

“Properly speaking, matter isn’t composed of constitutive unities(monads), but results from them, since matter, that is, extended mass is only a phenomenon grounded in things, like a rainbow or a perihelion, and all reality belongs only to unities.”

“Now this interconnection and accommodation of every created thing to every other, of all to each, gives every simple substance relations that express all the others so that each one is a perpetual living mirror of the universe.”¹¹

“Considering the matter carefully, we must say that there is nothing in things but simple substances, and in them, perception and appetition. Moreover, matter and motion are not substances or things as much as they are the phenomena of perceivers, the reality of which is situated in the harmony of the perceivers with themselves (at different times) and with other perceivers.”

“Time and space are not things, but orders of things.”

“Space and time taken together constitute the order of possibilities of one entire universe, so that these orders—space and time, that is—relate not only to what actually is but also to anything that could be put in its place.”¹²

“I hold space to be something merely relative, as time is, that I hold space to be an order of coexistences, just as time is an order of successions. For space denotes, in terms of possibility, an order of things that exist at the same time, considered as existing together, without entering into their particular manners of existing. And when many things are seen together, one perceives that order of things among themselves….If space was an absolute being, there would something happen, for which it would be impossible there should be a sufficient reason. Which is against my Axiom. And I prove it thus. Space is something absolutely uniform; and without the things placed in it, one point of space does not absolutely differ in any respect whatsoever from another point of space. Now from hence it follows (supposing space to be something in itself, besides the order of bodies among themselves), that it is impossible there should be a reason why God, preserving the same situation of bodies among themselves, should have placed them in space after one particular manner, and not otherwise; why everything was not placed the quite contrary way, for instance by changing east into west. But if space is nothing else but that order or relation; and is nothing at all without bodies, but the possibility of placing them; then those two states, the one such as it now is, the other supposed to be the quite contrary way, would not at all differ from one another. Their difference, therefore, is only to be found in our chimerical supposition of the reality of space in itself. But in truth the one would exactly be the same thing as the other, they being absolutely indiscernible; and consequently there is no room to enquire after a reason of the preference of the one to the other.”¹³

“The case is the same with respect to time….The same argument proves that instants, considered without the things, are nothing at all; and that they consist only in the successive order of things.”

“Time is the measure of motion, i.e. uniform motion is the measure of non-uniform motion.”

“I don’t say, therefore, that space is an order or situation, but an order of situations, or an order according to which situations are disposed, and that abstract space is that order of situations when they are conceived as being possible.”

“If place by itself does not make a change, it follows that there can be no change which is merely local. In general, place, position and quantity, such as number and proportion, are merely relations, and result from other things which by themselves either constitute or terminate a change. To be in a place seems, abstractly at any rate, to imply nothing but position. But in actuality, that which has a place must express place in itself, so that distance and degree of distance involves also a degree of expressing in the thing itself a remote thing, either of affecting it or of receiving an affection from it. So, in fact, situation really involves a degree of expressions.”

“If motion is nothing but the change of contact or of immediate vicinity, it follows that we can never define which thing is moved. For just as the same phenomena may be interpreted by different hypotheses in astronomy, so it will always be possible to attribute the real motion to either one or the other of the two bodies which change their mutual vicinity or position. Hence, since one of them is arbitrarily chosen to be at rest or moving at a given rate in a given line, we may define geometrically what motion or rest is to be ascribed to the other, so as to produce the given phenomena. Hence if there is nothing more in motion than this reciprocal change, it follows that there is no reason in nature to ascribe motion to one thing rather than to others. The consequence of this will be that there is no real motion.”

“No eye, wherever in matter it might be placed, has a sure criterion for telling from the phenomena where there is motion, how much motion there is and of what sort it is, or even whether God moves everything around it, or whether he moves that very eye itself.”

“In order to say that something is moving, we will require not only that it change its position with respect to other things but also that there be within itself a cause of change, a force, an action.”

“There is, moreover, a certain order in the transition of our perceptions when it passes from one to another through intervening ones. This order too we can call a path. But since this order can vary in infinite ways, there must necessarily be one simplest order, which would in fact be that order which proceeds according to the thing’s own nature through determinate intermediate perceptions, i.e. through those which are related as simply as possible to the two extrema. For unless this were so, there would be no order, and no reason for distinguishing among co-existing things, since one could pass from one given thing to another by any path whatever. And this simplest order is the shortest path from one to the other, whose magnitude is called distance.”

“The first thing to be properly grasped is that in substances, including created ones, force is absolutely real in a way in which space, time and motion are not. Those three are in a way beings of reason — they are not things in the world but upshots of certain ways of thinking about things in the world. The only truth and reality there is to them comes from their involving the divine attributes of immensity, eternity and activity, and the force of created substances. It follows immediately from this that there is no empty place or time, and also that if we set aside force and consider motion purely in terms of the geometric notions of size and shape, and changes in them, motion is really nothing more than change of place. So motion as we experience it is nothing but a relation.”

“There is no atom, which means that any body could be split. In fact, every body, however small, is actually subdivided. Because of that, each body, while it constantly changes because it is acted on by everything else in the universe in ways that make it alter, also preserves all the states that have been impressed on it in the past and contains in advance all that will be impressed on it in the future….Not only must there be effects produced in an atom from all the impacts of the universe upon it, but also conversely the state of the whole universe must be inferable from the states of the atom—the cause must be inferable from the effect. However, any given motion of an atom and any given shape could have come about through different impacts, so there is no way to infer from the present shape and motion of the atom what effects have been had upon it.”

“Something further that follows from my notions of body and forces is that whatever happens in a substance can be understood as happening spontaneously—not caused from outside the substance—and in an orderly way. Connected to this is the proposition that no change takes place in a jump.

Given this, it also follows that there cannot be atoms. To see how that follows, think about two bodies that collide and rebound away from each other. If these bodies were atoms—that is, bodies of maximal hardness and inflexibility—then clearly their change of motion would be taking place in a ‘jump’, i.e. instantaneously, for the forward motion would have to change to backward at the very moment of collision. ‘Perhaps the atoms might become stationary for an instant immediately after the collision, and then start moving in a different direction.’ That means that for a moment they lose all their force and then regain it! Anyway, as well as containing this and other absurdities, this proposal would again involve a change taking place in a single jump—an instantaneous change from motion to rest, with no intermediate stages.

So we have to recognise that when two bodies collide, from the point of collision onwards they are gradually compressed, like two balloons, and as their motion towards each other continues the pressure increases continuously; that makes the motion decrease as the force of striving is converted from a force for motion into a force for the elasticity of the bodies, until they come to a complete standstill. Then, their elasticity begins to restore them, and they rebound from each other in the opposite direction; their motion begins from rest and continuously increases until they finally reach the same speed they had when they came together but in the opposite direction….”

“In just the same way, one shape can’t be turned into another—e.g. a circle into an oval—except by passing through all the countless intermediate shapes, and nothing gets from one place to another, or from one time to another, without going through all the places and times in between.”

“No transition happens by a leap….This holds, I think, not only of transitions from place to place, but also of those from form to form, or from state to state.”

“Everything goes by degrees in nature, and nothing by leaps, and this rule as regards changes is part of my law of continuity. But the beauty of nature, which desires distinguished perceptions, demands the appearance of leaps.”

