Dear Sir,

Space, Time and coordinates arise from our concept of interval and sequence. When the interval is related to objects, we call it space. When the interval is related to events, we call it time. When we describe inter-relationship of objects or events, we describe the sequence by coordinates. Directions are arrangements of the sequences of intervals of objects in space. Dimension is the perception of differentiation between the internal structural space and external relational space of objects. Since we perceive through electromagnetic interaction, where the electric and magnetic fields are perpendicular to each other and both move perpendicularly, we have three mutually perpendicular dimensions. These are invariant under mutual transformation (if we treat length as breadth or height, the object is not affected) and can be resolved into 10 different combinations. Past, Present and future are segments of the sequences of intervals of time that are strictly ordered - future always follows present; present always follows past. However, while future always follows present in an ordered sequence, any past event can be linked to the present bypassing the specific sequence. This proves unidirectional time. Since past and future are not invariant under mutual transformation, time is not a dimension. Since the intervals are infinite, space and time are an infinite continuum. We use segments of this analog reality by choosing a fairly repeatable and easily intelligible interval as the unit like taking out water from the river by a pot. This is the empirical definition of space and time.

Number is one of the properties of all substances by which we differentiate between similars. If there is nothing similar at here-now, the number associated with the object is one. If there are similars, the number is many. Our sense organs and measuring instruments are capable of measuring only one at a time. Thus, 'many' is a collection of successive one's. Based on the sequence of perception of such one's, many can be 2, 3, 4....n. In a fraction, the denominator represents the one's, out of which some (numerator) are taken. Zero is the absence of something at here-now that is known to exist elsewhere (otherwise we will not perceive its absence at all). Infinity is like one: without similars. But whereas the dimensions of one are fully perceptible, the dimensions of infinity are not perceptible. There cannot be negative infinity to positive infinity through zero, as it will show one beginning or end of infinity at the zero point, which is non-existent at here-now. No mathematics is possible with infinity, as all operations involving it will have undefined dimensions - thus indistinguishable from each other. Irrational numbers were used in India since antiquity based on the above principle.

The simplest answer to Zeno's paradox is that velocity is related to the mass of the body that is moving, the energy used (force applied) to move it and the total density of and the totality of the energy operating on the field. These are all mobile units against the back drop of the field that is static with reference to these. Middle of the distance is related to the frame of reference, which is static, while the other aspects are relatively mobile. Thus, it is like comparing position and momentum. They do not commute. Hence there is no paradox, which is borne out of experience. While the middle of the distance is gradually reduced, the velocity is not reduced by the same proportion. Hence the runner will reach the end point.

Long before Euclid, geometry (called 'khila panjara' meaning closed continuum) was used in India (seen in Shulva Sootras). Since numbers are discrete, we can use scalar numbers with suitable units in geometry, but the analog field cannot represent numbers. This has misled modern mathematics. Further details can be seen in our essay.

Regards,

basudeba

    Your thoughtful work assumes the assumptions of systematically numbered system of the Universe in mathematical sense but it puts forth more permissible subscripts and not the numbering of the macro-world!

    Best Regards,

    Miss. Sujatha Jagannathan

      Dear Eckard,

      obviously I meant "leave", not "live". The mistake was mainly due to my absent-mindedness in a difficult moment.

      Much more serious is that I did not write "Fraenkel" correctly. My only "excuse" is that in a precedent book on similar arguments I wrote it corretly, but this can't clear my mistake.

      My present condition is very complicated. Last week, in addiction to my mother's illness, I lost a person who was very dear to me, and I am very sorry to say that I cannot follow the forum by now.

      Best regards for you!

      Giovanni

      LC,

      When Giovanni attributed the positive sign to time, I see this only reasonable on condition he referred to elapsed time, not to any event-related time scale, cf. Fig.1 in topic 1364.

      You wrote: "one can set zero where you want". Yes, one is even forced to arbitrarily choose an event of reference if one ignores the alternative of referring to the only natural point zero (of elapsed time), the now.

      Elapsed time exactly includes all past. Only elapsed time can be measured.

      Eckard

      I want to thank very much all those who have read my essay and have posted the welcome and interesting remarks I see here.

      Unfortunately I am not at home at present and I cannot answer the posts or follow the contest. My old mother, who lived alone, has had a serious accident for some days. Now she cannot move and I am engaged to help her constantly.

      I promise to answer to everyone who has written or will perhaps write to me as soon as it will be possible.

      I am very sorry for the delay and give you all my excuses!

