Essay Abstract

The most obvious explanation for the power of mathematics as the language of physics is that the physical world has the right sort of structure to be represented mathematically. But what this in turn means depends on the mathematical language being used. I first briefly review some of the physical characteristics required in order to unambiguously describe a physical situation using integers, and then take up the much more difficult question of what characteristics are required to describe a situation using geometrical concepts. In the case of geometry, and particularly for the most basic form of geometry-- topology--this is not clear. I discuss a new mathematical language for describing geometrical structure called the Theory of Linear Structures. This mathematical language is founded on a different primitive concept than standard topology, on the line rather than the open set. I explain how some other geometrical concepts can be defined in terms of lines, and how in a Relativistic setting time can be understood as the feature of physical reality that generates all geometrical facts. Whereas it is often said that Relativity spatializes time, from the perspective of the Theory of Linear Structures we can see instead that Relativity temporalizes space: all of the geometry flows from temporal structure. The Theory of Linear Structures also provides a mathematical language in which the fact that time is a fundamentally directed structure can be easily represented.

Author Bio

Tim Maudlin is Professor of Philosophy at NYU. He received his B.A. in physics and philosophy from Yale and his Ph.D. in History and Philosophy of Science from the University of Pittsburgh. His books include Quantum Non-Localtiy and Relativity (Blackwell), The Metaphysics Within Physics (Oxford), Philosophy of Physics: Space and Time (Princeton), and New Foundations for Physical Geometry: The Theory of Linear Structures (Oxford). He has been a Guggenheim Fellow.

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Dear Tim Maudlin,

You emphasize that associating a mathematical structure with physical items is not the same as postulating that they are mathematical entities. I fully agree with this.

As you note, boundaries and structural integrity through time is sufficient for enumeration, and, per Kronecker, given integers, the rest of math follows.

You then ask, what features must a physical entity have in order to display a geometrical structure? Perhaps this question has been less frequently asked because there is no obvious answer. You discuss topology as the standard basis of continuity based on open sets. I find open sets highly abstract and artificial, and difficult to relate to physical reality in the natural way that integers (counting) relate to discrete physical entities. It is therefore of some interest that you switch tools to your Theory of Linear Structures and ordered, i.e., directed lines. As this is, in my opinion, a preferred approach to geometry, it then suggests the question, "What physical feature of the universe might be responsible for creating lines?" Your suggestion that it is time that underlies geometry is both novel and fascinating.

Thank you for your essay presenting these ideas to the FQXi community. My own ideas seem compatible with your ideas. In my Automatic Theory of Physics (1979) I essentially identified directed lines as time. Once lines are associated with physical time as the basis of linear ordered events underlying geometry, one can then go to second-order and employ geometry to construct sequential switching (see my 'End Notes') to construct automata, which can then be mapped into any axiom-based physical theory, answering Wigner.

I invite you to read and comment on my current essay. I believe you'll find it compatible with your essay, while conflicting with your work on Bell. I'm interested in your comments.

Best,

Edwin Eugene Klingman

    Tim,

    Thank you for contributing an interesting essay. Perhaps I am misinterpreting your ideas, but for clarification, are you suggesting in an effort to understand the fundamentals of mathematics in physics, we use a different language or include more "descriptive fundamental concepts or words"? Furthermore, is it your argument that the physics we describe in our universe is somewhat coincidentally predisposed to an application of fundamental mathematics and if so, why is it? Perhaps I am a bit confused with my interpretation and will look again.

    Nevertheless, the universe is dynamic which is why relativity needed a serious upgrade from Einstein introducing GR. I find the definition of any abstract concept, event, or object leads to ambiguity (fuzziness), therefore the interpretation of these observations described in physics leads to relative descriptions in which we use mathematics to explain them quantitatively.

    Finally I am curious about your definition of time. (Excuse the pun, but only if you have time and don't mind?)

    Thanks for your essay - I look forward to your reply.

    Best Regards,

    D.C. Adams

      Dear Dr. Klingman,

      Thanks for your remarks. I will be reading other essays once the semester here settles down a bit and look forward to yours.

      Regards.

      Tim Maudlin

      Dear Demond,

      The issue is not so much words (although of course we have to use words to convey what we mean) but rather what the fundamental mathematical concepts are in a particular mathematical theory. The application of those concepts typically requires that certain axioms be satisfied, so the clearest way to understand why the mathematical concepts would apply usefully to the physical world is to postulate that the physical world contains physical structures that satisfy the axioms. Since different branches of mathematics use different basic concepts explicated by different axioms, the choice of one or another of these mathematical languages makes a difference to how we might postulate the sort of physical structure there is. None of this is controversial, I think, but we often use a particular set of mathematical tools just because they are the only ones around. By constructing different mathematical structures we have a broader set of choices, and also become more aware of the tacit decisions we make when employing one or another mathematical tool.

      The way to get ambiguity and fuzziness out of a description is to have the foundational concepts precisely defined via clear axioms. The particular set of mathematical concepts I define is just as exact and unambiguous as the standard concepts, but because they rest on different axioms they have different application to systems.