“It also follows that all rebounding arises from elasticity….Bodies change shape before they bounce off anything. And finally there is the most wonderful conclusion that each body, however small, is elastic, and is permeated by a fluid consisting of bodies that are even smaller than it is.”

“Nothing happens all at once, and it is one of my great maxims, and among the most completely verified, that nature never makes leaps: which I called the Law of Continuity….I have remarked also that, in virtue of insensible variations, two individual things cannot be perfectly similar, and must always differ more than numerically.”

“In every body there is some force or motion. No body is so small that it is not in turn actually divided into parts which are excited by different motions; and therefore in every body there are actually an infinite number of bodies. Every change of any body propagates its effect to bodies at any distance; i.e. all bodies act on all bodies and are acted upon by all. Every body is restrained by those which surround it so that its parts do not disperse, and therefore all bodies strive with each other reciprocally, and each body resists the whole universe of bodies.”

“Noticeable perceptions arise by degrees from ones which are too minute to be noticed.”

“When the difference of two cases can be diminished below every given magnitude in the data or in what is posited, it must also be possible to diminish it below every given magnitude in what is sought or in what results.”

“When the data form a series, so do the consequences.”

“If in a series of cases the inputs approach each other continuously and eventually become the same, the consequences or outcomes must do so also.”¹⁴

“Every particle in the universe contains a world of an infinity of creatures. However, the continuum is not divided into points, because points are not parts but boundaries; nor is it divided in all possible ways, because the contained creatures are not all separately there. It’s just that a series of divisions could go on ad infinitum separating some from others at each stage. But no such sequence separates out all the parts, all the ‘contained creatures’, because every division leaves some of them clumped together—just as someone who bisects a line leaves clumped together some parts of it that would be separated if the line were trisected.”

“There is no determinate shape in actual things, for no determinate shape can be appropriate for infinitely many effects. So neither a circle, nor an ellipse, nor any other definable line exists except in the intellect; lines don’t exist until they are drawn, and parts don’t exist until they are separated off.”

“It is to be observed that matter, taken as a complete being (i.e. secondary matter, as opposed to primary, which is something purely passive, and consequently incomplete) is nothing but a collection or what results from it, and that every real collection presupposes simple substances or real unities, and when we consider further what belongs to the nature of these real unities, i.e. perception and its consequences, we are transferred, so to speak, into another world, that is, into the intelligible world of substances, whereas before we were only among the phenomena of the senses.”

“Something unextended is required for the substance of bodies. Without that there would be no source for the reality of phenomena or for true unity….Since I have ruled out atoms, all that remains as a source of unity is something unextended, analogous to the soul, which they once called ‘form’ or ‘species’.”

“The forms that constitute substances must (except for ones that God still intends to create specially) have been created with the world and must always continue to exist.”

“All simple substances or created Monads might be called Entelechies, for they have in them a certain perfection; they have a certain self-sufficiency which makes them the sources of their internal activities and, so to speak, incorporeal automata.”

“A monad’s perceptions arise out of its other perceptions by the

• laws of appetites—the laws of the final causes of good and evil (these appetites are just conspicuous perceptions, whether orderly or disorderly),

just as changes in bodies or in external phenomena arise one from another by the

• laws of efficient causes—the laws governing the movements of bodies.

So there is perfect harmony between the perceptions of the monad and the movements of bodies, a harmony that was pre-established from the outset between the system of final causes and that of efficient causes.”

“In general, we must hold that everything in the world can be explained in two ways: through the kingdom of power, that is, through efficient causes, and through the kingdom of wisdom, that is, through final causes, through God, governing bodies for his glory, like an architect, governing them as machines that follow the laws of size or mathematics, governing them, indeed, for the use of souls, and through God governing for his glory souls capable of wisdom, governing them as his fellow citizens, members with him of a certain society, governing them like a prince, indeed like a father, through laws of goodness or moral laws.”

“The present state of body is born from the preceding state through the laws of efficient causes; the present state of the soul is born from its preceding state through the laws of final causes. The one is the place of the series of motion, the other of the series of appetites; the one is passed from cause to effect, the other from end to means. And in fact, it may be said that the representation of the end in the soul is the efficient cause of the representation in the same soul of the means.”

“The laws that connect the thoughts of the soul in the order of final causes and in accordance with the evolution of perceptions must produce pictures that meet and harmonize with the impressions of bodies on our organs; and likewise the laws of movements in the body, which follow one another in the order of efficient causes, meet and so harmonize with the thoughts of the soul that the body is induced to act at the time when the soul wills it.”

“There is no real influence of one created substance on another.”

“God originally created the soul (and any other real unity) in such a way that everything must arise for it from its own depths, through perfect spontaneity relative to itself, and yet with a perfect conformity relative to external things.”

“This is what makes every substance represent the whole universe exactly and in its own way, from a certain point of view, and makes the perceptions or expressions of external things occur in the soul at a given time, in virtue of its own laws, as if in a world apart, and as if there existed only God and itself.”

“The organized mass, in which the point of view of the soul lies, being expressed more closely by the soul, is in turn ready to act by itself, following the laws of the corporeal machine, at the moment when the soul wills it to act, without disturbing the laws of the other—the spirits and blood then having exactly the motions that they need to respond to the passions and perceptions of the soul.”

“It is this mutual relation, regulated in advance in each substance of the universe, which produces what we call their communication, and which alone brings about the union of soul and body.”

“For what happens with perception happens nonetheless mechanically, and to the passions of the soul there correspond bodily motions in the organs which always follow mechanical laws….Everything is by nature to be understood clearly and distinctly and could be manifested to our understanding by God if he willed to do so. And the operation of a body cannot be understood adequately unless we know what its parts contribute; hence we cannot hope for the explanation of any corporeal phenomenon without taking up the arrangement of its parts. But from this it does not at all follow that nothing can be understood as true in bodies save what happens materially and mechanically, nor does it follow that only extension is to be found in matter. For even though the confused attributes of bodies can be referred back to distinct ones, we must recognize that there are two kinds of distinct attributes, one of which must be sought in mathematics, the other in metaphysics. Mathematical science provides magnitude, figure, situation, and their variations, but metaphysics provides existence, duration, action and passion, force of acting, and end of action, or the perception of the agent. Hence I believe that there is in every body a kind of sense and appetite, or a soul, and furthermore, that to ascribe a substantial form and perception, or a soul, to man alone is as ridiculous as to believe that everything has been made for man alone and that the earth is the center of the universe. But on the other hand, I think that when once we have demonstrated the general mechanical laws from the wisdom of God and the nature of the soul, then it is as improper to revert to the soul or to substantial forms everywhere in explaining the particular phenomena of nature as it is to refer everything to the absolute will of God. For the action of the soul is determined by the state of the organ of the soul and its object, and the operation of God by the conditions of the individual things, and this not by the necessity of matter but by the impulsion of the final cause or the good.”

“Imagine two clocks or watches which are in perfect agreement. Now this can happen in three ways. The first is that of a natural influence….The second way of making two clocks, even poor ones, agree always is to assign a skilled craftsman to them who adjusts them and constantly sets them in agreement. The third way is to construct these two timepieces at the beginning with such skill and accuracy that one can be assured of their subsequent agreement.”

