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

    4 days later

    Dear Edwin Klingman,

    Thank you very much for reading my text carefully. I'll be very interested in reading your essay. There is a high probability that the universe started out in a condition of perfect symmetry but at present we observe imperfect, broken and hidden symmetry in nature. In my opinion, symmetry is a more general and fundamental notion compared to the concept of conservation. Symmetry is defined as a physical or mathematical feature of the system which is preserved or remains unchanged under some transformation. The conservation laws only apply to a particular measurable physical property which does not change in an isolated physical system. For example, the law of conservation of mass-energy, the laws of conservation of linear and angular momentum and the CPT theorem follow from the validity of Lorentz invariance symmetry.

    Best regards,

    Milen Velev

    Dear John,

    Thanks for the interesting question. I will read your essay carefully. I agree with your opinion that the politicians don't like the answers that match predictions. The choice of one macroeconomic policy or another could be dictated by a desired political effect. But the influence of the political factor is just one of a whole host of problems in this area. For example, in macroeconomics the concepts of potential aggregate output and natural rate of unemployment (NAIRU) are of great significance. The effectiveness of the applied macroeconomic policy depends on the correct determination of the values of these factors. But the problem of calculating them is difficult and not that clear. They can't be measured directly. There are a lot of differences and difficulties in determining these quantities. They can be evaluated by using various methods and means and each one would produce approximate, expert results. Another important issue is the time lag in implementing a macroeconomic policy. It takes a while between the time corrections are made to the macroeconomic policy and the time when it shows its effects, a period during which the economic conditions might change and the measures undertaken might prove inadequate. When a certain macroeconomic policy is implemented, the secondary effects that arise should also be considered. That's why economists often use the "Ceteris paribus" assumption.

    Best regards,

    Milen

    Dear Joe Fisher,

    Thanks for your comments. Your conception is very interesting. In my opinion, physics aims to describe the real universe, and to this end it uses mathematics as its principal instrument. The experiment is the criterion of the veracity of a theory.

    Best regards,

    Milen Velev

    Dear Basudeba,

    Thanks for reading and commenting on my essay. I agree with you that in a way mathematics is not entirely independent of human experience. Elementary arithmetic, Euclidean geometry, etc. can be given as examples. But some mathematical conceptions were introduced into mathematics without any relation to human experience, long before they were applied in physics (for example, imaginary numbers). Some mathematical conceptions are so abstract that initially, it wasn't clear where they could find any practical application (for example, Calabi-Yau manifold). Other conceptions have no analogues whatsoever in the physical reality (transfinite cardinal and ordinal numbers, the continuum hypothesis, etc.). Maybe at some point in the future they will find some practical application. So, it can be asserted that the development of mathematics is comparatively independent of the first-hand human experience.

    Best regards,

    Milen

    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.

        4 days later

        Dear Sujatha,

        Thanks for reading my essay and for the interesting comment.

        Best regards,

        Milen

        Dear Sylvain,

        I don't contend that in order for mathematics to be effective in describing a given system, this system has to possess symmetries. As you have rightly noted, mathematics has branches that aren't based on the ideas of symmetries. In those cases symmetries, if applicable at all, just make things simpler and easier, but nothing more. There are probably certain criteria which determine a system as more or less mathematical. I'd be really interested to read your essay.

        But the main problem I'm examining is different - why mathematics is so incredibly effective in describing physical reality (the entire universe known to us). At the beginning of my essay (p.1) I've pointed out the three aspects related to the success of mathematics in explaining the world around us. Your question concerns the first aspect only.

        In my opinion, the answer to the problem posed is related to the same way in which mathematics and physics are constructed. Both sciences are based on several axioms (principles) which are universal and generally applicable. This is the only way of constructing formal mathematical systems and there is nothing unusual and unexpected here. But what seems a bit puzzling is why nature would obey several universal and generally applicable principles and laws. The reason why the laws of nature are universal is symmetry intrinsic to space and time. And the reason why there are any natural laws and principles at all is related to the existence of a great number of abstract and hidden symmetries in nature, in which the physical laws originate. (Maybe in the universe there are a lot of symmetries, preserved or partially broken, which we'll discover and investigate in the future.) Where do these symmetries originate? In my opinion, in the fact that the universe started out in a condition of perfect symmetry. Why was the initial state of the universe characterized by perfect symmetry? Probably because the universe arose from 'nothing' (i.e., from the quantum fluctuations of the vacuum).

        Let me emphasize again - what has been said so far doesn't mean that if the universe didn't obey symmetries, it couldn't be described through the language of mathematics. (I must admit, I find it difficult to imagine a universe in which there are no symmetries at all.) All I'm saying is that in this case the mathematical description of the physical reality would by no means be as effective as it really turns out to be. In this case the superb physical theories which describe all familiar phenomena in the entire visible universe wouldn't take up a few pages only. And I've given the economic systems as an example and proof in this respect. They don't obey symmetries. Certain dependences may operate in a certain market, and in another market - other types of dependences. Therefore, we can't derive a generally applicable law of demand, for example, a law that would be applicable in all kinds of markets and for all kinds of commodities. To derive a particular quantitative dependence of the law of demand we have to investigate a particular market of a particular kind of commodity over a specified period of time. This quantitative dependence would be locally applicable and inapplicable in other kinds of markets.

        In economics the problem is not only that we have idealization of reality. Inaccurate and vague notions, the self-reflection of the economic systems, the irrational elements in human behavior, etc. are also among the problems.

        You've pointed out that mathematical models are hugely successful in engineering, meteorology and astrophysics despite the lack of symmetries there. But these are applied sciences rather than fundamental. In my opinion, the successful application of mathematical models in these sciences (insofar as it is successful) is the result of the effective application of mathematics in the fundamental physical theories.

        About the last question: In a closed universe the total positive energy, including the energy of the mass of the particles at rest, is exactly compensated by the negative gravitational energy, so that the total energy is exactly equal to zero. World-famous scientists share the same opinion. See, for example: Krauss, L. M. (2012), A Universe from Nothing (New York: Free Press) or Kaku, M. (2004), Parallel Worlds: A Journey Through Creation, Higher Dimensions, and the Future of the Cosmos (New York: Doubleday).

        Best regards,

        Milen

        10 days later

        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.

        17 days later

        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

        10 days later

        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.

        8 days later

        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