      Giovanni

      6 days later

      Dear Giovanni,

      Eckard Blumschein drew my attention to your essay and must I say I am glad I read it. In my opinion, it is one of the essays that must make the list of the Top 40 Finalists. The essay makes a good and sincere attempt to exorcise some outstanding fundamental devils in our physics and mathematics. Despite being such a good essay, I have some bones to pick and posers to raise, and I will do this by copying and pasting from your essay, then make my comment. Here goes...

      "...many of the greatest thinkers in history, such as Pythagoras, Plato, Aristotle, Descartes, Spinoza, Kant and Einstein"

      Isaac Newton's name is conspicuously missing. He has views that go deeper and are more reasonable in my opinion than the ones you name.

      "What stays behind Zeno's paradoxes"

      I agree with most of what you say here. I belong to the group that although mathematical tricks can resolve the paradox by seeming to make an infinite task completable in a finite time, most such methods still have the plague of having "seeming", "tending", "in the limit" dogging the claimed solutions.

      You rephrase Dedekind's axiom of continuity in the following terms: "If all points of the straight line fall into two classes so that every point of the first class lies on the left of every point of the second class, then one and only one point exists, which is common to both classes, thus producing the union of them in a linear continuum."

      Do you propose then that a line can only be divided at the 'unique' point and at no other place? The weakness of this Dedekind explanation is that it cannot explain the fact that a line can be divided in several places. I am therefore of the view that although interesting, Dedekind's proposal and its reformulation is inconsistent on the face of the fact that lines can be divided severally. Look at it this way, after dividing a line into two segments at your 'unique' point, are you now saying the segments become indivisible? Or is a line a living thing that can mutate and evolve another unique point after the first division? I propose my own hypothesis in my essay.

      Finally, may I ask if your "space as the set of all real numbers" is an eternally existing entity or concept? If the Universe collapses at a Big Crunch will all those real numbers be still expressed somehow as a mathematical concept or in physical reality? Or do they perish? If they can perish at some future time, can some not be perishing today? Give this some thought.

      I will not dwell much on the latter aspects of your well written essay mainly because I am of the opinion that the mathematical tricks therein have misled our physics in the last 100 years.

      Best regards,

      Akinbo

        Hi Lawrence,

        if time is - according to the definition I have given - the set of positive real numbers, you cannot set zero (or one) where you want. Zero (or one) is only where time begins.

        Things may be different for space: in it you can probably set zero where you choose.

        Kind regards,

        Giovanni

        Dear John Hodge,

        I thank you so much for your very interesting comments and apologize for the delay of my answer.

        I don't know if greek Logic and Mathematics are incompatible with the modern science of nature, but they are probably incomplete because, for example, they did not afford the developement of the infinitesimal calculus (with the partial exeptions of Eudoxus and Archimedes) and a precise understanding of the irrational numbers.

        When I use the terms "space" and "time" I refer to distance and duration, as in common language. I think also that my view is compatible with that of General Relativity, but I am not sure to be able to prove that.

        I agree with you in considering an abstraction the idea that a line is composed of infinite a-dimensional points. But I don't consider an abstraction the thesis that it is composed of infinite real numbers. In my view numbers really exist (they are not a mind construction) and they (or rather some particular sets of them) are space and time. On the contrary, the geometrical point probably does not exist phisically, but it is suitable to a geometrical representation of real numbers.

        I gladly accept yor remark about my use of the word "model". It gives me a chance to explain that by "model" I meant to refer to the logical or mathematical model theory, according to which a model is a structure that satisfies all the sentences of the theory. In particular cases this seems to involve "paradoxical" consequences, as in the Loewenheim-Skolem theorem, which states that every theory which has a model - e.g. the theory of real numbers - has a denumerable model. So, if information, aaccording to my point of view, is a logical model of reality, it holds all truths concerning the latter, although reality may be more complex than information.

        But in the essay I should not have used the expression "in a different and usually smaller scale", which is at least misleadind, I we use "model" in a mere logical meaning. That was a mistake for which I apologize.

        Finally I want to tell you that I shall very willingly read and comment your contest essay. But I need time, because I am living by now some difficult moments.

        With my best regards,

        Giovanni

        Dear Joe Fisher,

        Thank you for your attention to my essay and for your post, which contains also a summary of your worldview that I find fascinating, although it is different from mine.

        But I agree with you in thinking that differences are fundamental aspects of the cosmos and that each real object is unique.

        My best regards,

        Giovanni

        Dear Koorosh,

        Thank you for your kind comment!

        Your statement that the quantitative nature of physical objects is what permits a mathematical description of them is the winning idea of the science of nature. It goes back to Democritus (maybe also to Pythagoras), but has found its first effective achievement in Galilei's work.

        I will read your essay as soon as possible.

