Phil. Space and Time

Leibniz-Clarke Correspondence, Part 1

G. W. Leibniz (1646-1716); Samuel Clarke (1675-1729)

This famous correspondence touches on many issues in metaphysics, theology, and science; but since our main concern is the philosophy of space and time, we will extract their views as far as these are relevant to our problems. Further, it may be useful to arrange the quotations from this correspondence in the form of "point-to-point" dialogue, each answering to the other on the point made in the opponent's previous letter. In this Part 1, up to Leibniz's 4th paper and Clarke's 4th reply; however, I have omitted arguments for and against vacuum (Leibniz despises vacuum and endorses plenum). All quotations are from Alexander, ed., Leibniz-Clarke Correspondence, Manchester University Press, 1956.

Japanese translation of this correspondence is available:wŰCvjbcٓWA9xHٓܪA1989; the translator's comments are not particularly useful. However, Masahiko Yokoyama's expository essay at the end of the volume 3 (1999) is quite useful, touching on the communication between Huygens and Leibniz. But beware, these volumes are quite EXPENSIVE!


The Principle of a Sufficient Reason

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 requisite, as I have observed in my Theodicy: I mean, the principle of a sufficient reason, viz. that nothing happens without a reason why it should be so, rather than otherwise. And therefore Archimedes being to proceed from mathematics to natural philosophy, in his book De Aequilibrio, was obliged to make use of a particular case of the great principle of a sufficient reason. He takes it for granted, that if there be a balance, in which everything is alike on both sides, and if equal weighted are hung on the two ends of that balance, the whole will be at rest. 'Tis because no reason can be given, why one side should weigh down, rather than the other. Now, by that single principle, viz. that there ought to be a sufficient reason why things should be so, and not otherwise, one may demonstrate the being of God, and all the other parts of metaphysics or natural theology; and even, in some measure, those principles of natural philosophy, that are independent upon mathematics: I mean, the dynamical principles, or the principles of force. (Leibniz's 2nd paper, Alexander 1956, 15-6)

Uchii's Note: Notice that Leibniz says, at one place, that this principle is requisite for natural philosophy, and at other place, that this principle may be used for metaphysics and theology. The editor Alexander argues that there are at least three versions of this principle: (1) the causal principle, that nothing happens without a cause; (2) that God must always have a motive for acting; and (3) that God must always act for the best (Alexander 1956, xxii-xxiii).

'Tis very true, that nothing is, without a sufficient reason why it is, and why it is thus rather than otherwise. And therefore, where there is no cause, there can be no effect. But this sufficient reason is oft-times no other, than the mere will of God. For instance: why this particular system of mattter, should be created in one particular place, and that in another particular place; when, (all place being absolutely indifferent to all matter,) it would have been exactly the same thing vice versa, supposing the two systems (or the particles) of matter to be alike; there can be no other reason, but the mere will of God. Which if it could in no case act without a predetermining cause, any more than a balance can move without a prepoderating weight; this would tend to take away all power of choosing, and to introduce fatality. (Clarke's 2nd Reply, Alexander 1956, 20-1)

Uchii's Note: Clarke seems to admit the principle, but his interpretation (or application) of the principle is different from what Leibniz intends, as the following reply of Leibniz shows. [Source of Clarke's image: MacTutor History of Math., http://www-history.mcs.st-and.ac.uk/~history/Mathematicians/Clarke.html]

7. It appears from what I have said, that my axiom has not been well understood; and that the author denies it, tho' he seems to grant it. 'Tis true, says he, that there is nothing without a sufficient reason why it is, and why it is thus, rather than otherwise: but he adds, that this sufficient reason, is often the simple or mere will of God: as, when it is asked why matter was not placed otherwhere in space; the same situations of bodies among themselves being preserved. But this is plainly maintaining, that God wills something, without any sufficient reason for his will: against the axiom, or the general rule of whatever happens. This is falling back into the loose indifference, which I have confuted at large, and showed to be absolutely chimerical even in creatures, and contrary to the wisdom of God, as if he could operate without acting by reason. (Leibniz's 3rd paper. Alexander 1956, 27)

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2. Undoubtedly nothing is, without a sufficient reason why it is, rather than not; and why it is thus, rather than otherwise. But in things in their own nature indifferent; mere will, without any thing external to influence it, is alone that sufficient reason. As in the instance of God's creating or placing any particle of matter in one place rather than in another, when all places are originally alike. And the case is the same, even though space were nothing real, but only the mere order of bodies: for still it would be absolutely indifferent, and there could be no other reason but mere will, why three equal particles should be placed or ranged in the order a, b, c, rather than in the contrary order. And therefore no argument can be drawn from this indifferency of all places, to prove that no space is real. For different spaces are really different or distinct one from another, though they be perfectly alike. And there is this evident absurdity in supposing space not to be real, but to be merely the order of bodies; that, according to that notion, if the earth and sun and moon had been placed where the remotest fixed stars now are, (provided they were placed in the same order and distance they now are with regard one to another,) it would not only have been, (as this learned author rightly says,) la meme chose, the same thing in effect; which is very true: but it would also follow, that they would then have been in the same place too, as they are now: which is an express contradiction. (Clarke's 3rd Reply, Alexander 1956, 30-1)

Uchii's Note: Clarke now tries to apply the principle to the problem of space, confining himself to the question of natural philosophy. In particular, he tries to criticize Leibniz's relational theory of space on the principle of a sufficient reason; see Leibniz's view (below) first, and Clarke's criticism will be reproduced there.


