From the illustrations of the text of Baez one learns that a the 2-D surface in 4-D space-time is deformation known as third Reidermeister move. Physicists talk about Yang-Baxter equation (YBE) and it says that it does nothing for the topology. YBE tells that it does nothing to the quantum staet.
One can however assume that "doing nothing" is replaced with what is called 2-morphism. "Kind of gauge transformation" takes place would be the physicist's first attempt to assign to this something familiar. The outcome is unitarily equivalent with the original but not the same anymore. This actually requires a generalization of the notion of group to quantum group. Braid statistics emerges: the exchange of braid strands brings in phase or even non-commutative operation on two braid state.
Tetrahedral equation generalizes "Yang-Baxterator" so that it is not an identity anymore but becomes what is called 2-morphism. One however obtains an identity for two different combinations of 4 Reidemeister moves performed for 4 strands instead of 3. To make things really complicated one could give up also this identity and consider next level in the hierarchy.
What makes this so interesting that in 4-D context of TGD that also 2-knots formed by 2-D objects (such as string world sheets and partonic 2-surfaces) in 4-D space-time become possible. Quite generally: D-2 dimensional things get knotted in D dimensions. I have proposed that 2-knots could be crucial for information processing in living matter. Knots and braids would represent information such as topological quantum computer programs, 2-knots information processing such as developing of these programs.
In TGD one would something much more non-trivial than Reidermeister moves. The ordinary knots could really change in the operations represented by 2-knots unlike in Reidermeister moves. 2-knots/2-braids could represent genuine modifications of 1-knots since the reconnections at which knot strands can go through each other could open the knot partially or make it more complex (remember what Alexander the Great did to open the Gordion knot). The process of forming of knot invariant means gradual opening of knot in systematic stepwise manner. This kind of process could take in 4-D and be represented by string world sheet and corresponding evolution of quantum state in Zero Energy Ontology (ZEO) would represent opening of knot.
One of the basic questions in consciousness theory is whether problem solving could have as a universal physical or topological counterpart. A crazy question: could opening of 1-knot - a process defining 2-knot- serve as the topological counterpart of problem solving and give rise to its quantal counterpart in ZEO? Or could Reidermeister moves transforming trivial knot to manifestly trivial form correspond to problem solving. It would seem that the Alexandrian manner to solve problems is what happens in the real world;-).
What about higher-D knots? 4-D space-time surfaces can get knotted in 6-D space-times. If the twistorialization of TGD by lifting space-time surfaces to 6-D surface in the product of twistor spaces of Minkowski space and CP2 makes sense then space-time surfaces have representations as 4-surfaces in their 6-D twistor space. Could space-time surfaces get 4-knotted in twistor-space? If so, poor space-time surface - classical world - would be in really difficult situation!;-). By the way, also light-like 3-surfaces representing parton orbits could get knotted at the 5-D boundaries of 6-D twistor space regions assignable to space-time regions with Euclidian or Minkowskian signature!
For a summary of earlier postings see Links to the latest progress in TGD.
16 comments:
Let's remember also consciousness side of TGD and that observation events of 2D objects happen in 3+1 dimensions. And that 3n+1 with dividing by two forms the highly interesting Collatz Conjecture and its Unity.
As a side (or rather square!) note, the spread of tetrahedron's center's normals to the faces (cf bonds of tetrahedral methane molecule) is exactly 8/9, which is very nice rational result, compared to the very approximate "real" angle of standard trigonometry:
https://www.youtube.com/watch?v=QJP9Gn1Kzk4
The ratio of two cubed per three squared seems very basic symmetry here, with extra dimension of a cosmic joke... :)
I forgot to check why the equation is called tetrahedral. Presumably this relates to the four third Reidermeister moves (move crossing below strand) which can be done in two different manners for 4 braid strands and are assumed to produce same result. If I am not totally wrong modified YB means giving up strict associativity.
I dare to swear that the sines and cosines of the angled are algebraic numbers. From rationals to algebraics to reals and p-adics.
