The Bucket

Compare and contrast Newton’s and Leibniz’s opposing accounts of the nature of space. Prior to the development of General Relativity, which (if any) of the two accounts was better supported?

Introduction

The development of modern physics has led to greater conversation about the nature of space. However, General relativity and quantum mechanics have added more confusion than solutions. General relativity, particularly the Hole arguments, suggests that relationism is the better theory, whereas quantum mechanics indicates that substantivalism solves many problems with quantum entanglement. Consequently, there is merit in returning to the classical debate between Newton and Leibniz to assess which account of the nature of space has more virtues.
This essay will evaluate the virtues of their accounts by assessing their strongest argument for their view, its intellectual costs, and their resilience to objections. Both accounts hold some merit; therefore, the best-defended theory is better.

Newton was a Substantialist, meaning he believed that space could exist without material objects or, put another way, space is composed of real spatial locations independent of matter but where matter can exist (Huggett, 2002, pp148). For example, imagine two Trees; on the substantivalism picture, one Tree is located at space points x and the other at y. If the Tree located at y disappeared, Tree x would still be located at x and could move to y. Space is a real thing where objects exist within it.

Rationalism was famously held by Leibniz, who believed ‘space to be something merely relative’ (Huggett, 2002, pp146). There is matter, and then there are relationships between matter. However, there is no absolute space or area absence of material relation. What defines space are only relations between material objects. If all matter disappeared, then space would cease to exist. When objects move, they move closer or farther away from other matter, not to a different point in space. Again, imagine two trees; on the rationalist picture, it is not the case that those Trees are located at a space point, but rather, what defines their position is their relation between each other, i.e., Tree A is 30 metres from Tree B. If all that exists was Tree A, then it couldn’t move in a direction as there are no relations between objects which could change. Consequently, space is only a difference in relation to matter.

Newton's Non-inertial argument for Substantivalism

The Bucket Argument
Newton's strongest argument for substantivalism is its explanatory strength of inertial properties. This is most evident in Newton's Bucket argument, which discusses the properties of accelerating bodies with the example of a vessel filled with water (Newton, 1846, pp. 81). First, imagine a still bucket filled with water, handing on a twisted string. The water relative to the bucket is at rest and is flat across the bucket. Second, imagine the string being released, and the bucket begins to spin. The bucket is now spinning relative to the water, which also begins to spin. The bucket is in a non-inertial motion as the bucket sides rotate and speed up. Third, the bucket and water are now at rest relative to each other, but they will now have a new inertial property, which makes the water concave. Therefore, the water can be at rest relative to the bucket and have two states, namely, 1) The bucket without the inertial property, where the water is flat and 2) The bucket with the inertial property, making it concave. Newton's example presents a good case in which both the substantialist and relationalist must give an account. Specifically, why does one bucket of water have a concave property and the other does not when, in both cases, the water relative to the bucket is at rest?

Substantivalist Solution
The Substantivalist and their commitment to absolute space allows them to explain the two states of the bucket because the concaved water bucket is actually moving through spatial points. Bucket 1) is not moving through absolute space and thus is not infected with an inertial property. In contrast, Bucket 2 is moving though absolute space. Its angular velocity is changing, which gives rise to its inertial property of being concave. The water particles are pushed outward due to this non-inertial movement through spatial points. Therefore, there is an explanation for why one bucket is in a different state than the other, namely movement.

The relationalist problem
Motion, in relationalism, is true when there is a change in relation to other material objects (Ie. Distance, shape, size, etc.). Therefore, if the relations between material objects remain constant, then the relationalist can’t distinguish any movement.

First, take the Bucket Argument, excluding the external world. The water without the inertial property is not moving relative to the bucket, but the water with the inertial property is also not moving relative to the bucket relative. The two states, namely still and spinning, are relationally equivalent; both the water and bucket are at rest with each other and thus are not moving relative to each other. If the relationalist can't appeal to motion to explain the inertial property, they must find another difference to explain why the second state of the bucket has the inertial property. But there doesn’t seem to be an obvious one. Therefore, the relationalist has an inexplicable case of an inertial property appearing without any relative motion.

Second, there is a temptation to appeal to the external world for a relational explanation. Could it not be the case that the bucket is spinning relative to the observer and, thus, has an inertial property? However, the Substantivalist can argue that external relations are not a contributing cause of the inertial property in the state of the bucket. For example, you could imagine the surrounding walls and the observer moving around the bucket on a spinning table (Dasgupta, 2015, pp604). Relative to the observer, it would seem like the bucket is spinning, and relative to the bucket the room would be spinning. But intuitively, the bucket would not gain an inertial property even if, relative to the observer and room, the bucket is spinning.

