What's new inTGD as a Generalized Number TheoryNote: Newest contributions are at the top! 
Year 2007 
E_{8} theory of Garrett Lisi and TGDI have been a week in travel and during this time there has been a lot of fuss about the E_{8} theory proposed by Garrett Lisi in physics blogs such as NotEvenWrong and Reference Frame, in media, and even New Scientist wrote about the topic. I have been also asked to explain whether there is some connection between Lisi's theory and TGD. 1. Objections against Lisi�s theory The basic claim of Lisi is that one can understand the particle spectrum of standard model in terms of the adjoint representation of a noncompact version of E_{8} group. There are several objections against E_{8} gauge theory interpretation of Lisi.
2. Three attempts to save Lisi�s theory To my opinion, the shortcomings of E_{8} theory as a gauge theory are fatal but the possibility to put gauge bosons and fermions of the standard model to E_{8} multiplets is intriguing and motivatse the question whether the model could be somehow saved by replacing gauge theory with a theory based on extended fundamental objects possessing conformal invariance.
3. Could supersymmetry rescue the situation? E_{8} is unique among Lie algebras in that its adjoint rather than fundamental representation has the smallest dimension. One can decompose the 240 roots of E_{8} to 112 roots for which two components of SO(7,1) root vector are +/ 1 and to 128 vectors for which all components are +/ 1/2 such that the sum of components is even. The latter roots Lisi assigns to fermionic states. This is not consistent with spin and statistics although SO(3,1) spin is halfinteger in M^{8} picture. The first idea which comes in mind is that these states correspond to superpartners of the ordinary fermions. In TGD framework they might be obtained by just adding covariantly constant righthanded neutrino or antineutrino state to a given particle state. The simplest option is that fermionic superpartners are complex scalar fields and sbosons are spin 1/2 fermions. It however seems that the superconformal symmetries associated with the righthanded neutrino are strictly local in the sense that global supergenerators vanish. This would mean that superconformal supersymmetries change the color and angular momentum quantum numbers of states. This is a pity if indeed true since supersymmetry could be broken by different padic mass scale for super partners so that no explicit breaking would be needed. 4. Could Kac Moody variant of E_{8} make sense in TGD? One can leave gauge theory framework and consider stringy picture and its generalization in TGD framework obtained by replacing string orbits with 3D lightlike surfaces allowing a generalization of conformal symmetries. HHO duality is one of the speculative aspects of TGD. The duality states that one can either regard imbedding space as H=M^{4}×CP_{2} or as 8D Minkowski space M^{8} identifiable as the space HO of hyperoctonions which is a subspace of complexified octonions. Spontaneous compactification for M^{8} described as a phenomenon occurring at the level of KacMoody algebra would relate HOpicture to Hpicture which is definitely the fundamental picture. For instance, standard model symmetries have purely number theoretic meaning in the resulting picture. The question is whether the noncompact E_{8} could be replaced with the corresponding Kac Moody algebra and act as a stringy symmetry. Note that this would be by no means anything new. The KacMoody analogs of E_{10} and E_{11} algebras appear in Mtheory speculations. Very little is known about these algebras. Already E_{n}, n>8 is infinitedimensional as an analog of Lie algebra. The following argument shows that E_{8} representations do not work in TGD context unless one allows anyonic statistics.
Could one weaken the assumption that KacMoody generators act as symmetries and that spinstatistics relation would be satisfied?