Yang-Baxter equation is called also "triangle equation", generalizing it to 3D is most likely the motivation for name "tetrahedral equation".
Algebra is a matter of definition, and the ability to communicate math - and mathematical physics - depends from clear definitions that leave no room for ambiguity.
So, what does it mean when it is said that the parameter u of YBE "usually ranges over real numbers"? Does that include also non-computable real numbers (by Axiom of Choice) or just algorithmic (which includes algebraic, but is probably not limited to) numbers?
YBE is nice illustration of Bishop Berkeley's famous quote "two wrongs that make right", but in the social field the current non-communicating state of math and physics has strong parallel with medieval priests giving sermons in scholastic monk Latin to audience who don't understand a word, or if they do, shout like the little boy: "Emperor has no clothes!" (ie, is dressed up in ad-hoc Axioms only ;)). That was Berkeley's main criticism: that physics of Newton and Leibnitz (and then on...) is just as mystical and axiomatic and incommunicado as doctrines of Church. Cf. "Credo quid absurdum est" etc.
Of course it's too much of a task for a single person to go over not just all of physics, but also all of math, to develop a consistently communicating ToE, and given the framework one has been forced to work in by standard indoctrination it's understandable to resort to "against rigour" retorts, while at the same time complaining about general communication problems in the physics community.
It's not a personal issue or an issue of Credo, as long as we idealize that math and physics is or should be Discourse instead of Verbum from Authority, ability to communicate remains the crux of ethics and aesthetics of theory formulation, and a constant challenge where one can lead only by example.
Of course, ability to communicate does not involve only cutting through the intricate knot web of lies and half truths with sword of rigorous induction, or some other sword of choice (sic!), but even more so, tying up new and even more intricate riddles for future discourse to solve... :)
Probably you refer with u to velocity like parameter - I hope this is the case. One considers scattering in 2 dimensions in integrable QFT scattering is almost trivial. Particles are characterized by velocities. They go past each other and state changes. They can also "exchange" velocities. The S-matrix describe dynamical braiding as passing by: world lines cross as in a knot diagram defined by the projection of knot to plane. The original YB gives braiding only at the limit of vanishing velocities for which ratios are kept constant.
Physicists are rather practical people and do not consider the notion of real number when discussing YB;-).
Concerning appeal to authority I beg to disagree! There are periods when opportunism wins and authority of scientific instituation is misused. But in long time scale science is self correcting process.
I am just refreshing my tiny understanding about conformal field theories (CFTs) since I got an idea about how relate the basic vision about vacuum conformal weights of particles to existing CFT mathematics. I cannot but admire what these mathetmatical physicists have achieved and I feel myself hopelessly stupid: this what I try to understand is definitely not any monk latin.
This mathematics will certainly have marvellous applications sooner or later. And it already has. My attitude is definitely not any appeal to authority. Some humans have access to incredible mathematical intelligence and I am irritated that I do not have it!
RUu) is from the wiki article on YB, which says that u generally refers to "parameter" of parameter dependent YB. I can't comment on the physical interpretation of the parameter, just bring up the very general problem of what is math, what is physics, and how do they interrelate - in communicable way, if they do. Wiki, as usual, is not very helpful clarifying if e.g. YB is math, physics, both, neither, something else.
I don't expect there is definitive final answer to the general question, but perhaps it is not too much to ask that a ToE which includes also consciousness theory makes honest and ambitious enough effort to give theory dependent answer to the question of it's foundation, relation of math and physics, and what we mean or should mean by both concepts. Maybe the question could be formulated in terms of a basic math concept, does ToE (e.g. TGD) have an Identity, which defines "self" as "self"?
It is not enough to state, for example, that physics "reduces to number theory", when it's unclear which number theory of all possibilities is referred to. Peano Axioms and their IMO highly problematic approach to definition of identity? Some yet assumed, but not yet formulated number theory? Some general requirements for a kosher number theory? TGD makes bold attempt towards WCW, but what about similar "World of Number Theories"?
