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

          Dear Christian,

          Thank your for reading my essay and for your kind words, this is an honor for me! Allow me to reply to your points:

          1) It looks like someone has a little genius physicist in preparation ;) He is lucky to have you as he will learn a lot from you as he is growing up.

          2) Thank you! It is perhaps my main point, that regardless of what the past generations did not complete, the next generation is arriving and I trust their ingenuity.

          3) I think so too. There is a finite number of laws and seemingly always more math.

          4) I believe in scale invariance up to the extent where I would risk saying that a final theory should first guess the ratios and only after that the quantities. I hope I didn't seem to doubt it; if I did, I must have chosen an unfortunate expression.

          5) You are perfectly right from my point of view. Progress does not always equal benefits as the atomic bomb history shown. In this case, theories that are only conformally equivalent should be very well understood first and their limits of applicability rigorously established, so as to avoid false promise and dead ends.

          Thank you again for your generous comments and wish you the best of luck in the contest and in your research!

          Cheers,

          Alma

          Dear Alma,

          It was my pleasure. He is myself who thanks you for your kind words on my little son David. I think to have understood your point on scale invariance. It is nice and sharable.

          Cheers, Ch.

          Dear Alma,

          You essay and comments are insightful and you seem to be a charming person. I was also interested in Leifer's essay viewing the whole of knowledge as a scale-free network. Your idea of looking at possible phase transitions is developed in his Ref. [13], Sec. G, p. 63 where you can read that "the critical exponents of the phase transition equal the critical exponents of the infinite-dimensional percolation". On my side, in my Neuroquantology paper quant- ph/0403020, I wrote in the abstract "Time perception is shown to depend on the thermodynamics of a quantum algebra of number and phase operators already proposed for quantum computational tasks, and to evolve according to a Hamiltonian mimicking Fechner's law. The mathematics is Bost and Connes quantum model for prime numbers. The picture that emerges is a unique perception state above a critical temperature and plenty of them allowed below, which are parametrized by the symmetry group for the primitive roots of unity." We recently revisited the BC model in the context of Riemann hypothesis and quantum computation http://iopscience.iop.org/1751-8121/labtalk-article/45421. This is a good sign that a good mathematical theory may have many inequivalent applications.

          Today I have in mind to approach the subject of cognition with the tools I am advocating in my essay, it may take a while. I already mentioned that rivalry between the two cerebral hemisphres looks like a qubit.

          Thank you very much Alma for the stimulus you are giving me. My very best regards.

          Michel

          Dear Alma,

          A beautiful essay in all ways, original, insightful and with perfect English and logic, well argued.

          I feel I should reproduce the phrases that I felt hit the target most perfectly;

          'We can't expect math to work on its own'

          'You know you're missing something when there's just too much you can't explain.'

          'We've carefully chosen bits of math that resemble the phenomena we wanted to study'

          'In pure math, any inconsistency is shot dead on sight.'

          'Math requires us to be very careful when shaping a theory because the slightest false step will bring down the whole edifice that we've carefully built by creating a contradiction.'

          'when we lose the correct track we have no way to find it again without more new insights.'

          No time now, or even need, for detailed questions, time for scoring. i hope you may have time to read mine, which identifies a specific and important case of loosing the correct track but accepting illogicality by being satisfied that maths is enough. I'd love to hear your views.

          Very well done, and thank you.

          Peter

          Alma Ionescu wrote on Apr. 18, 2015 @ 15:08 GMT stub

          Dear En,

          This is nice and to the point. Your writing displays an interesting personal stance and I'm sure you enjoyed the exercise. I think that you're right when you're saying, in the third paragraph, that Wigner's expression should not be taken literally as it was more a metaphorical way of encouraging new lines of thought and maybe a feeling of delight in the face of the best known parts of the natural world.

          Warm regards,

          Alma

          report post as inappropriate

          Author En Passant replied on Apr. 21, 2015 @ 07:32 GMT stub

          Dear Alma,

          I actually have many contacts with Romanians. They are extremely good at programming, and (in fact) BitDefender (the best antivirus program) is programmed by Romanians.

          Don't worry, the NSA can "get in" anyway. But their interests are not what we worry about (banking, etc.).

          En

          Hi Alma--

          Your essay was a joy to read. There is a wonderful lyrical quality to your writing style. I loved your sayings, especially "You know you're missing something when there's just too much you can't explain". I shall incorporate that one into my daily lexicon.

          Quick question, admittedly somewhat off-point: Which philosopher has influenced you the most?

          Very best regards and luck in the contest,

          Bill.

            Dear Bill,

            Thank you very much for your kind words! Actually I like that question. As it stands, the philosopher in question would be Weinberg. I am half joking because his name sound German and imposing, but only half because besides Popper, Russell and newer philosophers like Maudlin and Earman I can't say I enjoy philosophy (to my embarrassment) and none of these four philosophers influenced me. I do however enjoy great thinkers; people like Weinberg and Susskind are thinkers, are to me they are greatly enjoyable. The writer that, if I'm lucky, influences my style, and who - I can say - taught me English, is Terry Pratchett. He was a greatly appreciated humourist from the UK. His writing is very mood lifting - and he too was a great thinker, writing about the human nature and combining it with obscure jokes about quantum mechanics.

            Wish you the best of luck in your work and in the contest!

            Alma

            Hi Alma--

            I admire your preferences for thinkers and philosophers. If you haven't already, I urge to read Jim Holt's book, "Why does the world exist"?. It's a wonderful book. Furthermore, Holt devotes an entire section to Weinberg (chapter 9). Reading Weinberg's views on philosophy, the multiverse, etc., was fascinating.

            Best regards,

            Bill.

            Oh, thank you for the recommendation, Bill! It sounds great! I just looked him up and watched his TED talk; I will definitely read the book.

            Alma,

            You're a winner. I do not exaggerate when I say that in my opinion, your essay ranks near Wigner's own, in its breadth of knowledge and depth of insight.

            A couple of confessions I have to make: I have followed and enjoyed your intelligent commentary in other forums, so I knew I would get to your essay eventually. And the other confession is that I checked on the internet to verify how young you are; I would never have believed it from your mature postings. I do hope your circumstances permit you to go on to higher academic pursuits -- it's obvious that you have the "right stuff."

            I didn't know the story of von Neumann and the engine. I do recall another one, though, of his lightning-fast calculating skill -- mathematicians like to try and fool each other with problems that seem complicated, but can be made simple with "tricks". One of them concerns two trains approaching each other, with a fly flying back and forth between the engines -- how far does the fly travel before getting squashed when the trains collide? One can get a quick answer by knowing the short cut of averaging -- otherwise, calculating the series by brute force is long and tedious. It is said that when the problem was put to von Neumann, he gave the solution almost immediately. Asked if he knew the short cut, he looked puzzled and replied, "What could be easier than summing the infinite series?" The Wolfram site has an article on the two-train problem.

            Highest marks, and good luck!

            Tom