Hi Gary,

Since I am proposing the Linear Structure as the most fundamental geometrical structure that exists, the objects over which it is defined are automatically geometrical points, by which I mean they have no intrinsic geometrical structure. I get that for free. But of course, they could differ in all kinds of other ways. They could have different intrinsic properties (as an illustrative example, think of them as having different electrical charges), but these would not be geometrical properties. The structure of these other properties would also be unconstrained by the Linear Structure over the elements. One could, I imagine, use quaternions to represent these intrinsic properties, but then the theory is going beyond geometry.

Thanks, and best wishes,

Tim

Hello. I have several remarks about your work, first about general topology, then about physics.

I also considered this problem, of how else topology might be formalized so as to be more convenient for different purposes, may it be easiness of use, restrictions or generalizations, or modified kinds of structures (usually more rigid than some topology, since topology is usually the lightest non-trivial interesting structure that may be found in a given space), and I came to consider other options, I will share with you.

First, I agree with you about the importance of linear orders: it is a remarkable property of lines that their topology is equivalently expressible in that very simple formalism (linear order), unlike other kinds of topological spaces, so we may consider to use this fact to define general topological spaces by somehow "reducing" them to this 1-dimensional case, explicitly involving lines somewhere in the general definition of topology. However there are different possible ways of doing so, and we need to see the larger picture of possible methods and how they may differ, in order to find out which method may be more relevant for given purposes, instead of picking up a fixed method just because it is the first method we happen to think of.

Let us consider this problem in its abstract generality : if there is a general kind of "spaces" we have intuitive ideas of, but we do not know how to formalize (by which kind of operation or relation, as none seems to fit), then can we still find a method to "anyway simply formalize it no matter what a subtle kind of system it is", that will be general enough to include topology ?

As surprising as it may seem, the answer is yes.

Let us explain this amazingly general and still simple toolbox.

First step is to look at the intended kind of "spaces" as forming a category: no matter how topology may be formalized, we expect the concept of "continuous function" between spaces to make sense. We have a clear intuition what it means for a function to be continuous, so there should exist a definite set of all continuous functions from a space to another, the only problem is that we cannot write a definition for it since we did not formalize what is the topological structure we want these functions to preserve. Never mind, we will define this later. At first, let us just admit this concept of continuous function as primitive, forming a category.

Now the problem is: starting with an arbitrary category, is there any general, systematic way to interpret its objects as systems with a kind of structures, so that the morphisms in this category will happen to preserve these structures, and that will still be interesting even if (as in usual topological spaces) there does not exist any non-trivial invariant algebraic operation in these spaces ? We can make it, and here is how.

First we need to pick up a fixed object K in the category, with the quality of being "flexible enough" to serve as the prototype from which the structure of other objects will be formalized. For the needs of expressing topology in physics, we can take K=R (the line of real numbers).

For even more general cases where one fixed object (space) is not "flexible enough", we can take a series of different spaces that would hopefully "complete each other" in their exhaustion of the different kinds of possible shapes.

Now with a fixed K, here is how to interpret any object M in the category as a system with structures. In fact there are 2 structures we can automatically define on all objects M, that will be preserved by all morphisms in the category:

1) The set Mor(K,M) of all morphisms from K to M;

2) The set Mor(M,K) of all morphisms from M to K.

Indeed, for any objects M and N, any morphism from M to N maps Mor(K,M) into Mor(K,N), and also Mor(N,K) into Mor(M,K). I wrote a page with a representation theorem that goes deeper than what I found usually done elsewhere on the topic.

Now as compared to this, your approach can be described as follows:

- You take as K not just the real line, but all possible linearly ordered sets;

- You only consider the embeddings of such K into all possible M, instead of any continuous functions. You refuse those mapping 2 non-neighbor points into neighbor points (in the case of finite graphs).

Some questions:

- It may look good to handle continuous lines as well as discrete lines in similar ways, however what is the use of "mixing" them by not formally assuming K as fixed, so as to admit the case of hybrid systems ? I imagine that sometimes we need to formalize continuous spaces, other times discrete graphs, but I hardly see the interest of hybrid systems. Anyway, there is no interesting continuous map in either way between a discrete and a continuous line.

