A Note for Students

For Brown students in PHYS0100: Flat Earth to Quantum Uncertainty: On the Nature and Meaning of Scientific Explanation, I have extracted some paragraphs from the Leibniz-Clarke Correspondence relevant to the study of space and time, leaving out much of the theologically motivated arguments.

At the very end, I provide an overly simplified summary of where the key arguments are on each side of the debate.

Leibniz' Third Letter

4. 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 have many Demonstrations, to confute the Fancy of Those who take Space to be a Substance, or at least an absolute Being. But I shall only use, at the present, one Demonstration, which the Author here gives me Occasion to insist upon. 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 it self, 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 it self. 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.

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.

Clarke's Reply to the Third Letter

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 placeing 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 fixt 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 même 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. The Ancients did not call All Space which is void of Bodies, but only extramundane Space, by the Name of Imaginary Space. The meaning of which, is not, that such Space is not real; but only that We are wholly ignorant what kinds of Things are in that Space. Those Writers, who by the Word, imaginary, meant at any time to affirm that Space was not real; did not thereby prove, that it was not real.

Lebniz' Fourth Letter

3. Tis a thing indifferent, to place three Bodies, equal and perfectly alike, in any order whatsoever; and consequently they will never be placed in Any order, by Him who does nothing without Wisdom. But then, He being the Author of things, no such things will be produced by him at all; and consequently there are no such things in Nature.

4. There is no such thing as Two Individuals indiscernible from each other. An Ingenious Gentleman of my Acquaintance, discoursing with me, in the presence of Her Electoral Highness the Princess Sophia, in the Garden of Herrenhausen; thought he could find two Leaves perfectly alike. The Princess defied him to do it, and he ran all over the Garden a long time to look for some; but it was to no purpose. Two Drops of Water, or Milk, viewed with a Microscope, will appear distinguishable from each other. This is an Argument against Atoms; which are confuted, as well as a Vacuum, by the Principles of true Metaphysicks.

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.

13. To say that God can cause the whole Universe to move forward in a Right Line, or in any other Line, without making otherwise any Alteration in it; is another Chimerical Supposition. For, two States indiscernible from each other, are the same State; and consequently, 'tis a change without any change. Besides, there is neither Rhime nor Reason in it. But God does nothing without Reason; And tis impossible there should be any here. Besides, it would be agendo nihil agere, as I have just now said, because of the Indiscernibility.

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.

18. Space being uniform, there can be neither any External nor Internal Reason, by which to distinguish its parts, and to make any choice among them. For, any External Reason to discern between them, can only be grounded upon some Internal one. Otherwise we should discern what is indiscernible, or chuse without discerning. A Will without Reason, would be the Chance of the Epicureans. A God, who should act by such a Will, would be a God only in Name. The cause of these Errors proceeds from want of care to avoid what derogates from the Divine Perfections.

19. When two things which cannot Both be together, are equally good; and neither in themselves, nor by their combination with other things, has the one any advantage over the other; God will produce Neither of them.

Lebniz' Fifth Letter

29. I have demonstrated, that Space is nothing else but an Order of the existence of things, observed as existing Together; And therefore the Fiction of a material finite Universe, moving forward in an infinite empty Space, [3] cannot be admitted. It is altogether unreasonable and impracticable. For, besides that there is no real Space out of the material Universe; such an Action would be without any Design in it: It would be working without doing any thing, agendo nihil agere. There would happen no Change, which could be observed by Any Person whatsoever. These are Imaginations of Philosophers who have incomplete notions, who make Space an absolute Reality. Mere Mathematicians, who are only taken up with the Conceits of Imagination, are apt to forge such Notions; but they are destroyed by superior Reasons.

A Modern Gloss on the Arguments

Relationists like Leibniz argued that space is a kind of fiction. They thought there are facts about how far apart material objects are from one another, but disagreed that these distance-facts were the consequence of facts about where objects were located. Leibniz takes the distance facts to be primitive, and thinks that all other spatial facts derive from these. Leibniz offers several arguments like the following: Suppose the positions of all objects were shifted one foot away in some direction. Such a change would not be distinguishable from the actual world. By the principle of identity of indiscernibles, the actual world and the shifted world are identical, which means that the shift did nothing at all. This backs up Leibniz’s theory of space in that fundamentally the only facts about space are distances between objects, and so the shift is a shift in name only, making no physical (or metaphysical) difference. If there were a space existing independent of matter, as Newton thought, then there would exist an objective (metaphysical) difference between the actual world and the shifted world. Newton is alleged to be wrong for positing such an in principle undetectable difference. This argument also comes in a theological variety, using the principle of sufficient reason, which claims that everything that happens happens for a reason. If the actual world and shifted world were distinct, God would have to choose which one to instantiate. But since they are alike in all detectible respects, God would have no reason to prefer one over the other. Nature is not allowed to put God in such a bind. Thus, there is no metaphysical difference between the two.

Similar arguments can be made using (velocity) boosts, twists (reorientation of the spatial direction), and temporal shifts (delays), but there are some some subtle differences between the boost and shift arguments.

Newton’s most important argument for the absolute nature of motion was that we need the concept of absolute space to explain rotation properly. Leibnizian theories of space arguably cannot explain why the water in a rotating bucket becomes concave because the relative distances among all the parts of the bucket and water remain constant whether the bucket is at rest or rotating. Specifically, the empirical phenomena requires us to postulate absolute (not relative to any material objects) acceleration. [Although Newton does not press the following argument explicitly, his conclusion that there exists absolute locations might be motivated by the following: Since absolute acceleration is a change in absolute velocity, and absolute velocity is a change in absolute position, we must postulate absolute positions and velocities even though we cannot in principle detect the true (absolute) positions or speeds of things.]

A subtle point is that Leibniz can express the observed phenomena (the concave surface of the water) in terms acceptable to him, but what is hard to do (perhaps impossible?) is to derive the degree of concavity using only relational quantities. Newton can explain the bucket phenomena because he can use facts about how fast each bit of water is traveling in the direction perpendicular to the radius. Leibniz cannot use such facts because motion perpendicular to the radius does not necessarily track changes in the distances between bits of water (or any other matter), or at least is hard to see how they could. The classical relationist is allowed to use facts like the spatial distances between objects and their time derivatives, so they can use r(t), r'(t), r''(t), etc. No one has formulated an account of classical mechanics using only relational quantities, but this is a research project a few people still work on.