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

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