- There is a diversity of non-isomorphic linear orders, beyond finite ones and that of R. Your formalism automatically admits them. Are you interested in such generalizations, despite the fact that only the case of R comes in usual theories of physics, namely in general relativity which seems to be what you focus on ? Examples : the set of rationals ; the set of irrationals ; the long line ; lines where all countable monotone sequences converge but that are incomplete since gaps have uncountable cofinality; Suslin lines, whose existence is undecidable in ZF.

- By only accepting embeddings of lines into your topological spaces, you make it straightforward to define embeddings between them, however it leaves the concept of other continuous functions less straightforward to define. Do you see the notion of embedding as a more important correspondence between topological spaces than continuous functions ?

I will write more comments later.

    Dear Sylvain Poirier,

    Thanks for the careful reading of the paper (which is, of course, very brief) and the comments. Let me try to advance the conversation.

    There are a few different threads of thought here. Let's separate them.

    One is how the Theory of Linear Structures interacts with category theory. It is an interesting question--when I taught a seminar on this, there was a math graduate student who wanted to look into the question. My own understanding of category theory is not at all strong, so at the moment I do not feel qualified to say much in detail, and everything here should be taken with a grain of salt. It does seem clear that there are enough similarities between the approaches to explore how they agree and differ. One obviously does not need to decide between them in any final way: the more tools in the toolbox the better, as long as one has a good understanding of them.

    As far as the structure of the conceptual analysis goes, I follow standard topology in this: I define the notion of a continuous function in terms of the function and the geometries of the domain and the range. If I follow, you want to take the notion of a continuous function as a primitive, not in need of further analysis. I don't think that there is any general method for evaluating whether a particular concept ought to be taken as primitive (and associated with some structural axioms, as happens with "open set" in the standard approach and "line" in my approach) or taken as definable in terms of some more conceptually primitive concept. So all I can ask is that one look at the actual definition of a continuous function that I give (in the book, not in the paper) and see whether it does a good job of capturing one's intuitions about continuous functions. I can say this: my definition of a continuous function is not the same as the standard definition in topology, and where the two approaches disagree about particular functions (whether they are continuous or not) mine better accords with intuition. But to discuss that we would have to get into the details of the definitions. I think that the ability to provide an explicit definition that works well does argue in favor of taking a concept as non-primitive. I could not tell from your description of your approach whether there are any structural features the set of continuous functions are required to have, or any axioms they must satisfy. If there are, one could look and see how intuitive they are.

    Of course, given a domain and a range whose geometries are otherwise undefined and set of functions that are to count as "continuous", one ought to be able to recover (or define) something about the geometries of the spaces. I guess I find this procedure a bit odd since the notion of a function obviously already requires the domain and the range, and in the standard approach (and mine) one specifies the geometries of the domain and range as spaces, completely independently of any considerations of functions between them. So I think there is a naturalness of this standard order of definition here. But again, one has to look at entire approaches to see what they accomplish.

    The advantage of taking the notion of a "line" as primitive is that the axioms governing lines make use only of the notion of a linear order, and that in turn is easily and cleanly defined.

    Let me answer your questions:

    1) It is true that my system handles "hybrid" geometries (single spaces with some continuous and some discrete lines, and even spaces with some lines whose linear orders are complete and some that are not). That is just a consequence of relying on the notion of a linear order, and the fact that there are linear orders that are dense, orders that are not, dense, orders that are complete, orders that are not complete, etc. It is not that I expect the hybrid spaces to come up in physics. It is easy to define a subclass of Linear Structures all of whose lines have the same sort of linear order--e.g. all of them are dense, or none are dense, or all are complete, or none are complete. I call these "uniform" Linear Structures, and my expectation is that the physical universe should be a uniform Linear Structure. But letting the definition range too widely is not a problem: if you want to restrict to some subclass, you just restrict. It also would handle, e.g., "infinitessimals" without any trouble, since they also should have a linear order. It is not beyond the realm of possibility that those have physical application. It is also obviously possible that physical space-time is discrete.

