Jekyll2026-08-24T10:59:36+00:00https://rickylixi.github.io/feed.xml李熙 | Xi LiAssociate Professor of Logic at Central South University. Research in Kolmogorov Complexity, Solomonoff Induction, Causal Inference, Rational Decision Theory, and AI Philosophy.Xi Li白居易眼中的音乐及后来的致敬者2025-10-27T03:22:00+00:002025-10-27T03:22:00+00:00https://rickylixi.github.io/blog/2025/10/27/music
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)
I, who perceive, exist.
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.”
“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.6
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.”7
“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.”8
“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.”
“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.”9
“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 is 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.”10
“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.”11
“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.”12
“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.”13
“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.”14
“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.”
“Any body can be imagined to be at rest, if only a sentient being is understood to be on it. But being thrown off along the tangent does not argue the real motion of the rotating thing, since it would be the same if everything moved around it. Of course, we humans are accustomed to attribute rest to larger bodies, and this for the sake of the abbreviation or ordering of thought.”
“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.”
“There is no precise and fixed shape in bodies, because of the actual
division of the parts to infinity.
A
B
C
Annotated line with labels A, B and C.
Let there be, for example, a straight line $ABC$: I say that it is not exactly straight. For with each part of the universe sympathizing with all the others, it is necessarily the case that if the point $A$ tends along the straight line $AB$, the point $B$ should have a tendency in another direction. For with each part $A$ striving to carry with it every other, but particularly the nearest $B$, the direction of $B$ will be composed of that of $A$ and some others; and it is not at all possible that $B$, which is indefinitely near to $A$, should be exposed to the whole universe in precisely the same fashion as $A$, in such a way that $AB$ composes one whole which has no subdivision.
It is true that it will always be possible to draw an imaginary line at each instant; but that line will endure in the same parts only for this instant, because each part has a motion different from every other, since it expresses the whole universe differently. Thus there is no body that has any shape for a definite time, however short it might be. Now I believe that what exists only at a moment has no existence, since it starts and finishes at the same time. I have proved elsewhere that there is no middle moment, or moment of change, but only the last moment of the preceding state and the first moment of the following state.”
“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.”15
“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 represents 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:
the primitive entelechy or soul;
the matter, namely, the primary matter or primitive passive power;
the monad made up of these two things;
the mass or secondary matter, or the organic machine in which
innumerable subordinate monads come together;
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.”16
“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.”17
“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.”18
“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.”19
“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.”20
“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.”21
“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. …22
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.”23
“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.24 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.”25
“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.”26
“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.”27
“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.”28
“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.”
“Freedom … consists in intelligence, which includes a distinct knowledge of the object of deliberation, in spontaneity, by which we determine ourselves, and in contingency, that is, in the exclusion of logical or metaphysical necessity. … The free substance determines itself by itself, following the motive of the good recognized by the understanding, which inclines it without necessitating it.”29
“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.”30
“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.”31
“Two hypotheses can be formed, one that nature is always equally
perfect, the other that it always increases in perfection.”32
“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.”
How to create the world?
Starting with the numerals $0$ and $1$, one obtains the set of
natural numbers.
Each of these numbers is interpreted as representing, or being
characteristic of, a specific primitive concept.
By way of logical combination the larger set of general concepts
is obtained.
Individual-concepts, i.e. the “ideas” corresponding to
individuals, will then be defined as maximally consistent
concepts.
Among the set of all possible individuals the relation of
compossibility is introduced.
Possible worlds are defined as certain maximal collections of
pairwise compossible individuals.
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.
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.” ↩
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.” ↩
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.) ↩
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.” ↩
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.” ↩
the coordination of knowledge in an encyclopedia — collect all present knowledge so we could sift through it for what is fundamental. With the set of ideas that it generated, we could formulate the characteristica universalis (which form the alphabet of human thought).
characteristica universalis — a universal ideal language whose rules of composition directly expresses the structure of the world.
\[\text{sign}\rightleftarrows\text{idea}\]
calculus ratiocinator — the arrangement of all true propositions in an axiomatic system.
decision procedure — an algorithm which, when applied to any formula of the characteristica universalis, would determine whether or not that formula were true. A procedure for the rapid enlargement of knowledge: replace reasoning by computation, the art of invention, and free mind from intuition.
A proof that the calculus ratiocinator is consistent.
Compute all descriptions of possible worlds that can be expressed with the primitive notions. And the possible worlds will all have some propensity to exist.
Compute the probabilities of disputed hypotheses relative to the available data. As we learn more our probability assignments will asymptotically tend to a maximum for the real world, i.e. the possibility with the highest actual propensity.
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.” ↩
Gottfried Leibniz and Jacob Bernoulli — Correspondence Regarding
the Art of Conjecturing.↩
Letter to Louis Bourguet. in: Die philosophischen schriften von
Gottfried Wilhelm Leibniz, vol. III C. I. Gerhardt (ed) pp 564-570. ↩
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.↩
Transfer Principle: A map from a standard model to a non-standard
model is an elementary embedding. ↩
Matsuo Basho: “Do not seek to follow in the footsteps of the wise;
seek what they sought.” ↩
Letter to Louis Bourguet. in: The Shorter Leibniz Texts: A
Collection of New Translations.↩
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