“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.”

“As very often there are diverse courses to choose from, one might, instead of the balance, compare the soul with a force which puts forth effort on various sides simultaneously, but which acts only at the spot where action is easiest or there is least resistance.”

“Our soul is also architectonic in its voluntary actions; and in discovering the sciences according to which God has regulated things (by weight, measure, number, etc.) it imitates in its realm and in the small world in which it is allowed to work, what God does in the large work.”

“I accept that all bodily phenomena can be traced back to mechanical efficient causes, but we are to understand that those mechanical laws as a whole derive from higher reasons.”

“There is a good distinction between

• perception $=$ the internal state of a monad that represents external things, and

• awareness $=$ consciousness, or the reflective knowledge of that internal state.

I think that the general name ‘monad’ or ‘entelechy’ is adequate for substances that have mere perception and nothing more, and that we should reserve ‘soul’ for the monads with perceptions that are more distinct and accompanied by memory.

What distinguishes us from the lower animals is our knowledge of necessary and eternal truths and, associated with that, our having a kind of ‘following from’ that involves necessity and depends on reason, rather than merely the ‘following from’ of the animals, which is wholly contingent and depends on memory. This is what gives us reason and science, raising us to the knowledge of ourselves and of God. And it’s what is called ‘rational soul’ or ‘mind’ in us.

Our knowledge of necessary truths, and our grasp of the abstractions they involve, raise us to the level of acts of reflection, which make each of us aware of the thing that is called ‘I’, and let us have thoughts about this or that thing in us. And by thinking of ourselves in this way we think of being, of substance, of simples and composites, of what is immaterial—and of God himself, through the thought that what is limited in us is limitless in him. And so these acts of reflection provide the principal objects of our reasonings.”

“Every soul is immortal; the mind, however, is not only immortal, but also always has some knowledge of itself, or memory of what has preceded, and therefore is capable of reward and punishment.”

“Each monad, together with its own body, constitutes a living substance. So every living substance is made up of smaller living substances which in their turn are made up of still smaller ones, and so on down to infinity. Thus, not only is there life everywhere—the life of organisms equipped with limbs or organs—but there are infinite levels of life among monads, some of which are more or less dominant over others.”

“God’s supreme wisdom made him choose, above all, the laws of motion that hang together the best, and that have the best fit with abstract or metaphysical reasoning. They conserve the same quantity of

• total or absolute force, i.e. of action, of

• relative force, i.e. of reaction, and of

• directional force.

Furthermore, adding to the wonderful simplicity of the basic laws of physics, action is always equal to reaction, and the complete effect is always equivalent to the total cause….If we want to explain why they are laws, it turns out, surprisingly, that we can’t do this purely in terms of efficient causes, that is, in terms of matter. I have found that to explain why the basic laws of physics are laws we have to bring in final causes, and that these laws don’t depend on the principle of necessity, as do the truths of logic, arithmetic and geometry, but on the principle of fitness, meaning that they depend on what God in his wisdom has chosen. For anyone who can look deeply into things, this is one of the most convincing and most evident proofs of the existence of God.”
“In phenomena, or in the resulting aggregate, everything is explained mechanically, and so masses are understood to impel each other. In these phenomena it is necessary to consider only derivative forces, once it is established whence these forces arise, namely, the phenomena of aggregates from the reality of the monads.”

“From the supreme Author’s perfection it follows not only that the order of the entire universe is the most perfect that could be, but also that, every living mirror that represents the universe according to its own point of view, that is to say, every monad, or, every substantial centre, must have its perceptions and its appetites ordered in the best way that is compatible with the perceptions and appetites of all the rest.”

“This interconnection, or this adapting of all created things to each one, and of each one to all the others, brings it about that each simple substance has relational properties that express all the others, so that each monad is a perpetual living mirror of the universe.”

“And just as the same town when seen from different sides will seem quite different—as though it were multiplied perspectively—the same thing happens here: because of the infinite multitude of simple substances it’s as though there were that many different universes; but they are all perspectives on the same one, differing according to the different points of view of the monads.”

“And that is the way to get the greatest possible variety, but with all the order there could be; i.e. it is the way to get as much perfection as there could be.”

“A created monad’s representation of the details of the whole universe is confused; it can be distinct only with respect to a small part of things, namely things that are either closest or largest in relation to it. Otherwise every monad would be divine! Monads are limited not in how widely their knowledge spreads, but in what kind of knowledge it is. They all reach confusedly to infinity, to everything; but they are limited and differentiated by their different levels of distinct perception.”

“It is true that things which happen in the soul must agree with those which happen outside of it. But for this it is enough for the things taking place in the one soul to correspond with each other as well as with those happening in any other soul, and it is not necessary to assume anything outside of all souls or monads.”

“I don’t really eliminate body but reduce it to what it is. For I show that corporeal mass, which is thought to have something over and above simple substances, is not a substance but a phenomenon resulting from simple substances which alone have unity and absolute reality.”

“Accurately speaking, matter is not composed of these constitutive unities but results from them….Substantial unities are not parts but foundations of phenomena.”

“The union which I find some difficulty explaining is that which joins the different simple substances or monads existing in our body with us, such that it makes one thing from them, nor is it sufficiently clear how, in addition to the existence of individual monads, there may arise a new existing thing, unless they are bound by a continuous bond …which the phenomena display to us.”

“Any mass contains innumerable monads, for although any one organic body in nature has its corresponding dominant monad, it nevertheless contains in its parts other monads endowed in the same way with organic bodies subservient to the primary one; and the whole of nature is nothing else, for it is necessary that every aggregate result from simple substances as if from elements.”

I distinguish:

1. the primitive entelechy or soul;

2. the matter, namely, the primary matter or primitive passive power;

3. the monad made up of these two things;

4. the mass or secondary matter, or the organic machine in which innumerable subordinate monads come together;

5. and the animal, that is, the corporeal substance, which the dominating monad makes into one machine.

“One suffices to derive all out of nothing.”

“All numbers are expressed by unity and by nothing.

Although there is no hope that men can, in this life, reach this hidden series of things, by which it will appear in what way everything comes from pure being and from nothing, it is sufficient to carry the analysis of ideas as far as the demonstrations of truths require.”

“A demonstration is a chain of definitions.”

“Every idea is analysed perfectly only when it is demonstrated a priori that it is possible.”

“It is useful to have definitions involving the generation of a thing, or if this is impossible, at least its constitution, that is, a method by which the thing appears to be producible or at least possible.”

“Obviously we cannot build a secure demonstration on any concept unless we know that this concept is possible, for from impossibles or concepts involving contradictions contradictory propositions can be demonstrated. This is an a priori reason why possibility is a requisite in a real definition.”

“Hence every real definition must contain at least the affirmation of some possibility. Furthermore, although names are arbitrary, once they are adopted, their consequences are necessary, and certain truths arise which are real even though they depend on the characters which have been imposed.”

“Moreover, to set up a hypothesis or to explain the method of production is merely to demonstrate the possibility of a thing, and this is useful even though the thing in question often has not been generated in that way.”