Furthermore, although the relative speed may be slow, planets, stars, and even galaxies are moving relative to the bucket and us right now (hubblesite.org, 2023). This shows us that the bucket can spin relative to lots of matter without gaining any observable inertial properties. Therefore, the relationalist is in the same position. Unable to give an account of why one bucket has an inertial property of concaveness, and the other one doesn’t when the bucket and water are at rest with each other and external relations to matter don’t seem to have any effect. Newtons concluded from this experiment that space must be substantival since relationalism can’t account for the inertial property.

Mach’s empirical objection
One line of objection which might weaken this argument is against its empirical claim. Newton Bucket's strongest virtue is empiricism, which makes it testable. Mach criticises what Newton's experiment can demonstrate by arguing ‘We cannot know how K [The bucket with the inertial property] would act in the absence of A, B, C [The rest of the matter in the universe]’ as Newton's experiment cannot be produced excluding the rest of the universe (Huggett, 2002, pp198). On a solely empirical basis, the bucket will always be spinning in relation to distant objects in the universe. It could be true that spinning in relation to absolute space leads to the inertial property. But the empirical experiment is equally compatible with saying spinning in relation to distant matter in the universe leads to the inertial property of the bucket. Even if we make the walls and observer spin around relative to the bucket, the bucket still spins relative to some distant matter in the universe. The relationalist can always appeal to this relational movement with distant fixed matter as the cause of the inertial property. Consequently, the bucket experiment exaggerates its empirical claim since absolute space is not the only explanation.

The substantivalist may admit that there is a relation between the spinning bucket and distant matter but argue that since objects like walls can spin relative to the bucket without infecting an inertial property this empirical objection is weak. The relationalist must explain why distant objects would infect the bucket with an inertial property when objects spinning relative to the bucket, such as a car driving around the bucket, have no observable effect.

Machian Space
Mach attempts to develop a theory which explains the inertial effect on a relational view while being equally compatible with the bucket experiment not gaining an inertial force from objects like the car (Huggett, 2002, pp183). He argues that the type of relative motion which produces the inertial effect is relative to the whole ‘Universe’ and the centre of mass it produces. The centre of mass is defined by ‘sufficiently distant and sufficiently large masses’. For example, imagine a universe with only three stars of equal weight and distance; the centre of mass would be in the centre of the triangle it would create. Those stars could still move, and there would always be a centre of mass somewhere. As a result, instead of motion relative to space points, it is relative to the centre of mass of all the matter in the universe, which causes an inertial force. Moving walls and the observer around the bucket does not infect an observable inertial property since this does little to move the centre of mass in the universe. This theory is incomplete, but it at least gives a possible non-Newtonian theory of inertia forces based on relationalism that also explains why some objects moving in relation to the bucket don’t cause observable inertial effects. As a result, Mach limits the strength of Newton's argument by restricting its empirical claim and providing a compatible theory of the inertial property.

Rebuttal to Machian space
Mach’s empirical objection is difficult to refute. No practical experiment excludes the rest of the universe, and so the relationalist can always appeal to relational movement. Although it is possible and empirically compatible, I argue that the substantivalist can demonstrate this as less plausible.

First, Machian mechanics is not as developed. Newtonian mechanics is a developed theory which has been constantly verified in experiments to predict inertial forces and motion of bodies. Machian laws of inertia are not used by physicists, and physicists who have tried to develop it have failed to create a working model (Dasgupta, 2015, pp622). This is a weakness of Mach's objection since, although it is compatible, it is not as strong as Newtonian mechanics. A similar analogy can be drawn with Geocentric vs Copernican models of the solar system. We can produce mathematics and laws, like Ptolemy of Alexandria did, to demonstrate that the world is at the centre of our solar system (Britannica, 2023). For example, it could be the case that the laws of the universe and forces produced from distant objects infect the planets to take a wiggly path around the Earth instead of circular orbits around the sun. Therefore, the Geocentric believer can produce a compatible account to experience. However, the Copernican model is generally thought of as more justified. This is due to the simplicity and completeness of its theory. As a result, the substantivalist is more justified with Newtonian substantivalist than machine relationalism.