The prediction of three fermion generations by E_{8} picture must be taken very seriously. In TGD three fermion generations correspond to three lowest genera g=0,1,2 (handle number) for which all 2surfaces have Z_{2} as global conformal symmetry (hyperellipticity). One can assign to the three genera a dynamical SU(3) symmetry. They are related by SU(3) triality, which brings in mind the triality symmetry acting on fermion generations in E_{8} model. SU(3) octet and singlet bosons correspond to pairs of lightlike 3surfaces defining the throats of a wormhole contact and since their genera can be different one has color singlet and octet bosons. Singlet corresponds to ordinary bosons. Color octet bosons must be heavy since they define neutral currents between fermion families. The three E_{8} anyonic boson families cannot represent family replication since these symmetries are not local conformal symmetries: it obviously does not make sense to assign a handle number to a given point of partonic 2surface! Also bosonic octet would be missing in E_{8} picture. One could of course say that in E_{8} picture based on fractional statistics, anyonic gauge bosons can mimic the dynamical symmetry associated with the family replication. This is in spirit with the idea that TGD Universe is able to emulate practically any gauge  or KacMoody symmetry and that TGD Universe is busily mimicking also itself. To sum up, the rank 8 KacMoody algebra  emerging naturally if one takes HOH duality seriously  corresponds very naturally to KacMoody representations in terms of free stringy fields for Poincare, color, and electroweak symmetries. One can however consider the possibility of anyonic symmetries and the emergence of noncompact version of E_{8} as a dynamical symmetry, and TGD suggests much more general dynamical symmetries if TGD Universe is able to act as the physics analog of the Universal Turing machine. For more details see the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts. References [1] G. Lisi (2007), An exceptionally simple theory of everything, [2] Z. Merali (1007), Is mathematical pattern the theory of everything?, New Scientist issue 2630. [3] E_{8} . [4] J. Baez (2002), The Octonions. 
A little crazy speculation about knots and infinite primesKea told about some mathematical results related to knots.
1. Do knots correspond to the hierarchy of infinite primes? I have been pondering the problem how to define the counterpart of zeta for infinite primes. The idea of replacing primes with prime polynomials would resolve the problem since infinite primes can be mapped to polynomials. For some reason this idea however did not occur to me. The correspondence of both knots and infinite primes with polynomials inspires the question whether d=1dimensional prime knots might be in correspondence (not necessarily 11) with infinite primes. Rational or Gaussian rational infinite primes would be naturally selected: these are also selected by physical considerations as representatives of physical states although quaternionic and octonionic variants of infinite primes can be considered. If so, knots could correspond to the subset of states of a supersymmetric arithmetic quantum field theory with bosonic single particle states and fermionic states labelled by quaternionic primes.
Some further comments about the proposed structure of all structures are in order.
All this looks nice and the question is how to give a death blow to all this reckless speculation. Torus knots are an excellent candidate for permorming this unpleasant task but the hypothesis survives!
One can consider a concrete construction of higherdimensional knots and braids in terms of the manysheeted spacetime concept.
The concrete construction would proceed as follows.
For details see the chapter TGD as a Generalized Number Theory III: Infinite Primes.

How to represent algebraic complex numbers as geometric objects?I already told about the idea of representing negative integers and even rationals as padic fractals. To gain additional understanding I decided to look at Weekly Finds (Week 102) of John Baez to which Kea gave link. Fascinating reading! Thanks Kea! The outcome was the realization that the notion of rig used to categorify the subset of algebraic numbers obtained as roots of polynomials with natural number valued coefficients generalizes trivially by replacing natural numbers by padic integers. As a consequence one obtains beautiful padicization of the generating function F(x) of structure as a function which converges padically for any rational x=q for which it has prime p as a positive power divisor. Effectively this generalization means the replacement of natural numbers as coefficients of the polynomial defining the rig with all rationals, also negative, and all complex algebraic numbers find a category theoretical representation as "cardinalities". These cardinalities have a dual interpretation as padic integers which in general correspond to infinite real numbers but are mappable to real numbers by canonical identification and have a geometric representation as fractals as discussed in the previous posting. 1. Mapping of objects to complex numbers and the notion of rig The idea of rig approach is to categorify the notion of cardinality in such a manner that one obtains a subset of algebraic complex numbers as cardinalities in the categorytheoretical sense. One can assign to an object a polynomial with coefficients, which are natural numbers and the condition Z=P(Z) says that P(Z) acts as an isomorphism of the object. One can interpret the equation also in terms of complex numbers. Hence the object is mapped to a complex number Z defining a root of the polynomial interpreted as an ordinary polynomial: it does not matter which root is chosen. The complex number Z is interpreted as the "cardinality" of the object but I do not really understand the motivation for this. The deep further result is that also more general polynomial equations R(Z)= Q(Z) satisfied by the generalized cardinality Z imply R(Z)= Q(Z) as isomorphism. This means that algebra is mapped to isomorphisms. I try to reproduce what looks the most essential in the explanation of John Baez and relate it to my own ideas but take this as my talk to myself and visit This Week's Finds to learn of this fascinating idea.
