Conceptions of Physical Space: Ideal, Absolute, Relational


People have been conjecturing on the true ontology of physical space since at least the pre-Socratic Greeks, and I shall not take the time here to trace the entire history of thought. Instead, I want to focus on three or four broad classes of conceptions about space (primarily absolute and relational, and mixtures of the two) that have been much discussed among European philosophers since the Seventeenth Century.

We measure space by measuring distances. When these distances change, we talk about motion. Change entails a notion of time; thus our notions of space and time are tangled up in, or perhaps derive from, the concept of motion. (Interestingly, Rene Descartes identified space with extension; he claimed that only matter can have the property of extension, and what we sometimes mistake for empty space is in fact filled with matter.) Descartes recognized relative motion–that is, motion of an object relative to other objects. However, he and many others supposed that “true motion” was relative not merely to other objects, but to a fixed background.

Isaac Newton proposed that this fixed background is three-dimensional Euclidean space, which he called absolute space. Newton’s space is not an attribute or a substance, but a “pseudo-substance” which is non-material but physical. Absolute space implies absolute speed, which would be the rate of change of position of an object relative to an arbitrary point in absolute space. This is the first concept of space I wish to highlight: the absolute conception.

Gottfried Leibniz disagreed with this description. He pointed out that Newton’s Laws of Motion assume Galilean Relativity, which means a closed system is unaffected by a constant change in position; in other words, absolute speed is never measurable by anyone, no matter how good their powers of perception. Thus Leibniz argued that absolute space is a fiction which no one can ever prove to exist through experiment. He went further, claiming space is a conception in the mind of the application of relations between objects (unlike Descartes and Newton, who thought space as objectively real, independent of any mind). Motion is possession of force, Leibniz theorized. Specifically, while relative motion is not real (in agreement with Newton), there is a quantity (mass times speed-squared–we can only measure its change, not any true initial value) which determines an object’s kinetic energy, and it is this non-spatial characteristic which defines true motion. Thus, Leibnizian motion actually has no “movement” in our everyday conception of the notion. Here, I shall refer to Leibniz’s description of space as the ideal conception.

In the latter half of the Twentieth Century, philosophers were able to develop a notion of neo-Newtonian spacetime (a.k.a. Galilean Spacetime) which solved the problem of absolute speed. In this geometric model, acceleration is well-defined, but velocity is not. Acceleration is absolute, defined in terms of the geometry of the spacetime rather than in terms of the relation between objects. Since this geometry has an independent existence, this implies that neo-Newtonian spacetime is a substance or pseudo-substance. This formulation also overcomes Leibniz’s objection about absolute speed. But then further questions arise: what sort of substance is it, and do we really need it as an explanation? Both the Newtonian and neo-Newtonian views of space are known as substantivalism.

It has been argued that rotation cannot be explained in terms of relative motion. Historically, this argument has been used in support not of Newtonian absolute space, but in support of Galilean spacetime substantivalism. Ernst Mach rejected this argument: he said that in reality, when an object rotates, it does so in relation to everything else around it; in a thought experiment where a “rotating” object is the only object in the Universe (and so has nothing to rotate relative to) we are not able to deduce anything, because we can never do such an experiment, and thus never know what would happen in such a situation. Therefore, the postulate of an absolute space is a metaphysical assertion unjustifiable by experiment, according to Mach. Instead of referring to absolute space, Mach said we can refer to a reference frame with respect to very distant objects averaged out over all directions. Mach’s idea is the paradigm for a true relational conception of space: he wanted to replace all notions of absolute space and times with relative ones. Such a theory would predict new effects due to relative accelerations (something like gravitomagnetic induction).

Is Machian relationism a consistent theory in classical mechanics? Albert Einstein attempted to create a Machian classical theory, but he was led to a non-classical notion of spacetime. In the 1970s, Barbour and Bertotti were able to create a Mach-style theory using Lagrangian mechanics (the language in which classical mechanics is modernly expressed), but it made empirically falsified predictions. They tried again in the 1980s, and that ended up being a subset of Newton’s theory.

In the theory of Special Relativity (SR), Einstein showed that acceleration and rotation are dependent on spacetime interval relations. This does not satisfy Mach’s full relationalism, because although there are no absolute positions or velocities in SR, there are absolute accelerations, and thus absolute motion. This is similar to the neo-Newtonian view mentioned earlier.

While acceleration is absolute in SR, Einstein relativized it in his theory of General Relativity (GR) with his Principle of Equivalence of Gravitation and Inertia. (In that principle, he postulated that there is no possible physical experiment which could ever determine whether a person in an accelerating spaceship was in fact accelerating or whether the ship was holding itself in a rest frame relative to a uniform gravitational field whilst the rest of the Universe was in free-fall around it.) Einstein introduced GR in 1915. Through it, he claimed to have eliminated any need for a notion of absolute motion; however, most physicists reject this claim. It was hoped that the equations of GR would demonstrate that gravitational / intertial forces are completely determined by the distribution of matter and their relative motions, thus constituting a completely relational theory; however, it turns out that these relations only partly determine the forces. Gravitational forces should be measurable in some situations even when matter “generating” the gravity is absent, according to GR. By 1918, Einstein realized that GR was not fully relational. (However, Julian Barbour has claimed that GR is, in fact, fully relational when reformulated as Shape Dynamics. Thus, whether GR can satisfy Mach’s vision is an open question of research.)

