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.

        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