The notions of generating function and rig generalize to the padic context.

Is it possible to have a set with 1 elements?I find Kea's blog interesting because it allows to get some grasp about very different styles of thinking of a mathematician and physicist. For mathematician it is very important that the result is obtained by a strict use of axioms and deduction rules. Physicist (at least me: I dare to count me as physicist) is a cognitive opportunist: it does not matter how the result is obtained by moving along axiomatically allowed paths or not, and the new result is often more like a discovery of a new axiom and physicist is evergrateful for Gödel for giving justification for what sometimes admittedly degenerates to a creative handwaving. For physicist ideas form a kind of bioshere and the fate of the individual idea depends on its ability to survive, which is determined by its ability to become generalized, its consistency with other ideas, and ability to interact with other ideas to produce new ideas. During last days we have had a little bit of discussion inspired by the problem related to the categorification of basic number theoretical structures. I have learned from Kea that sum and product are natural operations for objects of category but that subtraction and division are problematic. I dimly realize that this relates to the fact that negative numbers and inverses of integers do not have a realization as a number of elements for any set. The naive physicist inside me asks immediately: why not go from statics to dynamics and take operations (arrows with direction) as objects: couldn't this allow to define subtraction and division? Is the problem that the axiomatization of group theory requires something which purest categorification does not give? Or aren't the numbers representable in terms of operations of finite groups not enough? In any case cyclic groups would allow to realize roots of unity as operations (Z_{2} would give 1). I also wonder in my own simplistic manner why the algebraic numbers might not somehow result via the representations of permutation group of infinite number of elements containing all finite groups and thus Galois groups of algebraic extensions as subgroups? Why not take the elements of this group as objects of the basic category and continue by building group algebra and hyperfinite factors of type II_{1} isomorphic to spinors of world of classical worlds, and...yesyesyes, I must stop! This discussion led me to ask what the situation is in the case of padic numbers. Could it be possible to represent the negative and inverse of padic integer, and in fact any padic number, as a geometric object? In other words, does a set with 1 or 1/n elements exist? If this were in some sense true for all padic number fields, then all this wisdom combined together might provide something analogous to the adelic representation for the norm of a rational number as product of its padic norms. Of course, this representation might not help to define padics or reals categorically but might help to understand how padic cognitive representations defined as subsets for rational intersections of real and padic spacetime sheets could represent padic number as the number of points of padic fractal having infinite number of points in real sense but finite in the padic sense. This would also give a fundamental cognitive role for padic fractals as cognitive representations of numbers. 1. How to construct a set with 1 elements? The basic observation is that padic 1 has the representation 1=(p1)/(1p)=(p1)(1+p+p^{2}+p^{3}..) As a real number this number is infinite or 1 but as a padic number the series converges and has padic norm equal to 1. One can also map this number to a real number by canonical identification taking the powers of p to their inverses: one obtains p in this particular case. As a matter fact, any rational with padic norm equal to 1 has similar power series representation. The idea would be to represent a given padic number as the infinite number of points (in real sense) of a padic fractal such that padic topology is natural for this fractal. This kind of fractals can be constructed in a simple manner: from this more below. This construction allows to represent any padic number as a fractal and code the arithmetic operations to geometric operations for these fractals. These representations  interpreted as cognitive representations defined by intersections of real and padic spacetime sheets  are in practice approximate if real spacetime sheets are assumed to have a finite size: this is due to the finite padic cutoff implied by this assumption and the meaning a finite resolution. One can however say that the padic spacetime itself could by its necessarily infinite size represent the idea of given padic number faithfully. This representation applies also to the padic counterparts of algebraic numbers in case that they exist. For instance, roughly one half of padic numbers have square root as ordinary padic number and quite generally algebraic operations on padic numbers can give rise to padic numbers so that also these could have set theoretic representation. For p mod 4=1 also sqrt(1) exists: for instance, for p=5: 2^{2}=4=1 mod 5 guarantees this so that also imaginary unit and complex numbers would have a fractal representation. Also many transcendentals possess this kind of representation. For instance exp(xp) exists as a padic number if x has padic norm not larger than 1. log(1+xp) also. Hence a quite impressive repertoire of padic counterparts of real numbers would have representation as a padic fractal for some values of p. Adelic vision would suggest that combining these representations one might be able to represent quite a many real numbers. In the case of π I do not find any obvious padic representation (for instance sin(π/6)=1/2 does not help since the padic variant of the Taylor expansion of π/6;=arcsin(1/2) does not converge padically for any value of p). It might be that there are very many transcendentals not allowing fractal representation for any value of p. 2. Conditions on the fractal representations of padic numbers Consider now the construction of the fractal representations in terms of rational intersections of real real and padic spacetime sheets. The question is what conditions are natural for this representation if it corresponds to a cognitive representation is realized in the rational intersection of real and padic spacetime sheets obeying same algebraic equations.
