Dear Matt,

Congratulations on a well-written and timely essay! I particularly appreciate your recognition of the very strong conditions imposed even by admitting the rationals. A few remarks come to mind:

1. An important point to consider in regard to the number systems you mention is their order-theoretic structures. For example, the rationals, as you point out, are infinitely divisible, which corresponds to the interpolative property in order theory. One reason why this is fundamentally important is because of the prominence of order theory in describing the causal structures that arise in certain models of spacetime microstructure.

2. Another reason why the underlying number system is important is because the properties of particles in the standard model are determined by the symmetries of Lie groups, which live and die with the continuum. This is an example of how the decision to use the reals is not merely a philosophical issue of postulating structure that cannot be measured or "filling in the gaps."

3. There are a couple of essays on this thread (the ones by Torsten Asselmeyer-Maluga and Jerzy Krol) that explore surprising consequences of admitting nonstandard models of natural and real numbers. (This, generally speaking, is going in the opposite direction and admitting much "larger" systems such as in nonstandard analysis.) You may be interested in these if you have not yet read them.

My own essay here discusses the order theoretic and representation theoretic implications of rejecting the continuum and working with causal structures. If you are interested, I would appreciate your thoughts. Take care,

Ben Dribus

Dear Matt

Don't forget about induced gravity.

Unfortunately the proponents of this approach are very few.

This is one of them

Induced Gravity in Superfluid 3He

http://arxiv.org/abs/cond-mat/9806010

Grigori very smart person

When you read my essay?

Dear Matt,

I think your ideas could be better appreciated with the nuances that computability theory might bring to your notion of the differences between number systems. More particularly the distinction between computable numbers, and then make more obvious that in fact continuous math is only descriptive, but it becomes computable when numerically implemented, which is already making a stand about models and perhaps even about the physical world. I wonder if you have connected your ideas with computation rather than pure math.

    Dear Matt,

    a thought-provoking essay indeed. You seem to be arguing that we should choose the smallest possible set of numbers needed to described empirical reality. A couple of questions:

    a) Does such a set exist and is it unique? By that I mean that I can imagine the following situation: all of the numbers we "need" (based on our knowledge so far) belong in two (or more) set with different properties and both of these sets contain more numbers. What then? Suppose that both choices are equally small (or large).

    b) Is there any direct example that suggest that giving up the reals can make things better or solve a problem? I understand that one might loosely related giving up the reals, with differentials and fundamental discreteness, but one does not need to give up the reals altogether in order to write down a finite difference equation, for instance.

    Best of luck!

    Thomas

      Dear Matt,

      very good and useful resume on numbers which might be important in physics; I red it as fascinating story. When additionally we turn to model theory (which you already mentioned as non-standard analysis case) category theory, and topoi in particular, the spectrum of possibilities grows. We can have also spaces (smooth manifolds) as constructed from such reals. Let me mention only pointless topoi or many many other possibilities. Some clues from such spaces modeled on different number systems can be derived as feedback on the correctness of the numbers chosen. Also, different geometries can emerge here and these again serve as a top-down criterion for the selection of reals. This is just what I considered in my essay: http://fqxi.org/community/forum/topic/1443 Maybe it would be some interest to you.

      Good luck in the competition,

      Jerzy

        Dear Matt

        The best number system is mean by Einstein.

        "The philosophical significance of a complete set of units, is that it allows us to express any fundamental constant as a pure number. According to the ideal of theoretical physics expressed by Einstein

        "I would like to state a theorem which at present can not be based upon anything more than upon a faith in the simplicity, i.e., intelligibility, of nature: there are no arbitrary constants ... that is to say, nature is so constituted that it is possible logically to lay down such strongly determined laws that within these laws only rationally completely determined constants occur (not constants, therefore, whose numerical value could be changed without destroying the theory)."

        Listen to Frank Wilczek

        Dear Thomas:

        You ask:

        a) Does "the smallest possible set of numbers needed to described empirical reality" exist and is it unique? That depends on some subsidiary assumptions. If you assume or demand that the set of numbers you are interested in forms a field (in the sense of abstract algebra), then there is a procedure for forming the unique "algebraic closure" of that field. But this very much depends on the precise axioms of algebraic fields, and conceivably, (though I would not wish to encourage this), one might even be willing to abandon the algebraic field axioms - for instance working with algebraic rings or algebraic Euclidean domains.

        b) I think you are here distinguishing the possibility of abandoning the reals for the position coordinates x, but retaining the reals for the values of functions f(x)? I am not quite sure why you would do one and not the other. You also ask for an explicit example of the advantages of abandoning the reals? If I already had an explicit example, it would be a rather different essay. Here are some speculations: If you want flat space-time to be covered by a singular coordinate patch, then your number system should at least be countably infinite (otherwise I suspect there will be no hope of ever getting phase transitions, and other things would fail as well). It is the step from countably infinite coordinate locations to unaccountably infinite coordinate locations that is potentially the most interesting step to think about.

        Regards

        Matt

        Dear Jerzy:

        Yes indeed there are even more abstract and general things that can be done. For instance moving to "topi" and "categories", thereby leading to the process of "categorification".

        (Though I and most physicists would probably for "small categories", sometimes called "kittegories", so it should probably be "kittegorification".)

        As always there is a trade-off between abstract generality and pragmatic usefulness.

        I am unsure where exactly the best trade-off point is, but certainly agree that we should spend at least some time looking at these questions.

        Regards

        Matt

        Dear Hector:

        You state: "continuous math is only descriptive, but it becomes computable when numerically implemented". I take it you mean "continuous math" in the sense of abstract topology?

        In this context it is the step from a general abstract topology to a "locally Euclidean space" that sneaks the the real number system into the game - and then the step to a spacetime manifold adds a number of technical restrictions on the topology.

        One could then try to rephrase all of manifold theory in the language of computability theory, but it seems to me the key step has to do with one's choice of coordinates - the nature of the local coordinate charts, so one might just as well focus on exactly this point - the nature of the number system used to set up the local coordinate charts.

        The use of "computable numbers" (based on the rationals) is one possibility along these lines, but I have not looked any deeper into this particular route.

        Regards

        Matt

        4 days later

        If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

        Sergey Fedosin