Trick or Truth: The applicability of the mathematical language in physics and in economics by Milen Velchev Velev

Dear Milen Velev ,

I enjoyed reading your essay and found your contrast to Wigner's "effectiveness" in science with Velupillai's "ineffectiveness" in economics quite effective. I would suggest that the essential difference is the deterministic nature of most physics, complemented by the statistical 'determinism' of distributions implied by the partition function. Thus even apparently random phenomena are combinatorially predictable by entropic thermodynamics. This is opposed to those economic and social systems whose essential character is human free will. There is no "partition function" to describe distributions whose observations (say of the stock market) give rise to unpredictable (buy/sell) events that are manifestations of free will plus intelligence.

A major theme seems to be that symmetry is the underlying explanation for the effectiveness of mathematics. And I very much agree with you that in all likelihood the universe began as a state of perfect symmetry - until it broke. But, ignoring cosmological lack of perfect symmetry, in particle physics, from proton/neutron iso-symmetry to the failed SUSY symmetry, in which no superpartners are found with mass is equal to their Standard Model partners, there are no exact symmetries. They are all imperfect symmetries, useful, but not ideal. As you note, Noether's theorem implies conservation for every symmetry, but I suspect it is the opposite that is most basic. I propose that every conservation law yields an appropriate symmetry, and that it is conservation, not symmetry, that is at the root. Conservation would appear to be the physical reality, whereas symmetry appears as a mathematical overlay.

I would be interested in your thoughts on this.

I also invite you to read my essay and hope you will comment on my thread.

My very best regards,

Edwin Eugene Klingman

I think you got something. The reason math is less effective in economics or social sciences is the definition of terms and postulates is too vague.

But why is that? Do you have a suggestion as to why?

I suggest it is because the goals of these ranches of knowledge are to support a political model. The politics don't like the answers that match predictions. Therefore, the science gets diverted into vague relms that fail to make good predictions. My 2014 contest paper notes the Friedman economic (monetarist) model did make predictions. But the competing Keynesian model is accepted because it produces a politically accepted view but fails to predict observations. The difficulty caused by complexity is compounded by marginalizing knowledge that has successfully predicted events, but is politically awkward. For example, predicted stagflation that Keynesian derived doctrine said could not happen, predicted the collapse of the soviet system that Keynesian derived doctrine praised, and predicted negative results of big government. The Keynesian derived doctrine of big government has repeatedly been falsified. Friedman argued for a small national government, which the politicians vote against.

Dear Sir,

You have raised and brilliantly discussed several important aspects that provide food for thought for extending those lines. We thoroughly enjoyed your essay.

Mathematics is not independent of human experience, as numbers are perceived as a property of objects by which we differentiate between similars. If there are no similars, it is one. If there are similars, it is many; which can be 2,3....n depending upon the sequence of perception of 'one's. This has been experienced as true - hence accepted as the quantitative description of reality. We have discussed the fallacies of mathematics used in relativity and the views of Wigner in our essay in this forum. Tegmark extends the quantitative aspect of Nature to speculatively describe Nature itself. Obviously, none of these could be proved; which led Hamming to admit its limitations. Economics depends on regeneration and redistribution of goods and services, which are not purely quantitative aspects, but depends upon other factors including human nature, which are not always logically consistent. Thus, Velupillai was using the wrong analogy. You are absolutely right that "superb physical theories, whose basic equations will only take up a few pages". In fact, as Dr. Eckard Blumschein has written in his essay, physics suffers from unwarranted interpretations by self-centric individuals who perpetuate a cult of incomprehensibility to retain their importance.

Relativity is an operational concept, but not an existential concept. The equations apply to data and not to particles. When we approach a mountain from a distance, its volume appears to increase. What this means is that the visual perception of volume (scaling up of the angle of incoming radiation) changes at a particular rate. But locally, there is no such impact on the mountain. It exists as it was. The same principle applies to the perception of objects with high velocities. The changing volume is perceived at different times depending upon our relative velocity. If we move fast, it appears earlier. If we move slowly, it appears later. Our differential perception is related to changing angles of radiation and not the changing states of the object. It does not apply to locality. Einstein has also admitted this. But the Standard model treats these as absolute changes that not only change the perceptions, but change the particle also! We have discussed relativity at length in our essay in this forum.

Schrödinger equation in so-called one dimension Hψ = Eψ (it is a second order equation as it contains a term x^2, which is in two dimensions and mathematically implies area) is converted to three dimensional by addition of two similar factors for y and z axis. Three dimensions mathematically imply volume. Addition of three (two dimensional) areas does not generate (three dimensional) volume and x^2+y^2+z^2 ≠ (x.y.z). Hence, the Schrödinger equation could not be solved for other than hydrogen atoms. For many electron atoms, the so called solutions simply consider them as many one-electron atoms, ignoring the electrostatic energy of repulsion between the electrons and treating them as point charges frozen to some instantaneous position. Even then, the problem remains to be solved.

Symmetry also has been stretched unreasonably. Squaring is not multiplying with itself, but by its conjugates (with sign reversed)! But mathematical laws are different. Why divest mathematics from natural laws and after de-normalization, seek re-normalization!

Regards,

basudeba

Dear Milen Velev,

Reality does not need experimental verification.

Warm regards,

Joe Fisher

You've gathered lots of pieces to create a piece within piece which is symmetrically inclined.

