20th January, 2011.
E-infinity communication No. 74
NAMING E-INFINITY: Some aphorisms, quotations and remarkable historical events connected to the science of the infinite transfinite set theory and the theory of dimensions as well as the relation to religion and God.
1. In the beginning was the word and the word was with God and the word was God (The Gospel according to St. John, first verse).
2. “I have no need for this hypothesis”: This was the answer of Laplace to Napoleon’s question about why God did not appear in his work.
3. Philosophy is a battle against the bewitchment of our intelligence by means of language. (Wittgenstein – Philosophical Investigation).
4. Heisenberg “discontinuous” quantum mechanics was a giant leap forwards. Schrödinger’s continuity illusion of his differential equation was effectively a step backwards. (Mohamed El Naschie rephrasing words of A. Connes).
5. Not how the world is is the mystical but that it is. (Wittgenstein – Tractatus).
6. The name that can be named is not the eternal name. (Tao Te Ching).
7. In a fractal spacetime setting we must replace differentiation and integration with Weyl- Suslin scaling. In E-infinity the main scaling as well as the main renormalization semi group is the golden mean scaling and the golden mean renormalization group of M. Feigenbaum. (Mohamed El Naschie about E-infinity).
8. Dimension is a scale dependent phenomenon. (B.G. Sidharth about fractal spacetime).
9. Neither K. Gödel’s proof of the consistency of the continuum hypothesis with the axioms of set theory, nor mine of its independence from them was the final answer….but I think there is no answer other than the answer that it is undecidable. (P. Cohen).
10. The Cantor space c is the unique up to homeomorphism perfect non-empty compact, zero-dimensional space. (Text book definition of the Cantor space 2 to the power of Baire space used by El Naschie in E-infinity).
11. E-infinity theory is a weighted Borel hierarchy. (El Naschie following an idea due to J.A. Wheeler).
12. In Batz we walked by the sea …. It was there that Pavel Urysohn wrote his famous paper on countable connected Hausdorff spaces …. On August 17 out for another swim …. Urysohn was catapulted by a wave directly onto the rocks …. Urysohn was buried in Batz-sur-Mer… (Prof. Pavel Alexandrov remembering Pavel Urysohn, the discoverer (or inventor) of the transfinite deductive theory of dimensions and that the empty set has a topological dimension equal minus one ( ̶ 1)).
13. The Moscow School of Mathematics founded by Egorov and Luzin is one of the most important sources of information and inspiration for E-infinity theory of high energy physics. On the deep level of trying to understand quantum mechanics it is at best misguided and at worse childish and naïve to make a separation between mathematics and physics. (Mohamed El Naschie in one of his recent lectures 2009-2010).
14. For my work the most important members of the Russian (Moscow School of Mathematics) are Urysohn, Kolmogorov, Suslin, Gel’fand, Arnold, Alexandrov, Shinchin, Pontryagin and Sinai. (Mohamed El Naschie 2009).
15. Prof. Pavel Florensky was convinced that the nineteenth century was an intellectual disaster …. because of the concept of ‘continuity’ …. because of the strength of differential calculus with many practical application problems that could not be solved this way were ignored – essentially discontinuous phenomena (such as quantum mechanics). Florensky (member of the Moscow School of Mathematics) wanted to restore discontinuity to its rightful place in the “Weltanschaung”. (Graham-Kantor-El Naschie 2010).
16. Everything visible is connected to the invisible … the sensible to the nonsensible. Perhaps the thinkable to the unthinkable. (Novalis – fragment).
17. Anaxagoras conceived the infinite in the same way as Anaximander. He called the infinite (Apeiron) which is the primodal mixture of chaos. (P. Zellini).
18. von Neumann’s formulation of the question was adopted by Kurt Gödel. There, as in E- infinity, we distinguish between set and class. (Mohamed El Naschie).
19. In E-infinity we solve a measure problem by summing over infinitely many but countable Cantor sets. However every Cantor set has uncountably infinitely many points. That is how we arrive at finite expression for a completely wild situation as far as computations are concerned. (Mohamed El Naschie 2009).
20. If we went back to the most perfect image of the word soaring to the level of an invisible being, it would mean recalling the Hindu Vāc. In E-infinity the VAK is the vague attractor of Kolmogorov which is conjectured by R. Thom to be the stationary states of quantum mechanics and used by Mohamed El Naschie to calculate the mass spectrum of elementary high energy particles. The Logos of the Greeks is analogous to the Hindu Vāc. (P. Zellini and Mohamed El Naschie).
21. There is strictly speaking no such thing as mathematical proof. (G.H. Hardy).
22. It is obvious then that the mathematical problem of the infinite is automatically projected into the moral sphere …. This is clear in the work of F. Nietzsche and R. Musil. (P. Zellini and Mohamed El Naschie).
23. My fuzzy K3 as well as all the transfinitely corrected Betti numbers, Euler invariant and instanton numbers and curvatures as well as invariant dimensions are intuitive extensions of cohomology to the theory of fiber bundles. Without knowing, not being a mathematician, I extended cohomology in the same way as Grothendieck and Atiyah extended cohomology using K-theory. In other words my K3 and all the fuzzy manifolds are a product of a non-declared K-theory which was christened transfinite E-infinity theory. (Mohamed El Naschie 2010).