Doing mathematical physics at very high level of abstraction may be smiled upon benignely - and also amusedly - by Mother Nature, and Her numerological jokes could be taken as good signs (M89 is btw the 10th Mersenne prime), but in terms of mortal communication, if and when there are fundamental problems at any step (axiomatics, basic definitions, etc.) on which the highly abstracted levels of math are based, we can talk of self-deception - and the claimed infinite human potential in that field. So, maybe a honest and good ToE needs to be able to self-deceive, also... in addition to having unbreakable self-confidence. :)
PS: AFAIK, after Berkeley's crushing criticism, notion of continuum and analysis were refounded on theory of limits. Here's illustrative criticism of modern theory of limits and it's communicative problems, using Collatz Conjecture:
https://www.youtube.com/watch?v=Ek0URXLCZCE
I checked this and the parameter indeed corresponds to velocity or more precisely the additive parameter characterizing Lorentz boost allowing to achieve the velocity. Velocity is additive only in non-relativistic regime. This parameter is additive also relativistically. Hyperbolic angle is the interpretation. Hyperbolic rotation angle.
Geometric interpretation is following. When the velocity parameter is same, all strands are parallel as world-lines in Minkowskian plane. When velocities are not same, the world lines intersect. Same for ordinary braids which one thinks to be located in Euclidian plane and allows that they are parallel except in the regions where crossings occur.
One consider restriction of the values of the parameter u to rationals or algebraic extensions of rationals and I have proposed that in cosmology the quantization of cosmic recession velocity defining cosmic redshift corresponds to
a lattice defined by some tesselation of 3-D hyperbolic space. Similar quantization is possible in the case of Yang-Baxter. Allowed boosts would form a discrete subgroup of Lorentz group. Nothing in topology of braids would change since topology allows braids to have any form obtained with temporarily cutting the braid strands.
I hope that I have not said that physics reduces to number theory!;-). It is unpleasant to publicly disagree with some old me! It would be more realistic see number theoretic vision and geometric vision about physics as two aspects of it.
Everything of course depends also on what one calls number theory and what one calls geometry. TGD forces to
generalize these notions: world of classical worlds (WCW) emerges at the level of geometry. One can also define local p-adic variant of Riemann geometry but not global one. Infinite primes are new at the level of number theory. And adelic physics fusing real and p-adic geometries. One can also define geometric objects purely in terms of their symmetries (Erlangen program) about which finite geometries involving finite fields are a good example.
The most significant thing is that p-adic and real physics can be fused to together along common rationals, or along any algebraic extension of rationals. Originally I thought that common rational points of real and p-adic space-time surfaces would represent intersection of realities and p-adicities. This led to problems with fundamental symmetries.
Now I think that the intersection is much more abstract and relies on partonic 2-surfaces serving as space-time genes. They provide the data and strong form of holography allows to continue this data to 4-D preferred extremals of Kahler action. These 2-D surfaces are labelled by coordinates of WCW and these are rational or in some extension of rationals. WCW is discretized: space-time surfaces are however smooth and continuous. This discretization zillion times more elegant than the primitive attempts to replace space-time with lattice. Disgusting! Even I have considered this kind of possibility!
Returning to the basic problem of p-adicization. Purely p-adic geometry does not allow distances or areas naturally since p-adic definite integral is problematic. One can only speak about angles and lengths of vectors. p-Adic variant of metric makes sense as purely local notion. Conformal geometry seems natural p-adically.
What is important that in adelic vision one can define p-adic volumes as algebraic continuations from real sector. If say string world sheet has a real area expressible in terms of its WCW coordinates then also corresponding p-adic string world sheet - its cognitive representation - has p-adic valued area obtained by algebraic continuation of the WCW coordinates from reals to p-adics.
p-Adics are made for cognition and reals for sensory experience and these two aspects of experience do not reduce to each other although p-adics represent the real physics as cognition must do!
I am not speaking about infinite human potentials. I do not believe that we are at the top of hierarchy. We can however get in contact with higher levels. This explains why organisms with practically the same genes as apes have can do conformal field theory. The theory builder is entire Universe and our task is to entangle with higher levels.