Second, The substantivalist can adapt the bucket experiment to a thought experiment which prevents the relationalist appeal to the external universe. Imagine a universe with only the bucket experiment. If this universe were similar to ours, then we can conceptualise the bucket spinning. We can imagine it spinning in our world and, one by one, all the matter disappearing until just the bucket is left. As soon as the last particle disappears, the relationalist must believe the bucket has stopped spinning and has no inertial properties. Yet, a moment before the bucket could be spinning 60MPH and water concaved, the only difference being a relation to matter. Even if god impresses a great force upon the bucket, if the matter disappeared, then there would be no movement as the centre of mass of the universe would be solely dependent on the bucket. This thought experiment demonstrates the strangeness of relationalism and counterintuitive view of such a universe. Consequently, even if Mach provides a compatible account of inertial effects in the bucket experiment, I believe the substantivalist is more justified in their view by virtue of absurd and counterintuitive thought experiments.

Of course, thought experiments aren’t proofs, and a theory being underdeveloped doesn’t necessarily mean it is wrong. However, I believe they are reasons to prefer a theory if there are no other reasons to prefer one over another. The relationalist best defence against the bucket argument is finding ways a relational view is compatible. But as we have shown, they aren’t very convincing. If the relationalist can present an empirical or logical positive argument for substantivalism, then they are more justified. However, based on Macks's objections alone, substantivalism is more probable, ergo, the better theory.

Leibniz's metaphysical principles against Substantivalism

Leibniz's strongest positive argument for relationalism is based on two logical principles. I believe the strongest of his arguments is based on the Principle of Sufficient Reason (PSR). However, I think it is first important to show why the PSR is a stronger and more vital argument for Leibniz's account than his other argument based on the principles of indiscernibility (PII).

Leibniz PII
The Principle of indiscernibility is motivated by the PSR. Leibniz first suggests thinking about three bodies [a,b,c] that are ‘equal and perfect alike’ (Huggett, 2002, pp149). He then argues that there must be a reason for ordering or creating those objects a,b,c as opposed to c,a,b (PSR). But if they are equal and share all the same attributes, then there is no reason to order them one way or the other. Consequently, Leibniz argues that if god were to create such objects, he would not have a reason for deciding which order to put them in. Therefore, God's decision would be inexplicable and arbitrary. Leibniz argues god is ‘Wise’. Thus, this is not possible. Therefore, god would not have created two or more indiscernibility objects (PII). This can be adapted without the existence of god. If there is no explanation or difference in attributes of two objects, then there isn’t a reason for them being different. Consequently, they must be referring to the same object; therefore, there are no two indiscernible objects.

If the PSR was removed, two objects could exist without the need for an explanation for why they are separate. In fact, we can conceptualise two objects with identical predicates but that are separate (Black, 1952, pp156). Imagine a universe with only two spheres with all the same intrinsic properties like colour, mass, etc. Since they are the only two objectives in the universe, they all have the same extrinsic properties, such as relation to each other. This universe seems possible, and if we existed in the universe, we would see two spheres. Consequently, two indiscernible objects are conceptually possible but impossible to occur in reality if the PSR is true. For these reasons, any argument built on the PII is only as strong as the PSR.

Leibniz PSR Argument

The Principle of Sufficient reason states that for every fact, there must be a sufficient reason why it is that fact rather than another (Leibniz, 1714, pp5). This is a powerful ‘axiom’ as it lays the foundation for much rational thought. Intuitively, every action we make or situation we are in requires some explanation for why we are in that state rather than the other. If states could exist without explanations, then why did that state exist when another state could have just as much a reason, or lack of, to exist? Therefore, the PSR is necessary for explaining the states of the world and is fundamental to our reason.

Absolute space
It is important to note that absolute space is qualitatively identical to every other part of space (Huggett, 2002, pp147). There is no qualitative difference between a space point here and a space point over there because there are no distinguishable features between space points. It is only by virtue of being positioned at a space point one can distinguish a space point. As if we are situated at our current space points then we can know we aren’t at space points located separate from us. As a result, there is no way to demonstrate a difference in position based on substantival space points alone.

The Static Shift Argument
From this, Leibniz presents a reductio argument presuming absolute space with the PSR (Huggett, 2002, pp146). If facts need explanations (PSR), then facts about the Universe require an explanation. Our universe has matter, which exists at a regen of absolute space; therefore, there is an explanation for why that matter is located there rather than somewhere else. The nature of absolute space means that there is no qualitative difference between one region or the other. If there is no difference, then there is no explanation for favouring matter located at one region of space over the other. This violates the PSR. Therefore, absolute space is absurd.

