Dear Tejinder Singh and Anshu,

Your essay contains a numer of insights. Of course elementary counting, or number sense, is hardwired into the brain, but counting exists at almost every level of reality, from three quarks per baryon to the number of telomeres on chromosomes, and a significant number of lower lifeforms. Our own hardwired structures are extremely high level.

As for "primordial physics", I have employed a robot as a vehicle to eliminate bias and "baggage", while providing pattern recognition, learning algorithms (neural nets, self organizing maps, etc.) and have shown how counting, derived from logical physical structures, is essentially (along with simple arithmetic logic circuitry, silicon or biological) all that is required to go from raw measurement data to feature vectors of the quantum persuasion. This work is summarized in the first two pages of my essay.

It is appropriate to point out that mathematics is "an enterprise of the human mind, and not a universal Platonic truth 'out there'." This mystical misconception is a major problem for many physicist today. As you say, it is an "act of faith" (at best!) Lee Smolin provides an excellent analysis of this in his essay.

Your "three key developments in cognitive science" do bear crucially on the math-physics connection, and your hunter lighting the fire is very akin to my robot physicist, who notes changes and acts on or processes them. Comparison of differences is at the root of it all. Lack of change typically implies something can be ignored, while lack of change in a dynamic situation leads to conservation relations.

Most significant is your endnotes treatment of the 'oddities' of quantum mechanics:

1.) The theory has to rely on its own limits, i.e., classical mechanics.

2.) With initial state known precisely, yet the outcomes are probabilistic.

3.) The "collapse of the wave function" is a mystery, and has problems.

4.) The quantum theory depends on classical time for describing evolution.

You say "we physicists feel reluctant to modify quantum theory" (due to its successes). It may not be necessary to modify QM so much as to simply admit that it is incomplete. My essay describes a local model that accomplishes what Bell proclaims impossible, and analyzes how he correctly applies math to incorrect physical assumptions, leading to a false conclusion. My model should be experimentally verifiable, which, as you say 'reigns supreme in physics'.

I hope that you will find the time to read my essay and to grace me with your comments.

Best regards,

Edwin Eugene Klingman

    Dear Edwin,

    Thank you for your careful reading of our essay and your comments on it. We will definitely read your essay in the next few days and respond to it.

    Best regards,

    Tejinder, Anshu

    Dear Madam/Sir,

    Mathematics describes quantitative aspects of Nature; whereas physics describes its qualitative aspect (interaction is chemistry). Thus their relationship is like the chicken-egg problem or rather like electricity and magnetism. However, there is the danger of over-emphasizing some aspects like extra-dimensions, which could not be discovered even after more than a century, but still used by the physics community as Gospel truth. Or it may be limited observation like galactic red-shift that led to conclusions about expanding universe, dark matter, dark energy, etc., which concepts are now being questioned after discovery of galactic blue-shift and merger. But everyone has turned a blind eye to such questions. Your "observation that the product of electric and magnetic permeability equals the inverse square of the speed of light", read with e = mc^2 as discussed in our essay gives a very interesting picture over-looked till date.

    Inability of some 'theories' to explain certain phenomena do not mean everything about classical views is wrong. Instead of rectifying the theories, the baby is being thrown with the bath water. And where are we landing? In a fuzzy world of probabilities! Mathematics is all about certainties. It's unreasonable manipulation has led to the present state. The failure of the Michelson-Morley experiment to detect the motion of the earth through the hypothesized ether is one such wrong conclusion. The experiment was conducted with light, which is a transverse wave. All transverse waves are background invariant. Thus, the result was destined to be null. Yet, much fantasy has gone into building theories over such null result. Our essay is full of such examples, as well as a critical discussion on number theory, set theory, Gödel and Wigner.

    While science without technology is lame, technology without science is blind. With over-emphasis on the effectiveness of technology, its 'blindness' is increasing, which is manifest in various social and environmental problems. A very large number of people enjoy a cozy life in pursuing and teaching nothingness or self-destruction. We may enjoy temporarily, but ultimately everyone is going to suffer. There is a need to review and rewrite physics and un-mathematical mathematics.

    Regards,

    basudeba

      Dear Joe,

      Thank you for reading our essay. We are sorry you did not find it useful.

      We read your third paragraph above a few times but we do not understand it still. The latter part is at variance with what we learn in standard physics. What do you find wrong in the standard physics description, and do you have a mathematical formulation to support what you are proposing?

      Thanks and regards,

      Tejinder, Anshu

      Dear Basudeba,

      Thank you very much for reading our essay and for your comments above. There seems to be much that we disagree about! We cannot agree that physics describes *qualitative* aspects of nature, or that there is a chicken-egg problem here. Also, why do you say extra dimensions are a gospel truth? Surely they are till not accepted as confirmed by experiments. Also, we would not say that mathematics is all about certainties. Stochastic dynamics is very real, in various settings, at least in an emergent sense. Nor can we agree with what you conclude about the Michelson-Morley experiment.