3. Concrete representation Consider now a concrete candidate for a representation satisfying these constraints.

Intronic portions of genome code for RNA: for what purpose?The last issue of New Scientist contains an article about the discovery that only roughly one half of DNA expresses itself as aminoacid sequences. The article is published in Nature. The Encyclopedia of DNA Elements (ENCODE) project has quantified RNA transcription patterns and found that while the "standard" RNA copy of a gene gets translated into a protein as expected, for each copy of a gene cells also make RNA copies of many other sections of DNA. In particular, intron portions ("junk DNA", the portion of which increases as one climbs up in evolutionary hierarchy) are transcribed to RNA in large amounts. What is also interesting that the RNA fragments correspond to pieces from several genes which raises the question whether there is some fundamental unit smaller than gene. In particular, intron portions ("junk DNA", the portion of which increases as one climbs up in evolutionary hierarchy) are transcribed to RNA in large amounts. What is also interesting that the RNA fragments correspond to pieces from several genes which raises the question whether there is some fundamental unit smaller than gene. None of the extra RNA fragments gets translated into proteins, so the race is on to discover just what their function is. TGD proposal is that it gets braided and performs a lot of topological quantum computation (see this). Topologically quantum computing RNA fits nicely with replicating number theoretic braids associated with lightlike orbits of partonic 2surfaces and with their spatial "printed text" representations as linked and knotted partonic 2surfaces giving braids as a special case (see this). An interesting question is how printing and reading could take place. Is it something comparable to what occurs when we read consciously? Is the biological portion of our conscious life identifiable with this reading process accompanied by copying by cell replication and as secondary printing using aminoacid sequences? This picture conforms with TGD view about prebiotic evolution. Plasmoids [1], which are known to share many basic characteristics assigned with life, came first: high temperatures are not a problem in TGD Universe since given frequency corresponds to energy above thermal energy for large enough value of hbar. Plasmoids were followed by RNA, and DNA and aminoacid sequences emerged only after the fusion of 1 and 2letter codes fusing to the recent 3letter code. The cross like structure of tRNA molecules carries clear signatures supporting this vision. RNA would be still responsible for roughly half of intracellular life and perhaps for the core of "intelligent life". I have also proposed that this expression uses memetic code which would correspond to Mersenne M_{127}=2^{127}1 with 2^{126} codons whereas ordinary genetic code would correspond to M_{7}=2^{7}1 with 2^{6} codons. Memetic codons in DNA representations would consist of sequences of 21 ordinary codons. Also representations in terms of field patterns with duration of .1 seconds (secondary padic time scale associated with M_{127} defining a fundamental biorhythm) can be considered. A hypothesis worth of killing would be that the DNA coding for RNA has memetic codons scattered around genome as basic units. It is interesting to see whether the structure of DNA could give any hints that memetic codon appears as a basic unit.
