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FQXi FORUM
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CATEGORY: Questioning the Foundations Essay Contest (2012) [back]
TOPIC: Which Number System Is ''Best’’ for Describing Empirical Reality? by Matt Visser [refresh]

Author Matt Visser wrote on Aug. 27, 2012 @ 17:41 GMT
Essay Abstract

The ''unreasonable effectiveness of mathematics’’ in describing the physics of empirical reality is simultaneously both trivial and profound. After all, the relevant mathematics was, (in the first instance), originally developed in order to be useful in describing empirical reality. On the other hand, certain aspects of the mathematical superstructure have now taken on a life of their own, with some features of the mathematics greatly exceeding anything that can be directly probed or verified by experiment. Specifically, I wish to raise the possibility that the real number system, (with its pragmatically very useful tools of real analysis, and mathematically rigorous notions of differentiation and integration), may nevertheless constitute a ''wrong turn’’ when it comes to modelling empirical reality. I shall discuss several alternatives.

Author Bio

Professor Matt Visser is a Fellow of the American Physical Society, a Fellow of the Royal Society of New Zealand, and a Member of FQXi. He obtained his PhD at the University of California at Berkeley, and undertook postdoctoral research at the University of Southern California, Los Alamos, and Washington University in St Louis. He moved back to New Zealand 10 years ago. His research largely addresses the interface between quantum physics and general relativity, with particular emphasis on wormholes, analogue spacetimes, black holes, and cosmology. Trained as a physicist, he is currently based in a mathematics department.

Jeff Baugher wrote on Aug. 28, 2012 @ 01:22 GMT
Prof. Visser,

It would seem that this effectiveness breaks down when we get to certain processes where the mathematics become non-linear but the "real" world shows a finite outcome. Do you think we just do not understand the underlying mechanics or do you think mathematics might need new tools to help us describe these? I have in mind renormalization for one, but there are other aspects such as energy densities of particles in mind. What are your thoughts on how to handle non-linearities?

Regards,

Jeff Baugher

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Author Matt Visser replied on Aug. 28, 2012 @ 02:54 GMT
Dear Jeff:

There are certainly many well-understood situations where both mathematics and physics become non-linear, but do so in a controlled way. The occurrence of nonlinearity, by itself, is not necessarily a disaster for either mathematics or physics.

Some nonlinearities however, are very difficult to deal with, (turbulence for example), and it is far from clear how to handle that...

You also mention renormalization. There are two separate issues with renormalization that are often conflated: (1) renormalization as a tool for keeping perturbative infinities somewhat under control; and (2) renormalization as a way of re-expressing "bare" parameters in terms of physical measurements. Even if one has a finite field theory, without a single infinity anywhere in the system, one would still have to renormalize in the sense of (2).

Perturbative infinities are certainly tricky to deal with. We have some pragmatic techniques which work most of the time, but certainly not everyone is happy with the situation.

Regards

Matt Visser

Robert H McEachern wrote on Aug. 28, 2012 @ 01:33 GMT
Matt,

In your summary, you ask "Exactly which particular aspect of mathematics is it that is so unreasonably effective?" in describing empirical reality.

I would argue, that is not an aspect of mathematics at all, but rather, an aspect of physics. Specifically, some physical phenomenon are virtually devoid of information. That is, they can be completely described by a small number of symbols, such as mathematical symbols. Physics has merely "cherry picked" these sparse information-content phenomenon, as its subject matter, and left the job of describing high information-content phenomenon, to the other sciences. That is indeed both "trivial and profound", as noted in your abstract.

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Author Matt Visser replied on Aug. 28, 2012 @ 03:17 GMT
Dear Robert:

There are some tricky issues hiding in the phrase "virtually devoid of information".

In a highly technical information-theoretic sense any efficient axiom scheme (Euclid's axioms, the ZF axioms) can be argued to be "virtually devoid of information", since you are encoding large swathes of mathematics in "just a few symbols".

But very few mathematicians or physicists would agree that Euclidean geometry itself, or ZF set theory, is "virtually devoid of information".

Similarly - though the precise details are still an open question - the Navier-Stokes equations, a very compact set of equations which can be written down with very limited number of symbols, seem to encode all of turbulence. And an understanding of turbulence would hardly be "virtually devoid of information".

This suggests a need for a modified interpretation of the phrase "virtually devoid of information", one that not only considers the compactness of the axiom scheme itself, but also takes into account the size and complexity of the model one can deduce from the axiom scheme.

Without somehow taking this into account, one could easily fool oneself as to the information content of a specific model by considering a sloppy redundant axiom scheme.

