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.

          Dear Tim,

          I had a further thought about applications of the Theory of Linear Structures. Can you generalize the theory so that it can progress from dealing with sequential points on a line to points on a plane or curved surface and then on to points in three dimensions and the four dimensions of spacetime.

          The idea is to try to show a mathematical (or topological?) equivalence between String theory which models oscillating strings of one dimension in a hypothetical 12 dimensional spacetime (11 space plus one time) with the Spacetime Wave theory which proposes oscillations in spacetime (wave propagation of changes in spacetime curvature at speed c) as the description of photons and fundamental particles.

          This objective is referenced briefly in my essay on Solving the Mystery.

          Regards

          Richard

          Dear Richard,

          The Theory as it is deals with as high-dimensional spaces as you like. One specifies the Linear Structure of, say, a n-dimensional space-time by specifying those sets of events that constitute continuous lines. If the theory only worked for one-dimensional spaces it would not be worth much.

          Maybe you mean this: the 2-D world-sheet of an open string can be thought of as a sequence of lines. Can one extend the machinery used to describe sequences of points that constitute lines to cover sequences of lines that constitute worlds sheets? I have not thought about that. One problem, of course, is that the worlds sheet can be partitioned into sequences of lines in different ways, and none is "the right" war to do it. There is no similar ambiguity when resolving a line into a sequence of points.

          Regards,

          Tim

          "If you think you can change fundamental physics by your command, you are welcome to try"

          What I tried to say and maybe I did not say it correctly is that your comment " The physical world is as it is, and will not change at our command." involves a hypothesis that there is something we can call fundamental physics. You or anyone else have not proved that. I can assume that we live in a virtual reality in which the laws were changed by its creators. There is an infinite regression of fundamental physics founded on fundamental physics. Except if you are talking about fundamental assumptions like particles. I think Einstein changed fundamental physics by his command, meaning the fundamental physics that people thought they were. Speaking about absolute fundamental physics makes no sense imo. It's like speaking about unicorns. Because it is true that: Two unicorns imply 1+1 = 2

          Ah. By "the physical world" I mean the physical world, not our theories about the physical world (however widely accepted they may be). In this sense, Einstein did not change the physical world at all: he developed new theories about that world. Even if we live in a "virtual reality", then it is a physical fact about us that we do. Probably, we could never figure that out.

          If we do not separate sharply between the physical world as it is and our theories about it, our discussion will be very confused. Perhaps we will never develop a completely accurate theory of the physical world. But if we do, if will be framed in a mathematical language, so we should think about what languages are available and create more if needed.

          Thanks. I have a problem with this below you may want to give another shot at it although I understand that communicating through message boards is difficult especially in this area:

          "If we do not separate sharply between the physical world as it is and our theories about it... "

          Is it necessary that the physical world should be in a particular way or this is just an assumption to make our life easier?

            Dear Tim,

            I read your essay with great interest. I totally agree with you: "But we can change the mathematical language used to formulate physics, and we can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world." My high score.

            I think only that we must first to consider the proto-structure of the Universum (matter) from the point of view of eternity ("sub specie aeternitatis"), that is, to carry out the ontological structure of matter in the proto-era, "time before times began". When we "grab" (understand) the primordial (ontological) structure of space, then we will understand the nature of time. Therefore, the fundamental physics we must move from the concept of "space-time", to the onto-topological concept "space-matter-time". The primordial structure of matter determines the structure of the language in which Nature speaks to us, single language for mathematicians, physicists and poets , ie, language that contains all the meanings of the "LifeWorld"(E.Husserl). I invite you to read my essay .

            Kind regards,

            Vladimir

              I do think we are having some communication problems....Even if the laws of the universe change through time, for example, there is still some way it is (i.e. changing, and changing in a particular way), and some description of it. So I can;t see any substantial assumption is saying that there is some way the universe is, not subject to our control (in the relevant sense) that we are trying to describe.