      Nonetheless, thanks for pointing us to your essay, which we look forwarding to reading in due course.

      Best regards,

      Tejinder, Anshu

      Dear Tejinder, Anshu,

      I am deeply troubled that you would admit that you do not know what reality is.

      "Standard Physics" is an abstraction and that is what is wrong with it. All mathematical formulations are abstractions.

      I cannot help you,

      Dismayed,

      Joe Fisher

      Your work circulates in primordial features of mathematics which signifies the emergence of indistinguishable notions simmering hypothetical notions.

      Good Luck!

      Best Regards,

      Miss. Sujatha Jagannathan

        Thank you for your remark Sujatha. We wonder what you meant by your use of the word `simmering' here?

        Regards,

        Tejinder, Anshu

        Anshu and Tejinder,

        It might be fair to say that mathematics was born out of a qualitative or cognitive analogue that grouped objects together. It took probably a bit to figure out how to really count beyond one, two three, many. The tendency of the human brain is to count in a sort of logarithmic sense. People from cultures without arithmetic will often say the number 3 is the middle number of a set of 10 objects, or that 6 is the middle of 20 objects.

        You are correct I think in stating that our cognitive abilities underlie arithmetic. This might be compared to David Hume who said that reason is ultimately a slave to our passions.

        Cheers LC

        Dear Lawrence,

        Thanks for reading our essay and commenting on it. It is heartening that we agree on the role of cognition. Especially interesting is your remark on the tendency to count logarithmically...we learnt from Dehaene's book about Amazon tribes which tend to think of 1 and 2 being farther apart than 8 and 9 are [mental compression of a logarithmic nature]. It seems even young children, when asked to place say the number 10 on the number line, between 1 and 100, tend to put it near the middle of the line [like you suggest], and compress the larger numbers on the right half of the line. The linear equi-spaced ordering of numbers is learnt culturally through education as we grow up. This is perhaps another useful example of innate arithmetic versus learnt arithmetic.

        We look forward to reading your essay.

        Thanks and regards,

        Anshu, Tejinder

        Dear Tejinder Singh and Anshu,

        Thank you for your essay. It was a real pleasure to read it, especially when you described how mathematical and physical concepts were built on each other over time.

        I had great expectations on your essay and I felt a little bit disappointed. On one side, the perspective you introduced on mathematics and physics history is superb. On the other, I felt you went too much into relativism. Relativism and accepting that we know little or nothing might be accurate but it is not constructive.

        I wish you could have shown that by changing our cognitive axioms, we could have developed different views of the world (very simple examples would have been enough). Maybe this task is impossible for us, humans, to do.

        Wish you the best in this contest.

        Regards

        Christophe

          Dear Christophe,

          Many thanks for reading our essay and for your insightful comments. We would like to discuss further your constructive criticism. For that, it will be great if you could kindly elaborate and expand on your following remark:

          "On the other, I felt you went too much into relativism. Relativism and accepting that we know little or nothing might be accurate but it is not constructive."

          We are interested in understanding what you imply by relativism here.

          Your remarks about changing cognitive axioms to get a different view are also very interesting. We implicitly had in mind a unique set of axioms for theoretical physics, which should lead to a theory consistent with experiments. In the sense that physicists are inclined to believe there is only one correct theoretical description of a phenomenon. And if it seems there are more than one description, we make every effort to find out which is the right one. This is perhaps different from mathematics where one could start from differing sets of axioms.

          Did you have in mind different possible sets of starting axioms for theoretical physics?

          As for different cognitive axioms, we will be indeed hard put to come up with a proposal, having assumed that cognitive axioms draw intimately on our motor-sensory perceptions and are hence unique. But this needs further thought and discussion, which we are certainly happy to continue with you.

          Thanks and regards,

          Anshu, Tejinder

          Dear Anshu, Tejinder,

          I read your essay again and I find your case convincing.

          I suspect my hope is that understanding the connection between mathematics and physics would tell us something about the world. Reading your essay, it tells us something about us. That's probably where comes from my little disappointment, what I implied by relativism.

          I agree with you that evolution has shaped our abilities. For example, driving a car implies processing hundreds of dynamic variables. Everyone does it easily. On the other side, an equation with the same number of variables is inaccessible to us.

          If you have time, I have a short essay in this context. Your comments or criticisms will be appreciated. http://fqxi.org/community/forum/topic/2322

          I wish I would have introduced my arguments in more details. Funny how one can get caught in the game!

          Regards,

          Christophe

          Thanks Christophe,

          After reading your essay we understand your comments above better, and have posted a brief comment on your essay.

          Regards,

          Anshu, Tejinder

          Dear Profs. Majumdar and Singh,

          I read your essay with great interest. I noticed that you briefly address the foundations of quantum theory, by identifying four "oddities", essentially logical paradoxes and inconsistencies.

          In that regard, you might be interested in reading my essay: ("Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory"

          I argue that premature adoption of an abstract mathematical framework prevented consideration of a simple, consistent, realistic model of quantum mechanics, avoiding paradoxes of indeterminacy, entanglement, and non-locality. What's more, this realistic model is directly testable using little more than Stern-Gerlach magnets.

