CATEGORY:
Trick or Truth Essay Contest (2015)
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TOPIC:
Cognitive Science and the Connection between Physics and Mathematics by Anshu Gupta Mujumdar and Tejinder Singh
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Author Tejinder Pal Singh wrote on Feb. 9, 2015 @ 21:57 GMT
Essay AbstractThe human mind is endowed with innate primordial perceptions such as space, distance, motion, change, flow of time, matter. The field of cognitive science argues that the abstract concepts of mathematics are not Platonic, but are built in the brain from these primordial perceptions, using what are known as conceptual metaphors. Known cognitive mechanisms give rise to the extremely precise and logical language of mathematics. Thus all of the vastness of mathematics, with its beautiful theorems, is human mathematics. It resides in the mind, and is not `out there’. Physics is an experimental science in which results of experiments are described in terms of concrete concepts – these concepts are also built from our primordial perceptions. The goal of theoretical physics is to describe the experimentally observed regularity of the physical world in an unambiguous, precise and logical manner. To do so, the brain resorts to the well-defined abstract concepts which the mind has metaphored from our primordial perceptions. Since both the concrete and the abstract are derived from the primordial, the connection between physics and mathematics is not mysterious, but natural. This connection is established in the human brain, where a small subset of the vast human mathematics is cognitively fitted to describe the regularity of the universe. Theoretical physics should be thought of as a branch of mathematics, whose axioms are motivated by observations of the physical world. We use the example of quantum theory to demonstrate the all too human nature of the physics-mathematics connection: it is at times frail, and imperfect. Our resistance to take this imperfection sufficiently seriously [since no known experiment violates quantum theory] shows the fundamental importance of experiments in physics. This is unlike in mathematics, the goal there being to search for logical and elegant relations amongst abstract concepts which the mind creates.
Author BioAnshu Gupta Mujumdar is a freelance researcher and visiting faculty in Mathematics/Physics for IB diploma program at the Fazlani L'Academie Globale, Mumbai. She holds a doctorate from the Physical Research Laboratory, Ahmedabad (1997) in the field of general relativity. She has held several postdoctoral positions and was a recipient of a post-doctoral fellowship award in Mathematical Physics (in memory of S. Chandrasekhar, 1998) and Peter Gruber post-doctoral fellowship in 2001. Her research interests are in inflationary cosmology, quantum effects in biological systems, and history of Mathematics. Tejinder Singh is Professor of Physics at the Tata Institute of Fundamental Research, Mumbai.
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Edwin Eugene Klingman wrote on Feb. 11, 2015 @ 05:49 GMT
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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Author Tejinder Pal Singh replied on Feb. 11, 2015 @ 10:24 GMT
Dear Edwin,
Thank you for your careful reading of our essay and your comments on it. We will definitely read your essay in the next few days and respond to it.
Best regards,
Tejinder, Anshu
basudeba mishra wrote on Feb. 12, 2015 @ 18:01 GMT
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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Author Tejinder Pal Singh replied on Feb. 14, 2015 @ 09:23 GMT
Dear Basudeba,
Thank you very much for reading our essay and for your comments above. There seems to be much that we disagree about! We cannot agree that physics describes *qualitative* aspects of nature, or that there is a chicken-egg problem here. Also, why do you say extra dimensions are a gospel truth? Surely they are till not accepted as confirmed by experiments. Also, we would not say that mathematics is all about certainties. Stochastic dynamics is very real, in various settings, at least in an emergent sense. Nor can we agree with what you conclude about the Michelson-Morley experiment.
Nonetheless, thanks for pointing us to your essay, which we look forwarding to reading in due course.
Best regards,
Tejinder, Anshu
Sujatha Jagannathan wrote on Feb. 16, 2015 @ 06:25 GMT
Your work circulates in primordial features of mathematics which signifies the emergence of indistinguishable notions simmering hypothetical notions.
Good Luck!
Best Regards,
Miss. Sujatha Jagannathan
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Author Tejinder Pal Singh replied on Feb. 16, 2015 @ 13:43 GMT
Thank you for your remark Sujatha. We wonder what you meant by your use of the word `simmering' here?
Regards,
Tejinder, Anshu
Sujatha Jagannathan replied on Feb. 17, 2015 @ 08:55 GMT
Simmering in the sense 'Strong feeling'
Sincerely,
Miss. Sujatha Jagannathan
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Lawrence B Crowell wrote on Feb. 16, 2015 @ 17:03 GMT
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
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Author Tejinder Pal Singh wrote on Feb. 17, 2015 @ 07:55 GMT
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
Christophe Tournayre wrote on Feb. 17, 2015 @ 20:02 GMT
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
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Author Tejinder Pal Singh replied on Feb. 18, 2015 @ 12:49 GMT
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
Christophe Tournayre replied on Feb. 19, 2015 @ 20:10 GMT
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
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Author Tejinder Pal Singh replied on Feb. 21, 2015 @ 03:38 GMT
Thanks Christophe,
After reading your essay we understand your comments above better, and have posted a brief comment on your essay.
