Dear gentlemen,

Thank you for this greatly engaging comment! I like your formulation of conceptual unification and with your permission I will keep it.

I too think that theorems have emergent properties. One thing that seems to confirm this conjecture is that we have theorems that are very hard to prove. If all the properties were contained in the number theory we have so far, demonstrating or disproving the Riemann hypothesis should be a walk in the park. What leads to proofs is usually a little extra insight, a moment of intuition not unlike a critical point that acts a bit like a phase transition.

You're posing a difficult question with regards to emergence to which I'm inclined to answer yes. Emergence seems to be happening in both physical and mathematical systems, however it's difficult to ascertain at this point if this has a significance beyond philosophical considerations. Ideally such a link should help us determine more things about the world and have applications. For what I recall, there has been a recent breakthrough in modelling a sand pile avalanche, which is known to happen as a phase transition from complete stability to a state where sand flows down the slope much like a liquid. The breakthrough was achieved because the shape of the sand grains (somewhat like a quarter moon) was taken into consideration. I'm afraid I can't find the reference on self organized criticality right now, but my point is that - for me at least - there are hints that this direction of chaos and emergence is very promising. I think it is also very appetizing that the concept of chaotic attractors is seemingly so close to the perturbations used nowadays in QFT. Hopefully soon the answer to this question will be definitely affirmative.

Thank you again for your most thought provoking comments!

Warm regards,

Alma

Dear Alma,

Thank you for your comments. It will be great if someone took to investigating common emergence in physical phenomena and the in the mathematics used to describe it. For instance, there are concrete studies which derive quantum theory as an emergent phenomenon, and gravity as an emergent phenomenon, and they use precise mathematics at all levels. It will be interesting to explore if the corresponding mathematics can be described as emergent.

Just one small correction Alma ...one of us (Anshu) is female :-) We understand of course that it is not always easy to deduce gender from foreign names.

Kind regards,

Anshu, Tejinder

I don't know to what extent theorems have emergent properties as it might be the things theorem refer to. I have been working on how spacetime is a coarse grained manifestation of large N entanglements, which have a period 8 structure. In this way the basic SLOCC structure is 4-bit or 8-bit even if there are 2^n = N large number of quantum bits. The geometry of spacetime time then comes about from event horizons that are measures of these entanglements. This in part goes in favor of those who say that time is a sort of illusion, but as I think space is a bit of an illusion also.

On this basis we could say that distance measures, say the Pythagorean theorem, is not emergent so much as the underlying constructions of points, lines, planes and so forth. Given that those elements exist, even if they are emergent, then Pythagoras was right as is so much of other mathematics. Of course we are talking about physical emergence, not so much mathematical. Mathematics as an idealized system of models does not seem to my mind emerge as such, but rather the underlying natural system the mathematics models is emergent.

LC

Dear Torsten,

Thank you for taking the time to read my essay! I am very glad that you found our ideas similar. Actually after submitting, by reading other essays, I realized that I must have underestimated the level that was acceptable. I hope that the lightness of the style doesn't take much off the weight of the argument.

Warm regards,

Alma

Dear Vladimir,

Thank you for your kind words and for the link. She's very pretty and it was fun watching her :)

Wish you the best of luck!

Alma

Dear LC,

Thank you for taking the time to read my essay and for your kind words! It's very attention worthy that you are mentioning the Langlands program as a direction of your interest. I hope you do continue your research in this area and obtain the results you're hoping for. Considering your ideas, I think you'd be obtaining results nothing short than spectacular. Wish you good luck in your enterprise!

Cheers,

Alma

Dear Akinbo,

What I had in mind when I mentioned 'matchsticks' was simplicity. Matchsticks are small objects, completely similar and having the same properties and it takes some imagination to think that one can build things out of such seemingly inappropriate pieces.

That being said, I will add that I read your essay and will be commenting on your page shortly.

Warm regards,

Alma

Dear Anshu, Tejinder,

My sincerest apologies for the confusion! It is very nice of you to say it's understandable and thank you for that :)

Warm regards,

Alma

Dear LC,

That is a very subtle point you make there and I'm not sure I am skilled enough in this domain - I am probably not :) - to be able to bring the discussion further. But at least so as to clarify what I have in mind, I will bring the following example, which I realize is built on a different framework than the one you specified. There exists a mathematical proof for why and how phase transitions happen , but it relies on a lattice treatment (Sinai and Berezin). Until today, there is no treatment that can show the same thing for systems with more degrees of freedom. The emergence here is given by the extra idea that needs to exist in order to complete the proof. This extra idea is what adds insight and moves the pieces of the demonstration together in the right order, so they compose something that was not necessarily and sufficiently implied by the existing theory. That being said, i fully agree that the underlying natural models expressed by mathematics show emergent properties.

Cheers,

Alma

*I am copying these posts here as they are now hidden.

Dear Anshu, Tejinder,

My sincerest apologies for the confusion! It is very nice of you to say it's understandable and thank you for that :)

Warm regards,

Alma

    Dear LC,

    That is a very subtle point you make there and I'm not sure I am skilled enough in this domain - I am probably not :) - to be able to bring the discussion further. But at least so as to clarify what I have in mind, I will bring the following example, which I realize is built on a different framework than the one you specified. There exists a mathematical proof for why and how phase transitions happen , but it relies on a lattice treatment (Sinai and Berezin). Until today, there is no treatment that can show the same thing for systems with more degrees of freedom. The emergence here is given by the extra idea that needs to exist in order to complete the proof. This extra idea is what adds insight and moves the pieces of the demonstration together in the right order, so they compose something that was not necessarily and sufficiently implied by the existing theory. That being said, i fully agree that the underlying natural models expressed by mathematics show emergent properties.