Leibniz presents a static shift example to demonstrate this point (Huggett, 2002, pp146). On Substantivalism, all the matter in our universe is currently located at its own space point. Now imagine another Universe identical to ours in every single way except all the matter is shifted several space points in the same direction. This is entirely possible with substantival space being infinitely expansive spatial locations. We now have two possible ways the universe could have been, namely, 1) The state of our Universe located at its current spatial points and 2) The state of a Universe identical to ours except located at several spatial points from our own. The PSR demands an explanation of why the Universe is 1) and not 2). However, there is no explanation, as both states of the universe are quantitively identical. Universe 2) has the same type of matter, relations to other matter and properties, except it is located at different spatial points which are themselves identical to any other spatial point. Consequently, substantivalism cannot explain why it is located where it is, which violates the PSR. Therefore, substantivalism leads to absurd consequences.

This problem arises due to the commitment of Absolut space, but if space is merely relational then there doesn’t need to be an explanation for the location of the universe. There is no difference between Universe 1) and 2) since space and the relational properties of matter are identical. Consequently, if the PSR is right then relationalism is a far better account of space since it doesn’t violate the PSR. God's actions or the nature of the universe can prevent a lack of explanation by having relational space.

Brute Fact Objection
Leibniz presents a very strong argument. The position of the universe in substantival space is inexplicable. Therefore, I believe the only objection which could weaken Leibniz account is against the PSR. If we can demonstrate that this principle is not necessarily true, then this might allow enough room for it to be rejected in certain cases, such as the nature of space.
Russell notes that if the PSR is true, it needs an explanation for why it is true (Lin.M, 2007, pp296). It can't be a brut fact as this would violate itself, and it also can't appeal to the PSR without being a circular argument. There doesn’t seem to be an argument to demonstrate the PSR is necessarily true. Therefore, the PSR lacks an explanation for its own truth.
This blunts the edge of the PSR, but the relationalist can still motivate the PSR as a costly principle to ignore. Its strength in rational inquiry, which presupposes reasons for fact, and its constant verification in experience, with explanations for events, make the PSR incredibly practical. It is not enough to say the location of the universe doesn’t require the PSR, as the cost of rejection of the PSR degrades reason and it’s counterintuitive experience.

Inwagen Argument
On the other hand, I believe the Inwagen reductio argument demonstrates an equal or higher price to pay if we accept the PSR, namely the rejection of contingent truths (Pruss, 2006, pp97). His argument can be summarised as follows; ‘BCCF’ is the conjunction of all contingent truths. Every contingent proposition has an explanation (PSR). Necessary proportions can't explain contingent truth without making them necessary truths; therefore, they can't explain the BCCF. Thus, there must be a contingent proposition, X, which explains BCCF. X is one of the contingent truths in BCCF since BCCF contains all contingent truths. Therefore, X, which is part of the BCCF, explains the BCCF. This is absurd as it produces a circular argument. Therefore, the PSR must be false. Inwagen presents a strong argument as the set of all contingent truths requires explanations, which is also contingent and a part of the set of contingent truths. But this isn’t possible as the contingent truth would explain itself. This is a circular argument in which itself is explained by itself in an infinite circle. Consequently, if there are contingent truths, the SPR is violated. Therefore, I believe this shows that the intellectual cost of rejecting the PSR is as high as the cost of keeping it. It may be intuitive to experience and very practical in rational inquiry, but if it held in all cases, we would have to reject contingent truths.

Liebniz would find no problem with this argument since he rejects contingent truth (Pruss, 2006, pp97). I find this position very radical. If all facts are necessary, WW2 was unavoidable, future disasters will always happen and counterfactual statements are nonsensical. This is an extraordinary counterintuitive claim I believe few philosophers or people would accept. Consequently, I believe this gives us room to reject it if we have good reasons. I believe Clark presents a good reason for why the PSR doesn’t hold in the arrangement of the universe, which helps motivate substantivalism.

Clarkes Atom Argument
Clarke's argument is based on the view that composites can be broken down into fundamental matter which has the same qualitative properties (Huggett, 2002, pp152). This fits with modern physics, which suggests everything can be broken down into fundamental particles like quarks and electrons. Theses fundamental particles are qualitatively identical, much like space points are identical to each other. Consequently, we can imagine our universe 1) and a second universe 2) like our own, but one fundamental particle in one leaf is swooped with the fundamental particle in another. There would be no qualitative difference between one universe over the other. Therefore, there is no explanation for creating universe 1) over 2). This gives us an analogous case of an inexplicable universe being created. The only objection Leibniz presents against the arrangements of qualitatively identical atoms is the principle of indiscernibility, which we have already shown is only as strong as the PSR and presented an argument against it. Therefore, I believe this justifies the substantivalist account as we have an argument for the PSR not to hold concerning the ordering of matter in the space of qualitatively identical universes, which is analogous to substantivalism. If the position of matter can exist in an inexplicable order at creation, then the position of creation of the universe seems analogous. Therefore, Leibniz’s argument doesn’t do much damage to Neaton's account.