“Once a hypothesis or a manner of generation is found, one has a real definition from which others can also be derived, and from them those can be selected which best satisfy the other conditions, when a method of actually producing the thing is sought. Those real definitions are most perfect, furthermore, which are common to all the hypotheses or methods of generation and which involve the proximate cause of a thing, and from which the possibility of the thing is immediately apparent without presupposing any experiment or the demonstration of any further possibilities. In other words, those real definitions are most perfect which resolve the thing into simple primitive notions understood in themselves. Such knowledge I usually call adequate or intuitive, for, if there were any inconsistency, it would appear here at once, since no further resolution can take place.”¹⁵

“A true contingent proposition cannot be reduced to identical propositions, but is proved by showing that if the analysis is continued further and further, it constantly approaches identical propositions, but never reaches them. It is God alone, who grasps the entire infinite in His mind, who knows all contingent truths with certainty.”

“The laws of motion which actually occur in nature, and are verified by experiments, are not in truth absolutely demonstrable, as a geometrical proposition would be: but also it is not necessary that they should be so. They do not spring entirely from the principle of necessity, but they spring from the principle of perfection and order; they are an effect of the choice and wisdom of God.”

“Whatever is thought by us is either conceived through itself, or involves the concept of another….Thus one must proceed to infinity, or all thoughts are resolved into those which are conceived through themselves.”

“For in matters which do not possess metaphysical necessity, we must regard the agreement of phenomena as truth, since such agreement does not occur by chance but has a cause. Certainly it is only through this agreement among phenomena that we distinguish dreams from waking, and we predict that the sun will rise tomorrow only because it has fulfilled our faith so often. To this is added the great power of authority and of public testimony, since it is not likely that so many should conspire to deceive us. To these factors can be added what Saint Augustine has said on the utility of faith. The authority of the senses and of other witnesses once established, we may prepare a record of phenomena from which a mixed knowledge can be formed by combining with them truths abstracted from experience. But we need a particular art for arranging as well as for ordering and combining our experiments, so that useful inductions can be made from them, causes discovered, and general truths and postulates set up….Our human knowledge of nature seems to me at present like a shop well provided with all kinds of wares but without any order or inventory.”¹⁶

“Since the senses and inductions can never teach us perfectly universal truths, nor what is absolutely necessary, but only what is, and what is found in particular examples, and since we nevertheless know necessary and universal truths …it follows that we have derived these truths in part from what is within us.”

“The simpler a hypotheses is, the better it is.”

“Would that incomprehensibility were an attribute of God only! We should then have better hope of understanding nature. But it is too true that there is no part of nature which we can perfectly understand….No creature however noble can distinctly perceive or comprehend an infinity at one time; nay more, whoever understood one piece of matter, would understand the whole universe.”

“Synthesis is achieved when we begin from principles and run through truths in good order, thus discovering certain progressions and setting up tables, or sometimes general formulas, in which the answers to emerging questions can later be discovered. Analysis goes back to the principles in order to solve the given problems only, just as if neither we nor others had discovered anything before. It is more important to establish syntheses, because this work is of permanent value, while we often do work that has already been done in beginning the analysis of a particular problem.”¹⁷

“Since all belief consists in memory of past life, of proofs or of reasons, it is not in our power or in our free will to believe or not to believe, since memory is not a thing which depends on our will.”

“I have also used the analogy of a veined block of marble as opposed to an entirely homogeneous one or to an empty page….This is how ideas and truths are innate in us—as inclinations, dispositions, tendencies, or natural potentialities, and not as actual thinkings, though these potentialities are always accompanied by certain actual thinkings, often insensible ones, which correspond to them.”¹⁸

“I have always been, as I still am, in favour of the innate idea of God …and consequently of other innate ideas, which cannot come to us from the senses. Now I go still further, in conformity to the new system, and I even think that all the thoughts and actions of our soul come from its own nature, and that it is impossible they should be given to it by the senses….there are ideas and principles which do not come to us from the senses, which we find in us without forming them, though the senses give us occasion to notice them.”

“Although the senses are necessary for all our actual knowledge, they are not sufficient to provide it all, since they never give anything but instances, that is to say, particular or singular truths. But however many instances confirm a general truth, they do not suffice to establish its universal necessity, for it does not follow that what happened will always happen in the same way….From this it appears that necessary truths, such as we find in pure mathematics, and particularly in arithmetic and geometry, must have principles whose proof does not depend on instances, nor, consequently, on the testimony of the senses, even though without the senses it would never occur to us to think of them.”

“In fact our soul does always have in it the ability to represent to itself any nature or form when the occasion for thinking of it arises. And I believe that ability of our soul, insofar as it expresses some nature, form, or essence, is properly called an idea of the thing, and it is in us, and is always in us, whether we are thinking of the thing or not. For our soul expresses God and the universe, and all essences as well as all existences. This fits in with my principles, for nothing ever enters into our mind naturally from the outside; and we have a bad habit of thinking of our soul as if it received certain species as messengers and as if it has doors and windows. We have all these forms in our mind; we even have forms from all time, for the mind always expresses all its future thoughts and already thinks confusedly about everything it will ever think about distinctly.”

“Monads have no windows, through which anything could come in or go out. And accidents cannot detach themselves and stroll about outside of substances.”

“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.”

“One must hold as certain that the more a mind desires to know order, reason, the beauty of things which God has produced, and the more it is moved to imitate this order in the things which God has left to his direction, the happier it will be.”

“There is an old saying that God created everything according to weight, measure, and number. But there are things which cannot be weighed, those namely which have no force or power. There are also things which have no parts and hence admit of no measure. But there is nothing which is not subordinate to number. Number is thus a basic metaphysical figure, as it were, and arithmetic is a kind of statics of the universe by which the powers of things are discovered.”

“Our characteristic, however, will subject everything to numbers and, in order to let reasons be pondered too, it will provide a sort of statics. Even probabilities are subject to reckoning and proof, since it is always possible to estimate, from the given circumstances, what will more probably happen.”¹⁹

“But we must go further than words! Since the admirable connection of things makes it most difficult to give the characteristic numbers of a few things separated from others, I have thought of an elegant device, if I am not mistaken, by which to show that ratiocination can be proved through numbers. Thus I imagine that these most remarkable characteristic numbers are already given, and, having observed a certain general property to be true of them, I set up such numbers as are somehow consistent with this property, and applying these, I at once demonstrate through numbers, in wonderful order, all the rules of logic and show how we can know whether certain arguments are in good form. But the material soundness or truth of an argument can be judged without much mental effort and danger of error only when we have the true characteristic numbers of things themselves.”

“I admit that no more beautiful example of the art of combinations can be found anywhere than in algebra and that therefore he who masters algebra will the more easily establish the general art of combinations, because it is always easier to arrive at a general science a posteriori from particular instances than a priori. But there can be no doubt that the general art of combinations or characteristics contains much greater things than algebra has given, for by its use all our thoughts can be pictured and as it were, fixed, abridged, and ordered; pictured to others in teaching them, fixed for ourselves in order to remember them; abridged so that they may be reduced to a few; ordered so that all of them can be present in our thinking.”

