Dear Lawrence,
I finally got to read your essay, and I loved it! As usual, you make excellent and deep connection between various things, connections that allow us to see relevant subtleties. You made interesting connections between computation, quantum theory, homotopy, black holes, and proved that HOTT may be very well the way to the next stage of physics.
Best wishes,
Dear Dr. Crowell,
I thought that your engrossing essay was exceptionally well written and I do hope that it fares well in the competition.
I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.
All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.
Joe Fisher
Christinel,
Thanks for the positive assessment of my paper. I gave your paper a pretty high score a few weeks ago. I did this while I was on travel and I don't think I had time to write a post on your blog page. I will try to write a comment, which will probably require rereading your paper.
There is a paper by Schreiber on directly applying HOTT to physics. This is a difficult and in some ways foreign way of doing physics. I am less sure about the role of HOTT directly in physics, but rather that a simplified form of mathematics that connects to HOTT will become more important. It is in much the same way that physicists do not employ set theory a whole lot in theoretical physics. However, behind the analysis used by physicist there is point-set topology. We generally reduce the complexity of this mathematics. If I were to actually engage in this I would study the HOTT, and an introduction to HOTT with physics and related web pages on this site, are worth going through.
To be honest it has been a while since I have studied this. I have been working on a homotopy approach to quantum gravity. I mention some of that in my essay. This concerns Bott periodicity with respect to holography. The connection though is rather apparent. There are also some similarities to C* algebra. This work of mine connects with what is called magma, which constructs spacetimes as the product on RвЉ•V, for V a vector space,
(a, x)в--¦( b, y) = (au + bv, [x|y] - ab)
where the square bracket is an inner product. This is a Jordan product and the right component is a Lorentz metric distance. This is also the basis for magma, which leads to groupoids and ultimately topos. A more convenient "working man's" approach to HOTT is needed.
There is my sense that mathematics has a body and a soul. The body concerns things that are computed, such as what can run on a computer. The soul concerns matters with infinity, infinitesimals, abstract sets such as all the integers or reals and so forth. If you crack open a book on differential geometry or related mathematics you read in the introduction something like, "The set of all possible manifolds that are C^в€ћ with an atlas of charts with a G(n,C) group action ... ." The thing is that you are faced with ideas here that seem compelling, but from a practical calculation perspective this is infinite and in its entirety unknowable. This along with infinitesimals, or even the Peano theory result for an infinite number of natural numbers, all appears "true," but much of it is completely uncomputable.
Cheers LC
I will take a look at it in the near future.
LC
Dear Lawrence,
Thank you for the links, and for the explanations.
Best wishes,
Cristi
Dear Laurence,
As always, you submitted a challenging and thought provoking essay, and I am glad it is doing very well so far in the ratings. I particularly enjoyed your discussion of numbers too big for a Turing machine in our universe to count to.
In your conclusion, you write:
"Chaitan has advanced ideas that mathematics is not something that exists in any sort of coherent wholeness. It is more a sort of archipelago of logically consistent systems that sit in an ocean of chaos. [...] Possibly the quantum vacuum is similar. It may be a tangle of self-referential quantum bits, where some sets of these exist in logical coherent forms. These zones of logical coherence might form a type of universe. These logical coherent forms are then accidents similar to Chaitan's philosophy of mathematics. It is very difficult to understand how this could be scienti fically demonstratedｬ yet maybe regularities in physics described by mathematics exist for no reason at allｮ｢ｼｯpｾｼpｾﾁs you found out when your read ｼﾕﾒﾌ urlｽ｢httpsｺｯｯfqxiｮorgｯcommunityｯforumｯtopicｯｲｴｹｷ｢ｾｼsｾﾛｼｯsｾ my essayｼeｾﾝｨhttpsｺｯｯfqxiｮorgｯcommunityｯforumｯtopicｯｲｴｹｷｩｼｯeｾｼｯﾕﾒﾌｾｬ this is pretty much how ﾉ see our universe in relation to the ﾍaxiverse that results from the ﾍathematical ﾕniverse ﾈypothesisｮ ﾏur universe exists for no specific reasonｬ because all possible universes doｬ and the regularities that we observe between our physics and known mathematics is simply a necessary condition for the existence of selfｭaware substructuresｮ ｼｯpｾｼpｾﾁll the bestｬｼｯpｾｼpｾﾍarc
Marc,
Thanks for the encouraging word here, and the upward boost.