Absolute vs. Relational Theory

As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is; that I hold it to be an order of coexistences, as time is an order of successions. For space denotes, in terms of possibility, an order of things which exist at the same time, considered as existing together; without enquiring into their manner of existing. And when many things are seen together, one perceives that order of things among themselves.

5. ... I say then, that 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 'tis impossible there should be a reason, why God, preserving the same situations of bodies among themselves, should have placed them in space after one certain particular manner, and not otherwise; why every thing 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.

6. The case is the same with respect to time. Supposing any one should ask, why God did not create every thing a year sooner; and the same person should infer from thence, that God has done something, concerning which 'tis not possible there should be a reason, why he did it so, and not otherwise: the answer is, that his inference would be right, if time was any thing distinct from things existing in time. For it would be impossible there should be any reason, why things should be applied to such particular instants, rather than to others, their succession continuing the same. But then the same argument proves, that instants, consider'd without the things, are nothing at all; and that they consist only in the successive order of things; which order remaining the same, one of the two states, viz. that of a supposed anticipation would not at all differ, nor could be discerned from, the other which now is. (Leibniz's 3rd Paper, Alexander 1956, 25-7)

Uchii's Note: Leibniz now states his own relational theory of space and time (as against the Newtonian absolute theory). Then he criticizes the absolute theory, on the principle of a suficient reason, presuming that his own relational theory is exempt from this sort of difficulty. Notice the qualifying words "in terms of possibility".

2. Undoubtedly nothing is, without a sufficient reason why it is, rather than not; and why it is thus, rather than otherwise. But in things in their own nature indifferent; mere will, without any thing external to influence it, is alone that sufficient reason. As in the instance of God's creating or placing any particle of matter in one place rather than in another, when all places are originally alike. And the case is the same, even though space were nothing real, but only the mere order of bodies: for still it would be absolutely indifferent, and there could be no other reason but mere will, why three equal particles should be placed or ranged in the order a, b, c, rather than in the contrary order. And therefore no argument can be drawn from this indifferency of all places, to prove that no space is real. For different spaces are really different or distinct one from another, though they be perfectly alike. And there is this evident absurdity in supposing space not to be real, but to be merely the order of bodies; that, according to that notion, if the earth and sun and moon had been placed where the remotest fixed stars now are, (provided they were placed in the same order and distance they now are with regard one to another,) it would not only have been, (as this learned author rightly says,) la meme chose, the same thing in effect; which is very true: but it would also follow, that they would then have been in the same place too, as they are now: which is an express contradiction. (Clarke's 3rd Reply, Alexander 1956, 30-1)

4. If space was nothing but the order of things coexisting; it would follow, that if God should remove in a straight line the whole material world entire, with any swiftness whatsoever; yet it would still always continue in the same place: and that nothing would receive any shock upon the most sudden stopping of that motion. And if time was nothing but the order of succession of created things; it would follow, that if God had created the world millions of ages sooner than he did, yet it would not have been created at all the sooner. Further: space and time are quantities; which situation and order are not.

5. The argument in this paragraph, is; that because space is uniform or alike, and one part does not differ from another; therefore the bodies created in one place, if they had been created in another place, (supposing them to keep the same situation with regard to each other,) would still have been created in the same place as before: which is a manifest contradiction. The uniformity of space, does indeed prove, that there could be no (external) reason why God should create things in one place rather than in another: but does that hinder his own will, from being to itself a sufficient reason of acting in any place, when all places are indifferent or alike, and there be good reason to act in some place? (Clarke's 3rd Reply, Alexander 1956, 32)

Uchii's Note: In these three paragraphs, Clarke seems to have almost entirely missed the point of Leibniz's relational theory. For instance, Clarke's argument in the first paragraph already assumes that "the remotest stars" have "places" independently of their relations with other bodies in the universe; whereas according to Leibniz's relational theory, it does not make sense to speak of their "places" aside from their relations with others. Thus Clarke's argument is nothing but question-begging, and Leibniz points this out in the following reply. However, in the second paragraph, Clarke raises an important question: how does the relationism explain quantitative (metrical) aspects of space and time? In the third paragraph, it is clear that Clarke's and Leibniz's understanding of "sufficient reason" are still quite different.

5. Those great principles of a sufficient reason, and of the identity of indiscernibles, change the state of metaphysics. That science becomes real and demonstrative by means of these principles; whereas before, it did generally consist in empty words.

