Can Mathematics Reasonably Represent Nature? by Donald Palmer

Well done. I agree and have suggested this in 2014 contest paper and in Survival is the only moral goal of life] Survival is the only moral goal of life.

For inanimate matter, the survival measure takes a form of growth such as mass attracts mass to grow in size and strength. But also note the politics reject the predictions goal of physics. This is the cause of not applying math to these areas.

This is an interesting and thought provoking essay. In particular, I find myself provoked to make the following points: (1) There is a difference between fundamental laws and useful approximations. Even if we have a fundamental law describing the motion of particles, the direct application of that law to 10^23 particles won't be something we can do. If the motions are random then a statistical analysis (as you point out, throwing out lots of information) may be the best we can do. Thus, it might be better to think of Wigner's question as applied to fundamental laws than to the various approximations we might make in various situations. (2) there used to be a debate about "vitalism" that is, whether there was some mysterious "life-force" or whether what goes on inside living organisms is essentially chemical reactions (which are fundamentally described by quantum physics). You don't hear much of that debate anymore because the anti-vitalist/reductionist/biochemical approach has been so successful. We probably don't want to take a step backward to how science was before the successes of modern biochemistry. (3) On the fundamental side, what is striking is how non-unique objects are: all electrons have the same mass and charge, etc. So if we emphasize a fundamental approach, perhaps we don't have to worry so much about non-uniqueness. (4) As you know, given any real number x and any epsilon there is a rational number within epsilon of x. But all measurements come with an uncertainty due to the properties of the measuring apparatus. So the rational numbers should certainly be adequate to express the results of all measurements, and indeed when we quote measurements by giving a finite number of decimals, we are expressing the measurement as a rational number. So why do we need real numbers in physics at all? It's not just because we want to have the square root of 2 (though that's nice to have) but instead because the real numbers are complete with respect to the absolute value norm. And one can express the real numbers as that completion (or in terms of Dedekind cuts) without having to invoke any type of representation.

(and one can make similar remarks about the complex numbers and algebraic completeness). So, it's not so clear to me that physics would need any new numbers or that we are somehow hamstrung by our current representations of numbers.

Dear Donald Palmer,

It looks like we may have a similar computer background so it is perhaps not surprising that I loved your essay. Your comment "animate objects need to be studied in a 'timely' manner, where the growth and movement of an animate object is crucial to understanding it." match my view.

Your argument leading to the comment "to achieve a full description of nature means to be able to describe individual situations, in all their individuality and uniqueness" was very solid.

I have written a lot of computer modelling and animation software using very small distances and time steps, so I would be very interested in "a short book on modeling of scale in the physical world" if it is something that is easy to post online.

Hope you get a chance to have a look at my essay and I would be interested in your comments on my models of the particles of the standard model. The models are meant to be built on and to allow to more complex computer modelling of the interaction of these particles in both chemistry and solid state physics.

Best of luck in the essay contest and you deserve a great rating.

Regards, Ed Unverricht

I would like to respectfully disagree with your first point. It may be difficult to model 10^23 particles at once, but why cant we develop proper animations of say 10^3 particles and test these animation by proving they properly model nature? Would not proper models of small groups of particles provide additional insight into these particles?

Dear Donald Palmer,

Perhaps you were mainly motivated by curiosity as a teacher when you dealt e.g. with numbers. I am an old engineer and when I will read your essay, I expect that you might have arrived at similar ideas and also at different conclusions.

For instance, I don't see any reason to distinguish between fractions like 0.5, 1/2 and 4/8. Likewise I don't distinguish between, 100, 2 times 50 and 4 times 25.

I even expect experts like you having difficulties to accept my reasoning. Please don't forget, I was motivated by practical problems and found valuable historic details already in the second edition of Mückenheim's History of Infinity.

David Joyce commented on an earlier essay of mine that I made some interesting points. I hope you will agree.

Sincerely,

Eckard Blumschein

Dear Donald,

Very interesting analytical essays in the spirit of deep Cartesian doubt. I note your particular important ideas, conclusions and fundamental questions for the "fundamental science" that I liked:

"If we use Wigner's limitation of the study of nature to that of physics, then we immediately remove a huge swath of nature and knowledge - that of animate (or living) nature. In response to Mr. Wigner's original statement (the title of his article) we must expand our view of natural science to more than physics and include animate nature. We must also consider how reasonable (or unreasonable) mathematics is in representing all of nature - not simply physics. Let us first expand our view of 'natural science' beyond Wigner's admittedly limited view. "

"The distinction between animate and inanimate nature still remains a mystery, as the question of what is lost when a living being stops living is still unknown."

"There is another aspect of the animate world we should consider - the importance of uniqueness to animate objects."

"Should the goal of science be simply about developing 'invariant' laws or are we attempting to describe nature? Can invariant laws, even theoretically, completely describe nature? This is an assumption, even presumption, of the scientific method and certainly of some 'Theory of Everything'. However, consider a hypothesis (more properly a 'conjecture') that a universal aspect of nature both animate and inanimate, is the uniqueness of objects, beings, and events. It would seem an obvious conjecture, since it is a common experience for us. However, this conjecture would mean that no two objects have precisely the same characteristics and the more accurate (precise) we care to be the more we will see these differences."

"If science is about finding a set of universal laws, which are invariant across all objects and could (theoretically) predict all motion and change, then this seems uniquely at odds with a universal property that all objects are unique. More to the point, to achieve a full description of nature means to be able to describe individual situations, in all their individuality and uniqueness. A methodology, like the scientific method, that removes individual characteristics would seem at odds with this goal. As well, the unique character of animate objects might undermine a scientific methodology that removes the uniqueness of the objects it studies.