Given the primacy of the notion of a geometric space in GR, many have assumed its ontological significance, similar to neo-Newtonian substantivalism. However, this is not necessarily the case. It has been suggested that the metric be considered as a field element. The resulting metric field does not endow space with properties; rather it is a description that results from other laws of physics which govern matter fields. (This appears to be Nobel laureate Frank Wilczek’s view in his pop-sci book, The Lightness of Being.)

Another Nobel laureate in Physics, Steven Weinberg, agrees that “spacetime geometry” is merely a convenient language. In the preface to his classic textbook, Gravitation and Cosmology he writes that students of GR may ask why a free-falling object follows a geodesic. It is insufficient to answer that “because spacetime is a Riemannian manifold” because the geometric picture obscures deep connections between gravity and the rest of physics; we already know that the non-gravitational interactions (electroweak, strong) cannot be consequences of geometry. Instead, Weinberg prefers to build the concepts of GR off the experimental fact of the Principle of Equivalence. Geometry is introduced only as a mathematical tool when needed. The question then becomes “why is the Equivalence Principle true?” Weinberg believes the answer lies in a quantum theory of gravitation.


Huggett, Nick and Hoefer, Carl. “Absolute and Relational Theories of Space and Motion”, The Stanford Encyclopedia of Philosophy (Spring 2018 Edition), Edward N. Zalta (ed.), URL: <>.

Weinberg, Steven. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley & Sons, Inc., 1972.

Wilczek, Frank. The Lightness of Being: Mass, Ether, and the Unification of Forces. Basic Books, 2008.


[The two pictures in this post are original drawings by Hermann Minkowski.]


“The Physics of Infinity”–a review


This article is a review of an essay in Nature Physics called “The Physics of Infinity” by Ellis, Meissner, and Nicolai. I have put my own thoughts in italics and in parentheses to distinguish them from the ideas of the authors.

The authors give examples where the word “infinity” is sometimes used in modern science: number of galaxies; number of cosmoi in “multiverse” models; the small distances where infinities appear and cause quantities to diverge. They then ask the question: if you want a theory which is always applicable at arbitrarily large and arbitrarily small distances, does this imply an actual infinity in nature?

Just because we have a symbol for it, and it can be manipulated in equations, does not mean infinity is real. They state: “Infinity is not just a very big number: it is not a number at all; rather, it is bigger than every number.”

There is a duality between zero and infinity which can be expressed by “one divided by zero equals infinity”. If one element of the duality does not occur in nature, then the other one ought not as well. The authors point out that we know from theory and experiment that “nothing” does not and cannot exist in nature. Then they “interpret” the number zero as a measure of physical existence, thus showing that this zero cannot actually refer to a physical state of nothingness. The implication is that infinity cannot refer to any physical state, either. (I am skeptical on this point. Zero apples is not the same measurement as zero oranges. If they are restricting their meaning of zero to refer to the amount of energy in a given arbitrary but finite volume, then the argument can only mean there can never be infinite energy in an arbitrary finite volume, which I think most physicists would already agree about. If I’m wrong, please explain in the comments below.)

The authors highlight two concepts of potential infinity: “very large number” and “essential”. Very large number infinity is a placeholder for an unimaginably large but finite number, whilst essential infinity is … (is what? The authors never define it except to say that it has a paradoxical nature. However, given the way they use these terms in the essay, it seems as if by “essential infinity” they mean actual infinity, whereas by “very large number infinity” they mean potential infinity. If that is the case, then it would not be correct to say that the two infinities they discuss are both potential.)

Traditionally, when physicists refer to infinity, they mean the “very large number” sense: it is a number so large they never have to worry about its exact–or even ballpark–value. Infinity in their calculations is a mathematical idealization which helps to calculate limits. However, essential infinity cannot exist physically because it violates conservation laws, as demontrated by the fact that if you take any number and add it to or multiply it by essential infinity, the result is still infinity.

The aforementioned duality implies there is an “essential zero” which is dual to essential infinity. Its relations demonstrate that it can never have a physical effect, and is therefore an unphysical concept.

But there is another zero which is dual to the very large number infinity. This is a very small number which operates as an effective zero. The authors don’t explain how such a zero would work, but they refer to a paper: “Are Real Numbers Really Real?” by Nicolas Gisin.

Since essential infinity and essential zero cannot exist in physics, this should be a criterion to guide theory choice. Infinity should be viewed merely as a mathematical idealization, a placeholder which refers to a very large number.

There follows a discussion about continuum assumptions and treating particles as points, which cause problems with infinity in Quantum Field Theory. Theories which attempt to resolve this issue have their own problems. The authors claim, for example, that Loop Quantum Gravity is not a viable theory because it uses essential infinity in a basic, built-in assumption of the theory.