6. To suppose two things indiscernible, is to suppose the same thing under two names. And therefore to suppose that the universe could have had at first another position of time and place, than that which it actually had; and yet that all the parts of the universe should have had the same situation among themselves, as that which they actually had; such a supposition, I say, is an impossible fiction. (Leibniz's 4th Paper, Alexander 1956, 37)

Uchii's Note: Here, Leibniz alludes to the principle of the identity of indiscernibles. This principle, as well as that of sufficient reason, allows ambiguous interpretations.

15. It is a like fiction, (that is) an impossible one, to suppose that God might have created the world some millions of years sooner. They who run into such kind of fictions, can give no answer to one that should argue for the eternity of the world. For since God does nothing without reason, and no reason can be given why he did not create the world sooner; it will follow, either that he has created nothing at all, or that he created the world before any assignable time, that is, that the world is eternal. But when once it has been shown, that the beginning, whenever it was, is always the same thing; the question, why it was not otherwise ordered, becomes needless and insignificant.

16. If space and time were any thing absolute, that is, if they were any thing else, besides certain orders of things; then indeed my assertion would be a contradiction. But since it is not so, the hypothesis [that space and time are any thing absolute] is contradictory, that is, 'tis an impossible fiction.

17. And the case is the same as in geometry; where by the very supposition that a figure is greater than it really is, we sometimes prove that it is not greater. This indeed is a contradiction; but it lies in the hypothesis, which appears to be false for that very reason. (Leibniz's 4th Paper, Alexander 1956, 38-9)

Uchii's Note: Leibniz explains that his argument against the absolute theory takes the form of reductio ad absurdum, assuming the principle of a sufficient reason. If you take the absolute theory, then such absurdity; if the relational theory, no difficulty.

5 and 6. Two things, by being exactly alike, do not cease to be two. The parts of time, are as exactly like to each other, as those of space: yet two points of time, are not the same point of time, nor are they two names of only the same point of time. Had God created the world but this moment, it would not have been created at the time it was created. And if God has made (or can make) matter finite in dimensions, the material universe must consequently be in its nature moveable; for nothing that is finite, is immoveable. To say therefore that God could not have altered the time or place of the existence of matter, is making matter to be necessarily infinite and eternal, and reducing all things to necessity and fate.

13. If the world be finite in dimensions, it is moveavble by the power of God and therefore my argument drawn from that moveableness is conclusive. Two places, though exactly alike, are not the same place. Nor is the motion or rest of the universe, the same state; any more than the motion or rest of a ship, is the same state, because a man shut up in the cabin cannot perceive whether the ship sails or not, so long as it moves uniformly. The motion of the ship, though the man perceives it not, is a real different state, and has real different effects; and, upon a sudden stop, it would have other real effects; and so likewise would an indiscernible motion of the universe. To this argument, no answer has ever been given. It is largely insisted on by Sir Isaac Newton in his Mathematical Principles, (Definit. 8.) where, from the consideration of the properties, causes, and effects of motionl he shows the difference between real motion, or a body's being carried from one part of space to another; and relative motion, which is merely a change of the order or situation of bodies with respect to each other. This argument is a mathematical one; showing, from real effects, that there may be real motion whee there is none relative; and relative motion, where there is none real: and is not to be answered, by barely asserting the contrary.

14. The reality of space is not a supposition, but is proved by the foregoing arguments, to which no answer has been given. Nor is any answer given to that other argument, that space and time are quantities, which situationa and order are not.

15. It was no impossibility for God to make the world sooner or later than he did: nor is it at all impossible for him to destroy it sooner or later than it shall actually be destroyed. As to the notion of the world's eternity; they who suppose matter and space to be the same, must indeed suppose the world to be not only infinite and eternal, but necessarily so: even as necessarily as space and duration, which depend not on the will, but on the existence of God. But they who believe that God created matter in what quantity, and at what particular time, and in what particular spaces he pleaseed, are here under no difficulty. For the wisdom of God may have very good reasons for creating this world, at that particular time he did; and may have made other kinds of things before this material world began, and may make other kinds of things after this world is destroyed.

16 and 17. That space and time are not the mere order of things, but real quantities (which order and situation are not;) has been proved above (See Third Reply, 4; and in this paper, 13,) and no answer yet given to those proofs. And till an answer be given to those proofs, this learned author's assertion is (by his own confession in this place) a contradiction. (Clarke's 4th Reply, Alexander 1956, 46-9)

Uchii's Note: Clarke's objection suggests that the principle of the identity of indiscernibles is ambiguous, since what Clarke says is not a contradiction; but still, Clarke repeats essentially the same point as that in his previous letter. But then, after presenting what he takes to be "conclusive" argument, he introduces a new objection, drawing on Newton's work. He is saying that there are empirical phenomena which show the distinction between relative and absolutte motion. Since he claims he gave proofs whereas Leibniz gave no answers, this elicits a much longer reply (5th Paper) from Leibniz.

[To be continued to Part 2]

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Last modified October 11, 2001.
(c) Soshichi Uchii suchii@bun.kyoto-u.ac.jp