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.

Dear Alex,

Thank you for reading our essay and for your comments, though it is not clear to us whether your last remark

"Thank you for the effort but completely disagree with both the motivation and the conclusions."

pertains to the entire essay or only to the example of planetary orbits. If you have a different view on the connection between physics and mathematics we will be glad to learn from you about it.

We are certainly aware of the controversial history of the inverse square law prior to Newton, and involving Hooke, Wren, Halley, Bullialdus, and Borelli. It is our impression that the inverse square law was certainly not an established truth [although Hooke undoubtedly made an important contribution] prior to its application by Newton to the data and analysis of Brahe and Kepler, but only an idea and a suggestion. Also, there is historical evidence that prior to receiving correspondence from Hooke in 1679-80 on the inverse square law, Newton already in the 1660s had inferred an inverse square law for circular planetary orbits. It is also known that before the the publication of the Principia, Newton himself had doubts as to the accuracy of the inverse square law, especially near a massive sphere. This of course refers, among other things, to Newton's very significant proof that if a point mass produces a gravitational field which varies inversely as the square of the distance from the point, so does a spherical body, outside of itself. Surely it is common knowledge that the discovery of this proof held Newton back for many years from announcing his findings.

We could not have said all this in nine pages, keeping in view that the topic of the essay is not the history of the inverse square law. Nonetheless, let us grant, for the sake of argument, that the inverse square law was an established fact before Newton. Does it make a difference whether he fitted the force law to Kepler's data, or Kepler's data to the force law? We do not understand your remark "Newton made many abstract considerations to reach the final result that could not be made by mathematicians." In this work of his, we certainly are not thinking of Newton as a mathematician, but as a great theoretical physicist. In any case, it is our understanding that the inverse square law gained universal acceptance only after the orbits were explained. Had it turned out to be the case that the orbits are explained instead by a force law where the force varies as say, the inverse fourth power of the distance, the inverse square law would have been given a decent burial.

And as students of physics, may we humbly submit that we do know that the inverse square law admits three (why do you say four) conic sections, and that the bound elliptical orbit is picked out when the total energy of the orbiting body is negative? :-) We did not think it necessary to state this elementary fact, in the limited space available.

Best regards,

Anshu, Tejinder

6 days later

Dear Tejinder and Anshu,

You have done a good review of the theme of this year's essay. Even though it can be difficult to fault your finding that mathematics originates "from the brain", and is not "out there". However, if you will entertain my alternative view, I believe it cannot be ruled out that mathematical objects are 'out there', and the human brain evolved to meet them. It is my opinion, that there is no difference between the objects of mathematics (when properly and unambiguously defined) and physical objects.

Take Euclid's point for example as defined in Elements, Book 1, Definition 1. If defined as a zero-dimensional object it cannot correspond to a physical point. But if 'point' is mathematically defined as an extended indivisible object of the smallest possible dimension, it can unambiguously correspond to physical reality. Same with definition of a line having length and of zero width and thickness. It cannot correspond to physical reality. Anything that is mathematically zero in any of its dimensions does not and cannot physically exist.

In your essay appears this statement, "there is no place in mathematics for matter (material substance), and by extension, for light! This to us is the biggest difference between physics and mathematics, from which all other differences germinate". I agree with this. If we are therefore to rephrase this statement for the search for a unifying theory for math and physics, and eliminate this biggest difference, we must find a place for light velocity in mathematics! That is, we must treat light velocity like all other velocities. All other velocities are vector quantities whose resultant values depend on the observer's motion. We can therefore not turn a vector into a scalar, whose value is constant between frames of reference merely because it has a value of 3x108m/s.

It is at this point I wish to comment on the statement, "The failure of the Michelson-Morley experiment to detect the motion of the earth through the hypothesized ether led Einstein and others to abandon the ether, and look for a set of mathematical coordinate transformations which allow the speed of light to be the same for all inertial observers...". As you mentioned, in that experiment motion of the earth had no effect on light arrival time, i.e. the resultant velocity of light was constant despite observer motion, i.e. c v = c.

Looking for a "new" set of mathematical coordinate transformation would be valid only if there no findings where light arrival time is influenced by earth motion. But there are! Some of these are seen in Pulsar light signal records, Lunar laser ranging, Cosmic microwave dipole anisotropy and the Global Positioning System. In these, the earth's motion can be detected from observing changes in the resultant speed of light and the "old" set of mathematical coordinate transformation is applicable.