“No one should fear that the contemplation of characters will lead us away from the things themselves; on the contrary, it leads us into the interior of things. For we often have confused notions today because the characters we use are badly arranged; but then, with the aid of characters, we will easily have the most distinct notions, for we will have at hand a mechanical thread of meditation, as it were, with whose aid we can very easily resolve any idea whatever into those of which it is composed. In fact, if the character expressing any concept is considered attentively, the simpler concepts into which it is resolvable will at once come to mind. Since the analysis of concepts thus corresponds exactly to the analysis of a character, we need merely to see the characters in order to have adequate notions brought to our mind freely and without effort. We can hope for no greater aid than this in the perfection of the mind.”

“If we had an established characteristic we might reason as safely in metaphysics as in mathematics.

You say that it is difficult to set up definitions of things; perhaps you mean in the most simple and the primitive concepts, so to speak. These, I admit, it is difficult to give. We must realize, indeed, that there are several definitions of the same thing, that is, reciprocal properties which distinguish one thing from all other things and that from each one we can derive all the other properties of the thing defined. You are not unaware of this, but some of these definitions are more perfect than others, that is, they come nearer to the primary and adequate notions. Indeed, I hold this to be a certain criterion of a perfect and adequate definition: that when the definition is once grasped, we cannot further doubt whether the thing defined in it is possible or not.

Besides, anyone who wishes to construct a characteristic or universal analytic can use any definitions whatever in the beginning, since all will eventually lead to the same result when the analysis is continued.”

“When what is searched is not determinable or expressible from the data, one can do one of two things: either to approximate ad infinitum, or else, when dealing with conjectures, at least to determine, by means of demonstrative reason, the degree of probability itself that it is possible to obtain from the data...I cultivate a certain part of logic dealing with the estimation of the degrees of probability, and with the balance of proofs, conjectures, and indications.”

“I have said more than once that we need a new kind of logic, concerned with degrees of probability, providing us with balances which are needed to weigh likelihoods and to arrive at sound judgments regarding them.”

“I should venture to add that if I had been less distracted, or if I were younger or had talented young men to help me, I should still hope to create a kind of universal symbolistic in which all truths of reason would be reduced to a kind of calculus. At the same time this could be a kind of universal language or writing, though infinitely different from all such languages which have thus far been proposed, for the characters and the words themselves would give directions to reason, and the errors—except those of fact—would be only mistakes in calculation. It would be very difficult to form or invent this language or characteristic but very easy to learn it without any dictionaries. When we lack sufficient data to arrive at certainty in our truths, it would also serve to estimate degrees of probability and to see what is needed to provide this certainty. Such an estimate would be most important for the problems of life and for practical considerations, where our errors in estimating probabilities often amount to more than a half….

It is true that this part of useful logic has not yet been found, but it will be of marvellous use in practical matters when it is a question of presumptions, indices and conjectures, in order to know the degrees of probability when there are a number of apparent reasons in favour of one or the other in some important deliberation. Thus when there are not enough given conditions to demonstrate certainty, the subject being merely probable, one can always give at least demonstrations concerning the probability itself.”²⁰

“If demonstration shows the connection of ideas, probability is nothing else than the appearance of this connection based upon proofs in which immutable connection is not seen.”

“Probability is degree of possibility.”

“Probability is grounded in its conformity with what we know, or in the testimony of those who know.”

“This is the axiom: aequalibus aequalia, equal suppositions must have equal consideration. But when the suppositions are unequal they compare them with each other….A new kind of logic would be required which would treat of the degrees of probability, since Aristotle in his ‘Topics’ has done nothing less than this, and has contented himself with putting in a certain order certain popular rules distributed according to the topics, which may be of use on some occasion where the question concerns the amplification of the discourse and the giving to it probability without putting it to the trouble of furnishing us a necessary balance for weighing probabilities and forming thereupon a solid judgment. It would be well for him who should treat of this matter to pursue the examination of games of chance; and in general I wish that some skillful mathematician would produce an ample work with full details and thoroughly reasoned upon all sorts of games, which would be very useful in perfecting the art of invention, the human mind appearing to better advantage in games than in the most serious matters.”

“Given any number of points, an infinite number of curves can be found passing through them.²¹ Thus, I show the following: I postulate (and this can be demonstrated) that given any number of points, some regular curve can be found passing through these points….It may be added that, although a perfect estimation cannot be had empirically, an empirical estimate would nonetheless be useful and sufficient in practice.”²²

“The art of conjecture is founded on what is more or less easy, or rather more or less feasible, for the Latin facilis (easy), derived from faciendo (from what is to be done), literally means feasible….One may still estimate likelihoods a posteriori, by experience, to which one must have recourse in default of a priori reasons….It could be said that what happens more or less often is also more or less feasible in the present state of affairs, putting all the considerations together that have to combine to bring about the production of a fact.”²³

“I prefer to maintain that it is always grounded in likelihood or in conformity with the truth; and the testimony of another is also a thing which the truth has been wont to have for itself as regards the facts that are within reach. It may be said then that the similarity of the probable and the truth is taken either from the thing itself, or from some extraneous thing. The rhetoricians employ two kinds of arguments: the artificial, drawn from things by reasoning, and the non-artificial, based only upon the express testimony either of man or perhaps also of the thing itself. But there are mixed arguments also, for testimony may itself furnish a fact which serves to form an artificial argument.”

“Men are never more ingenious than in games.”

“Games of chance serve especially in the estimation of probabilities.”

“The games mixed of fortune and combinations represent very well human life, and especially military actions and the practice of medicine, things in which it is necessary to give one part to science and the other to fortune….It is thence that it is necessary to unite certain consequences to those that one evaluates by probability and verisimilitude.”²⁴

“All magnitudes being infinitely divisible, there is none so small but that we can conceive in it an infinity of divisions, which will never be exhausted.”

“Properly speaking, it is true that there are an infinity of things, i.e. that there are always more of them than can be assigned. But there is no infinite number, or line or any other infinite quantity, if these are understood as true wholes, as it is easy to prove….The true infinite exists, strictly speaking, only in the Absolute, which is anterior to all composition, and is not formed by addition of parts.”

“The idea of the infinite is not formed by extension of finite ideas.”

“It is found that the rules of the finite succeed in the infinite.”²⁵

“It is not necessary to make mathematical analysis depend upon metaphysical controversies, nor to make sure that there are in nature strictly infinitesimal lines….This is why, in order to avoid these subtleties, thought that, to render the reasoning intelligible to everybody, it sufficed in this to explain the infinite by the incomparable, i.e. to conceive quantities incomparably greater or smaller than ours.”

“But free or intelligent substances possess something greater and more marvellous, in a kind of imitation of God. For they are not bound by any certain subordinate laws of the universe, but act as it were by a private miracle, on the sole initiative of their own power, and by looking towards a final cause they interrupt the connection and the course of the efficient causes that act on their will. So it is true that there is no creature ‘which knows the heart’ which could predict with certainty how some mind will choose in accordance with the laws of nature; as it could be predicted (at any rate by an angel) how some body will act, provided that the course of nature is not interrupted. For just as the course of the universe is changed by the free will of God, so the course of the mind’s thoughts is changed by its free will, so that in the case of minds no subordinate laws can be established as is possible in the case of bodies which are sufficient for predicting a mind’s choice.”