This touches in many ways on the issue of unification of physics. In particular this concerns the plethora of unification schemes. I tend to think that they may all, or most of them, are correct. They may have some probability assignment, but they all may well manifest themselves. There may be cosmologies with very different particles and interactions than what exists in this observable cosmology. I base this on part with what I am working on, which seems to be deriving a type of landscape of string/M-theory. That is of course a good thing that I can recover something known.
I think the many worlds account of QM has some connection to the string landscape. If the spatial surface of this cosmology is infinite there are an infinite number of us sufficiently far out there. In a cosmology that is infinite, vast distance and lack of causal connection may imply quantum entanglement. This in particular would apply across the particle horizon.
Cheers LC
Dear Lawrence,
I am giving your essay a second read. In addition to my earlier comments above, I wish to take you up on a part of your essay. You said, "...in point set topology there are an infinite number of points between any two points on the real number line with a finite distance between them. This means if they exist in some meaning according to computation there must be a machine that performs any calculation of points separated by any tiny finite set of intervals segmenting the distance between these points"
If we may interrogate this, I wish to ask:
1 - Can distance be what separates two points, when distance itself is constituted of points? Or are there some distances constituted of points and other types of distances not so constituted and not having points as their extremities of their extension or segments thereof.
2 - What is an interval made of? Is it spatial or temporal?
3 - How many intervals, if such exist can be on a real number line? I ask because of the 'finite' set of intervals in the quote above.
4 - How can a real number line with an infinite number of points be divided, if points cannot be divided into parts and there is always a point at the incidence of cutting?
5 - Finally, talking about "existing in some meaning", are points eternally existing objects or can they perish? If the Universe can perish, will points outlive it? If there was a Big bang Universe creation from Nothing, did points precede it?
Regards,
Akinbo
*If you don't mind you may drop me a note on my forum so I get email notice. That is if you are inclined to discuss the above.
Thanks Lawrence for dropping your comments at my forum. Appreciated.
If you have the time, you may wish to volunteer direct opinion to the 5 questions I attempted to raise here.
Regards,
Akinbo
I can answer some of these. An interval in relativity is the measure of a clock on a frame bundle and on a certain path. It is the integration of the path length.
The matter of infinitesimals, a length or displacement along a certain direction that is arbitrarily small, has been a subject of debate and research for a long time. This matter has only been somewhat resolved with the so called Robinson numbers, which have underlying it set theoretic concerns of forcing and the continuum. I am not a great expert on this subject, so I can really only make mention of this in a short post like this. In the end it only works, as I understand, within a certain continuum model. The underpinnings of calculus and questions surrounding the Dedekind cut do not seem to be derived according to a complete axiomatic system. However, with a few basic ideas you can develop a lot of calculus.
Mathematics in the objective or in some ways the Platonic perspective does not perish with the heat death or end of the universe. If mathematics is nothing more than a pattern system derived from the natural world then in that model it might perish. I am not terribly committed to either perspective. There are troubles with either viewpoint.
Garrison Keillor has his "Guy Noir," who "On the tenth floor of the Atlas building still seeks answers to life's persistent questions." If you have ever listened to his "Prairie Home Companion" you know this well. There are persistent questions, such as "Does God exist," that will probably never be conclusively answered.
LC
Hi Lawrence,
If it hasn't been brought up yet -- I want to make sure that we get the spelling of Gregory Chaitin's name right. He's among my favorite mathematicians/computer scientists, so the typo jumps out at me.
There's no getting around the issue of how we differ in our views of foundations. I do not think classical physics is either finished, or emergent from conventional quantum theory -- in fact, I think it is the other way around. Although it is commonly believed , as you say, that "The classical picture of the universe is a continuum of flows [3] ..." this is not true. Continuous functions as described by differential equations or topological methods do not support a physical continuum of space independent of Minkowski spacetime, because space has no physical reality independent of time.