While physics has by and large certainly picked problems that are relatively clean, and to some extent the relevant mathematics has been developed to address these clean problems, the phrase "virtually devoid of information" is perhaps overkill. Physicists certainly are interested in complex systems, and the techniques of physics are increasingly being used to address complex systems.

Regards

Matt Visser

Robert H McEachern replied on Aug. 28, 2012 @ 05:40 GMT
Matt,

I understand your point. There are indeed "tricky issues hiding in the phrase "virtually devoid of information." That is precisely my point. I am using the term "information" as it is used in Information Theory, not Physics or Math. Physicists, have completely misunderstood just how "tricky" knowledgeable observers can be, when it comes to extracting information from observations. Consequently, attempting to use equations, "devoid of information", to describe what observers are doing is "THE" source of all the weirdness in interpretations of quantum theory. Observers behave "symbolically" towards observations, not "physically." They can (and often do, without even realizing it) treat a measurement not as a datum, but as a "serial number", used to label the set of "behaviors" that need to be "looked up" and performed whenever that symbol is observed. Such types of behavior are vastly more non-linear than anything dealt with by the equations of physics, or phenomenon like turbulence. For such observers, it is not the equations, but rather the initial conditions, in the observers memory, that dictates behaviors.

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Yuri Danoyan wrote on Aug. 28, 2012 @ 02:04 GMT
Dear Dr Visser

It so happened that my essay contains a reference to your article

http://fqxi.org/community/forum/topic/1413

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Author Matt Visser replied on Aug. 28, 2012 @ 03:22 GMT
Dear Yuri:

Thanks. From time to time I still periodically think about Sakharov's induced gravity, and how to get a little more out of Sakharov's proposal...

Regards

Matt Visser

Yuri Danoyan replied on Aug. 29, 2012 @ 00:53 GMT
Which Number System Is ''Best’?

For example Mp/Me=1836 is a true dimensionless constant. I found that it is a beautiful symmetric number because 1+8=3+6=9, after it is converted to numerological addition. In the binary code 1001 present mirror symmetry.

See my essay

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Yuri Danoyan replied on Sep. 8, 2012 @ 00:41 GMT
Dear Matt

Is it time to read my essay more closely?

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Joe Fisher wrote on Aug. 28, 2012 @ 15:20 GMT
Dear Professor Vissar,

Although I am not a mathematician, I quite enjoyed reading your exceptionally well written essay. “The “unreasonable effectiveness” in describing the physics of empirical reality is simultaneously both trivial and profound” In my essay, Sequence Consequence, I mention the fact that the number 1 is often considered to be the most important number in a competitive list, yet it is always accorded the least whole value in currency systems. I believe that one real Universe having one real appearance can only be perpetually occurring once in real here and real now in one real dimension. For this reason, I believe that only 1 of anything can ever exist once. All real stuff has to be kept in one dimension. I suppose it would be more sensible of me to believe that there could be three abstract spatial dimensions, however, if that was the case, how is the abstract stuff in the three abstract dimensions distributed? Is the heavy abstract stuff in dimension A, the average abstract stuff in dimension B and the light abstraction stuff in dimension C? Does the abstract stuff stay in the relative dimensions, or does it intermingle in a measurably orderly fashion that you could attach an accurate number to?

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Author Matt Visser replied on Aug. 28, 2012 @ 20:41 GMT
Dear Joe:

Thanks for the feedback.

You mention: "I believe that only 1 of anything can ever exist [at] once".

There is a sense in which (assuming classical physics) this is almost axiomatic - in classical physics, particles (and more generally, objects) are distinguishable with well-defined locations, and by adding enough qualifiers to your description of an object you can indeed in principle uniquely specify the object. So, essentially by definition, with a detailled enough description there's only 1 of any object you might wish to consider...

In quantum physics this feature seems to go away: Quantum elementary particles of the same type are, as far as we can tell, utterly indistinguishable from each other, which is a key feature leading to (for instance) Bose-Einstein condensation, which we certainly do see experimentally...

So there are very good reasons for being extremely careful and cautious when discussing issues of "uniqueness" in the physical universe we inhabit.

Regards

Matt Visser

Joe Fisher replied on Aug. 29, 2012 @ 14:08 GMT
Dear Professor Vissar,

Thank you for your considerate answer. All real stuff is unique. All human abstract thoughts attempt to impose fixed systems of duplication on all physicality finding congenial mental security in commonly collective shared practicality. “Quantum elementary particles of the same type are, as far as we can tell indistinguishable, which we certainly do see experimentally.” Naturally formed snowflakes are indistinguishable to the naked eye, yet experimentation has proven that no two snowflakes of the trillions that have fallen have ever been found to have been identical. One real Universe can only have one real law. If there have never been identical natural snowflakes, it is physically impossible for any particles of any type to be identical just because they were fabricated. Every one of the sparks created by CERN has to be unique.