          Alan Kadin

            Thank you Alan, for making time to read our essay. We had a first read of your essay, and look forward to reading it again for better understanding, and discussing it with you on your page.

            In this context, we wonder if it might be of help for the sake of comparing your quantum viewpoint with ours, if you could critically examine the popular video `Does nature play dice'?' which one of us posted in the recent FQXi video contest. Understandably, we have quite different outlooks, but I am sure the comparative discussion will be stimulating. In particular, it would be interesting to know how you evaluate your proposal with the other modifications / reinterpretations of quantum theory discussed in the video.

            Thanks and best regards,

            Anshu, Tejinder

            Dear Authors,

            I've read two times your essay, and what I find very nice in it is the perceivable 'pleasure' by which you move up and down the history of physics and mathematics, mentioning the most important milestones in both areas.

            At a first sight, I also found quite interesting the idea to make the 'unreasonable' effectiveness of maths in physics become reasonable, or even inevitable, by identifying the roots of both in human primordial perceptions. But on a second though I am still left with much doubts about the validity of this explanation.

            Imagine an other universe similar to ours, with galaxies, stars, planets, but where the phenomenon of conscious life (humans) has not emerged. Planet trajectories still follow the beautiful equation of the ellipsis and most phenomena still match the beautiful and simple patterns described so effectively by math. How could you explain this match in that case?

            One could exclude this scenario, claiming that there is no reality without a conscious entity (say a human) that perceives it, but I had the impression that you are not a follower of this (rather extreme) school of thought. Then we are left with a universe nicely describable by compact math formulas (Tegmark's External Reality Hypothesis) - although no brain is there to formulate and enjoy its mathematical description. But in case a conscious alien came to visit it from a parallel universe, he would probably enjoy the matching between math and physics, and find it 'unreasonable' indeed.

            How to fix the problem? Or did I miss some crucial element in your reasoning?

            Thanks

            Tommaso

              Dear Tommaso,

              Thanks so much for reading our essay and thanks for your interesting comments. In particular, you say:

              "Imagine an other universe similar to ours, with galaxies, stars, planets, but where the phenomenon of conscious life (humans) has not emerged. Planet trajectories still follow the beautiful equation of the ellipsis and most phenomena still match the beautiful and simple patterns described so effectively by math. How could you explain this match in that case? "

              We make a distinction between the physical world [which we believe exists even when conscious humans are not there] and the mathematical description of the physical world [which we believe is only possible when conscious humans are there]. That the physical world exists and existed when humans are/were not there, seems provable by scientific methods [radioactive dating of historical records for instance] and we do not call this belief an act of faith. However, we do not see how to scientifically establish that a mathematical description exists / existed when humans are not there. Thus, in your example above we would agree that in the absence of humans the planet still goes around the star, but in our absence there would not be the truth `the elliptical orbit of the planet is being caused because its acceleration falls as inverse square of its distance from the star'. This last bit [elliptical...acceleration...inverse square law] is to our understanding a very human `description' of reality, which is distinct from the reality itself. We find it very hard to understand how the maths can `live' in the material substance, i.e. the planet. Maybe one day there will be an experiment based scientific proof that the maths that describes the thing is the same as the thing itself, but this has not happened thus far and we consider it unlikely. How to put matter into mathematics?

              Furthermore, we feel that the formalism of mathematics that the human brain has created is based on the level of complexity the brain has evolved to, and depends strongly on conceptual metaphors. Suppose, a non-human intelligent alien studies the physical universe/nature, what formalism "it" will create would likely depend on its ability to observe nature, creating metaphors etc. Would it overlap with human mathematics or would it be different is for us hard to predict. Thus one might say that the existence of the physical universe is observer independent (invariant); however description of nature is observer dependent, and It depends on the observer's perception, its ability and so on.

              We look forward to another reading of your mathematical thriller :-) and to reading your views on the Mathematical Universe Hypothesis.

              Best regards,

              Anshu, Tejinder

              Dear Sirs,

              You claim in your essay the following:

              "Next, mathematics is introduced by way of the second law, which encodes the experimentally verified inter-relation between the concrete concepts of mass and force, and the abstract entity from calculus (acceleration as the second time derivative of position). The second equality, the force law of gravitation, is motivated, amongst other things, by the necessity to deduce Kepler's empirical inference that the orbit is an ellipse."

              The task of Newton, assigned to him by his peers, was to prove that given the law, the known orbits hold, called the "inverse problem". The law was known long ago before Newton. He proved that if the law is true, the orbit can be one of four conical sections, not only an ellipse. This cannot be reduced to some "motivation". This was a difficult task that changed science forever. Your approach to this subject is emotive. Newton made many abstract considerations to reach the final result that could not be made by mathematicians. You obviously have not read his books. I suggest you do so. Thank you for the effort but completely disagree with both the motivation and the conclusions.