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

Dear Joe,

Thank you for reading our essay. We are sorry you did not find it useful.

We read your third paragraph above a few times but we do not understand it still. The latter part is at variance with what we learn in standard physics. What do you find wrong in the standard physics description, and do you have a mathematical formulation to support what you are proposing?

Thanks and regards,

Tejinder, Anshu

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

Thank you for your remark Sujatha. We wonder what you meant by your use of the word `simmering' here?

Regards,

Tejinder, Anshu

Dear Lawrence,

Thanks for reading our essay and commenting on it. It is heartening that we agree on the role of cognition. Especially interesting is your remark on the tendency to count logarithmically...we learnt from Dehaene's book about Amazon tribes which tend to think of 1 and 2 being farther apart than 8 and 9 are [mental compression of a logarithmic nature]. It seems even young children, when asked to place say the number 10 on the number line, between 1 and 100, tend to put it near the middle of the line [like you suggest], and compress the larger numbers on the right half of the line. The linear equi-spaced ordering of numbers is learnt culturally through education as we grow up. This is perhaps another useful example of innate arithmetic versus learnt arithmetic.

We look forward to reading your essay.

Thanks and regards,

Anshu, Tejinder

Dear 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

Thanks Christophe,

After reading your essay we understand your comments above better, and have posted a brief comment on your essay.

Regards,

Anshu, Tejinder

Thank you Alan, for making time to read our essay. We had a first read of your essay, and look forward to reading it again for better understanding, and discussing it with you on your page.

In this context, we wonder if it might be of help for the sake of comparing your quantum viewpoint with ours, if you could critically examine the popular video `Does nature play dice'?' which one of us posted in the recent FQXi video contest. Understandably, we have quite different outlooks, but I am sure the comparative discussion will be stimulating. In particular, it would be interesting to know how you evaluate your proposal with the other modifications / reinterpretations of quantum theory discussed in the video.

Thanks and best regards,

Anshu, Tejinder

Dear Tommaso,

Thanks so much for reading our essay and thanks for your interesting comments. In particular, you say:

"Imagine an other universe similar to ours, with galaxies, stars, planets, but where the phenomenon of conscious life (humans) has not emerged. Planet trajectories still follow the beautiful equation of the ellipsis and most phenomena still match the beautiful and simple patterns described so effectively by math. How could you explain this match in that case? "

We make a distinction between the physical world [which we believe exists even when conscious humans are not there] and the mathematical description of the physical world [which we believe is only possible when conscious humans are there]. That the physical world exists and existed when humans are/were not there, seems provable by scientific methods [radioactive dating of historical records for instance] and we do not call this belief an act of faith. However, we do not see how to scientifically establish that a mathematical description exists / existed when humans are not there. Thus, in your example above we would agree that in the absence of humans the planet still goes around the star, but in our absence there would not be the truth `the elliptical orbit of the planet is being caused because its acceleration falls as inverse square of its distance from the star'. This last bit [elliptical...acceleration...inverse square law] is to our understanding a very human `description' of reality, which is distinct from the reality itself. We find it very hard to understand how the maths can `live' in the material substance, i.e. the planet. Maybe one day there will be an experiment based scientific proof that the maths that describes the thing is the same as the thing itself, but this has not happened thus far and we consider it unlikely. How to put matter into mathematics?

Furthermore, we feel that the formalism of mathematics that the human brain has created is based on the level of complexity the brain has evolved to, and depends strongly on conceptual metaphors. Suppose, a non-human intelligent alien studies the physical universe/nature, what formalism "it" will create would likely depend on its ability to observe nature, creating metaphors etc. Would it overlap with human mathematics or would it be different is for us hard to predict. Thus one might say that the existence of the physical universe is observer independent (invariant); however description of nature is observer dependent, and It depends on the observer's perception, its ability and so on.

We look forward to another reading of your mathematical thriller :-) and to reading your views on the Mathematical Universe Hypothesis.

Best regards,

Anshu, Tejinder

Dear 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

Thank you'