      I do not try to "define" time: it seems like a very good choice of a physically primitive feature of the world that is not analyzed in terms of anything else. The point is rather than if time orders events, and if furthermore the way it does is well described by Relativity, then the particular geometrical language I have developed has clear application to the physical world.

      Regards,

      Tim

      I'm afraid that you did not read the paper with any attention. The paper contains a completely novel mathematical approach to describing the geometry of a space and an application of that mathematics to describing the structure of physical space-time, as well as a thesis about the physical structure that makes the mathematics applicable.

      Hi Tim,

      Thanks for the enjoyable essay. Although i share your view that "The聽physics ... physical world has the right sort of structure to be represented mathematically", but I think it is not end of the story, It is to say, in same physical world we have e.g. rise of self-consciousness in particles, that can't be represented聽mathematically. The other words, there are structures that can't fit into math either partially or entirely.聽

      The essential fact is that whatever thing that has a quantity can fit into math, for example the true mechanism behind the quantum entanglement can't be observed and hence doesn't have a quantity and can't be fitted into.math. These are what I had to say in my article.

      Kind regards

      Koorosh

        Dear Koorosh,

        I agree that consciousness raises a very difficult problem, which I do not pretend to be able to solve. Since physics has gotten along so well dealing with non-conscious systems, there should be a way to understand that which is independent of that issue. That is all I was trying to discuss.

        Regards,

        Tim

        Dear Professor Maudlin,

        You make excellent point in your essay. I want to comment on three ideas. First, on "The

        Unreasonable Relevance of Some Branches of Mathematics to Other Branches", I would recommend Connes' essay "A view of mathematics: "http://www.alainconnes.org/docs/maths.pdf (see page 3): "there is just "one" mathematical world, whose exploration is the task of all mathematicians and they are all in the same boat somehow." "Where things get really interesting is when unexpected bridges emerge between parts of the mathematical world that were previously believed to be very far remote from each other in the natural mental picture that a generation had elaborated. At that point one gets the feeling that a sudden wind has blown out the fog that was hiding parts of a beautiful landscape."

        Then on "Why should any mathematical concepts that do not fall into the class of naturally suited ones nonetheless be of use in physics?" I can offer a modest suggestion: all mathematical structures are "unique" but only some of them are "distinguished", meaning they are used by nature. For example, SO(3,1) is the group that describes special theory of relativity, and this is distinguished from say SO(345,2411) which has no role in nature. Why are some mathematical structures distinguished? Because they are consequences of physical principles (see my essay on this http://www.fqxi.org/community/essay/winners/2009.1#Moldoveanu ).

        Last, on continuous and discrete spaces, the proper formalism is that of non-commutative geometry (see the large table in http://fmoldove.blogspot.com/2014/01/solving-hilberts-sixth-problem_17.html from http://arxiv.org/pdf/math/0408416v1.pdf which shows how non-commutative mathematical structures generalize the commutative ones). This has extremely deep implications for the Standard Model:

        http://fmoldove.blogspot.com/2014/11/clothes-for-standard-model-beggar.html where instead of quantizing gravity Connes is geometrizing the theory from a strange space: a Cartesian product of a manifold with a set of two points. See also http://www.mth.kcl.ac.uk/~streater/lostcauses.html#XXII of why the other way around does not work.

        Sincerely,

        Florin Moldoveanu

          Dear Prof. Moldoveneanu,

          Thanks for the message. The comment by Connes is on target, of course. I recently was discussing my work with Shahn Majid, who also works on non-commutative geometry, and he did find commonalities and points of contact between the two. But I doubt that the two approaches will yield exactly the same formalism or results, so there is more to explore about the similarities and differences. In the most radical version of the non-commutative approach you do not recover an underlying point set, but in my approach there always is one. Whether one regards that as an advantage or a disadvantage would require a long discussion.

          Regards,

          Tim Maudlin

          I was wondering in relationship to your question: "If it is correct, then we might see how the time itself creates the geometry of space-time" if you are aware of 1963 Zeeman's result: causality implies the Lorentz group: https://download.wpsoftware.net/causality-lorentz-group-zeeman.pdf

          Time demands causality and Zeeman's result (which was independently discovered by 2 more people) is quite powerful and I think it might represent the end goal of your program. I also see natural links with Sorkin's causal set program and with efforts to derive special relativity in the framework of first order logic.

          Florin

          Dear Florin,

          Yes, the Zeeman result is familiar to me, although it is actually much too restrictive for my purposes, as it presupposes a Minkowski metric, and we want to have a theory that can describe all of the the solutions to the General Relativistic Field Equations, and even other possible geometries (e.g. discrete geometries) as well. What is easy to show in my setting is that the Linear Structure of a General Relativistic space-time already contains the complete conformal structure. Contrast this with the standard approach, in which the topology is just a four-dimensional manifold, that contains no information about the light-cone structure. To recover the entire Relativistic metric, in the continuum case, one needs to add a temporal metric. All of this is uncontroversial.