Regards,
Anshu, Tejinder
Alan M. Kadin wrote on Feb. 21, 2015 @ 17:09 GMT
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
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Author Tejinder Pal Singh replied on Feb. 22, 2015 @ 05:21 GMT
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
Tommaso Bolognesi wrote on Feb. 24, 2015 @ 18:30 GMT
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
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Author Tejinder Pal Singh replied on Feb. 25, 2015 @ 09:17 GMT
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...
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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
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Alex Newman wrote on Mar. 1, 2015 @ 08:34 GMT
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.
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Author Tejinder Pal Singh wrote on Mar. 1, 2015 @ 14:42 GMT
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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George Rajna wrote on Mar. 2, 2015 @ 08:51 GMT
Congratulation for such a brilliant essay. You deserve the best.
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Author Tejinder Pal Singh replied on Mar. 2, 2015 @ 10:00 GMT
Akinbo Ojo wrote on Mar. 8, 2015 @ 16:04 GMT
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 3x10
8m/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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Author Tejinder Pal Singh replied on Mar. 9, 2015 @ 06:09 GMT
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
Akinbo Ojo replied on Mar. 9, 2015 @ 10:16 GMT
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...
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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
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Michel Planat wrote on Mar. 10, 2015 @ 10:47 GMT
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
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Author Tejinder Pal Singh replied on Mar. 10, 2015 @ 18:19 GMT
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
Ed Unverricht wrote on Mar. 15, 2015 @ 05:49 GMT
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
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Author Tejinder Pal Singh replied on Mar. 15, 2015 @ 12:13 GMT
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
Dipak Kumar Bhunia wrote on Mar. 15, 2015 @ 07:49 GMT
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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Author Tejinder Pal Singh replied on Mar. 15, 2015 @ 12:33 GMT
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
Pentcho Valev wrote on Mar. 15, 2015 @ 20:45 GMT
"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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Author Tejinder Pal Singh replied on Mar. 16, 2015 @ 05:06 GMT
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
Pentcho Valev replied on Mar. 16, 2015 @ 09:31 GMT
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
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Author Tejinder Pal Singh replied on Mar. 17, 2015 @ 07:36 GMT
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...
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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
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Michel Planat wrote on Mar. 21, 2015 @ 19:28 GMT
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
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Author Tejinder Pal Singh replied on Mar. 29, 2015 @ 05:11 GMT
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
James Lee Hoover wrote on Mar. 28, 2015 @ 00:00 GMT
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
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Author Tejinder Pal Singh replied on Mar. 29, 2015 @ 05:15 GMT
Dear Jim.
Thank you for reading our essay and for your kind comments. Happy to know we are in agreement.
Best regards,
Anshu, Tejinder
James Lee Hoover replied on Apr. 10, 2015 @ 03:29 GMT
Anshu and Tejinder,
Time grows short and I am revisiting essays I have read to determine if I've rated them. Yours I did on 3/28.
I would like to see your thoughts on mine.
Jim
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James Lee Hoover replied on Apr. 11, 2015 @ 05:20 GMT
Anshu and Tejinder,
"And Jim we do not seem to find in your essay an explanation for the central question as to why mathematics is so successfully employed in physics. Wonder what your thoughts on this are."
First there was the equation on page 3 that represent P, M, and B (physics, math and the human brain), showing their integral interconnection. I provide examples on page 3 & 5 in "Math's Applications" and "Math's Quantum Modeling" section on how math's use in modeling and algorithms, including lines of programming code containing the algorithms that mathematically tie physics concepts together with the LHC, DNA and quantum biology studies and successes.
My conclusion on page 7 shows how these connections of the brain, math and physics are vital in the stellar progress we've had in all physics but especially in those areas I cite.
Thank you both for reading my essay.
Jim
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Author Tejinder Pal Singh replied on Apr. 11, 2015 @ 14:07 GMT
Thanks for the clarification Jim.
Anshu, Tejinder
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Peter Martin Punin wrote on Mar. 31, 2015 @ 10:24 GMT
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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Author Tejinder Pal Singh replied on Apr. 2, 2015 @ 01:33 GMT
Dear Peter,
Many thanks for your kind remarks, and your detailed and incisive comments. We will respond after understanding them carefully and reading your essay, by later next week.