I think this is easier to see by critical study of Perelman's solution to the Thurston geometrization conjecture -- all singularities on S^3 are extinguished in finite time by continuation (via surgery) of the Ricci flow, on the half open interval [0, oo). This is the mathematical advantage that any simply connected 4-dimensional world -- including Minkowski space-time -- has over a multiply connected space of random functions in 3 dimensions (or in fact, Hilbert space of any dimensionality).
Nevertheless -- my highest score goes to your essay, for setting up the issues in thoughtful and highly readable terms, even though I couldn't be more opposed to the notion that "Spacetime is built up from entanglements [13]" Classical orientation entanglement explains the phenomenon just as well, when a time parameter (such as that of Hess-Philipp) is included in the dynamics.
I hope you get a chance to check out my essay as well.
All best wishes,
Tom
Thanks for the positive vote or score.
Before 1900 it was commonly thought the universe was a continuum, and the idea of atoms was under attack, as this was thought to not conform to the continuum picture of reality. Of course Planck assumed that energy occurred in discrete steps, and Planck and Bohr assumed discrete values of angular momentum as well to model the atom. Quantum physics does have continuum structure, such as the dynamics of the wave function or the system of paths in a Feynman path integral. However, these no longer have the sort of ontology that continuum structures have in classical physics. The existential aspects of the quantum wave function is not longer ontological, and recently it is being found that the epistemological foundation of the quantum wave is not satisfactory either.
How classical physics emerges is tough to understand. How an einselected basis occurs so that a particular eigenvalues corresponds to a measurement or is associated with a classical value is not solved. The paper by Sax proposes that Goedel's incompleteness theorem plays a role. I had some discussions with him on this on his essay blog page. This is curiously important with D-branes, for these are classical or macroscopic structures. While they are ultimately made of strings, or are similar to Fermi surfaces of electrons or condensates of quantum states, they are nonetheless classical and important for foundations.
Sorry about the Chaitan for Chaitin. That is a regrettable typo. I don't remember if I read your paper or not. I will try to take a look at it soon.
Cheers LC
Dear Lawrence,
An assumption at the end of your article
"Mathematics and physics have this curious relationship to each other for purely stochastic or accidental reasons; there ultimately is no reason for this"
provokes me to note that this possibility is refuted in our essay on the scientific ground.
Best regards,
Alexey Burov.
Dear Alexy and Lev.
Your paper is well argued. I will admit to being very agnostic about these sorts of ideas. In particular I am very agnostic about Tegmark's hypothesis, which seems not mathematically provable, nor scientifically testable. Even string theory is only at best indirectly testable, but Tegmark's Mathematical Universe Hypothesis seems impossible to test.
A couple of points I mention first. The WAP as I understand it is the statement that the universe observed must be of sufficient complexity and structure to permit such observers. It does not mean that any cosmology that exists must admit observers. I think that is the strong AP (SAP). The other point is that chaos, at least within the meaning of Hamiltonian chaos or strange attractor physics, means that a system can execute a vast number of complex dynamics, all of them separated by very small initial conditions. This means that phase space is specified to a very small fine grained detail. Given this is cut into N boxes or pieces, and in each is one of the possible states (0, 1), the degree of complexity is 2^N = e^{S/k}. This is the dimension of the Hilbert space corresponding to this classical setting and the entropy S = k ln(2)N = k ln(dim H), H = Hilbert space. Chaos then in fact implies a high level of complexity.
I did not make much mention of this in my essay. It could be said that mathematics has a body and soul. The body concerns things that are numerically computed and can in fact be computed on a computer. The soul involves things that involve infinitesimals and continua. These tend to be at the foundations of calculus with limits and related arguments. Even though my essay discusses homotopy, this is argued on the basis of continuous diffeomorphisms of loops or paths. However, in the end this is not what we directly compute in mathematics. We are interested in numbers, such as indices or topological numbers, and in physics that is much the same.