          However, if one uses these resources to build a discrete space-time structure, then one can use the counting measure to complete the metric...but you have to be careful to count the right thing! It is natural to see similarities to Causal Sets here, but the actual implementation is quite different. The way I do it, the fundamental lines that constitute the geometry of a discrete space-time are all light-like. But in Causal Sets, the chance of any of the event being light-like related is zero (using the usual sprinkling method). I did not have space to go into all this in the essay.

          Cheers,

          Tim

          Here is another reference, a work by Jochen Rau http://arxiv.org/pdf/1009.5523.pdf on reconstructing general relativity (see Fig 1 for his line of argument). I do have a question about your framework: how do you get to curvature? Are your "lines" really straight lines, or geodesics?

          Can I also ask you a question in a different area? If one introduces a new quantum mechanics interpretation, what questions must one address? For reference here is a draft of a paper on a new QM interpretation I am working on: http://fmoldove.blogspot.com/2015/01/the-composability-interpretation-before.html

          Florin

          PS: I don't know if you associate the face with the name, I go every year to the New Directions in the Foundations of Physics conference in DC.

          Tim,

          I thought your essay was interesting, though I have somewhat different ideas about things. I think spacetime is emergent from quantum entanglement. The emergence of time occurs in the Wheeler De-Witt equation HОЁ[g] = 0. The wave functional is defined on an entire spatial manifold, but in general spatial slices only have diffeomorphisms with each other that define time on a local chart or patch. We may then consider the projection onto Hilbert subspaces H_i вЉ‚ H, P_i so that

          P_i|ОЁ[g]> = e^{Оё_i}|П€_i>,

          which may be accomplished with a sum over other states

          P_i = sum_{j=!i}|П€_j>

            I forgot that this editor does not like carrots, so I use parentheses instead.

            P_i = sum_{j=!i}|ψ_j)(ψ_j|. It is simplest of course to consider i = 1 or 2 for two different regions. The relative phase θ_i = ω_it, and defines a local time.

            A simple case of this is the de Sitter spacetime with two patches in static coordinates or standard dS slicing, such as seen in the diagram below. In this case there are two patches with different time "arrows," or better put two sets of diffeomorphisms that correspond to local spacetime.

            Cheers LC

            Dear Lawrence,

            Your proposal is a bit compressed here, but one obvious question, apart from others, is where the ω you mention comes from. Without it, you do not define a t. In the usual case, ω is E, the eigenvalue in an energy eigenstate, and the energy eigenstates are the eigenstates of H. But of course, in Wheeler-deWitt H annihilates the state, so the eigenvalue is zero. That is the classic problem of time in Wheeler-deWitt.

            Also, if you are somehow dealing with deSitter, the you already have a space-time structure (since the Ψ[g] is defined over the entire space of metrics, how did we get to de Sitter?), so the "emergence" problem must already have been solved to make sense of the rest of the construction. How, in that case, does time "emerge"?

            Regards,

            Tim

            Dear Joe Fisher,

            Thanks for the note. In the sense of "abstract" as I use it, the physical world is by definition not abstract but rather concrete. It has some real structure, which appears to be accurately describable in some mathematical way. I suppose you can tie my use of "concrete" to be close to yours of "material" or "real object". So to understand why mathematics is so efficient and powerful for describing the real, material, concrete world, we have to understand both what the best mathematical description is and what sort of the thing the real world is to admit of such a description.

            Regards,

            Tim Maudlin

            The WDW equation does not have any time, for spacetimes in general relativity most often do not have a single set of diffeomorphisms that include the entire manifold. The conjugate meaning of this is that with E conjugate to t there is no manner in most manifolds by which one can form a Gaussian surface to define mass-energy. Mass-energy is not localizable. This only happen for stationary spacetimes with some asymptotic flatness.

            What I mentioned above is a situation where the Hilbert space for the WDWE is partitioned into two parts so that locally in the manifold corresponding to either of them time can be maybe defined. The states for the two Hilbert space subsets are entangled and this entanglement is exchanged with the occurrence of horizons separating the two regions. The horizons, such as in a black hole, are entangled with states in the exterior. This is in effect a sort of coarse graining, and spacetime is a coarse graining of entangled states. This coarse graining reflects a lack of information about the nature of the entanglement.

            Cheers LC

            Dear Tim Maudlin,

            Every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once.

            Abstract mathematical systems do not explain reality, they confound it.

            Glad to be so indisputably informative,

            Joe Fisher

            Tim,

            This is a very interesting essay. I have one minor criticism. The first four pages are very philosophical. That's ok, no problem. You lay out some history, some limitations, etc ... Basically counting and geometry. Then at the end of page four you introduce the true objective of your essay. A lazy reader would be denied this new knowledge:-)

            Regarding your new topology, is it necessary to use two of the same thing to create the linear structure? By this I mean can you only form a linear structure using geometric points? The reason for my question is that it appears to me that you are applying topology with its vast legacy knowledge to Hamilton's quaternions.

            All in all, very well done. Thank you for sharing these new ideas.

            Best Regards and Good Luck,

            Gary Simpson