Conclusion

In conclusion, the classical debate prior to the development of general relativity has shown both arguments have some merit. However, I believe Newton’s account is more justified. His positive argument after objections still shows his account as more intuitively compelling in thought experiments and a more complete theory of explaining inertial properties than Mach’s. This led to us assessing the strength of the Leibniz argument centred on the PSR, which we found reasons to reject due to the cost of rejecting contingent truths and the PSR producing inexplicable cases with the arrangement of fundamental particles. For these reasons, I believe Newton's account is slightly more justified, ergo, the better theory.

Bibliography:

Black, M. (1952). The identity of Indiscernables. Oxford University Press, 61(242), 153-164 (12 pages). URL: https://www.jstor.org/stable/2252291. Accessed: 11/01/2024
Britannica, The Editors of Encyclopaedia. "geocentric model". Encyclopedia Britannica, 15 Dec. 2023. https://www.britannica.com/science/geocentric-model. Accessed: 11 January 2024.
Dasgupta, S. (2015). ‘Substantivalism vs Relationalism About Space in Classical Physics’. Philosophy Compass, 10(9), 601-624. URL: https://doi-org.libproxy.york.ac.uk/10.1111/phc3.12219. Accessed: 11/01/2024
Earman, J. (1989). World Enough and Space-Time (chaps. 4-5). Cambridge, MA: MIT Press.
Forrest, P. (2020). "The Identity of Indiscernibles". The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.). URL: https://plato.stanford.edu/archives/win2020/entries/identity-indiscernible/. Accessed: 11/01/2024
Hubblesite.org (2023). ‘One of Hubble’s Key Projects Nails Down Nearly a Century of Uncertainty’. URL: https://hubblesite.org/mission-and-telescope/hubble-30th-anniversary/hubbles-exciting-universe/measuring-the-universes-expansion-rate#:~:text=Using%20the%20largest%20telescope%20of,expanding%20uniformly%20in%20all%20directions. Accessed: 11/01/2024
Leibniz, G. W. (1714). ‘The Principles of Philosophy known as Monadology’. URL: https://www.earlymoderntexts.com/assets/pdfs/leibniz1714b.pdf. Accessed: 11/01/2024
Leibniz, G. W. (1686). ‘Discourse on Metaphysics’. URL: https://www.earlymoderntexts.com/assets/pdfs/leibniz1686d.pdf. Accessed: 11/01/2024
Leng, M. (2024). ‘Philosophy of Physics: Absolute vs. Relational Theories of Space’ Lecture at University of York.
Lin, M. (2007). “Spinoza’s Arguments for the Existence of God.” Philosophy and Phenomenological Research, 75(2), 269–97. http://www.jstor.org/stable/40041109. Accessed 11/01/2024.
Maudlin, T. (1993). “Buckets of Water and Waves of Space: Why Spacetime Is Probably a Substance.” Philosophy of Science, 60(2), 183–203. http://www.jstor.org/stable/188350. Accessed: 11/01/2024
Melamed, Y. Y., & Lin, M. (2023). "Principle of Sufficient Reason". The Stanford Encyclopedia of Philosophy, Edward N. Zalta & Uri Nodelman (eds.). URL: https://plato.stanford.edu/archives/sum2023/entries/sufficient-reason/. Accessed: 11/01/2024
Newton, I. (1846). Mathematical Principles of Natural Philosophy. Published by Daniel Adee, 45 Liberty Street NY. URL: https://redlightrobber.com/red/links_pdf/Isaac-Newton-Principia-English-1846.pdf. Accessed: 11/01/2024
O’Connor, T., & Franklin, C. (2022). "Free Will". The Stanford Encyclopedia of Philosophy. URL: https://plato.stanford.edu/archives/win2022/entries/freewill/.
Pruss, A. R. (2006). ‘The Principle of Sufficient Reason. Cambridge University Press.
Williams, B. (1981). ‘Internal and external reasons’. In Moral Luck: Philosophical Papers 1973–1980 (pp. 101–113). Cambridge: Cambridge University Press.