Best regards,

Miss. Sujatha Jagannathan

Hello. You seem to identify the mathematical character with the presence of symmetries. I agree to see a connection between both, but not as much as you do. One way is clear : a symmetry means an exact correspondence between parts, and such an exactness is characteristic of the mathematical nature of a system. However the converse is not true: a system can be mathematical but still not symmetric.

In my essay I explained how I analyze what it means for systems to be more (remarkably) or less mathematical. The basic condition is having a simple foundation, but then simplicity may qualify more or less important aspects of the systems (may these aspects be "foundational" or not). Symmetry contributes to wide aspects of simplicity for big systems, and thus to this mathematical character.

Systems without symmetry can still be completely mathematical, they are only less remarkably so. For example, mathematical models can be extremely successful (or anyway the only available method even if moderately successful) in engineering, meteorology and astrophysics, despite the usual absence (or lack) of symmetries there. The problem with big systems without symmetries is that they are usually quite complex (because, of course, they have many elements not isomorphic to each other, thus having all different properties), which usually does not fit very well with the possibility of a complete mathematical description.

Still there are possible exceptions. For example, in my essay I pointed out a way other than the presence of symmetries, by which quantum physics is also remarkably mathematical (simple) in a wide range of aspects. Outside physics, we can also develop elegant mathematics without any symmetry, for example the theory of ordinals. We can also consider algorithmic as a branch of mathematics that does not have any symmetry in its foundation.

You seem to see physics and economics as very different fields, as concerns the kind of concepts they involve. I don't. On the contrary I see a remarkable similarity there : the fundamental theorems of welfare economics describe the conditions of market equilibrium and stability of this equilibrium, in much the same way as the equilibrium of classical mechanics. Moreover, in both physics and economics, this concept of equilibrium should ultimately be applied not to a specific instant but to whole histories, regarding time as a space dimension like others (relativistic mechanics does so with the least action principle).

Of course the problem with the economic case is that this model is only an idealization that misses more aspects of relevant phenomena than in the physics case. However the mathematical concepts of these models are remarkably similar, and also have in common to not contain any symmetries (there are many possible physical problems to be analyzed by the least action principle and not concerned by any symmetry; they can be very complex too by the choice of the systems to consider, you can find plenty of such in astrophysics and mechanical engineering).

On my side, I work to analyze economic problems and possible solutions by means of high-level, elegant mathematical concepts, such as how to make a better money system, and other solutions I did not all publish yet. This does not involve any symmetry.

Something odd in your essay: you wrote "If the universe is closed, then its total energy will be zero." Where did you get that ? On the contrary, the measure of the thing related to energy that is conserved (that is not exactly the energy but something related to the difference between the energy density and the square of the expansion speed), is nonzero in a closed (spherical) universe. More details in my page on cosmology. Of course, the total electric charge in a closed universe is zero.

I don't need to refer to what "World-famous scientists" wrote in popularization books to discover how things go because I perfectly understand General Relativity myself, so I know very well what I'm talking about. The "disagreement" between us purely comes from the fact that you are yourself ignorant in General Relativity so that you did not understand what I meant, and your only way of replying is to refer to popularization books in guise of an argument. Please know that scientific knowledge in highly mathematical theories is not properly expressed in the language of popularization. So when scientists write popularization books, they may put together some words to look like it means something, but it cannot properly express what the theory really means. In particular the fact is that the expression "total energy of the universe" is basically senseless in General Relativity, so it may be possible to metaphorically use this expression to mean something or something else, but whenever you read it, even from "world-famous scientists", be warned that, due to the fundamental senselessness of this expression, it cannot be meant to be taken seriously anyway. This is metaphor, not science. Several physicists can use a same English expression in popularization books to mean different things when this expression does not have a natural meaning in the theory, so an ignorant reader may have the illusion that they disagree, but in fact they don't. So I'm sure these "world-famous scientists" would agree with me if you ask them.

Dear Professor Velev,

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

Simplicity is a result of long term evolution in a close system. The resulting equilibrium gives rise to simplicity. The infinite possibilities of any member of the system have been largely reduced to a highly confined options. Most of the possibilities are prohibited due to forces that have long been cancelled out during the long evolution. Because of this simplicity, there appears to be causal effect. In other words, causal effect is a direct product of simplicity. Take our universe as an example, the universe is in equilibrium by and large. Only a handful forces remain. Because there are relatively few forces and laws, the universe appears to be orderly and thereby allows mathematics to even exist and work. Mathematics owes its existence to the equilibrium of the universe. Equilibrium brings orderliness and slowness to change. Just imagine, if one puts one stone by another stone, and because the stones decay so fast, by the end of this action of moving them together, one counts zero stone. The law of addition will be forever different from what we know today. In this sense, math and physics have 'this worldliness' feature, and is a localized knowledge to this universe at this phase of equilibrium. It could be vastly different in other possible states of the universe or other universes.

One notable exception to the simplicity in universe is the complexity in bio-sphere. Because the bio-sphere is inherently expansive and interactive, we cannot reduce the theories to a few laws and mathematics models. The bio-sphere is NOT an equilibrium system. Therefore it is very hard to apply causal effect to explain human society for instance. It is very hard to generalize theories or apply mathematics in bio-sphere or human society, as we are able to in cosmology.

Dear Milen,

Thank you for the excellent and informative essay. It raises a lot of questions and I hope to come back with some specific comments.

Regarding "Roger Penrose thinks that mathematical ideas exist in a separate "Platonic" world", I wonder whether his view fully reflects Bertrand Russell's view on the same issue. In fact, this is not really a question to you - I myself am interested in answering it.

Best wishes,

Vesselin Petkov