[1] E. Lozneanu and M. Sanduloviciu (2003), Minimalcell system created in laboratory by selforganization, Chaos, Solitons and Fractals, Volume 18, Issue 2, October, p. 335. See also Plasma blobs hint at new form of life, New Scientist vol. 179 issue 2413  20 September 2003, page 16. For details see the new chapter DNA as Topological Quantum Computer.

Farey sequences, Riemann Hypothesis, tangles, and TGDFarey sequences allow an alternative formulation of Riemann Hypothesis and subsequent pairs in Farey sequence characterize so called rational 2tangles. In TGD framework Farey sequences relate very closely to dark matter hierarchy, which inspires "Platonia as the best possible world in the sense that cognitive representations are optimal" as the basic variational principle of mathematics. This variational principle supports RH. Possible TGD realizations of tangles, which are considerably more general objects than braids, are considered. One can assign to a given rational tangle a rational number a/b and the tangles labelled by a/b and c/d are equivalent if adbc=+/1 holds true. This means that the rationals in question are neighboring members of Farey sequence. Very lighthearted guesses about possible generalization of these invariants to the case of general Ntangles are made. For details see the chapter Category Theory, Quantum TGD, and TGD Inspired Theory of Consciousness".

Quantum quandariesFor long time it has been clear that category theory might provide a fundamental formulation of quantum TGD. The problem has been that category theory seems to postulate quite too many objects. The reading of Quantum Quandaries by John Baez helped to see the situation in all its simplicity.

Platonism, Constructivism, and Quantum PlatonismI have been trying to understand how Category Theory and Set Theory relate to quantum TGD inspired view about fundamentals of mathematics. I managed to clarify my thoughts about what these theories are by reading the article Structuralism, Category Theory and Philosophy of Mathematics by Richard Stefanik (Washington: MSG Press, 1994). The reactions to postings in Kea's blog and email correspondence with Sampo Vesterinen have been very stimulating and inspired the attempt to represent TGD based vision about the unification of mathematics, physics, and consciousness theory in a more systematic manner. The basic ideas behind TGD vision are following. One cannot understand mathematics without understanding mathematical consciousness. Mathematical consciousness and its evolution must have direct quantum physical correlates and by quantum classical correspondence these correlates must appear also at spacetime level. Quantum physics must allow to realize number as a conscious experience analogous to a sensory quale. In TGD based ontology there is no need to postulate physical world behind the quantum states as mathematical entities (theory is the reality). Hence number cannot be any physical object, but can be identified as a quantum state or its label and its number theoretical anatomy is revealed by the conscious experiences induced by the number theoretic variants of particle reactions. Mathematical systems and their axiomatics are dynamical evolving systems and physics is number theoretically universal selecting rationals and their extensions in a special role as numbers, which can can be regarded elements of several number fields simultaneously. For details see the last section of the chapter Category Theory, Quantum TGD, and TGD Inspired Theory of Consciousness or the article Platonism, Constructivism, and Quantum Platonism. 
Langlands Program and TGDNumber theoretic Langlands program can be seen as an attempt to unify number theory on one hand and theory of representations of reductive Lie groups on the other hand. So called automorphic functions to which various zeta functions are closely related define the common denominator. Geometric Langlands program tries to achieve a similar conceptual unification in the case of function fields. This program has caught the interest of physicists during last years. TGD can be seen as an attempt to reduce physics to infinitedimensional Kähler geometry and spinor structure of the "world of classical worlds" (WCW). Since TGD ce be regarded also as a generalized number theory, it is difficult to escape the idea that the interaction of Langlands program with TGD could be fruitful. More concretely, TGD leads to a generalization of number concept based on the fusion of reals and various padic number fields and their extensions implying also generalization of manifold concept, which inspires the notion of number theoretic braid crucial for the formulation of quantum TGD. TGD leads also naturally to the notion of infinite primes and rationals. The identification of Clifford algebra of WCW as a hyperfinite factors of type II_{1} in turn inspires further generalization of the notion of imbedding space and the idea that quantum TGD as a whole emerges from number theory. The ensuing generalization of the notion of imbedding space predicts a hierarchy of macroscopic quantum phases characterized by finite subgroups of SU(2) and by quantized Planck constant. All these new elements serve as potential sources of fresh insights. 1. The Galois group for the algebraic closure of rationals as infinite symmetric group? The naive identification of the Galois groups for the algebraic closure of rationals would be as infinite symmetric group S_{∞} consisting of finite permutations of the roots of a polynomial of infinite degree having infinite number of roots. What puts bells ringing is that the corresponding group algebra is nothing but the hyperfinite factor of type II_{1} (HFF). One of the many avatars of this algebra is infinitedimensional Clifford algebra playing key role in Quantum TGD. The projective representations of this algebra can be interpreted as representations of braid algebra B_{∞} meaning a connection with the notion of number theoretical braid. 2. Representations of finite subgroups of S_{∞} as outer automorphisms of HFFs Finitedimensional representations of Gal(\overline{Q}/Q) are crucial for Langlands program. Apart from onedimensional representations complex finitedimensional representations are not possible if S_{∞} identification is accepted (there might exist finitedimensional ladic representations). This suggests that the finitedimensional representations correspond to those for finite Galois groups and result through some kind of spontaneous breaking of S_{∞} symmetry.