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Don Limuti wrote on Aug. 28, 2012 @ 18:54 GMT
Hi Matt,

An important essay, at the core of our models of physics, the "unreasonable e ectiveness of mathematics". Which number system to use is a question I would have not though of. You wrote a clear understandable and concise essay.

Changing the subject a little, I find in my own work two areas where the effectiveness of mathematics breaks down. These are addition, and calculus. See: http://www.digitalwavetheory.com/DWT/5_Math-Physics_Connecti

If you should care to comment I would appreciate it very much.

Thanks,

Don Limuti

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Cristinel Stoica wrote on Aug. 29, 2012 @ 19:32 GMT
Dear Prof. Visser,

This is a very intriguing essay, and it approaches both the theme of this edition of the FQXi essay contest, as well as the previous one. It is very catchy, and pleasant to read. I think the study of the various number systems may be useful for many reasons. It would be useful to have a monograph, or maybe a branch of mathematics, dealing with such number systems and what...

view entire post

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Eckard Blumschein wrote on Aug. 29, 2012 @ 22:41 GMT
Dear Matt Visser,

Are all constructible numbers really countable? Didn't ancient mathematicians count {2,3,...}? Weren't the algebraic irrationals already treated by Cantor? Aren't computable reals just rational approximations of reals? Is there really something between countable and uncountable? Galileo did definitely not agree on that there are more hyper-reals than reals. So don't I.

Maybe I mistook you. I largely agree with some of your intentions. While you are living in New Zealand and I in the old center of arbitrary mathematics, I have different antipodes of mine in mathematics: Weierstrass, Dedekind, Cantor, and Hilbert. Do you dare commenting on my essay?

Regards,

Eckard

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Roger Granet wrote on Aug. 30, 2012 @ 03:32 GMT
My vote for which number system that best models reality is that it depends on the reference frame of the observer (ie, the physicist or mathematician trying to model reality) relative to the reality that he/she's trying to model. For example, a finite observer within an infinite set of finite balls might view each ball as an integer. But, a hypothetical, infinite-sized observer outside this set would not be able to see the boundaries of each of these balls (because they're infinitely small relative to him) and so the set would appear to him to be smooth and continuous. Therefore, he/she might like to use the real numbers to describe the set. I put these thoughts into my entry for the last FQXi contest and they're also at the below if anyone is interested. Thanks!

inet/infinite-sets-ii

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Eckard Blumschein replied on Sep. 1, 2012 @ 13:19 GMT
Hi Roger,

I cannot see who is the author of the linked statements on infinite-sets.

What about your claim that what is the best model of reality depends on the reference frame of the observer, I strongly disagree. Does it matter for a model of the moon whether it is observed from the earth or from somewhere else?

Also, I would like to object against sloppy use of "infinitely small relative to" something. Aren't Leibniz's infinitesimals strictly speaking finite?

I never came across in serious literature with an infinite-sized observer.

Regards,

Eckard

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Roger replied on Sep. 5, 2012 @ 20:49 GMT
Eckard,

Hi. The author of the link was me as indicated by the preceding sentence:

"I put these thoughts into my entry for the last FQXi contest and they're also at the below if anyone is interested."

Everything we observe in reality depends on the perspective of the observer relative to the thing being observed. In regard to your example of the moon, an observer on Earth would view it as an orbiting satellite. But, a hypothetical microbe living in the interior of the moon might view it as an almost infinitely big rock and see the Earth as the satellite.

In regards to the comment that you've never seen anything in the "serious literature with an infinite-sized observer.": First, I used the word "hypothetical" to show that this was an imaginary observer and did this just to show what a set might look like from his viewpoint. Second, whenever mathematicians discuss infinite sets, the mathematician can be the observer describing what is going on in the set.

Thanks.

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Peter Jackson wrote on Sep. 2, 2012 @ 17:21 GMT
Matt

An interesting resume on number systems. Can you comment on the suggestion by Smolin, and Ken Wharton here, that after abstracting to numbers from reality to predict evolution of a system (nature), when renormalised of 'mapped' back to reality, we have no guarantee that similarities found mean there is any real physical relationship of the algorithms to the natural processes.