The recent problems in theoretical physics are due to forgetting this humble attitude- a direct consequence of the physicalist view locating consciousness to human brain. The worst thing to do is to collect a committee which decides that some school has reached the final truth. We did even this- superstrings.
I must say that Wildbergers arguments look circular to me. If mathematician find nothig against the definition of limit, he has not understood the problems that Wilderger claims to be involved. I do believe that mathematicians are not cheating when they say that there are no problems involved.
W seems to demand that the mathematics of reals should be performable by computers and this is is certainly not possible. If we are going to do real number based mathematics we must accept the existence of transcendentals and that at least for our consciousness they are indeed transcendent. We cannot base mathematics on our cognitive restrictions or on what computers can do.
It is good if W is able to generate revolutionary attitude: it helps real learning. I do not however agree with W. I remember that as first year students we were always talking about the lack of rigor! It is good to be critical but much better attitude is the creative one. I would be really happy if I saw young people bursting with ideas also at theoretical physics department of Helsinki University.
I agree that (math of) classical physics cannot restrict number theory and computability, not least because of quantum theory and computation. I don't know what our cognitive restrictions are, if there are any. A good theory opens up cognitive potential.
The revolutionary attitude is very enjoyable, and rational trigonometry succeeds in revealing beautiful relations that have so far remained hidden under the hood of reals. 8/9 is so much more pleasant to our aesthetic sense than a string with three dots.
I think the healthiest criticism is that current definition of reals hopelessly mixes up (finite) algorithmic numbers and (infinite) noisy strings by Axiom of Choice - to pretend to have solved the fundamental big problem of mathematical continuity with "real line" - and that leads to huge problems at most fundamental level of logic and computability, also for quantum computation, or in a word: communication.
So, when a mathematician says e.g. that "there are more transcendentals than any other numbers (but we basically know only pi and e of those)", he is speaking of purely axiomatic construct that mixes up apples and oranges.
As you say, it's important not to throw away the baby with the washwater, but creatively nourish and raise the baby. W gives presents his own approach to limit here, the new stuff starts roughly half way:
https://www.youtube.com/watch?v=K4eAyn-oK4M
The comment section had link to paper discussing Cantor and Kroenecer and Dedekind, which clarifies beautifully the issue, and touches qualities of "primeness" in a way that is central and critical also to your own approach, not least for your notion of infinite primes:
http://mcps.umn.edu/philosophy/11_5Edwards.pdf
Should we lend some sympathy to the intuitionist position, we can ask what possibilities does emerging quantum computation, and more specifically, TGD view of quantum computation, open up to intuitionist criterion of 'counting'?
http://mathoverflow.net/questions/26549/is-there-much-difference-between-kroneckers-and-dedekinds-methods-in-algebraic
It is good to add here Wikipedia definition of intuitionism.
"In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality."
It is perhaps needless to tell my view. To me intuitionism is hopelessly limited approach. If we would apply the intuitionism as apes apply it we would have three numbers: 1, 2 and many. That would be the whole mathematics.
Harsh :). Never the less, your response does not do justice to the real issue. What is more "constructivist" than ad hoc axiomatics and ill defined set theory of infinite sets of the Cantor/Dedekind approach? It's pure constructivism, just very bad constructivism, deceitful human distortion of what is presumably be (in the) mind of God of Platonia. Not the light brought by Prometheus, but the sack of ills opened by his lesser brother Epimetheus.
The issue at heart is countability, and constructivism of intuitionism, as the word suggests, can very well include communication between various states of mind, including "normal" state with mathematically more evolved states of mind. This way there is no need to presuppose any external and objective "Platonia", and the process remains dynamic and evolutionary.
In this sense the intuitionist position is that we participate in the Constructive and Evolutionary God of Platonia (cf. dynamic holography) in creative way. To clarify further, intuitionist wants God of Platonia to be fully negentropic, where all parts of construction communicate with each other in harmonious and consistent way, instead of entropic noise of Axiom of Choice "numbers" that don't compute in any way, etc ad hoc axiomatics. Look at the child who tries to run with continuum by completion axiom, before learning to walk, face in the mud.