Kind regards,
Anshu, Tejinder
Author Tejinder Pal Singh replied on Apr. 11, 2015 @ 14:03 GMT
Dear Peter,
After leaving a post on your essay, we returned again to your comments above and re-read them...you undoubtedly express your stance very clearly - namely that using cognition to understand cognition is also metaphysics. We respectfully tend to disagree, and made some remarks to this effect on your page and perhaps one could avoid repeating them here. Principally we are saying that in considering a cognitive basis for the physics-maths connection there is at least hope for making a scientific model. As opposed to when one half, the mathematical half, is an immaterial reality - at least for now, until and unless, as you say, brains develop channels to communicate directly with such mathematical reality.
One further remark ... if we ask why does the world NOT exist rather than existing, we feel one day physics will give us an answer to that. Thus instead of relegating this question to a metaphysical realm, we want to think of it as a currently unsolved problem in physics.
Your criticism of the cognitive approach as being metaphysical is incisive and very clear, and trying to defend it has helped us understand our position better. We appreciate the dialogue you have initiated and will be happy to carry it further. Thank you.
Best regards,
Anshu, Tejinder
Richard Lewis wrote on Mar. 31, 2015 @ 13:10 GMT
I did enjoy reading this essay which covers a wide ranging scope of topics in Mathematics, Physics and Cognitive Science.
I also enjoyed reading your technical end notes on Quantum Theory and completely agree with your assessment that quantum theory might be incomplete. I think you will enjoy reading my essay: 'solving the mystery' which addresses the four oddities that you mention in your essay. The problem is treated from a different conceptual viewpoint (the spacetime wave theory).
Best wishes
Richard Lewis
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Author Tejinder Pal Singh replied on Apr. 2, 2015 @ 01:35 GMT
Many thanks Richard, for your kind comments. We are looking forward to reading your essay and hope to respond in a week or so.
Kind regards,
Anshu, Tejinder
Joe Fisher wrote on Apr. 2, 2015 @ 14:42 GMT
Dear Anshu & Tejinder,
I thought that your engrossing essay was exceptionally well written and I do hope that it fares well in the competition.
I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.
All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.
Joe Fisher
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En Passant wrote on Apr. 12, 2015 @ 07:11 GMT
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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Author Tejinder Pal Singh replied on Apr. 12, 2015 @ 09:54 GMT
Dear En,
Thank you for your candid remarks, and for pointing out the typos. We regret these errors and will correct them in a subsequent version. [We edited the essay together; and you are absolutely right - we were educated in India, with the medium of instruction being English (which is essentially British English), and then of course followed by lot of US texts in higher education! How you figure out something like that is beyond us :-)].
We are glad that we agree on the primitive origin of the force concept, and indeed we appreciate your remark that this captures the essence of our stance.
Kind regards,
Anshu, Tejinder
En Passant replied on Apr. 12, 2015 @ 15:00 GMT
Dear Anshu and Tejinder,
I have read your reply, and can now see that I did a terrible job of promoting your essay.
It was not my intention to make any subsequent readers think (and hopefully they will not) that your essay’s message could be abridged to something like what my (the relevant) paragraph says.
On the contrary. Your essay offers interesting and valuable insights, and yes, there is a ‘need to go into further details.’ Much is to be gained from reading every line of your essay, and consider the thoughts “contained” therein.
You were too polite in saying “…indeed we appreciate your remark that this captures the essence of our stance.” A less “polished” Westerner might have told me to go take a hike.
There is one question that you may still want to consider (and answer it to yourselves, or in a comment). It concerns this quote taken from page 1 of your essay: “Physics, on the other hand, is an experimental science (hence dependent on technology) of the world we observe, where experiments couple with great leaps of conceptual unification. The mathematics used in physics comes in only at a later stage, when we seek a precise language to describe the observed physical phenomena.”
The question that I had in mind is this. When you talk about physics in the quoted segment, are you thinking about physics as it is actually practiced, or as an idealized discipline (“that’s what physics ought to be”)? I am only asking whether you would like to make this distinction explicit.
I enjoyed your essay. It keeps the reader keen to learn more from each new observation you make.
En
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Author Tejinder Pal Singh replied on Apr. 13, 2015 @ 05:15 GMT
Dear En,
Greetings. No, we were not being polite! :-) We certainly thought you made a very good point by highlighting (using the example of force) that physical and mathematical concepts are built using metaphors based on primordial perceptions, and are not out there. But yes indeed we do expect and hope that an interested reader will read other parts of the essay too.
Regarding your latter question, we only had / have in mind physics as it is actually practiced, and not an idealized discipline. We honestly do not have much thought on what the idealised discipline should / would be like. Same for mathematics. It is more like: what is, is.
