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...

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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

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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,...

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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

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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...

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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

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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...

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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

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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...

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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

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"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...

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"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

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Dear Anshu,

Dear Tejinder,

You are anti-Platonists (or rather non-Platonists), and I am a Platonist aware of all difficulties this option represents. But similar difficulties do undermine ALL options that could be discussed within the framework of this contest. Personally I see (i) this contest as a philosophical one and (ii) philosophy as the choice to focus on issues that do not...

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Dear Anshu,

Dear Tejinder,

You are anti-Platonists (or rather non-Platonists), and I am a Platonist aware of all difficulties this option represents. But similar difficulties do undermine ALL options that could be discussed within the framework of this contest. Personally I see (i) this contest as a philosophical one and (ii) philosophy as the choice to focus on issues that do not allow one single answer. Of course sometime an initially philosophical question finally gets a definitive answer, but in this case it ceases to be a philosophical question. Anyway, regarding foundational research about mathematics and/or links between mathematics and physics, we must “empirically” recognize that for n people debating about, there are at least n + 1 opinions. Under these conditions, the BEST we could do is to try a mutually benefit full constructive dialogue, and this primarily with colleagues defending “opposite standpoints.” You agree probably that such a constructive dialogue should be for everyone the main motivation to participate in this contest.

The calm, circumspect, and courteous background tone of your well written, highly interesting essay expressing a wide deepened culture about history of mathematics and physics make me think that you are seeking in turn dialogue.

Before going further, I would state – even repeating myself – that it has been an enormous pleasure for me to read your really well written elegant essay with its beautiful symmetry between sections I and II. You are absolutely right to emphasize that mathematics and (the interminable stammering of) physics initially knew essentially separate paths, and that the manifest collusion between physics and (a relatively modest part of) mathematics represents at least at first glance a mystery to elucidate; otherwise the subject of this contest would not have any reason to be proposed.

Well, and now let us discuss some essential points of your essay, with the sole aim to exchange ideas.

You are wondering how are we ever going to be able to scientifically prove Platonism. Here, my position as a Platonist – I expose it at length in my own essay – is clear. Platonism IS metaphysics and still metaphysics and nothing but metaphysics. Yes, but all competing theories of Platonism are as well metaphysical as Platonism they are trying to “refute”. If such a refutation was possible, the competing theory in question would transform Platonism in a no longer metaphysical but scientifically refutable approach being effectively refuted, just like it can happen to other entirely scientific approaches. At first sight it seems certainly strange to postulate that “there is a mathematical universe, which we somehow grasp in an extra-sensory manner”. But in my own essay, I quote a reference text of R. Carnap where this categorical anti-metaphysical main representative of logical positivism relegates material realism to metaphysics, making no difference with other ontological theories like idealism. Of course, we receive information on material reality by our sensory channels before cognition processes this information. But it seems to me that the problem is just THERE: To evaluate (i) objectively and (ii) non-metaphysically the degree of adequation between the image we have of reality following its cognitive processing and reality “as such” (??), we should transmute us in pure immaterial spirits able to go out/beyond of our cognition. The fact that all our information about material reality run through our sensory channels, whereas there is nothing equivalent concerning our access to an immaterial mathematical world being objectively given, this only fact does not refute Platonism. First, nothing allows us to predict that humans will “never” discover brain channels able to receiving directly immaterial information. At present we do not know anything about that issue - at least to my knowledge – but anyway, scientists who by definition do not have the vocation to utter metaphysical and /or arbitrary propositions should not play the prophets of what “will or will not be discovered.” I would add that the (too) famous “Benacerraf argument against Platonism” – “we have good knowledge about cognition dedicated to the empirical reality, whereas there is no equivalent for an immaterial world” - has not at all “definitely refuted Platonism.” Some authors like Penelope Maddy even do not think that Benacerraf would advance an anti-Platonist evidence. Anyway, discussions about “Benacerraf's argument” are continuing for 43 years, and nobody sees the end.