        My best regards,

        Giovanni

        Dear Tommaso,

        Thank you for your kind and thoughtful comment! It was a pleasure for me to receive it and I regret not having been able to answer before.

        I substantially agree with your remark that discreteness is what we need to describe the physical world (as evidenced by the power of fomalised languages and Turing machines), but I hold the opinion that space and time - considered from our reference-frame - are continuous. In continuous space-time it is possible to place not only the existing events - which are at most countably infinite - but also all not-existing but logically possible states of affairs - which are uncountably infinite. (I considered this topic in the book, that I will be pleased to send you, if you like; within the contest essay bound it was impossible for me to explain all my points).

        As regards my definition of "model", I was referring to the logical model theory. It affirms that a model of a theory is a structure in which all sentences of the theory are satisfied. In certain cases that seems to have "paradoxical" consequences, as in the Loewenheim-Skolem theorem, because a denumerable model (e. g. the set of natural numbers) is able to represent the sentences which are true for uncountably infinite sets. Nevertheless some words I use in the essay about the notion of model are surely misleading: for example, I should not have written "in a different and usually smaller scale", because the scale, from a strictly logical point of view, is totally irrelevant.

        I can agree with your remarks on the distinction between subjective duration and objective order in space-time. If time is an ordered set of numbers, also the order of the events is objective and immutable. What can change - depending on the speed of the observer's reference-frame - are lengths and lapses of time, not the position of the events in space-time.

        I read today your brilliant, original and inspired essay. I find it one of the best (if not the best one) I've seen in the contest. I hope to be able to comment on it soon.

        Ciao Tommaso, un saluto a te e a Pisa, dove ho trascorso parte importante della vita!

        Giovanni

        Dear Sir,

        thanks for your detailed post.

        It contains a lot of interisting suggestions, but I am not sure of having correctly grasped all its points. I shall endeavour to read your essay carefully.

        Best regards,

        Giovanni

        Dear Miss Sujatha,

        Actually I think that macro-world too can be numbered as well. Once we state that things (objects, or facts) are placed in a space-time made of numbers, it is possible tu put them in a one-to-one correspondence with numbers themselves. Moreover I think that the number of existing objects or facts amounts to a countably infinite set, while the set of state of affairs which are possible in a purely logical way in uncountably infinite.

        I could not develop these points within a brief essay.

        Kind regards

        Giovanni

        Dear Akinbo,

        I thank you so much for your kind post and for your appreciation of my essay! (AndI have to thank again Eckard for having been the first who resolutely has found it worth of attention.)

        I cannot but subscribe you remark aboot Newton, who is absolutely a giant of thought, but I don't know if his views about space and time go deeper than those of all others I mention.

        With regards to Dedekind's axiom of continuity, I specify that the dividing point is unique only in the sense that the line has to be divided in two part, if we want to view it as a geometrical representation of time. The left side of it represents the past, the right side the future, and the dividing or, better, uniting point, represents the present. This latter is unique for each reference-frame, because every reference-frame has its own present, but different observers, or reference-frames, can have a different present.

        Finally I think that space, as well as the set of all real numbers, are not merely concepts, but are eternally existing entities, which cannot perish, as long as there is an Universe. Numbers (or space-time) are fundamental aspects of the Universe and are inseparable from it. I cannot say if it is possible for numbers to exist outside the Universe, because it is not possibile for me to conceive anything outside Universe.

        All the best,

        Giovanni

        23 days later

        Dear Giovanni,

        I hope your old mum has fully recovered. In your response, "that space, as well as the set of all real numbers, are not merely concepts, but are eternally existing entities, which cannot perish, as long as there is an Universe.", I assume that this holds only if the Universe too is also eternally existing. Your statement suggests that if the Universe is perishable, then space is also perishable. I put this idea to good use in my essay.

        Regards,

        Akinbo

        Dear Akinbo,

        Mom has improved, but a full recover is difficult, because she is very old. Thanks a lot for your kind interest!

        Your supposition is right: I cannot take for granted that space and time, as long as the Universe, are eternally existent beings.

        I have already read your essay and I find your hypothesis that spatial distances can perish very original and interesting, as well as are unconventional and interesting your approach to the Parmenidean "spell" and your solution to Zeno'a paradoxes of motion. I find also several points of similarity between your perspective and mine, but I have to re-read carefully your essay, if I want to comment on it properly, as you deserve.

        All my best regards,

        Giovanni

        9 days later

        Dear Dr. Prisinzano,

        I thought that your engrossing essay was exceptionally well written and I do hope that it fares well in the competition.

        I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

        All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

        Joe Fisher