If you crack open a book on differential geometry or related mathematics you read in the introduction something like, "The set of all possible manifolds that are C^в€ћ with an atlas of charts with a G(n,C) group action ... ." The thing is that you are faced with ideas here that seem compelling, but from a practical calculation perspective this is infinite and in its entirety unknowable. This along with infinitesimals, or even the Peano theory result for an infinite number of natural numbers, all appears "true," but much of it is completely uncomputable. This is because the soul of mathematics touches on infinity, or infinitesimals.
The soul also involves things that quantum mechanically are not strictly ontological. These are wave functions or paths in a Feynman path integral. The existential status of these is not known, for the standard idea of epistemic interpretation is now found to be not complete. This differs from classical physics, where the physics is continuous, with perfectly sharply defined paths and energy values and so forth.
I am somewhat agnostic about the existential status of the soul of mathematics. In some sense it seems compelling to say it exists, but on the other hand this leads one into something mystical that takes one away from science. So it is not possible as I see it now to make any hard statement about this. We seem to be a bit like Garrison Keillor's Guy Noir, "At the tenth floor of the Atlas building on a dark night in a city that knows how to keep its secrets, one man searches for answers to life's persistent questions, Guy Noir private eye."
I will give your essay a vote in the 7 to 10 range. I have to ponder this for a while.
Cheers LC
There is a profound principal in the universe that says there is no central entity or notion anywhere, and that everything has no special significance than any other things in physics terms. This principal dispelled 'earth-centric' idea and later the Newtonian absolute time-space concept. It is a universally accepted principal in modern science. If math and physics are intertwined inextricably then it seems natural numbers ought to have an equal standing as any other numbers, irrational, complex, or even numbers yet to be invented.
Is there any physical underlying reason for natural numbers' special status? Or are the natural numbers just a convenient way for people to count and were invented by macro intelligent beings like us?
Since all natural numbers are mere derivatives of the number '1', so let's look closely at what this number one really means. There are two broad meaning of the number one. First it registers a definitive state of some physical attribute, such as 'presence' or 'non-presence'. We can find its application in information theory, statistical physics, counting and etc. The second interpretation of number one is that it denotes the 'wholeness' of an entity.
In physics, natural numbers virtually have no sacred places prior to the establishment of quantum mechanics. After all, we don't need any natural numbers in our gravity functions or the Maxwell electro-magnetic wave functions. Some sharp observers would argue that the 'R squared' contains a natural number 2. However on close examination the number 2 is merely a mathematical notation for a number multiplying by itself, and it has no actual physical corresponding object or attribute. The fact that there is no natural number in the formulae represents the idea that time-space is fundamentally smooth. For instance there is no such law in physics that requires 7 bodies (non quantum mechanical) to form a system in equilibrium.
Had we obtained calculus capability before we can count our fingers, we probably would have been more familiar with the number e than 1-2-3. We might have used e/2.718 to represent the mundane singletons. There is no logical requirement that we couldn't or shouldn't do it. It is all due to the accident that people happened to need to count their fingers earlier than the invention of calculus. There is no physical evidence that the number '2' is more significant than the any other numbers in the natural world.
However with the standard model of quantum mechanics, energy is quantized, that is, it can only take natural numbers. This idea profoundly altered the status of natural numbers in physics and is a direct contradictory of the notion of 'no center in the universe' principal. In this sense it is far more unorthodox than the two relativity theories combined because the latter in fact enhance the 'no center in the universe' law. Why does the quantum have to be integer times of a certain energy level, and not an irrational number like square root of 17, or the quantity e? Does it really mean there are aristocrats in the number world, where some are nobler than others? Were the ancient Greek mathematicians right after all, who worshiped the sacredness of natural numbers and even threw the irrational number discoverer into the sea?
From this standpoint we can almost say that quantum theory has some bad taste among all branches of natural science.
Before the quantum theory got its germination, actually people should have noticed the unusual role natural numbers play in rudimentary chemistry. For instance, why two hydrogen atoms and not five, are supposed to combine with one oxygen atom to form a water molecule? If scientists are sharp enough back then they ought to be able to be alarmed by the oddity underlying the strange status of natural numbers. It could almost be an indirect way to deduce the quantized nature of electrons.