“Free will...consists in the view that the strongest reasons or impressions which the understanding presents to the will do not prevent the act of the will from being contingent, and do not confer upon it an absolute or (so to speak) metaphysical necessity. It is in this way that I have the habit to say that the understanding can determine the will, in accordance with which perceptions and reasons prevail, in a manner which, although it is certain and infallible, inclines without necessitating.”

“It must be confessed, moreover, that perception and what depends on it are inexplicable by mechanical reasons, that is, by figures and motions. If we pretend that there is a machine whose structure enables it to think, feel, and have perception, one could think of it as enlarged yet preserving its same proportions, so that one could enter it as one does a mill. If we did this, we should find nothing within but parts which push upon each other; we should never see anything which would explain a perception. So it is in the simple substance, and not in the composite substance or machine, that perception must be sought. Furthermore, this is the only thing—namely, perceptions and their changes—that can be found in simple substance. It is in this alone that the internal actions of simple substances can consist.”

“Miracles are not to be multiplied beyond necessity.”

“Wisdom is merely the science of happiness or that science which teaches us to achieve happiness….Pleasure is the feeling of a perfection or an excellence, whether in ourselves or in something else….Thus we see that happiness, pleasure, love, perfection, being, power, freedom, harmony, order, and beauty are all tied to each other, a truth which is rightly perceived by few. Now when the soul feels within itself a great harmony, order, freedom, power, or perfection, and hence feels pleasure in this, the result is joy, as these explanations show. Such joy is permanent and cannot deceive, nor can it cause a future unhappiness if it arises from knowledge and is accompanied by a light which kindles an inclination to the good in the will, that is, virtue. It follows from this that nothing serves our happiness better than the illumination of our understanding and the exercise of our will to act always according to our understanding, and that this illumination is to be sought especially in the knowledge of such things as can bring our understanding ever further into a higher light. For there springs from such knowledge an enduring progress in wisdom and virtue, and therefore also in perfection and joy, the advantage of which remains with the soul even after this life. Such joy , which a person can always create for himself when his mind is well ordered, consists in the perception of pleasure in himself and in the powers of his mind, when a man feels within himself a strong inclination and readiness for the good and the true, and particularly through the profound knowledge which an enlightened understanding provides for us, namely, that we experience the chief source, the course, and the purpose of everything, and the incomprehensible excellence of that Supreme Nature which comprises all things within it. Thus are we lifted above the unknowing, just as if we were looking down from the stars and could see all earthly things under our feet. Then at last we learn that we have reason to find the highest joy in all things that have happened and are yet to happen, but that we must also seek, as far as is in our power, to direct what has not yet happened for the best. For it is one of the eternal laws of nature that we shall enjoy the perfection of things and the pleasure which results from it, only in the measure of our knowledge, our good will, and our contribution to this perfection.”²⁶

“Wisdom is the science of happiness or of the means of attaining the lasting contentment which consists in the continual achievement of a greater perfection or at least in variations of the same degree of perfection.”

“Happiness is a lasting state of joy.”

“Joy is the total pleasure that results from everything the soul feels at once.”

“Pleasure is a knowledge or feeling of perfection not only in us, but also in others, because that then excites some pleasure in us.”

“Perfection is the harmony of things, or the state where everything is worthy of being observed, that is, the state of agreement or identity in variety; you can even say that it is the degree of contemplatibility.”

“To love is to find pleasure in the perfection of others.”

“There are two kinds of knowledge, that of facts, and that of reasons. That of facts is perception, that of reasons is intelligence.”

“Knowledge of reasons perfects us because it teaches us of universal and eternal truths, which express the perfect being. However knowledge of facts is like that of the streets of a town, which is useful to us while we remain there, after which we do not want to weigh the memory down with it.”

“We could not love God without knowing his perfections or his beauty. And as we would only know him through his emanations, there are two ways of seeing his beauty, namely in the knowledge of eternal truths, explaining the reasons in themselves, and in the knowledge of the harmony of the universe, by applying reasons to facts. That is to say, we must know the wonders of reason and the wonders of nature.”

“We must hold for certain that the more a mind desires to know the order, the reason, the beauty of things that God has produced, and the more it is moved to imitate this order in the things that God has left to its management, the more it will be happy.”

“We are freer in proportion as we are further removed from indifference and more self-determined.”

“The more substances are determined by themselves, and removed from indifference, the more perfect they are.”

“The more we act according to reason the more we are free, For the more we act according to reason, the more we act conformably to the perfections of our own nature.”

“All actions are contingent, or without necessity. But also everything is determined or regular, and there is no indifference. We may even say that substances are freer in proportion as they are further removed from indifference and more self-determined. And that the less they have need of external determination, the nearer they approach to the divine perfection. For God, being the freest and most perfect substance, is also the most completely determined by himself to do the most perfect. So that Nothing (le Rien), which is the most imperfect and the furthest removed from God, is also the most indifferent and the least determined. Now in so far as we have lights, and act according to reason, we shall be determined by the perfections of our own nature, and consequently we shall be freer in proportion as we are less embarrassed as to our choice. It is true that all our perfections, and those of all nature, come from God, but this, far from being contrary to liberty, is rather the very reason why we are free, because God has communicated to us a certain degree of his perfection and of his liberty. Let us, then, content ourselves with a liberty which is desirable, and approaches that of God, which makes us the most disposed to choose well and act well; and let us not pretend to that harmful, not to say chimerical liberty, of being in uncertainty and perpetual embarrassment, like that Ass of Buridan, famous in the schools, who, being placed at an equal distance between two sacks of wheat, and having nothing that determined him to go to one rather than the other, allowed himself to die of hunger.”

“The Supreme happiness of man consists in the greatest possible increase of his perfection.”

“When God chooses, it is through his knowledge of the best; when man does so, it will be the alternative which has struck him the most.”²⁷

“Two hypotheses can be formed, one that nature is always equally perfect, the other that it always increases in perfection.”²⁸

“To employ the art of consequences, we need an art of bringing things to mind, another of estimating probabilities and, in addition, knowledge of how to evaluate goods and ills; and we need to be attentive, and, on top of all that, to have the patience to carry our calculations through. Finally, we need to be firmly and steadily resolved to act on our conclusions; and we need skills, methods, rules of thumb, and well-entrenched habits to make us true to our resolve later on, when the considerations that led us to it are no longer present in our minds.”

“Parts are not always simpler than wholes, though they are always less than the whole.”

“If this happens, it must follow that those minds which are not yet sufficiently capable will become more capable so that they can comprehend and invent such great theorems, which are necessary to understand nature more deeply and to reduce physical truths to mathematics, for example, to understand the mechanical functioning of animals, to foresee certain future contingencies with a certain degree of accuracy, and to do certain wonderful things in nature, which are now beyond our capacity….

Every mind has a horizon in respect to its present intellectual capacity but not in respect to its future intellectual capacity.”

“There is hardly any paradox without utility.”

“Thomas Hobbes, everywhere a profound examiner of principles, rightly stated that everything done by our mind is a computation.”

“He who knows me only through what I have published does not know me.”