    Cheers,

    Alma

    Dear Joe,

    I must say that I did read your essay but I am not sure I am able to understand it properly as it is departing from the previously established theory.

    Alma

    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

    Dear Alma,

    I spent a few hours this morning reading your essay and preparing a non trivial comment for you. You will have a very good comment and appreciation from me by the end of the day. May be this is an instance of distant entanglement between brains at least not a pure coincidence. My thoughts during the two hour fast walk I just had was about rivalry between the two hemispheres that you might be call Phys and Math., a kind of quantum superposition that collapses one side or the other depending on context.

    Have a good afternoon.

    Michel

      Dear Alma,

      Great essay! You have an engaging writing style that is enjoyable and accurately expresses your arguments. I particularly liked the section "Simplicity and What Follows" and "The Shape of Things to Come" and I share your optimism. I also discussed the role of simplicity in my essay; I would like to take your opinion.

      All the best,

      Mohammed

        Dear Mohammed,

        Thank you for your kind words! I will give your essay a read and I hope to be able to leave you my comments today

        Warm regards,

        Alma

        Thank you Michel! I'm looking forward to find out what you have in mind :)

        Dear Alma,

        In simple sentences and well chosen pieces you gave a relevant hint of what relates maths and physics, a proof of maturity that some experts either never had or have lost. Let me quote you "When one makes the (upper-half) complex plane, fractals grow in it uninvited", "Einstein's cosmological constant and black holes - mere second-handunwanted consequences of his theory - and Higgs' field were firstly miraculous ideas based on physical intuition, not lucky attempts to search the whole of math in the hope of a chance finding" and "It may very well be a gap in our total knowledge of the world, not of the physical, or biological or mathematical fields in isolation."

        I had more time than you to think about the power of mathematical physics and always found this remarkable coincidence between maths and physics in several fields from electronic engineering to quantum information. About 20 years ago, I have been fascinated by the possible connection between cognition and the quantum: I gave a few references at Vincent Douzal blog. At the the same time, I attended a school on Grothendieck's dessin d'enfants that remained "dormant" until very recently.

        You are the first to acknowledge me of introducing modular forms in the program! May the moonshine topic is no longer as fashionable as it was, or just too abstract!

        I don't want to be too long, but reading you revitalized my forgotten quest of putting our cognitive ability into the picture. I remembered me reading the Nobel winner in Physiology or Medicine: John Eccles "I here express my efforts to understand with deep humility a self, myself, as an experiencing being. I offer it in the hope that we human selves may discover a transforming faith in the meaning and significance of this wonderful adventure that each of us is given on this salubrious Earth of ours, each with our wonderful brain, which is ours to control and use for our memory and enjoyment and creativity and with love for other human selves." --How the Self Controls Its Brain, pp. 180-1 (1994).

        I suspect that the gap in our knowledge of the world lies in our neglect of other fields such as those concerning ourselves. I also suspect that maths can be as much effective there with enough imagination.

        I wish you all the best and keep ready for further interaction.

        Michel

          Dear Alma,

          As I told you in my Essay page, I have read your intriguing Essay. You made an excellent and original work. Here are some comments/questions:

          1) I see your pretty statement that "the belief that math has an independent existence beyond our daily lives is based on the observation that even a child can intuitively understand math" confirmed every day by my son David, 4 years old, who plays with numbers...

          2) Your aphorisms "Physics is the only science that can work with spherical cows in a void", "As opposed to an engine, one can't fix math", "Mathematical physics is only as good as physical insight.", "We can't expect math to work on its own", "Physics is simple", "There is still time for math", "You know you're missing something when there's just too much you can't explain" and "Just wait to see our children" are fantastic! For the last one, see my point 1).

          3) Your stress we don't have a quantitative match between theories in pure math and the description of nature. I think we will never have it.

          4) Do you think physics is not scale invariant?

          5) I agree with you that the Langlands program sounds like good news for physics, but it must be handled very carefully. I know two possibilities to translate an intractable problem into another framework which sometimes generate confusion: the Maldacena conjecture, which, in my opinion, does nor resolve the black hole information paradox as it is often claimed, and the "Einstein frame versus Jordan frame" controversy in astrophysics observations.

          Finally, I found the reading of your beautiful Essay very interesting and enjoyable. Thus, I am giving you a deserved highest rate.

          I wish you best luck in the contest.

          Cheers, Ch.

            Dear Michel,

            It brings me great joy that a scientist of your caliber has found things to appreciate in my essay. For your comment alone and it was well worth participating into this contest.

            To me it is natural to speak about modular forms because in my opinion they are the Langlands program, first and foremost. The most famous achievement of the program lies with modular forms. I don't think they are forgotten, but probably very difficult even for skilled mathematicians. Moonshine is not often mentioned today much like the prime gap was not in focus before Zhang made his breakthrough.

            I too find interesting the way humans are capable of working with complex categories instead of exhaustive search to push knowledge further. There are many things we don't understand in detail about how our minds and brains operate and to be honest, it wouldn't be very surprising if quantum effects were found at the scale at which neurons operate. Regarding knowledge itself, another essay in this contest made me think the other day about how new ideas are generated. If knowledge can be modeled as information points in a network, a new idea may be thought of as the minimum number of information points needed to deduce a new piece of the puzzle, as related to the complexity dimension of the concept that needs to be deduced and occurs as a phase transition. Since you considered the cognitive ability in your work, it would be of great interest to me to know your thoughts and your approach to the subject.

            Many many thanks for your words! You made my day!

            I realize I didn't include any contact information that is visible of my profile, so I am adding my personal address here alma.ionescu83@gmail.com

            My warmest regards and my profound appreciation!

            Alma