Finally, I thank you for submitting an essay that was quite enjoyable to read. You discussed briefly about the continuum in your essay. You may wish to read my essay and answer the question: How can you cut a line, either the one that is "out there" or the one in the "brain"?

Best regards,

Akinbo

    Dear Akinbo,

    Thanks so much for reading our essay and for your kind comments. We respect the Platonic view even though it is essentially the opposite of ours. It is that we do not yet see how one could scientifically establish, without appealing to some yet unknown extra-sensory perceptions, the brain making contact with a Platonic `out there' mathematics. Maybe when the field of neurobiology has made further significant advances we will know the answer.

    Regarding your comments on the extended point: we readily agree that an extended point can represent a physical reality. But we make a clear distinction between the `thing itself' and `mathematical representation of the thing'. The former is out there and the latter is in our mind, according to our viewpoint.

    Unfortunately we could not understand your remarks about velocity of light. We certainly agree that the motion of the earth through space can be detected say via the cosmic microwave background dipole, but you will agree that such a detection does not imply that the speed of light is not a universal invariant independent of the choice of inertial frames.

    Thanks for pointing us to your essay - we look forward to reading it in the next few days.

    Best regards,

    Anshu, Tejinder

    While looking forward to your comments on my essay, I wish to clarify my remarks regarding the velocity of light. It is a very common and very crucial misinterpretation what the statement "the speed of light is a universal invariant/ constant" means.

    What is implied in special relativity is that the resultant velocity of light is invariant, c v = c or c - v = c, where c is the velocity of light in vacuum and v is the velocity of the observer towards () or away (-) from the observer respectively. Thus, unlike Galilean relativity where resultant velocity can be c v or c - v and so cause earlier or later arrival time of light due to the observer's velocity, v towards or away from light respectively, in Special relativity such observer motion cannot alter light arrival time, hence the SR statement that arrival time over a given distance is independent of the choice of inertial frame (observer frame moving towards or away from incoming light). RESULTANT is the key word. Just as for Sound that has a velocity 340m/s in air, its resultant velocity can be dependent on the choice of inertial frame, i.e the observer's frame. Not so for Special relativity and this was based on the experimental finding of Michelson and Morley, who found no change in light arrival time no matter the direction of earth motion. To buttress let us hear from Einstein himself from his book, The Meaning of Relativity, pg. 27/28.

    "But all experiments have shown that electromagnetic and optical phenomena, relatively to the earth as the body of reference, are not influenced by the translational velocity of the earth. The most important of these experiments are those of Michelson and Morley, which I shall assume are known". It is on this that the validity of the principle of special relativity rests.

    As you point out in your reply, there are cases where earth motion influences optical phenomena.

    That is why there has been a struggle among physicists over the past 100 years. You can also read Herbert Dingle's book, Science at the crossroads when you have the time. I was pointed to this free copy by Pentcho Valev, a contributor on this forum.

    Best regards,

    Akinbo

    Dear Anshu and Tejinder,

    I am intrigued to know where mathematics and our language of physics comes from. That this starts with the human brain is a very reasonable working hypothesis and some scientists like Roger Penrose and Stuart Hameroff have even tried to locate our consciousness in brain microtubules. I am curious to learn your opinion about such a controversial subject.

    My own mathematical impregnation led me to favour concepts that simultaneously and potentially contain maths and physical aspects like non linear models (and the resulting chaos, fractals, solitons...). For QM and its paradoxes, I was pushed to Grothendieck's "dessins d'enfants" (two-permutation groups possessing cosets, topology, a Riemann surface, algebra over the rationals and geometry). You mention some of these aspects in your paper and you may be interest to read what I have to say in connection to the Monstrous Moonshine.

    To conclude, I found your essay stimulating and very well written. I agree with the goal of relating cognition and mathematical physics. I like that you insist that a basic level is that of complex numbers.

    Michel

      Dear Michel,

      Thank you for reading our essay and for your kind remarks.