Even if there were an “infinite number” of galaxies in the essential infinity sense, and even if there were no horizons so that we could count them, we would never be able to count enough to prove it, so the claim is not scientific. (Given that there really are observation horizons, the claim that “there are a finite number of galaxies in existence” is also not falsifiable. However, in his pop-sci book Time Reborn, Lee Smolin claims that an infinite Universe implies boundedness, which leads to various logical inconsistencies; the only logically coherent model of the Universe, he claims, is one which is finite and unbounded.)

Because visual horizons do exist, a “multiverse” claim (which is the proposition that “multiple cosmoi exist”) is not verifiable through observation. The implication of the existence of a “multiverse” is derived from Inflation Theory, which is not established physics. Probability calculations do not lend support to the idea, because there is no unambiguous probability measure under such circumstances.


Ellis, George F.R. et al. “The Physics of Infinity.” Nature Physics, 23 July 2018.*

*Thanks to Sabine Hossenfelder for sharing this link on her Twitter.

Streaks and stripes
This picturesque view from the NASA/ESA Hubble Space Telescope peers into the distant Universe to reveal a galaxy cluster called Abell 2537. Galaxy clusters such as this one contain thousands of galaxies of all ages, shapes and sizes, together totalling a mass thousands of times greater than that of the Milky Way. These groupings of galaxies are colossal — they are the largest structures in the Universe to be held together by their own gravity. Clusters are useful in probing mysterious cosmic phenomena like dark matter and dark energy, the latter of which is thought to define the geometry of the entire Universe. There is so much matter stuffed into a cluster like Abell 2537 that its gravity has visible effects on its surroundings. Abell 2537’s gravity warps the very structure of its environment (spacetime), causing light to travel along distorted paths through space. This phenomenon can produce a magnifying effect, allowing us to see objects that lie behind the cluster and are thus otherwise unobservable from Earth. Abell 2537 is a particularly efficient lens, as demonstrated by the stretched stripes and streaking arcs visible in the frame. These smeared shapes are in fact galaxies, their light heavily distorted by the gravitational field of Abell 2537. This spectacular scene was captured by Hubble’s Advanced Camera for Surveys and Wide-Field Camera 3 as part of an observing programme called RELICS.

k-Dimensional Tree Visualization

A k-d tree is a bifurcating arborescence (a type of directed acyclic graph) used for making data structures. It recursively divides a k-dimensional space. A kind of algorithm for it has been made by Ricardo Ponce, implemented with Houdini nodes. I followed his guidance to make the visualization you see below. It could be used to create a cityscape or any number of things. I might experiment with it further, and if I do, I’ll update this post with the new material.


Computation in Nature

The other day I happened to stumble across a transcript of a 2008 talk given by Oxford physicist David Deutsch at Indiana University Bloomington. I don’t know how long IU will keep it on their website, so I am re-posting it here. Some highlights:

… the laws of physics refer only to computable functions—either directly or via computable differential equations.

… most mathematical functions are not computable—in fact, the set of computable functions is of measure zero in the class of all mathematical functions, let alone in the class of all mathematical relationships.

Mathematics is about absolutely necessary truths. Such truths are all abstract and essentially they are truths about what is or isn’t logically implied by particular axioms, but science isn’t about what’s implied by anything. It’s about what it is really out there in the physical world. Laws of nature do therefore have to be consistent but unlike mathematical axioms they also have to correspond to reality, so that’s the fundamental difference between mathematics and science, between theories and theorems.


… whether a mathematical proposition is true or false is completely, that is, indeed completely independent of physics but proof is 100% physics: proofs are not abstract ….

A computation is a physical process in which physical objects like computers, or slide rules or brains are used to discover, or to demonstrate or to harness properties of abstract objects ….

There is nothing deeper known about the physical world than the laws of physics. And, I think, there is nothing deeper known within physics than the quantum theory of computation. And for that reason I entirely agree that it’s likely to be fruitful to recast our conception of the world and of the laws of physics and physical processes in computational terms, and to connect fully with reality it would have to be in quantum computational terms. But computers have to be conceived as being inside the universe, subject to its laws, not somehow prior to the universe, generating its laws.

To get the full arguments, please read the full transcript.

Nonfiction Writing Support

If you would like to support the nonfiction writing you read here, there is now a donation button on the right-hand side. As you can see from the menu options above, I write essays as well as reviews of books and papers concerning science and philosophy. Coming soon is a section devoted to my notes taken from textbooks and academic courses. Over the coming months I would like to deepen the content of this blog, and your support is greatly appreciated. Thank you.



The purpose of this brief post is merely to announce that I am no longer on Facebook.

I did not use it that much anyway, except to have my tweets posted automatically to my wall. I log in about once a year to check my privacy settings and make sure everything looks alright. But today when I attempted my annual login, Facebook said my account had been locked. It said I could contact three “friends” for help, and gave me a list people I do not recognize. There was one alternative: to upload an official photo ID, such as a driver’s license. I would have to be a special kind of fool to give them my personal information.

Thus I shall not be posting on Facebook for the foreseeable future. If you see anyone posting under the name Ander Nesser, that is not me. It’s okay, though. I would rather be bookfaced.