Cognitive Science and the Connection between Physics and Mathematics by Anshu Gupta Mujumdar and Tejinder Singh

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.

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

Dear Anshu,

Dear Tejinder,

I just have answered your post on my own page.

Best regards

Peter

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

Oh, I see, thank you very much for explaining it, now it makes perfect sense! And thank you for the reference!

Dear Tejinder, Anshu,

Thank you for your insightful comment! I enjoyed very much answering to your question!

Warm regards,

Alma

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

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

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

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,