Best regards,
Anshu, tejinder
Peter Martin Punin wrote on Apr. 13, 2015 @ 17:46 GMT
Dear Anshu,
Dear Tejinder,
I just have answered your post on my own page.
Best regards
Peter
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Alma Ionescu wrote on Apr. 16, 2015 @ 13:25 GMT
Dear Tejinder, Anshu,
This was definitely one of my favorite essays in the contest. Although I'd say - and I'm sure you agree - that the patterns underlying the natural world are observer independent, I agree that the written part of math which makes up the totality of human research in this domain, can only become manifest through development brought by intelligent beings. I enjoyed a lot the part in which you bring evidence about the pattern recognition hard-wiring in the brain from cognitive science as I was unaware by some of the research you mentioned, research which is extremely interesting.
However for me the icing on the cake were the technical notes. With those alone and you would have had, in my opinion, more than enough material to participate in this contest. In there I found a very mature and original treatment of long lingering problems. I will have to read at least a couple of references, namely 21 and 22 as they sound extremely interesting. One naive question if I may, can I ask which theorem is referenced here: "
However, a no-go theorem forbids that, so long as X is an ordinary (commutative) manifold"?
Thank you again for a most interesting read and wish you good luck in the contest! Should you have the time to read my essay, your comments are much appreciated.
Warm regards,
Alma
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Author Tejinder Pal Singh wrote on Apr. 16, 2015 @ 16:29 GMT
Dear Alma,
Thank you for reading our essay, and for your very kind remarks. Yes, we very much agree with you on the observer independence of the physical world.
The no-go theorem is due to John Mather - the original reference is his paper
"Simplicity of certain groups of diffeomorphisms" Bulletin of the American Mathematical Society 80, 271 (1974).
It is briefly discussed in context by Connes on p. 39-40 of his elegant review (our Ref. 29):
http://arxiv.org/pdf/math/0011193v1.pdf
The theorem's content being that the diffeorphism group of a connected ordinary manifold is simple, and hence cannot have a nontrivial normal subgroup, thereby disallowing the desired semi-direct product structure one is seeking.
We look forward to reading your essay within the next few days, and if possible, leave our comments on your page.
Thank you again, and kind regards,
Anshu, Tejinder
Alma Ionescu replied on Apr. 16, 2015 @ 20:10 GMT
Oh, I see, thank you very much for explaining it, now it makes perfect sense! And thank you for the reference!
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Alma Ionescu replied on Apr. 17, 2015 @ 12:44 GMT
Dear Tejinder, Anshu,
Thank you for your insightful comment! I enjoyed very much
answering to your question!
Warm regards,
Alma
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Alma Ionescu replied on Apr. 18, 2015 @ 15:24 GMT
Dear Anshu,
I'm so sorry for the confusion I made and I'm very glad you realize it was a slip. I'm especially sorry for it since I appreciate your work. Thank you very much for being so nice and understanding :)
Alma
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Author Tejinder Pal Singh replied on Apr. 19, 2015 @ 05:48 GMT
Dear Alma,
It is fine, no need to feel sorry. We brought to your notice in order to avoid further confusion. As is reflected from our earlier correspondence, I too have enjoyed your insightful essay.
Regards,
Anshu
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Cristinel Stoica wrote on Apr. 20, 2015 @ 07:18 GMT
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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Author Tejinder Pal Singh replied on Apr. 20, 2015 @ 17:55 GMT
Dear Cristi,
Greetings! It is a pleasure to meet you here again.
Thank you so much for reading our essay, and for your kind comments and detailed evaluation. We agree with your analysis above, and cannot really think of adding anything more to it at present.
With kind regards,
Anshu, Tejinder
ABDELWAHED BANNOURI wrote on Apr. 21, 2015 @ 17:33 GMT
Dear Sing Tejinder :
Certainly, your essay shows a great comprehension of the world of mathematics and physics. I agree with you on many points.
The human brain is divided into two hemispheres, the left is masculine, active, extroverted. called "rational". while, the right is female, passive, introverted, called "irrational,"
the same thing can be said about courage and fear, which are two primitive moods .
The most important thing is that these two opposite positions, are always present.
in mathematical term (X + 1) and (X - 1) are two limits, represent the needle of the scale.
You wrote: "Einstein and Bohr on a firm mathematical footing, in their extremely elegant and universal equations".
The standard Bohr's atomic model is not complete, because it does not explain the origin of the polarìzation, pace, time and force..... The General relativity, in addition to this, explains the GRAVITY incorrectly, "the mere presence of a massive body can not bend the space".
The answer to this question leads us inside of the "theory of everything".
The Bi-iterative model has already the answer, the theory of everything exist and real.
Sincerly yours
Bannouri
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vincent douzal wrote on Apr. 22, 2015 @ 09:24 GMT
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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