The above discussion should bring into question whether the methodology and tools being applied to inanimate nature are adequate for completely describing nature. Further, do the (and how do the) methodology and underlying tools impact the study of all of nature? If the goal of science is to describe nature, including the unique aspects of nature, then are (mathematical) tools, which remove the unique aspects of nature, short-circuiting this goal? If this is the case, then mathematics (and science using this methodology) cannot represent nature in its entirety."

"So there is an historic precedent we might need to consider: We are using a concept of number and representational system for numbers that is limiting our models - in both mathematics and science. What evidence is there for this situation, especially given how amazingly (unreasonably?) accurate our current theories appear to be? The place to start might be with our assumption that Real numbers, especially as represented today (via decimals, logarithms, and various bases) are the end-all of linear numbers (comprising a line or continuum). What if we have limited our conception of number, and of a geometric continuum, to our conception of a 'Real' number - because representations like decimals for Real numbers are all we know? This would be similar to the situation of the Pythagoreans, where ratios (Rational numbers) were all they knew and comprised a continuum that 'irrational' numbers did not have a seat at. What if 'more' numbers than Real numbers can exist on a continuum?"

"We need to question the adequacy of the underlying mathematics we apply to nature. We generally presume that our current representation of complex numbers (and quaternions) is adequate to the task of modeling nature. Do we have evidence of this? We are aware that we do not have fully formed values for complex or quaternion numbers. Our current representation always involves unknowns (i.e.. 'i', 'k', 'j'). So how can a theory built on these unknowns be 'real'? This situation is like pre-decimal times, when a Real number (at least an 'irrational' one) was not considered a real number. Only rational numbers (represented as integer ratios) were considered 'numbers'. Some people saw that solutions to certain algebraic problems could not be solved using ratios, but they could not properly represent these irrational (and 'transcendental') values - so the numbers were somehow phantom numbers. Only after decimals came into common use was representing them as values possible. We still do not have an adequate representation of complex or quaternion values. Since such numbers pervade many physical models, how can we expect these models to be adequate representations of nature?"

Mathematics overcome their crises through the dialectic breakthroughs, connecting the unconnected, thereby overcoming the regular limit on the path of knowledge. Modern Mathematics and Physics- sciences without ontological justification (basification), as well as all "the fundamental knowledge". How to choose a path? Deeper limit ontology and dialectic in the spirit of Plato - Cusa - Hegel: "coincidence of opposites." We have now a lot of logic. To grasp the Universum as a whole (to connect Cartesian "res cogitans" and "res extensa") need the dialectical logic. Requires the deepest synthesis of all the accumulated knowledge, including Tradition. The "Occam's razor" should be extremely sharp. Necessary unification "matter" at all levels in the spirit of Plato's, ontological justification (basification) and extension "space" based on absolute (unconditional, extreme) forms of existence of matter: "continnum" + "discretuum" + "dis-continuum". We need to understand the ontology and dialectic "three in one" in Nature.

E.Husserl gave good tips in "Origin of Geometry": "Only to the extent, to which in case of idealization, the general content of spatio-temporal sphere is apodictically taken into account, which is invariant in all imaginable variations, ideal formation may arise , that will be clear in any future for all generations and in such form will be transferable by the tradition and reproducible in identical intersubjective sense."

I believe that only the deepest ontological turn of the fundamental science will provide an opportunity to get out of the "crisis of understanding", "crisis of interpretation and representation" to the new heuristics.

I invite you to see my analysis of the philosophical foundations of mathematics and physics, the method of ontological constructing of the primordial generating structure, "La Structure mère" as the ontological framework, carcass and foundation of knowledge, the core of which - the ontological (structural, cosmic) memory and information - polyvalent phenomenon of the ontological (structural) memory of Universum as a whole. I believe that the scientific picture of the world should be the same rich senses of the "LifeWorld» (E.Husserl), as a picture of the world lyricists , poets and philosophers.

Kind regards,

Vladimir

Dear Donald Palmer,

I enjoyed your essay, and some of your comments have helped clarify your points.

You focus on number 'systems' [i.e., decimal versus fraction, and real versus complex versus quaternion, etc.], pointing out that actual calculations depend strongly on the system chosen. You also note that modeling 'animated' systems shifts the focus to 'uniqueness'. You discuss using mathematical systems to model nature and seem to suggest that even more appropriate number systems may be discoverable/inventable. In my essay I discuss the use of AND and NOT gates (at all scales, from RNA/DNA to neural organism, to silicon machines) to evolve theories based on measurement data. It would seem that any future numeric system would still be implemented as combinations of these gates, regardless of the specific numeric representation (a sort of 'existence theorem'.)

You focus on the power of decimals for modeling reality, noting that

"Even if we had some concept of real and complex numbers, we could not produce our technology (and science) without a numeric system like the decimals."

That is an excellent point that is often overlooked in discussions of Wigner. As a result of reading your essay and your comments I also have new awareness of the distinction between the symbolic aspects of math and the numerical aspects of math, a distinction that is usually glossed over.

I invite you to read my essay and welcome your comments.

Best regards,

Edwin Eugene Klingman

You're absolutely right, Donald. Can be called "trialektika". But the "wave" is like "Figaro" - she carries two basic states of matter, "linear" and "vortex".

I totally agree with you - "a current crisis of understanding ... It is imperative that we resolve this crisis, however such a resolution will not provide an underlying understanding of why we get into these crises and how to resolve or avoid them in future."

Thanks for your comment on my forum - I gave my answer and explanation of my ontological structure.

With great respect,

Vladimir

Dear Donald,

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

Donald,

A truly great essay. Massive shame I missed it earlier, and tragedy it's not a finalist. My score would have been 10.

Thanks for your extraordinarily perceptive comments on mine. I've replied in detail there.

Great to make contact.

Peter