On the other hand, the material reality which exists – let us admit that it exists – could also not exist. A scientific is generally not interested in the issue “why exist the material reality, instead of not existing.” Well, but this issue equivalent to this other: “why should an immaterial objective mathematical world NOT exist instead of existing?” Everywhere we encounter metaphysics, much more metaphysics then we would admit it spontaneously. In other terms: If we deny the existence of material world, we must assume an infinite number of epistemological problems. But we must also ask – without any a priori - if it is not the same for Platonism, and more specifically, if the denial of Platonism does not cause in turn a jolly good number of epistemological problems regarding for example the links between mathematics and physics. In my own essay, I try to raise such problems, leaving it to the reader to appreciate these problems in her/his personal manner.

In your essay you say that “in their work physicists and mathematicians generally prefer to ignore or 'forget' the brain (...).” Personally, I don't think (i) that it is a question of ignoring or forgetting and (ii) that this point is not a detail. Let me use a metaphor. Imagine you are resolving a mathematical or physical problem, using a computer to facilitate your task. Doing this, you don't “ignore or forget” your computer and its internal functioning. But in this context, the computer is accessory. Perhaps and even probably it is an unavoidable mediator between your work and the given object of your work, may it be mathematics or physical reality. This certainly does not hinder you to be interested about hardware oriented computer science computer science. However, making mathematics or physics as such, you tacitly are bracketing all issues related to computer science or hardware. Now it is clear that your computer and more generally each information processing system must be equipped so that it can carry out its task. Hence the system is supposed to share the logic (and other factors) of (i) the task to be effectuated and (ii) the object that this task concerns. But this does not necessarily mean that information treating system “GENERATE” or “CREATE” the object of its task. Moreover, in the case of a computer or other artificial information processing systems, we can KNOW that the system does not “generate” or “create” the object of its task, since we are outside the circuit linking the system to the object of its task. But it is not the same for our brains, or, more generally, for our cognition.

Of course, to be able to make mathematics or physics, our brains must be equipped for. Concerning mathematics, it is often said that elementary mathematical structures, principally counting, are innate and do pre-exist in infants to all effectively effectuated operations. Given the complexity of the issue, I am not sure that this point can be categorically asserted, but it is not so important. Not only it is obvious that our brains are disposed for mathematics – otherwise there would be no mathematics, at least no mathematics effectively written on paper – but we can still do some interesting overlaps between the theories of "constructivist" philosophy (Brouwer, Heything ea) reducing any form of mathematics effectively constructable to “intuition of counting”, and, on the other hand, modern computational mathematics. Regarding physical reality (in a very vast sense), it is in turn clear that our cognition is able not only to register it, but also to investigate it. Otherwise there would be neither physics, nor other sciences.

But now complications arise.

To respond – in cognitive terms – to the following issue “How to explain that (a part of) physical phenomena correspond to (a part of) mathematics being (at least potentially) pre-programmed in / by the human brain?”, we need very heavy hypotheses. In a cognitive perspective, if we assume that mathematics is at least potentially rooted in the brain, we must also assume that our brains organize the physical reality in the way that it aligns with given part of mathematics. This lead to a kind of Kantian or neo-Kantian philosophy which inevitably will be the subject of endless controversies. But regardless the option we would adopt in this area, the fundamental problem remains the same: unlike the information processing system mentioned above, where humans being outside the circuit linking the system to the reality can compare the treated information to its original, human cognition CAN NOT “get out from itself” in order to check the degree of adequacy between the reality as such and the reality as it is conditioned by cognition. In other terms, there is CIRCULARITY, and each tentative to break an objectively given circularity leads to metaphysics.

Nevertheless it is not a deadlock, but the solution can not be classical.