In this sense, intuitionism is the humble position that bad math is bad theology, while "objective Platonism" or what ever projects it's own sins to externalized God. The Blind God.
For intuitionist holographic theology, computability is of course not a fixed limit, but dynamically evolving potentiality, both quantitavely and qualitatively. Intuitionist takes problem of computability honestly and humbly, fully accepting his responsibility for participatory constructivism, and doesn't shove problems under the carpet just show off an Emperor of Continuum clad in nothing but ad hoc axioms etc. make-believe. Emperor who does not compute and communicate with the means we have available, but preaches like hell.
Following standard approach, you apparently claim that adeles are way to combine p-adics and reals, not just algebraic numbers but the whole "completion", but with my very limited understanding, I remain highly skeptical of taking tensor products of those real numbers that by defintion, are so far non-computable not-even-algorithmic-processes.
More precisely, following wiki, meaning of "restricted product" remains mystery to me, I don't know if tensor product is supposed to play role in terms of Adele ring of completions of rationals, as it does with ring of rational adeles. So the claim that "The ring of (rational) adeles can also be defined as the restricted product of all the p-adic completions Qp and the real numbers" remains highly dubious mystery.
Hope you can clarify this issue, and the role of computability here. As it seems, the real problem here is not algebraic numbers, but the notion of "completion" related to "real line" and dream of mathematical contimuum.
As I have explained so many times I find difficult to understand the idea about a secret plot of bad mathematicians do destroy civilization by all the evil produced by real numbers. Our mathematic and physics is a result of long and merciless selection process for ideas - essentially evolution of consciousness. If one does not accept this one must introduce really heavy arguments against it or even propose something better.
Intuitionalists refuse to realize that most of understanding is something which is not a result of computation representable numerically. The idea of computer catches however vanishingly small part of mathematical consciousness. Intuitionists should think seriously what we understand about consciousess.
I am not able to get emotional with these issues: I find much more interesting to participate the flow of ideas and see how they help to understand this world in which we live - this posting was a documentary about this magical explanatory power - rather than fight about whether reals exist or not. If someone wants to destroy my theory by claiming that the real numbers do not exist (this is certainly the most original objection against TGD: congratulations!), this is completely ok for me and does not spoil my piece of mind. I am just a little droplet in the big flow and the existence or non-existence of reals is not my responsibility.
I agree with Matti, i never consider the reality or nonreality of numbers when doing math, thinking of physics, or programming computers or trading the stock market..
There is no secret plot, e.g. Horkheimer's 'Critique of Instrumental Reason' explains the issue clearly enough, in this context submission of theoretical physics and pure mathematics in service of applied sciences, with the cost of logical consistency.
Notion of "real line continuum" works well enough for certain engineering etc. applied purposes, and it's main motivation derives from there, which should be obvious for any historian, not from ideals of logical consistency that more "formalistic" sciences still largely claim to adhere to.
The question is about both keeping your cake and eating it, in field which to this day remains dominated by bivalent logic. The axiomatic construction of real line continuum means that most of the stuff on the line consists of non-algorithmic strings of "numbers". Finite "pinary" strings is OK idea, but even it doesn't solve the basic issue, choice between Computability _OR_ Continuum. By "computability" in this context I'm not referring to classic computers etc., but the all important aspect of communication, that others too can compute and verify the results of new theoretical concepts, based on accumulating logical consistency of the self-correcting social theoretical process.
Some more higher lever abstractions are less dependent from fundamental structures than others, and can survive independent of them. But e.g. notion of 'ring', as far as I know, is by definition based on computability of it's elements. If this is not so, please feel free to correct me.
So here again the very simple question by a simple student, if most points on real line are non-algorithmic strings that don't compute even with each other, how can they be elements of an adele ring?
And another question, the motivation for dichotomy between "cognitive" and "sensory" is not at all clear to me, care to elaborate?
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