      We have a little bit [though not much] familiarity with the work of Penrose and Hameroff, proposing that quantum coherence and collapse of the wave function in the microtubule environment is the source of consciousness, somehow. With due respect to these great scientists it is our opinion that much more work needs to be done to make this proposal credible. The first issue has to do with the proposal by Penrose, and others, that gravity is responsible for the collapse of the wave-function during a quantum measurement. If you would like to have a look, a recent review can be found here. We find this a very attractive idea but it needs to be developed into a concrete mathematical model first and tested in the laboratory. Only then can we consider applying it to an environment as complex and sophisticated as the brain. Even then, one would have to make a sound mathematical model of the quantum to classical transition in a microtubule. We did not find something like that in the works of Penrose and Hameroff. From what we know, biologists are probably not even agreed on whether microtubules are at all involved in consciousness (open question). Thus while definitely worthy of further study, the idea has a long way to go before becoming believable - so we think.

      We had a first look at your very elegant essay on the maths-physics dialogue, and hope to get back to you soon.

      Best regards,

      Anshu, Tejinder

      4 days later

      Dear Anshu and Tejinder,

      Thank you for the easy to read and thought provoking essay. I like how you start with a definition of math "Mathematics is a precise language in which true statements can be proved starting from a set of axioms, using logic." and a definition of physics "Physics, on the other hand, is an experimental science of the world we observe, where experiments couple with great leaps of conceptual unification." and make a solid argument towards "... it is experiments and concepts first, and then the mathematical formulation."

      I have a small question where you point out for both math and physics, "shape, pattern recognition, counting, space, time, and change." but missing in math is ".. no place in mathematics for matter (material substance), and by extension, for light!". What about the Dirac electron where the mass of the electron is the coupling constant of a pair of spinors, or the more recent work on defining mass through mathematical properties of the Higgs Boson?

      That said, I very much enjoyed how you tied mathematics and physics to cognitive mechanisms in our own human brain. You left lots of things for the reader to ponder, a mark of a well done essay.

      Best of luck in the contest and I hope you get a chance to have a look at my essay here where I take a different approach to our minds understanding of the concepts behind the standard model.

      Thank you for the enjoyable read.

      Regards, Ed Unverricht

        Dear Dr. Anshu Gupta Mujumda

        & Dr.Tejinder Singh

        It's really a pleasure for me to read your nice essay.

        You probably emphasized on a "primordial" logical connection which links both physic and mathematics originated from the "brain perceptions" of "at least one planet full of intelligent beings".

        You also wrote: "The mathematics used in physics comes in only at a later stage, when we seek a precise language to describe the observed physical phenomena..."; and for the mathematics you rightly justified, " Remarkably enough. the primordial roots of mathematics are in the same human perceptions as in physics: shape, pattern recognition, counting, shape, and change. with no significant difference: there is no place in mathematics for matter (material substance), and by extension of light!"

        Whether we can realize such Physics and Mathematics, in broader terms, respectively as 'Hardware' and 'Software' of the nature (including the universe in itself) where those "intelligent beings" are inseparable part in that nature. Is it not true, such biological "intelligent beings" are basically formed by both of those natural hardwares and softwares?

        I also like to add you, is not such an "intelligent" being's centric cognition in this "planet" are fundamentally limited up to any kind of quantized form of message exchanging in-between observers-objects? And any thing, which if exists, conceptually, beyond such messaging limit for that intelligence could be ever rest beyond perceivable limits of that Quantized Cognitive Intelligence (QCI)? Therefore, to such a QCI, up to which it perceives nature through quantized limits of signalling, it's cognition would encourage to believe in some basic axioms of 'casualty' in nature to predict all futures casually within that limit of nature. Beyond that limit as if the nature might appear as a zone of all broken casualties to that QCI.

        Then the same nature looks like having two folds: one as 'Casual' and another as 'non-casual' or 'deterministic' and 'probabilistic'; and the Physics & Mathematics (also being the tools to study that nature) have two similar corresponding folds: 'deterministic' and 'probabilistic'. Therefore, why not, there would be fundamentally two prototype or primordial logics (instead of one) respect to those two folds of nature? And such two prototype logics would connect two such corresponding sets of Physics and Mathematics or Hardware and Software to study as well as fold and unfold of that nature?

        Once again thanks for the essay.

        I invite you also in my submitted essay "A tale of two logics"

        http://fqxi.org/community/forum/topic/2393

        Regrads

        Dipak Kumar Bhunia

          Dear Ed,

          Thanks so much for reading our essay and for your kind remarks.