    2) I think our mind should be open to all these possibilities. The reason that R is used so much is that R is familiar and R has nice algebraic properties. I think that a lot of the algebraic properties (i.e. ones that depend on the fact that the elements of R can be added and multiplied) evidently have nothing to do with geometrical structure as such. R may keep coming up in physics just because it is the tool we always pull out of the toolbox! In the book, I discuss the "rational line" in great detail, and the set of irrationals (with the usual linear order) as well. Your other examples are more sophisticated, and I have not considered them. But again, having cast the net too wide (allowing one to define geometries that will not come up in physics) is not a sin: having cast too narrowly (so you miss the right structure) is.

    3) I provide an explicit definition of continuity that covers all functions from one space to another. I also define various weaker notions (continuous at a point, continuous at a point in a direction, etc.) I also define both weaker and stronger properties of functions. For example, I define what it is for a function to be "convergent" (which actually corresponds most closely to the standard topological definition of continuity), and is a weaker notion than continuity. I also define what it is for a function to be "lineal", which is a stronger notion than continuity. All of these are defined using just the resources provided by the theory. I would have to refer you to the book to see how straightforward the definitions are.

    By the way, the definition does provide for continuous functions from a continuum to a discrete space. For example, the floor and ceiling functions from R to Z are continuous by my definition. So when you say that there is no interesting continuous function between a continuous line and a discrete space I disagree! Not every such function, by my definition, is continuous, but those are. The only continuous functions from a discrete space to a continuum, though are constant functions. So in that direction it is trivial.

    Regards,

    Tim Maudlin

    When I mentioned to consider the sets of morphisms as primitive, I meant only : in a first draft of consideration, until fixing a definition of the structures that the morphisms will preserve. Also, while not strictly necessary, I generally admit that the objects in the category are given as sets (with structures to be introduced later), and the morphisms are maps between these sets.

    Finally, the only sets of morphisms I take as primitive are the sets M0=Mor(K,M) and/or M*=Mor(M,K) with fixed K, which I take to play the roles of structures for M.

    The condition of "preserving the structure" is thus defined as follows:

    Putting on each M the structure M0, A function f from M to N is said to preserve that structure if тИАx тИИ M0, foxтИИN0.

    Example : M is an affine space and M0 is the set of all affine maps from R to M.

    This condition on f means that f preserves every operation of barycenter between 2 points with any coefficients (because the barycenter of points x and y with coefficients (1-u) and u is the image of u by the unique affine map from R to M which sends 0 to x and 1 to y), so that it is an affine map, mapping every straight line into a straight line.

    Now if we put on each M the dual structure M*, that is the notion of affine function with real values, the preservation condition for this structure by a map f from M to N says : тИАy тИИ N*, yofтИИM*, that is, the pullback of an affine form is an affine form. Then, since the preimage of a singleton by an affine form is either an hyperplane or the empty set or the whole space, this directly implies that the preimage of an hyperplane by such a preserving function, is either an hyperplane, the empty set or the whole space. The advantage here is that morphisms so defined between infinite dimensional topological affine spaces are automatically continuous, a condition which the algebraic definition by barycenters does not ensure (while both concepts of morphism are equivalent in finite dimensional spaces).

    The very same tools can as well define the structure of vector space (with fewer axioms than usual by taking the dual space as primitive), topological spaces with the particular case of topological manifolds, then Lipschitz structures on topological manifolds, and also differential manifolds with whatever degree of smoothness you choose.

    It is very simple to introduce the notion of measure on a topological manifold M : take M* the vector space of continuous functions with real values, then the space of measures on M is the vector space of linear forms on M* that is "generated by M", i.e. the set of limits of sequences of linear combinations of elements of M in the dual of M*. Now if you take as M a differential manifold and M* the set of smooth functions on M, then what you get in this construction (closed vector space generated by M) is the space of distributions on M.

