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

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      • 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

            Dear Richard,

            What I have developed is not itself a physical theory but rather a mathematical language in which physical theories can be written. The language provides clues about, for example, how to describe a discrete space-time that has Relativistic characteristics. But no complete novel theories have as yet been formulated in this mathematical language, since it has just been developed. Nonetheless, one can see how this language could be well-adapted to describing the physical world on account of its temporal structure.

            Regards,

            Tim

            " The physical world is as it is, and will not change at our command. But we can change the mathematical language used to formulate physics, "

            I'm not sure about either one of these two hypotheses. It is a realist view that does not conform with the standard model of quantum mechanics. It complies with the tensless view of relativity but these are different theories and definitively not fully united other than in QED. Can we be sure that the laws of physics will not change one million years from now? Obviously, if we say they will not, this is an axiom and far from a common sense truth.

            I can't imagine what you have in mid by this comment. If you think you can change fundamental physics by your command, you are welcome to try. And if you are unable to change the mathematical language you use to write physical theories, then you are oddly constrained. I did not say that the laws won't change (I do not mention laws anywhere here), but in any case they will not change at our command. so if we want our mathematical theories to describe the sort of structure the world has, we have to change the theories to fit the world rather than the other way around.