In other words, the maths used is 'unrealistically representative' of the renormalised mathematical model, but not necessarily also of reality itself. This then would be the reason for all the problems and anomalies.

Also, do you consider that conceptual ontology must be correct before abstraction is used? And might limiting abstraction to, perhaps, rational numbers and rigorously applying the rules and structures of logic to matters of 'process' or 'mechanisms', then allow more precise quantitative modelling?

I've explored the logic and ontology route, particularly regarding kinetics, and believe found some success. I'm not however competent to now evolve the mathematical structures to accompany it.

If you have the time I'd be grateful if you'd also read my essay and comment.

Thank you for yours, and very best of luck in the competition.

Peter

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Hoang cao Hai wrote on Sep. 19, 2012 @ 15:19 GMT
Dear

Very interesting to see your essay.

Perhaps all of us are convinced that: the choice of yourself is right!That of course is reasonable.

So may be we should work together to let's the consider clearly defined for the basis foundations theoretical as the most challenging with intellectual of all of us.

Why we do not try to start with a real challenge is very close and are the focus of interest of the human science: it is a matter of mass and grain Higg boson of the standard model.

Knowledge and belief reasoning of you will to express an opinion on this matter:

You have think that: the Mass is the expression of the impact force to material - so no impact force, we do not feel the Higg boson - similar to the case of no weight outside the Earth's atmosphere.

Does there need to be a particle with mass for everything have volume? If so, then why the mass of everything change when moving from the Earth to the Moon? Higg boson is lighter by the Moon's gravity is weaker than of Earth?

The LHC particle accelerator used to "Smashed" until "Ejected" Higg boson, but why only when the "Smashed" can see it,and when off then not see it ?

Can be "locked" Higg particles? so when "released" if we do not force to it by any the Force, how to know that it is "out" or not?

You are should be boldly to give a definition of weight that you think is right for us to enjoy, or oppose my opinion.

Because in the process of research, the value of "failure" or "success" is the similar with science. The purpose of a correct theory be must is without any a wrong point ?

Glad to see from you comments soon,because still have too many of the same problems.

Regards !

Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY

August 23, 2012 - 11:51 GMT on this essay contest.

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Anonymous wrote on Sep. 20, 2012 @ 19:03 GMT
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

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Member Hector Zenil wrote on Sep. 25, 2012 @ 05:23 GMT
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.

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Author Matt Visser replied on Sep. 25, 2012 @ 23:10 GMT
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

Member Thomas Sotiriou wrote on Sep. 25, 2012 @ 10:05 GMT
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

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Matt Visser replied on Sep. 25, 2012 @ 22:02 GMT
Dear Thomas:

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

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Yuri Danoyan wrote on Sep. 25, 2012 @ 10:59 GMT
See my discussion with George Ellis

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Jerzy Krol wrote on Sep. 25, 2012 @ 14:04 GMT
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

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Matt Visser replied on Sep. 25, 2012 @ 22:17 GMT
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

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Yuri Danoyan wrote on Sep. 25, 2012 @ 15:02 GMT
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

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Yuri Danoyan wrote on Sep. 25, 2012 @ 23:00 GMT

Real numbers

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Yuri Danoyan wrote on Sep. 25, 2012 @ 23:18 GMT
+ Angles

Summary: Real numbers + Angles

Enough for Pythagoras "World is a number".

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Yuri Danoyan wrote on Sep. 25, 2012 @ 23:24 GMT
For example my articles:

This one http://www.fqxi.org/community/forum/topic/946

http://vixra.or
g/abs/0907.0014

and this one

http://vixra.org/abs/0907.0012

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Yuri Danoyan wrote on Sep. 30, 2012 @ 10:41 GMT
There is no one way to describe the world, it is possible to describe different but ultimately finite number of equations.

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Sergey G Fedosin wrote on Oct. 4, 2012 @ 06:06 GMT
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
$R_1$
and
$N_1$
was the quantity of people which gave you ratings. Then you have
$S_1=R_1 N_1$
of points. After it anyone give you
$dS$
of points so you have
$S_2=S_1+ dS$
of points and
$N_2=N_1+1$
is the common quantity of the people which gave you ratings. At the same time you will have
$S_2=R_2 N_2$
of points. From here, if you want to be R2 > R1 there must be:
$S_2/ N_2>S_1/ N_1$
or
$(S_1+ dS) / (N_1+1) >S_1/ N_1$
or
$dS >S_1/ N_1 =R_1$
In other words if you want to increase rating of anyone you must give him more points
$dS$
then the participant`s rating
$R_1$
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

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