Fundamentally if natural numbers indeed play a very unusual role in nature, then nature resembles a codebook not just from a coarse analogy standpoint. It is the ultimate codebook filled with rules for a limited number of building block codes. The DNA code is an excellent example.
If it's a codebook, inevitably it takes us to surmise if information itself is the ultimate being in the universe. It is probably not electrons, strings, quarks or whatever 'entities' people have claimed. It is the information that is the only tangible and verifiable entity out there. Everything else is a mirage or manifestation of some underlying information, the codebook.
In this sense physics has somewhat gone awry by focusing on the wrong things, the 'attributes' such as momentum, position and etc. Instead, information is what contemporary physicists talk about and experiment with. Otherwise, the physicists would have no right to laugh at the medieval scholars who based their intellectual work on the measurement of the distance between a subject and God's throne.
The nature has revealed her latest hand of cards to us. It looks like it's the final hand but no one can be sure of course.
I think the world around us gives us all type of CLUEs to our intellectual pursuit. Great thinkers are great observers of hints and clues. Ultimately the physical world (not the bio-sphere or human society) is far simpler than anyone can think of, because of the equilibrium status of the universe. Or read differently the static nature of the universe. The universe is a dead body waiting for eons to be dissected. Complexity is largely suppressed because of the cancelling effect of forces, as well the elimination of complexity by evolution of a closed system. The complexity is decreasing as a function of increasing approximation to an equilibrium.
The trick is to change our human centric mind set and think out our own box. I pointed the natural number conundrum. It is an example of homo sapiens obsession. New type of understanding is needed and can be achieved by shedding unconscious constraints posed upon ourselves.
Dear Lawrence, I am following your good idea to double the comment.
*****
Dear Lawrence,
Thank you so much for your generous compliments to our essay. As you see, we are showing there how Tegmark's MUH is refuted on the scientific ground. Yes, it goes against the dominating opinion of the community of cosmologists (and your own), that the full-blown MUH is unfalsifiable, but our refutation looks very solid for me.
About your 'couple of points'. First, your distinction of WAP and SAP fully agree with the conventional one, as I may judge. It isn't clear to me what point you were trying to make about them. Second, we use the word "chaos" in its ancient meaning, as we stress it when this word is used the first time, pointing there to Platonic philosophy. This meaning sometimes is expressed by such words as "nothingness" or "nothing". This formless entity, chaos/nothingness, is a source of pure accidental, random, causeless factors. It has little to do with the mathematical concept of "dynamical chaos" you mention, which assumes certain mathematical forms already given.
Your ideas about "the soul of mathematics" sound very interesting to me, and I would very much wish to discuss them with you in much more detail than this specific place and occasion allows. You know how to find my email. Please be assured that I would highly value communication with you on these and other questions.
All the best,
Alexey.
Math is all about mapping from one artificial domain to another artificial domain. The mapping works well but does not give rise to the legit or such mappings.
I propose that quantum mechanics needs an entirely new type of math. The current approach to quantum mechanics is bewildering and confusing. The fundamental rules of math need change to adapt to the quantum mechanics world.
Hi Lawrence,
I have read your essay and many of your posts and appreciate what you are trying to do. But please take a look at my essay if you haven't already and give it some effort since you are a good programmer, it should not take much of your time. You will see that physics can be done based on the most elementary mathematics known(addition, comparison ...etc). So fundamentally no need for exotic math or to worry about infinity, compute-able ... etc. The system can be put on more formal level(fundamentally it is a geometric probability problem), but through simulation the origin of the design of reality is so clear and makes a lot of sense. Then MUH is established(confirmed) with minimum effort, we start by postulating it and end by confirming it. Our reality's existence is the proof of the Platonic realm of mathematics.
Thanks and good luck
P.S. But please read some of the first post in my thread about running the programs if you like to delve in them more.
Congratulations Lawrence on your high standing..
I'm just finishing reading your excellent paper, and I'll have some comments after I catch some sleep. This is one of your best essays so far, and it appears that high marks are well deserved.
All the Best,
Jonathan