    I do not need to check your book to know that you define continuous functions as functions f such that the image of any line with endpoints x and y either contains a line with endpoints f(x) and f(y), or f(x)=f(y); and that f is continuous at a point x if for any line with endpoint x, either f is constant near x on this line or the image by f of this line contains a line with endpoint f(x). So it is indeed less straightforward than with the general tool I told you, where (in the direct version) the condition implies that the direct image of any figure of some kind in M (conceived as the direct image of K by some morphism from K to M) is already (instead of : contains) that kind of figure, while, in the dual version, the condition implies that the preimage of any figure of some kind is a figure of that kind.

    I see some differences with usual topological concepts, which you did not specify in your essay.

    For example, in the set of (x,y)тИИR2 such that (y=0тЙдx) or (0 < y тЙд x and 1/x is an integer), the line (y=0) is a neighborhood of (0,0) according to your definition but not in the standard topological definition, where any neighborhood of (0,0) in this set must contain the whole subset of points with x smaller than some positive value. Does your definition of "neighborhood" fit your intuitive idea of this concept in this case ? Note that if, instead of only admitting lines as subsets, you worked with the tools I gave you, allowing squeezed lines, defined as continuous maps from a totally ordered set into the space but not required to be an embedding, then the resulting concept of neighborhood would coincide with the traditional one in this particular case.

    I do see a specific problem with your concept of neighborhood : according to the classical definitions, if f is continuous in x then the preimage of a neighborhood of f(x) by f, is a neighborhood of x. I can imagine a continuous idempotent function that squeezes {(x,y) | 0 тЙд y тЙд x} onto the above set, however the preimage of the neighborhood (y=0) of (0,0) by this map is not a neighborhood of (0,0).

    Another difference, is that your topology does not admit any Cantor space, unless you give it a very different topology that makes it... connected, and thus no more feeling like a Cantor space in the usual topological sense.

    Well, I won't try to comment on your approach, given that there is obviously not room to explicate it properly, and in any case it is not directly relevant to my paper.

    There is quite a bit I did not specify in my paper, as it is a less-than-9-page introduction to something that takes hundreds of pages to present!

    But to your questions: Yes, by my definitions, y=0 is indeed a neighborhood of (0,0) in that set, and that does completely align with my intuitions. Any continuous line that arrives a (0,0) has a segment in y = 0. In that sense, which is the only obvious one, the set of points y=0 completely surrounds (0,0) in that set, which is what we want a neighborhood to do. You seem to think that it is a problem that my definitions yield different results than the standard ones. In this case, mine in the more intuitive. As for Cantor sets, neither I nor anyone else has any real intuitions about them, so it is hard to argue that any result about them is either good or bad.

    I should also note that you simply repeat certain properties of the standard definition as if they are desirable but without any argument. This is particularly the case with pre-images. The most natural thing is to define properties of functions by the geometrical characteristics that they preserve under their action: that is, the geometrical characteristics of the domain that are preserved in the range. The whole idea of looking at what is preserved backwards, i.e. by the pre-images. is just strange. You are used to it because it is what has been done, but it is just peculiar.

    There are lots of differences between my definitions and the standard ones. You seem to think that these differences are per se objectionable. But there is no argument to that effect.

    I am also a bit mystified by your assertion that you do not need to check my book to see how I have defined things. Are you psychic?

    Hi Tim,

    Clearly one of the best essays yet :) I have one issue to pick. You state that the physical world has the right structure to be describable as math as if that was a fact, but forget to question whether that is indeed so. It is arguably true that some of it is describable as math. But is all of it describable by math? What if not? That's what I've addressed in my essay.

    -- Sophia

      No, I'm not psychic :) I believe psychic abilities exist but I'm not sure if it would work for this purpose. What I meant is that I find only one possible definition that mathematically fits. Now I understand that your continuous functions from continuous spaces into finite graphs are those that make only one step at a time.

      Well if you like your definitions... like tastes these cannot be discussed.