3. Correspondence between finite groups and Lie groups The correspondence between finite and Lie group is a basic aspect of Langlands.
4. Could there exist a universal rational function giving rise to the algebraic closure of rationals? One could wonder whether there exists a universal generalized rational function having all units of the algebraic closure of rationals as roots so that S_{∞} would permute these roots. Most naturally it would be a ratio of infinitedegree polynomials. With motivations coming from physics I have proposed that zeros of zeta and also the factors of zeta in product expansion of zeta are algebraic numbers. Complete story might be that nontrivial zeros of Zeta define the closure of rationals. A good candidate for this function is given by (ξ(s)/ξ(1s))× (s1)/s), where ξ(s)= ξ(1s) is the symmetrized variant of zeta function having same zeros. It has zeros of zeta as its zeros and poles and product expansion in terms of ratios (ss_{n})/(1s+s_{n}) converges everywhere. Of course, this might be too simplistic and might give only the algebraic extension involving the roots of unity given by exp(iπ/n). Also products of these functions with shifts in real argument might be considered and one could consider some limiting procedure containing very many factors in the product of shifted zeta functions yielding the universal rational function giving the closure. 5. What does one mean with S_{∞}? There is also the question about the meaning of S_{∞}. The hierarchy of infinite primes suggests that there is entire infinity of infinities in number theoretical sense. Any group can be formally regarded as a permutation group. A possible interpretation would be in terms of algebraic closure of rationals and algebraic closures for an infinite hierarchy of polynomials to which infinite primes can be mapped. The question concerns the interpretation of these higher Galois groups and HFF:s. Could one regard these as local variants of S_{∞} and does this hierarchy give all algebraic groups, in particular algebraic subgroups of Lie groups, as Galois groups so that almost all of group theory would reduce to number theory even at this level? Be it as it may, the expressive power of HFF:s seem to be absolutely marvellous. Together with the notion of infinite rational and generalization of number concept they might unify both mathematics and physics! For more details see the new chapter TGD and Langlands Program.

The idea that configuration space CH of 3surfaces, "the world of classical worlds", could be realized in terms of number theoretic anatomies of single spacetime point using the real units formed from infinite rationals, is very attractive.
The correspondence of CH points with infinite primes and thus with infinite number of real units determined by them realizing Platonia at single spacetime point, can be understood if one assume that the points of CH correspond to infinite rationals via their mapping to hyperoctonion realanalytic rational functions conjectured to define foliations of HO to hyperquaternionic 4surfaces inducing corresponding foliations of H.
The correspondence of CH spinors with the real units identified as infinite rationals with varying number theoretical anatomies is not so obvious. It is good to approach the problem by making questions.
Updated vision about infinite primesI have updated the chapter about infinite primes so that it conforms with the recent general view about number theoretic aspects of quantum TGD. A lot of obsoletia have been thrown away and new insights have emerged.
For more details see the revised chapter TGD as a Generalized Number Theory III:Infinite Primes. 