The position I defend in my own essay is the following: (i) All foundational approaches concerning mathematics and/or links between mathematics and physics are ultimately metaphysical. (ii) Platonism belongs to metaphysics as well as anti-Platonism, whatever it would be. (iii) Even cognitive approaches ultimately DO NOT escape to metaphysics: all investigations of human cognition are conditioned by human cognition. To appreciate this latter “objectively”, human cognitive scientists should be able to get out from their cognition, to go “beyond” of all links between cognition and its objects, and this remains a genuine metaphysical idea. (iv) All we can do is to compare several competing metaphysical theories under EPISTEMOLOGICAL criteria such as simplicity in the logical sense of this term, complexity of primary and secondary hypotheses, internal consistency, consistency on the level of consequences and so on. On these bases, I try to show that a necessarily metaphysical Platonism is more plausible than its (also necessarily metaphysical) competing theories. Further details are in my own essay. Here, I try to summarize it as follows: Platonism is based on assumptions neither provable nor refutable. But it's the same for anti-Platonism. On the other hand, anti-Platonism must additionally assume some forms of circularity that Platonism can - at least in purely logical terms - avoid.

I would be glad to know your counter-propositions about my vision I tried to summarize here above and also to know the criticism that you would not fail to address to my own contribution. It would be precious to of further reformulations on the foundations of a broader vision. Ultimately, this the raison d'être of the present contest.

With best regards

Peter

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Dear Anshu and Tejinder,

I will start off with what will appear to be irrelevant observations (at least irrelevant to the shared subject matter).

Your writing style reflects a commitment to writing “flawlessly.” I don’t correct grammar or spelling of people who are committed to their thought, and let their writing be as it may.

In future issues of your essay (or parts...

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Dear Anshu and Tejinder,

I will start off with what will appear to be irrelevant observations (at least irrelevant to the shared subject matter).

Your writing style reflects a commitment to writing “flawlessly.” I don’t correct grammar or spelling of people who are committed to their thought, and let their writing be as it may.

In future issues of your essay (or parts thereof), you may want to correct a typo now found on page 3, 4th line from the bottom: you will want to replace “word” with “world.”

On page 8, about ¼ of the page down, you will find “Riemannean,” which is usually spelled as “Riemannian.” I think your choice of spelling was influenced by the spelling of a word that preceded it (Euclidean).

To most people such details will seem inconsequential, but they help me understand the thinking of the writers. Your essay appears to be edited by someone (and it could be one of you two, or both) educated in the U.K., and subsequently influenced by reading a lot of US texts.

Your essay says very reasonable things. My favorite observation is this one: “Force, for instance, could be metaphorically related to the primordial human perception of the muscular exertion in throwing a stone at a prey or a threat.” You are right. That is precisely where our concept of force came from. As people tried to push or lift bigger and bigger rocks, they realized that the required effort increased with the size of the rock, along with a corresponding increase in “pain” felt in the muscles. They called it “force.” Obviously then, you must believe that there isn’t any such thing as force out there, but F is a convenient “shorthand” (abstraction) for various things (“m x a” being one of them).

After this, there is no need to go into further details. The above paragraph “captures” the essence of your message.

And please do not go to my essay page (and don’t feel obligated to read or rate it). It will feel too “mercenary” if you do that.

En

P.S. Your essay deserves a high rating.

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Dear Anshu and Tejinder,

Thank you for the beautiful and insightful essay. While most essays discussed the unreasonable effectiveness of mathematics in physics, your essay comes with the fresh view that the effectiveness in both math and physics is due to the human mind. I fully agree with what you said, "Theoretical physics should be thought of as a branch of mathematics, whose axioms are...