      For example according to your definitions, the identity map (inclusion) of the rationals (or the irrationals) into the reals is discontinuous ; this seems odd to me.

      If your ideas take hundreds of pages to present, I'm afraid this means they are not as simple and intuitive as you are trying to advertise.

      You may fail to have an intuition about Cantor sets, however please do not project your failure on others. I have been quite interested in the Mandelbrot set and Julia sets, and this provides direct visual images of Cantor sets.

      Now about the physics. What physics ? The only physics clearly related with your topology, is that of General Relativity (as for quantum gravity, a main candidate is Loop Quantum Gravity, that, if I understood well, does not fit with your topological concepts with time orientation). But for this, what you do is that you start with a time order taken as primitive to define oriented lines, and then you say that this concept of oriented lines can be taken as a basis for topology and thus geometry. But if in this case, the linear structure is equivalent with the time order (i.e. each is definable from the other), then what is the interest of developing the concept of linear structure, rather than just looking at the time order, which is formally simpler ?

      Then, this linear structure, or time order, defines the conformal structure of space-time... though, even using the linear structure, it seems complicated to me to define the tangent space at each point, to be able to express that it indeed forms a vector space and the light cone near a point is actually a quadratic cone. But what is the use to point out that conformal structure, intermediate between those of topological manifold and pseudo-Riemannian manifold ? I may have seen that it was considered in some works, however I fail to figure out any physical context where that structure naturally remains fixed while the pseudo-Riemannian structure may arbitrarily vary (multiplied by arbitrary scalar fields).

      You may point out that particles or planets, whose mass affect the space-time curvature, follow time-like lines in space-time. However they do not follow any such lines but only geodesics if they are isolated, which involves the whole metric and not just the conformal structure ; moreover, the electromagnetic field contributes to the space-time curvature without being contained in any line.

      Finally, the main equation of General Relativity (the Einstein field equations) is a tensor formula on the tangent space at each point, that involves the fields and the metric in a way formally independent of the particular signature of the metric; the same equation may be written, keeping much of its properties, in a manifold with any signature, thus making the particular time-oriented linear structure irrelevant to the understanding of this equation.

      Hi Sophia,

      So what we know for sure is that certain aspects of the physical world have structures that can be described to very high accuracy by mathematically formulated theories: everyone uses the anomalous magnetic moment of the electron as the example of precision, and the entire standard model is tremendously successful, as is the General Theory of Relativity. It does not follow that everything can be captured mathematically, but whatever cannot must not play a very noticeable role in producing all of the phenomena that physics has been able to predict so accurately. If some aspect of the world cannot be represented mathematically, then no mathematically formulated theory will succeed predictively for it.

      In any case, I did not claim that it is all describable by math (probably no mathematical theory will account for consciousness, for example) but that the geometry of space-time is. That's a hard enough problem all by itself!

      Regards,

      Tiim

      Dear Tim,

      I just read your interesting essay, and also your comments elsewhere on the Heraclitean and Parmenidean views of reality. Your essay touches on fundamental questions in geometry and how to correct the perceived wrong conceptual foundation. I am sure others will have other questions on your interesting contribution. For me since I discuss similar interest in my essay, I will have two questions for you:

      Question 1. You state and I quote: "The fundamental structural characteristic of an open line is this: given the points in an open line, there is a linear order among its points such that all and only the intervals of that linear order are themselves open lines. This basic structural characteristic of the open line holds for lines with infinitely many points (such as lines in Euclidean space) and lines with only finitely many elements... So the Theory of Linear Structures is capable of describing the geometry of continua and of discrete spaces (such as lattices) using the same conceptual and definitional resources."

      How can either of the two varieties of lines be cut? In the first variety, there will always be a point at the incidence of cutting, and a point is uncuttable, so how does cutting proceed? In the second, the elements, whatever they may be are uncuttable, being fundamental and if the interval between them is a distance, distance also consists of points, so where can you cut successfully?