          Regarding your question, you of course have a good point. However we are making a distinction between physical reality (in this case the mass), and its mathematical representation. It would be like saying that if we hold a ball in our hand and squeeze it, we can feel and appreciate its `materialness' through our senses. On the other hand when we make the statement `We are holding a ball of mass M in our hand' this sort of lingual / mathematical representation of the physical reality is lacking in `materialness' even though we can perceive in our mind what we mean. We do not question the possibility that there can be an elegant mathematical explanation for the origin of mass. We hope (not sure though) this addresses your enquiry.

          We enjoyed your essay and left a post on your page.

          Best regards,

          Anshu, Tejinder

          Dear Dipak,

          Thank you for reading our essay and for your kind remarks.

          You have raised some very nice points. We surely agree that the natural evolution of intelligent beings involves physical hardware (the brain) in which the software (mind, cognition) are operational. For the purpose of the essay, we are compelled to take the brain / mind as given; you will agree perhaps that not enough is known in neurobiology to answer how and why nature evolves hardware and software which then acts back to `understand' nature.

          We agree there maybe fundamental limitations to the efficiency of the observer - observed interaction, but how to explore that scientifically? We also did not quite follow what you meant by `quantised' in this context. Hope to understand this from your essay.

          You also suggest that cognition has a causal / deterministic aspect and a non-causal / probabilistic aspect. Is there a formal construction of this kind in cognitive science? We would have thought that probabilities are attributed to randomness / ignorance of initial conditions, rather than being a limitation of cognition. But you raise an interesting aspect which we need to think more about, and perhaps learn from your essay.

          Best regards,

          Anshu, Tejinder

          "The failure of the Michelson-Morley experiment to detect the motion of the earth through the hypothesised ether led Einstein and others to abandon the ether, and look for a set of mathematical coordinate transformations which allow the speed of light to be the same for all inertial observers."

          That was a dishonest step that eventually ruined physics. In 1887 (prior to FitzGerald and Lorentz advancing the ad hoc length contraction hypothesis), the Michelson-Morley experiment unequivocally confirmed the variable speed of light predicted by Newton's emission theory of light and refuted the constant (independent of the speed of the source) speed of light predicted by the immobile ether theory and later adopted by Einstein as his special relativity's second postulate:

          Alberto Martinez: "In sum, Einstein rejected the emission hypothesis prior to 1905 not because of any direct empirical evidence against it, but because it seemed to involve too many theoretical and mathematical complications. By contrast, Ritz was impressed by the lack of empirical evidence against the emission hypothesis, and he was not deterred by the mathematical difficulties it involved. It seemed to Ritz far more reasonable to assume, in the interest of the "economy" of scientific concepts, that the speed of light depends on the speed of its source, like any other projectile, rather than to assume or believe, with Einstein, that its speed is independent of the motion of its source even though it is not a wave in a medium; that nothing can go faster than light; that the length and mass of any body varies with its velocity; that there exist no rigid bodies; that duration and simultaneity are relative concepts; that the basic parallelogram law for the addition of velocities is not exactly valid; and so forth. Ritz commented that "it is a curious thing, worthy of remark, that only a few years ago one would have thought it sufficient to refute a theory to show that it entails even one or another of these consequences...."

          John Norton: "These efforts were long misled by an exaggeration of the importance of one experiment, the Michelson-Morley experiment, even though Einstein later had trouble recalling if he even knew of the experiment prior to his 1905 paper. This one experiment, in isolation, has little force. Its null result happened to be fully compatible with Newton's own emission theory of light. Located in the context of late 19th century electrodynamics when ether-based, wave theories of light predominated, however, it presented a serious problem that exercised the greatest theoretician of the day."

          John Norton: "In addition to his work as editor of the Einstein papers in finding source material, Stachel assembled the many small clues that reveal Einstein's serious consideration of an emission theory of light; and he gave us the crucial insight that Einstein regarded the Michelson-Morley experiment as evidence for the principle of relativity, whereas later writers almost universally use it as support for the light postulate of special relativity. Even today, this point needs emphasis. The Michelson-Morley experiment is fully compatible with an emission theory of light that CONTRADICTS THE LIGHT POSTULATE."