22nd January, 2011
E-infinity Communication No. 75
Fake R(4) and exotic Milnor seven Spheres S(7) in the fuzzy or average knot Yang-Mills instantons of E-infinity
Donaldson fake R(4) was considered in the work of El Naschie in E-infinity quite early on. A little later he considered the exotic Milnor seven spheres. In a paper published in CS&F6, 19 (2004), pp. 17-25 influenced by the work of El Naschie entitled “On Milnor seven dimensional sphere, El Naschie E-infinity theory and energy of a Bianchi universe” by Gamal Nashed of Ainshams University in Cairo, Egypt the particular relation between exotic geometry and E-infinity was discussed and an interesting summary was given in a very nice illustrative form in Fig. 1 on page 23. Also following El Naschie, Nashed made important use of the maximum sphere surface area and maximum sphere volume given in his figures 2 and 3 on page 24. El Naschie remarked that Nash formula gives a seven sphere for an Euclidean embedding of a one dimensional object because D = (0.5)(n)(3n ̶ 11) = 14/2 = 7. In addition he introduced the fractal seven dimensional sphere with the dimension 7 plus phi to the power 3, i.e. 7.23606799 which played a role in his fractal black hole theory. We recommend reading the paper entitled “Fractal black holes and information” by M.S .El Naschie, CS&F 29, (2006), pp. 23-35 and consider the explanation of Fig. 1 on page 25 and Fig. 3 on page 27. The most important conclusion of all these attempts for E-infinity research was the deep realization that the idea of moving from the factorial function to the gamma function should be generalized as done in moving from a topological dimension to a Hausdorff dimension. In fact doing this systematically one moves from classical quantum field theory to K-theory which is the mathematical realization of E-infinity theory. El Naschie proclaimed that Nottale’s idea of giving up classical differentiability and replacing it with Robinson’s non-standard analysis should be considered much deeper. El Naschie was familiar with non-standard analysis from his work on the canard of catastrophe theory. Therefore he was convinced that moving to Nottale’s frame work is a first step. The second step was to move to exotic ‘differentiability’. However this was not sufficient in his view and that is when he moved to point set geometry with cardinality equal to that of the continuum and that is how he arrived at Cantor sets and Cantorian spacetime of E-infinity theory.
It follows then that Yang-Mills theory must be modified to account for the true transfinite nature of high energy particle physics. This modification is what most probably inspired ‘t Hooft recently to include a dilaton field in his quantum field theory while hoping to refine classical calculus which is in principle of course possible when accepting some difficulties as the price. On the other hand random Cantor sets with their golden mean Hausdorff dimension offers natural quantization coupled with incredible computational ease due to the inbuilt golden mean number system which we explained in many previous communications. The usual mathematical way of thinking about fiber bundle theory is that we start with point set then move to a topological manifold, then smooth manifold, then geometric manifold, then bundle. We may start before point set and end beyond bundles. E-infinity is both the prior point set and the beyond bundle. Let us argue the case for an E-infinity action which is far more physical than ‘t Hooft’s S = 82 and at the same time much easier to hand, all apart from the unexpected fact that using E-infinity, hidden connection which would have passed totally unnoticed become obvious and trivially visible.
We reconsider again 82. This is obviously exactly 16 four dimensional sphere volumes. The volume of a four dimensional sphere with unit radius is as is well known, vol S(4) = 2/2. Consequently (16)( 2/2) = 8p2 = S, the action of ‘t Hooft’s Yang-Mills instanton. In E-infinity however we make a much richer relation when we take average everything. This average is a transfinite average. You could call it fuzzy values if you want. First we replace the volume of the spheres with the fuzzy hyperbolic volume of knot. We take K(82). For this knot the hyperbolic fuzzy volume is 5 ̶ ϕ4 where ϕ is the golden mean 0.618033989. Instead of taking 16 spheres we take an average of 16 + k = 16.18033989 knots of the 82 type. That way the total volume is exactly SF = ( /2) + 10 = 78.5419966. This is the value corresponding to 8p2 = 78.95683521 of ‘t Hooft. However we see here relations which we cannot see when using the classical analysis of ‘t Hooft. In particular we see that SF ̶ 10 when multiplied with 2 gives the exact theoretical inverse electromagnetic fine structure constant namely = 137 + ko = 137.082039325. From that a plethora of other relations follow, for instance (3 + ϕ)( ) = E8 E8 and remembering that E6 = 78 is already an integer approximation to S = (3)(26) = 78 we see that the net of interrelations with the exceptional Lie symmetry groups and not only SO(3) where we noted in a previous communication that volume SO(3) = 8p2. In general we can say the El Naschie fuzzy K3 is a K-theory K3 and that the transfinite Feynman diagrams of E-infinity are the equivalent of Feynman motives which was developed recently. Thus E-infinity could be called K-infinity and El Naschie’s fuzzy golden field theory is nothing else but a Grothendieck motives applied to the theory behind the standard model of high energy physics. In this sense Mohamed El Naschie was deadly right that ‘t Hooft’s dimensional regularization implies E-infinity spacetime which means noncommutative physical spacetime. The same conclusion was recently made by A. Connes. It is interesting to note that ‘t Hooft did not agree initially but he may reconsider the situation in view of the compelling E-infinity results.