      Question 2: You seem to dismiss the Parmenidean view. I believe it should not be dismissed but be reviewed appropriately as I do in my essay. My question: If the universe itself can perish, can your Linear Structures similarly perish or are they eternally existing objects immune to perishing? If in a region, the 'Linear Structures' therein perish and in some other region, 'Linear Structures' appear spontaneously or a mixture of the two is occurring in a rhythmic pattern in some region, what could be physically manifest?

      Best regards and good luck in the competition,

      Akinbo

        Dear Sir,

        We have discussed Wigner's paper in our essay to show that the puzzle is the result of unreasonable manipulation to present an un-decidable proposition and impose the unreasonableness on mathematics. We have specifically discussed complex numbers (since he has given that example) and other examples. You are welcome to read and comment on it.

        Your statement: "A physical world completely described by fluid mechanics would contain no such objects, so the physics does make a crucial contribution" ignores bosons, which also behave like fluids and are "uncuttable". The problem with your example of the child with square tiles and Fermat's last theorem are put in an un-decidable format by equating integers with area (tiles) and volume (cubes). The integers are scalar quantities that are related to differentiation between similars as repeat perception of 'one's. The "similars" can have various units. While the value of the integer; say 3, remains same, 3 apples or 3 square meters is not the same as 3 cubic meters. While apples are discrete, areas or volumes are analog. There is no puzzle here. We have discussed these in detail in our essay.

        Regarding number of atoms in DNA and number of mountains in Europe, there is no puzzle in principle. It is only a matter of interest. If we want, we can easily count all. However, if we fail to define something precisely, as is done in most branches of physics (including space, time, dimension, wave-function, etc. so that there is scope for manipulation), then we cannot say it is puzzling. In our essay, we have defined each term precisely to avoid ambiguity. Due to conservation laws, cell number does not become indeterminate during cell division - it is our inability to count precisely that creates the problem. Further, name dropping is regarded as a sign of superiority and views are presented piecemeal to suit one's own requirements - to prove that particular point anyhow. We have given one example from one of the essays here in Dr. Lee Smolin's thread. There is a need to reassess and rewrite physics.

        Regards,

        basudeba

        Dear Akinbo,

        Let me try to answer your questions. If "cutting" a line just means partitioning it into two parts, each of which are themselves lines, that is partitioning a line into two segments, then this can be done for any linear order: it is exactly what Dedekind called a "Schnitt". Now Dedekind wanted something more in order to define a continuum: for every Schnitt, there should be either one or two points that correspond to the Schnitt. Think of this as either the greatest or least element of one of the two parts of the partition. Some linear orders can be cut in this way without there being such an element. For example, the set of positive rational numbers can be partitioned in two groups: those whose squares are greater than 2 and those whose squares are less than two. That is a perfectly good Schnitt, but there is no greatest element of the one part or least element of the other. So, by Dedekind's definition, the set of rational numbers is not a continuum.

        I have no problem with Dedekind's definition. It just shows that lines can be defined--and cut--even if the space is not a continuum.

        As for perishing: the physical lines I have in mind are sets of events, ordered by a temporal order. The universe could have a maximal element in time--a last event. That is a claim about the overall geometry of the universe. If you mean by "perishing" that any object in the past has "perished", then lines do indeed perish: lines made of events in my past have, from my present perspective, perished. That is just the same sense in which we generally talk about things in time perishing: no longer existing.

        Regards,

        Tim Maudin

        Dear Tim,

        Thanks for finding the time to reply. Following your response, I checked on 'Schnitt', which is German for 'cut'. So as not to confuse issues, by cutting of 'a line', I do not mean mathematical cutting of the number line in Dedekind's sense. By line, I mean extension in Euclid's sense. A point cannot be cut by definition, and unlike the number line where an irrational number can be invented as a 'trick' to provide a "gap" in order for cutting to take place, on an extended line "gap" itself will connote either an extension, distance or space and therefore consist of points, all of which similarly cannot be cut. Probably, if you later read my essay you may get my meaning.

        If the physical lines you have in mind are sets of events, rather than extension that I mean, then of course events cannot be cut in two.

