Dear Alma,
thanks for reading my essay and your words. I want to make the unification of science using math very clear.
But I gave the complement back. I also read your essay and it is really great. Much easier to read then my essay (and maybe also easier to uderstand for any reader). I'm glad that the conclusion of our essay are (in principle) the same.
I wish you also the best luck for the contest (for that I gave a high rate).
Torsten
Dear Alma,
Excellent analysis in the spirit of Cartesian doubt. Deep, all-encompassing questions on fundamental questions of Mathematics and Physics and the direction of finding the answers to them:
"That our knowledge manifests gaps around emergent phenomena seems to be an indication that we lack some insight of the mathematical description itself, not just of what happens physically. 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. Are we just lacking the right mathematics to treat other fields with similar power and rigor as physics? You know you're missing something when there's just too much you can't explain. "
"Our math does not make sense in a million ways, 7 but always choses a certain correct path and discards everything else. We woke up in a place where so many things might have headed in a different direction, yet our universe is very well constrained.
The constants are not changing and reality seems sturdy, like it will last forever. Everything insists to make perfect sense. "
Indeed: "But there's still time. Just wait to see our children."... My high appreciation.
It's time we start the path...
I am sure that current problems of Mathematics and Physics - a problems with the ontology. Dialectics too, do not be afraid, it is a good helper for a deeper vision of the dialogic Nature. Dialectics and ontology help to see Universum as a whole, understand the nature of the information, time, consciousness. The Information age requires revision and updating of the basic concepts of fundamental science. The new paradigm and a new vision of the world, including the Foundations of Mathematics and Physics, filling them with the sense of "life-world" is possible only on the basis of the broadest synthesis, taking into account all accumulated knowledge, teamwork «ratio», «intuition», «emotion».
Good luck in the contest,
Kind regards,
Vladimir
Dear Alma,
Don't know for what reason your essay had not caught my eye. Probably because of the number of essays. Better late than never. Certainly a brilliant contribution and will reflect in my rating.
Under the section, Simplicity and What Follows, you wrote, "My room and everything in it is made from one type of lepton and two types of quarks... The universe takes a bunch of matchsticks, assembles them in the shape of various Rube Goldberg machines and obtains an overwhelming level of diversity". This is an area of great interest to me and I discuss this as well in my essay.
If we assume the matchsticks to be indivisible lengths, if you break down the various Rube Goldberg machines, the lepton and quarks into their matchsticks, what will distinguish one matchstick from another? Or framed another way, what can distinguish one fundamental length from another assuming space either of distance or of matter is not infinitely divisible? Certainly, what will separate the matchsticks into discreteness cannot be of same nature as the matchstick or can it?
The other issue I have to ask here, is whether these matchsticks are eternally existing? Whether they have the same date of manufacture? Whether they have the same expiration date? If your answer is no, can "time" distinguish the universe's matchsticks?
All the best in the competition. Well done.
Regards,
Akinbo
*If you don't mind could you leave me a note on my forum that you have responded so I can be alerted.
Alma,
You mention the Langlands program, which is pretty close to what I am looking into. The generalization of the Tanyama-Shimura conjecture into algebraic varieties and categories is one reason why I have been looking into this homotopy approach to the foundations of mathematics.
Your essay was interesting and was free of obvious problems or errors. Mathematics has qualities as you say with the complex plane and fractals that make it appear less of an invention and more as something discovered.
Cheers LC
Dear Alma,
It was a pleasure reading your well-written and enjoyable essay, with many points to take home, and commonalities with the ideas in our essay. Especially the emphasis that mathematical physics is as good as the physical insight preceding it [what we called conceptual unification].
A noteworthy remark you make concerns thinking of mathematics as an emergent process. Very interesting. Now on the one hand of course elegant theorems follow from axioms. Can one think of this as emergence in the same sense as emergence in physical and biological systems? Does a theorem possess emergent properties which the axioms do not have, and which could not have been anticipated just by staring at the axioms. We suppose yes, as for instance features we see in number theory - properties possessed by a collection of numbers which simply cannot be guessed by looking at individual numbers. Perhaps you make a very important point here. Can emergence in physical systems be in any way linked to the corresponding emergence in the maths used to describe them? We will appreciate any thoughts you might have on this.
Kind regards,
Anshu, Tejinder
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