          Relativity and Its Roots, Banesh Hoffmann, p.92: "There are various remarks to be made about this second principle. For instance, if it is so obvious, how could it turn out to be part of a revolution - especially when the first principle is also a natural one? Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether. If it was so obvious, though, why did he need to state it as a principle? Because, having taken from the idea of light waves in the ether the one aspect that he needed, he declared early in his paper, to quote his own words, that "the introduction of a 'luminiferous ether' will prove to be superfluous."

          Pentcho Valev

            Dear Pentcho,

            Thank you for your comments. It was my understanding that many many different experiments, carried out independently and using different set-ups, terrestrial as well as astronomical, rule out the emission theory of light to a very high precision.

            With regards,

            Tejinder

            Dear Tejinder,

            Your statement "many many different experiments (...) rule out the emission theory of light to a very high precision" is unfalsifiable - how can I oppose it? We can only discuss the experiments one by one. I hope you agree now that the Michelson-Morley experiment did confirm the variable speed of light predicted by the emission theory, and refuted the constant (independent of the speed of the light source) speed of light predicted by the ether theory and adopted by Einstein as his second postulate. The Pound-Rebka experiment also confirmed the variable (in a gravitational field) speed of light predicted by Newton's emission theory of light:

            Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."

            Pentcho Valev

            Dear Pentcho,

            My understanding is that both emission theory and special relativity are consistent with the Michelson Morley experiment, but the former is refuted by subsequent experiments. From what I know, Einstein himself did consider an emission theory of his own, before discarding it in favour of special relativity. As regards the multitude of experiments that refute emission theory, I myself of course do not have the expertise to judge them one by one; I am satisfied that different experiments were reported and published. I would find it extremely hard to believe that a group of experimentalists have over decades indulged in a conspiracy of sorts to deliberately discredit emission theory. I do understand that there are physicists who continue to support the emission theory and probably that is your stance too. I can only say that I respectfully disagree with this stance.

            I wish to add that I am not an all out pro-establishment theorist! :-) I disagree with the establishment view on quantum theory, and I think the theory needs better understanding. However, until a decisive experiment comes along and shows quantum theory to be approximate, the debate is not going to be settled one way or the other. Same holds for the proposal of dark matter: we cannot be sure of its existence until a candidate is found, and I am sympathetic to alternate explanations such as modified gravity and MOND, even though the establishment is strongly pro-dark matter. Another good example I thnk is cosmological inflation: despite all claims to success it is still a hypothesis. I also feel the pro-establishment community is open to considering concrete alternatives, even if these meet pockets of resistance. Thus I am very reluctant to believe that the emission theory is being deliberately suppressed by vested experimentalists and theorists. On the other hand, we do find a healthy response to concrete proposals to modify special relativity.

            Those are my two cents :-)

            Best regards,

            Tejinder

            4 days later

            Dear Anshu and Tejinder,

            I just red your reply following my very positive appreciation of your essay.

            There is another essay about science and cognition by Vincent Douzal that you should not miss.

            Me too I am not yet convinced by Hameroff and Penrose, I met them sometimes ago at a Tucson conference.

            I am now rating your essay highly. I hope it will not be balanced by a stupid 1 as usual. Myself I already got 1 three times.

            Best,

            Michel

              6 days later

              Tejinder and Anshu,

              Congratulations on a weighty discussion. As a modeller in the urbane, offensive and defensive weapons cost and support, and as a teacher of English, I see human mathematics as building on metaphors, not being an entity itself. My background lends that prejudice.

              You do not burden yourself with this question, but do state your feeling: "tempting but erroneous to conclude that the beautiful math description is resident in the physical world."

              I also believe that Cognition draws on the physical world to invent the stable human language of math. Such modelling has led us to discoveries quantum biology, DNA, and LHC through what I see as the connections of math, mind, and physics.

              Thanks for the opportunity to share your views.

              Jim

                Dear Michel,

                Greetings, and thanks for pointing us to Vincent Douzal's paper, which we read and liked. Indeed there are a few papers in the contest emphasising the importance of cognition in the present context (though perhaps too few!). The research works of Lukaff, Nunez, Dehaene, Hestenes, amongst others, are noteworthy.

                Kind regards,

                Anshu, Tejinder