        Then on "perishing" and the possibility of your own type of line perishing, please give a thought of the implication of this in resolving Zeno's Dichotomy paradox, even though Calculus is mathematically used to find a solution to it. However, the 'infinitesimal' of calculus or "ghost of departed quantities" as is famously called challenges aspects of physical reality. Calculus does not tell us the size of the last dx in the race. Calculus cannot also explain how to cut a line of the extended type. Thanks for the exchange.

        All the best,

        Akinbo

        • [deleted]

        • Post #33

        "Our understanding of the structure of time has been revolutionized by the Theory of Relativity. Intriguingly, the change from a classical to a Relativistic account of temporal structure is of exactly the right sort to promote time into the sole creator of physical geometry."

        Even if the speed of light were constant, as Einstein postulated in 1905, it can be shown that no new temporal structure arises - rather, the concept of time becomes "not even wrong". But that is an obsolete argument because, as a recent experiment showed, the speed of light is not constant:

        "The work demonstrates that, after passing the light beam through a mask, photons move more slowly through space."

        Pentcho Valev

          The question of the "speed" of light does not even arise at this level of geometrical analysis. What gets built into the Linear Structure of a Relativistic space-time is just the conformal (light cone) structure. That structure has no classical analog at all, and so constitutes something completely new in relation to classical absolute time and absolute simultaneity.

          Strange argument. What could

          "The question of the "speed" of light does not even arise at this level"

          mean? You don't want to think of it? The postulate of the constancy of the speed of light is essential, even if all heads are in the sand, and since it is false, the return to "classical absolute time and absolute simultaneity" is unavoidable.

          Pentcho Valev

          The notion of the speed of anything is a metrical notion. Topology describes geometrical features of a space that are not tied so closely the metrical features: a topological feature is invariant under transformations that change metrical relations. So no speed is definable at this level of description.

          Consider just the conformal structure of a Relativistic space-time. This does not contain the sort of metrical information one would need to define a speed, but does define a notion of temporal precedence and a by that a light-cone structure. Those structures are not classical. Even more, they are inconsistent with the classical account of temporal structure.

          It is not a matter of what I want to think about, but of what is formally definable at a certain level of geometrical description.

          Tim Maudlin

          Topology of Linear Structures is in all you've generalised, it would be more explainable if you diverse from geometrically conceived prospects, which would bring more light to the subject.

          Great job & luck!

          Sincerely,

          Miss. Sujatha Jagannathan

          Dear Tim,

          I posted this elsewhere in conversation and I thought I would share this with you to add to our previous conversation. Also as something you can confirm directly from Roger Penrose being a fellow FQXi member...

          Here is what Roger Penrose has to say in his book, The Emperor's New Mind, p.113... "The system of real numbers has the property for example, that between any two of them, no matter how close, there lies a third. It is not at all clear that physical distances or times can realistically be said to have this property. If we continue to divide up the physical distance between two points, we should eventually reach scales so small that the very concept of distance, in the ordinary sense, could cease to have meaning. It is anticipated that at the 'quantum gravity' scale (...10-35m), this would indeed be the case".

          Hence, my asking assuming, without conceding that the system of real numbers applies to distance, how can a distance be divided if there is always a third element between two elements and going by geometrical considerations these elements are uncuttable into parts?

          Regards,

          Akinbo

            Dear Tim,

            I do like your theory of linear structures and it does look as if it could have application in four dimensional spacetime in which a series of points on a line in four dimensional spacetime can include variability in the space and time dimensions.

            Can you describe any applications or experiments in which this theory has been used?

            Regards

            Richard

              Dear Akinbo,

              Geometrical points have no geometrical parts, by definition. Hence, a single point cannot be further divided. That is true whether the points on a line are dense (there is always a point between any other two) or not dense. Roger Penrose (and George Ellis, for example) think that in this sense physical space or space-time is not dense. My own mathematical language can handle spaces that are dense and spaces that are not dense.

              A distance can be divided, as Dedekind shows, by partitioning a line into two sets of points in certain way. This does not require dividing any individual point in two.

              Regards,

              Tim