1. How to create the world?

   1. Starting with the numerals $0$ and $1$, one obtains the set of natural numbers.

   2. Each of these numbers is interpreted as representing, or being characteristic of, a specific primitive concept.

   3. By way of logical combination the larger set of general concepts is obtained.

   4. Individual-concepts, i.e. the “ideas” corresponding to individuals, will then be defined as maximally consistent concepts.

   5. Among the set of all possible individuals the relation of compossibility is introduced.

   6. Possible worlds are defined as certain maximal collections of pairwise compossible individuals.

   7. The real world is distinguished from its rivals by being the richest, i.e. most numerous and, perhaps, also in some other respect the best of all possible worlds.

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2. 庞加莱: 在如此多的废墟中间, 还有什么东西屹立长存呢? 最小作用量原理迄今未经触动, 人们似乎相信他会比其他原理更久长. 事实上, 它是更加模糊, 更加抽象. ↩

3. Plato(Timaeus): “First then, in my judgment, we must make a distinction and ask, what is that which always is and has no becoming; and what is that which is always becoming and never is? That which is apprehended by intelligence and reason is always in the same state; but that which is conceived by opinion with the help of sensation and without reason, is always in a process of becoming and perishing and never really is. Now everything that becomes or is created must of necessity be created by some cause, for without a cause nothing can be created. The work of the creator, whenever he looks to the unchangeable and fashions the form and nature of his work after an unchangeable pattern, must necessarily be made fair and perfect; but when he looks to the created only, and uses a created pattern, it is not fair or perfect. Was the heaven then or the world, whether called by this or by any other more appropriate name—assuming the name, I am asking a question which has to be asked at the beginning of an enquiry about anything—was the world, I say, always in existence and without beginning? or created, and had it a beginning? Created, I reply, being visible and tangible and having a body, and therefore sensible; and all sensible things are apprehended by opinion and sense and are in a process of creation and created. Now that which is created must, as we affirm, of necessity be created by a cause. But the father and maker of all this universe is past finding out; and even if we found him, to tell of him to all men would be impossible. And there is still a question to be asked about him: Which of the patterns had the artificer in view when he made the world—the pattern of the unchangeable, or of that which is created? If the world be indeed fair and the artificer good, it is manifest that he must have looked to that which is eternal; but if what cannot be said without blasphemy is true, then to the created pattern. Every one will see that he must have looked to, the eternal; for the world is the fairest of creations and he is the best of causes. And having been created in this way, the world has been framed in the likeness of that which is apprehended by reason and mind and is unchangeable, and must therefore of necessity, if this is admitted, be a copy of something. Now it is all-important that the beginning of everything should be according to nature. And in speaking of the copy and the original we may assume that words are akin to the matter which they describe; when they relate to the lasting and permanent and intelligible, they ought to be lasting and unalterable, and, as far as their nature allows, irrefutable and immovable—nothing less. But when they express only the copy or likeness and not the eternal things themselves, they need only be likely and analogous to the real words. As being is to becoming, so is truth to belief.”

   “He was good, and the good can never have any jealousy of anything. And being free from jealousy, he desired that all things should be as like himself as they could be. This is in the truest sense the origin of creation and of the world, as we shall do well in believing on the testimony of wise men: God desired that all things should be good and nothing bad, so far as this was attainable. Wherefore also finding the whole visible sphere not at rest, but moving in an irregular and disorderly fashion, out of disorder he brought order, considering that this was in every way better than the other. Now the deeds of the best could never be or have been other than the fairest; and the creator, reflecting on the things which are by nature visible, found that no unintelligent creature taken as a whole was fairer than the intelligent taken as a whole; and that intelligence could not be present in anything which was devoid of soul. For which reason, when he was framing the universe, he put intelligence in soul, and soul in body, that he might be the creator of a work which was by nature fairest and best. Wherefore, using the language of probability, we may say that the world became a living creature truly endowed with soul and intelligence by the providence of God.”

   Plato(Meno): “The soul, then, as being immortal, and having been born again many times, and having seen all things that exist, whether in this world or in the world below, has knowledge of them all; and it is no wonder that she should be able to call to remembrance all that she ever knew about virtue, and about everything; for as all nature is akin, and the soul has learned all things; there is no difficulty in her eliciting or as men say learning, out of a single recollection all the rest, if a man is strenuous and does not faint; for all enquiry and all learning is but recollection.” ↩

4. Richard Feynman: “Is it true that the particle doesn’t just ‘take the right path’ but that it looks at all the other possible trajectories? ... The miracle of it all is, of course, that it does just that. ... It isn’t that a particle takes the path of least action but that it smells all the paths in the neighborhood and chooses the one that has the least action.”

   The Principle of Maximal Variety: Among the set of all possible universes compatible with an irreducibly minimal set of structural constraints, the actual universe is the one which maximizes the variety.

   Ernst Mach: “Science is a form of business. Its purpose is to find the maximum amount of the infinite eternal truth with the minimum amount of work, in the minimum expenditure of time and with the minimum amount of thought effort.” ↩

5. Principles of Nature and Grace, Based on Reason. ↩

6. On the Radical Origination of Things. ↩

7. A Résumé of Metaphysics. ↩

8. Principle of sufficient reason: Every question of the form of why is the universe like $X$ rather than $Y$ has a reason sufficient to explain why.

   Laplace’s principle of insufficient reason (indifference): If there is no reason to think differently than that all cases are equally possible, then their probabilities are equal. (Equivalent states of knowledge should be assigned equivalent epistemic probabilities.) ↩

9. Bertrand Russell: “Leibniz based his philosophy upon two logical premises, the law of contradiction and the law of sufficient reason. Both depend upon the notion of an ‘analytic’ proposition, which is one in which the predicate is contained in the subject—for instance, ‘all white men are men.’ The law of contradiction states that all analytic propositions are true. The law of sufficient reason (in the esoteric system only) states that all true propositions are analytic.”

   “The principle of sufficient reason …is usually supposed to be, by itself, adequate to the deduction of what actually exists. To this supposition, it must be confessed, Leibniz’s words often lend colour. But we shall find that there are really two principles included under the same name, the one general, and applying to all possible worlds, the other special, and applying only to the actual world. Both differ from the law of contradiction, by the fact that they apply specially—the former, however, not exclusively—to existents, possible or actual. The former, as we shall see, is a form of the law of causality, asserting all possible causes to be desires or appetites; the latter, on the other hand, is the assertion that all actual causation is determined by desire for the good. The former we shall find to be metaphysically necessary, while the latter is contingent, and applies only to contingents. The former is a principle of possible contingents, the latter a principle of actual contingents only.”

   “Thus the law of sufficient reason, as applied to actual existents, reduces itself definitely to the assertion of final causes, in the sense that actual desires are always directed towards what appears the best. In all actual changes, the consequent can only be deduced from the antecedent by using the notion of the good. Where the change depends only upon God, it really is for the best; where it depends upon a free creature, it is such as seems best to the creature, but is often, owing to confused perception, not really the best possible change. Such a connection can only be regarded as contingent by admitting, as Leibniz does, that a law may be general, i.e. may apply to every part of time, without being necessary, i.e. without being capable of a statement in which no actual part of time is referred to.”