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Dear Anshu and Tejinder,

Thank you for the beautiful and insightful essay. While most essays discussed the unreasonable effectiveness of mathematics in physics, your essay comes with the fresh view that the effectiveness in both math and physics is due to the human mind. I fully agree with what you said, "Theoretical physics should be thought of as a branch of mathematics, whose axioms are motivated by observations of the physical world." I think that there is still a lot of work to be done on some already existing branches of physics, to make them satisfy the rigour of mathematics. But every time we managed to mathematize a piece of physics, the reward was great, since apparently independent concepts become more logically connected, and new predictions are made (as you exemplified with Dirac's and Einstein's predictions). It is true that you approached the questions regarding what human constructed math and physics are, and how are they related and why. Of course, since we are talking about constructions of our minds, we can't avoid the major role of the human mind here. This leads to the question: are these subjective constructions about an objective reality? The questions about the reality of the universe, its objective existence, its independence of our mind, questions about the reality of mathematical structures, or of their manifestation as a physical universe, these fall in another category than that you addressed. But the reality and our descriptions have to be related though, so we can ask in what measure the constructs of the human mind are reconstructions/approximations/rediscoveries of the real physical world (and of course not extra-sensorial perceptions of the Platonic world, as you well said and probably no serious person believes). It is true that believing in the reality of mathematical structures, either Platonic, or even as subjacent to the physical world, is an act of faith, which is motivated by the success in making predictions. If all that there is is just the human mind inventing connections of the dots, then how can this explain the predictive power? I also enjoyed very much the technical endnotes, and the criticism to the standard procedure of quantization, and the other "oddities" of quantum mechanics. I fully agree that there is much to be understood about quantum mechanics, and I think the approach on you are working is very promising in this direction.

Best wishes,

Cristi Stoica

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Dears Anshu and Tejinder,

I fully agree that a connection between physics and mathematics, if to be explained, must be rooted in cognitive science.



Your essay makes very important remarks, not often seen, notably that the mathematics involved in physics are relatively simple (and many current theoretical physics explorations seem to be just picking randomly in the...

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Dears Anshu and Tejinder,

I fully agree that a connection between physics and mathematics, if to be explained, must be rooted in cognitive science.



Your essay makes very important remarks, not often seen, notably that the mathematics involved in physics are relatively simple (and many current theoretical physics explorations seem to be just picking randomly in the toolbox of established mathematics!).

I would not completely adhere to your claim that ``primordial perceptions such as object, size, shape, pattern and change'' are ``hard-wired'' in humans. I see what you mean, and I agree. But I would not build a whole theory with the present aims, upon these precise primordial terms, as if I could fix them and forget about them. The main reason is that it is extremely problematic to fix a bottom layer once for all, in the faith that it will work universally. We have to live with the tension between the need for fixed basic elements —formal— to be able to reason with certainty, and the permanently renewed experimental fact that, whenever we dig further into reality, whatever the modality, we find always new structures, without ever finding a bottom layer. Thus different situation require different formalisations. What seems a bottom layer is better viewed as a horizon, an intrinsic limitation of our particular mode of investigation. I am being elusive here, because it would take too long to make the fully the case, so I would warmly recommend Gilles Cohen-Tannoudji's Universal constants in physics (Mcgraw Hill, 1992), for his illuminating interpretations of universal constants as such horizons to physics (to knowledge), and not some absolute, universal constants of nature.

I have approached the case of perception more abstractly. I would have been curious to read your comments on how I have addressed this precise point. I took a starting point very close to your remark that ``physicists ignore or `forget' the brain, treating it as a perfect passive agent''. In addition to having unscientific aspects, this stance —which has been very fruitful, though— completely neglects that knowledge (included physical) is relative to cognitive subjects. This relativity can be expressed very precisely, in the terms of the cognitive subject being a frame of reference. Since physics has often advanced by discovering new relativities, it should not durably eschew this one. A wider scientific framework must include the cognitive subject as a constitutive part, the key actor of the building of knowledge. And perception should occupy a central position in the framework. Most philosophical traditions have made perception a pivotal phenomenon in the edification of knowledge; by banishing the cognitive subject, physics has forgotten much of the ancient wisdom. Again, this banishing, has had fruitful consequences, but also, inherent limitations. The so-called von Neumann-Wigner interpretation of quantum mechanics is, in my view, an all too clear case of such limitation: when you reach the limits of what your theoretical framework can do, you suddenly call the banished and hold-in-contempt subject to the rescue, to help you collapse the wave function: what you have no way to do from inside the theory. Thus, suddenly, you appeal to perception.

Regards,

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