   “The sufficient reason for one change rather than another is to be found in the nature of activity. In substances which are not free, this activity is regulated by general laws, which themselves have a sufficient reason in God’s perception of fitness; in free substances, the sufficient reason lies in the more or less confused perception of the good on the part of the substance itself. But in no case is the connection between two states in itself necessary; it always arises from the perception, either in God or in the creature (if this be free), that the change is good.” ↩

10. Bertrand Russell: “From what has been said of activity, it is plain that those predicates of a given substance which are existents in time form one causal series. Leibniz sometimes goes so far in this direction as to approach very near to Lotze’s doctrine that things are laws. All singular things, he says, are subject to succession, nor is there anything permanent but the law itself, involving continual succession. Successions, he continues, like such series as numbers, have the property that, given the first term and the law of progression, the remaining terms arise in order. The only difference is, that in successions the order is temporal, but in numbers the order is that of logical priority. Further, the persistence of the same law is the ground for asserting that a new temporal existent belongs to the same substance as a past existent. The identity of a substance at different times is recognized, he says, ‘by the persistence of the same law of the series, or of continuous simple transition, which leads us to the opinion that one and the same subject or monad is changing. That there should be a persistent law, involving the future states of that which we conceive as the same, is just what I assert to constitute it the same substance’. These passages explain very definitely what Leibniz means by his phrase, that each monad contains in its nature the law of the continuation of the series of its operations. They enable us, also, to see what would remain of the doctrine of monads if the appeal to substance were dropped. All the predicates of a given substance form one causal series: this series might, therefore, be taken as defining what we are to mean by one substance, and the reference to subject and predicate might be dropped. The plurality of substances would then consist in the doctrine, that a given existent at a given moment is caused, not by the whole preceding state of the universe, but by some one definite existent in the preceding moment. This assumption is involved in the ordinary search for causes of particulars.” ↩

11. 莱布尼茨: 所有事物在任何可能世界里都是相互关联的. 即使最微小的运动都会将其结果传递到无论多么遥远的地方, 尽管这个结果随着距离的增大而相应地变小.

    每个微粒都受到宇宙中所有物体的作用, 以这种方式, 全知者便能够从每个微粒中认识宇宙中的一切.

    灵魂按其与身体的关系将整个宇宙的其余部分都表象给它自身, 从而成为宇宙的一面镜子.

    灵魂按照它自己的观点表象宇宙, 就像同一个城市从不同角度看便会形成不同的景观. ↩

12. Relative Space and Handedness:

    • An object is an incongruent counterpart of another if they cannot be made to occupy the same place by rigid motions in a local region of space.

    • An object is said to possess handedness (chirality) just when it and its mirror image are incongruent counterparts.

    • If the space is 3-dimensional, or 2-dimensional but non-orientable, then “F” and its mirror image are congruent counterparts (e.g. a Möbius strip).

    • Whether or not a mirror image is an incongruent counterpart depends on the properties of the space it is located in.

    ↩

13. 李·斯莫林:

    • 宇宙之外一无所有.

      • 因果完备性: 宇宙是因果封闭的. (万事皆有因, 但因果链不能追溯到宇宙之外.)

      • 时空相对性: 空间是事物之间的相互关系; 时间是事物关系变化的度量. (如果空间是绝对的, 那么将宇宙往右平移一米将与原宇宙不可区分; 如果时间是绝对的, 那么将宇宙往后平移一分钟将与原宇宙不可区分. 违背”不可分辨者的同一性”.)

      • 宇宙不是由物质构成, 而是由事件的因果过程构成. 宇宙是一个因果关系的网络. 其中每一部分的性质由其与其它部分的关系决定.

      • 相互作用: 如果A作用于B, B也一定会作用于A. (时空告诉物质如何运动; 反过来, 物质告诉时空如何弯曲. 而牛顿的绝对时空是单向的, 时空指引物质运动, 物质却无法影响时空.)

      • 背景独立性: 基础物理理论不应该依赖于固定不变的背景, 即只建模宇宙的一部分, 而把其余部分视为不随其它要素变化的背景.

      • 对称性是背景依赖的性质. 由于”不可分辨者的同一性”, 不存在终极的对称性. 对称性, 要么是近似的, 要么是破缺的.

      • 没有任何两个瞬间完全相同, 从这个意义上说, 时间是真实的.

      • 开放系统的能量流, 倾向于驱动系统向组织度更高的结构演化. (在熵最大的平衡态, 许多瞬间会不断复现.)

      • 观察者是宇宙的一部分.

      • 观察者只能观测到宇宙的部分信息. 未来的观察者可以看到更多信息.

      • 逻辑与观察者有关. 不能假定每个陈述非真即假. 至少有三类: 一类可判断为真, 一类可判断为假, 一类无法判断.

    ↩

14. Continuity: \(\forall x\in I\forall\varepsilon>0\exists\delta>0\forall y\in I \big(|x-y|<\delta\to|f(x)-f(y)|<\varepsilon\big)\) \(\textcolor{gray}{\forall (x_n)_{n\in\mathbb{N}}\subset I: \lim\limits_{n\to\infty}f(x_n)=f(\lim\limits_{n\to\infty}x_n)}\) ↩

15. On Universal Synthesis and Analysis, or the Art of Discovery and Judgment. ↩

16. On Universal Synthesis and Analysis, or the Art of Discovery and Judgment. ↩

17. On Universal Synthesis and Analysis, or the Art of Discovery and Judgment. ↩

18. New Essays on Human Understanding. ↩

19. On the General Characteristic. ↩

20. Letters to Nicolas Remond. ↩

21. Jacob Bernoulli: “It is nevertheless known that there are levels in infinity, and that the ratio of one infinity to another infinity is still a finite number, and can be expressed either precisely or sufficiently precisely for practical use….If five points have been observed, all of which are perceived to lie along a parabola, the notion of a parabola will be stronger than if only four points had been observed: for although there are an infinite number of curves which may pass through five points, there is nevertheless beyond this infinite number another infinite number — rather, an infinitely times more infinite number — of curves which may pass through only the first four points and not through the fifth point, all of which are excluded by this fifth observation. And yet, I admit that every conjecture which is deduced by observations of this sort would be quite flimsy and uncertain if it were not conceded that the curve sought is one of the class of simple curves; this indeed seems quite correct to me, since we see everywhere that nature follows the simplest paths.” ↩

22. Gottfried Leibniz and Jacob Bernoulli—Correspondence Regarding the Art of Conjecturing. ↩

23. Letter to Louis Bourguet. in: Die philophischen schriften von Gottfried Wilheim Leibniz, vol. III C. I. Gerhardt (ed) pp 564-570. ↩

24. Notes on Some Games and Principally on the Chinese Game, on the Difference of the Game of Chess from the Littler Robbers, and on a New Kind of Naval Game. ↩

25. Transfer Principle: A map from a standard model to a non-standard model is an elementary embedding. ↩

26. On Wisdom. ↩

27. 莱布尼茨: “遵循事物的本性, 而非其他人的榜样.”

    Matsuo Basho: “Do not seek to follow in the footsteps of the wise; seek what they sought.” ↩

28. Letter to Louis Bourguet. in: The Shorter Leibniz Texts: A Collection of New Translations. ↩