Langlands' program seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Langlands program is described by Edward Frenkel as a kind of grand unified theory of mathematics. In the TGD framework, $M^8-M^4\times CP_2$ duality assigns to a rational polynomial a set of mass shells $H^3$ in $M^4\subset M^8$ and by associativity condition a 4-D surface in $M^8$, and its it to $H=M^4\times CP_2$. $M^8-M^4\times CP_2$ means that (...) number theoretic vision and geometric vision of physics are dual or at least complementary. This vision could extend to a trinity of number theoretic, geometric and topological views since geometric invariants defined by the space-time surfaces as Bohr orbit-like preferred extremals could serve as topological invariants. Concerning the concretization of the basic ideas of Langlands program in TGD, the basic principle would be quantum classical correspondence (QCC), which is formulated as a correspondence between the quantum states in the "world of classical worlds" (WCW) characterized by analogs of partition functions as modular forms and classical representations realized as space-time surfaces. L-function as a counter part of the partition function would define as its roots space-time surfaces and these in turn would define via Galois group representation partition function. QCC would define a kind of closed loop giving rise to a hierarchy. If Riemann hypothesis (RH) is true and the roots of L-functions are algebraic numbers, L-functions are in many aspects like rational polynomials and motivate the idea that, besides rationals polynomials, also L-functions could define space-time surfaces as kinds of higher level classical representations of physics. One concretization of Langlands program would be the extension of the representations of the Galois group to the polynomials $P$ to the representations of reductive groups appearing naturally in the TGD framework. Elementary particle vacuum functionals are defined as modular invariant forms of Teichm\"uller parameters. Multiple residue integral is proposed as a manner to obtain L-functions defining space-time surfaces. One challenge is to construct Riemann zeta and the associated $\xi$ function and the Hadamard product leads to a proposal for the Taylor coefficients $c_k$ of $\xi(s)$ as a function of $s(s-1)$. One would have $c_k= \sum_{i,j}c_{k,ij}e^{i/k}e^{\sqrt{-1}2\pi j/n}$, $c_{k,ij}\in \{0,\pm 1\}$. $e^{1/k}$ is the hyperbolic analogy for a root of unity and defines a finite-D transcendental extension of p-adic numbers and together with $n:$th roots of unity powers of $e^{1/k}$ define a discrete tessellation of the hyperbolic space $H^2$. This construction leads to the question whether also finite fields could play a fundamental role in the number theoretic vision. Prime polynomial with prime order $n=p$ and integer coefficients smaller than $n=p$ can be regarded as a polynomial in a finite field. If it satisfies the condition that the integer coefficients have no common prime factors, it defines an infinite prime. The proposal is that all physically allowed polynomials are constructible as functional composites of these. (shrink)

The item is the handout for a seminar session held on 21.09.2021 at the Dpt. of Philosophy of the University of Navarre, Pamplona, Spain. The presentation approaches the question "what is the main reason for the lack of harmony between Physics and Christian theology". This presentation defends the view that the main reason for the lack of harmony is that the epistemological stances are opposed to each other. That opposition can be reduced by developing a conceptual frame that is free (...) of three intrinsic limitations of Physics. This frame is intimately linked to Physics. It does not alter Physics, but gives to Physics an epistemological outlook closer to that of Christian theology, when the latter speaks of the material world. (shrink)

The item is a handout for a seminar session held on 21.09.2021 at the Dpt. of Philosophy of the University of Navarre, Pamplona, Spain The presentation approaches the question "Why is there Mathematics in Physics" from the starting point of reductionisms in Physics. If these reductionisms are omitted, the consequences drawn from "prescientific" experience turn out to be the beginning of answering the said question.

This article was inspired by the inverse problem of Galois theory. Galois groups are realized as number theoretic symmetry groups realized physically in TGD a symmetries of space-time surfaces. Galois confinement as an analog of color confinement is proposed in TGD inspired quantum biology. Galois groups, in particular simple Galois groups, play a fundamental role in the TGD view of cognition. The TGD based model of the genetic code involves in an essential manner the groups A5, which is the smallest (...) non-abelian simple group, and A4. The identification of these groups as Galois groups leads to a more precise view about genetic code. The question why the genetic code is a fusion of 3 icosahedral codes and of only a single tetrahedral code remained however poorly understood. The identification of the symmetry groups of the I, O, and T as Galois groups makes it possible to answer this question. Icosa-tetrahedral tesselation of 3-D hyperbolic space H3, playing central role in TGD, can be replaced with its 3-fold covering replacing I/O/T with the corresponding symmetry group acting as a Galois group. T has only a single Hamiltonian cycle and its 3-fold covering behaves effectively as a single cycle. Octahedral codons can be regarded as icosahedral and tetrahedral codons so they do not contribute to the code. (shrink)

Herein, I review the shattering moment in my life when I awoke from the dream of self to find being as part of the living world and not in my head, discovering my perspectival center to be literally everywhere. Since awakening to the world takes one beyond thought and language thus also beyond the symbolic construction of time, it is strange to place this event and its aftermath as happening long ago in my life. It is forever present. This fact (...) puts into question the reality of my daily journey from dawn to dusk with all the mundane tasks I must complete. My linear march into aging and death inexorably continues, yet it seems somehow unreal, worth a smile as the inevitable changes ensue. Still, I write of the events leading up to my time out of mind and then review the serious repercussions that followed when I was drawn back into ego only to find I did not have the conceptual tools or the maturity to understand what had occurred. I close by looking back with theories that might explain what happened. I am now ready to allow the memory to sink into peaceful oblivion and reference it from within my mind no more. Ironically, the memory itself with its façade of knowledge may prevent me from a new, unexpected mystical experience. Only by forgetting can I hope to leave a crack in the verbal armament of self, so the world soul may break through and free me once again. (shrink)

This article was inspired by the inverse problem of Galois theory. Galois groups are realized as number theoretic symmetry groups realized physically in TGD a symmetries of space-time surfaces. Galois confinement is as analog of color confinement is proposed in TGD inspired quantum biology. Two instances of the inverse Galois problem, which are especially interesting in TGD, are following: Q1: Can a given finite group appear as Galois group over Q? The answer is not known. Q2: Can a given finite (...) group G appear as a Galois group over some EQ? Answer to Q2 is positive as will be found and the extensions for a given G can be explicitly constructed. (shrink)

In the TGD Universe space-time surfaces within causal diamonds are fundamental objects. 1. M8 − H duality means that one can interpret the space-time surfaces in two manners: either as an algebraic surface in complexified M8 or as minimal surfaces in H = M4×CP2. M8 − H duality maps these surfaces to each other. 2. Minimal surface property holds true outside the frame spanning minimal surface as 4-D soap film and since also extremal of K¨ahler action is in question, the (...) surface is analog of complex surface. The frame is fixed at the boundaries of the CD and dynamically generated in its interior. At frame the isometry currents of volume term and K¨ahler action have infinite divergences which however cancel so that conservation laws coded by field equations are true. The frames serve as seats of non-determinism. 3. At the level of M8 the frames correspond to singularities of the space-time surface. The quaternionic normal space is not unique at the points of a d-dimensional singularity and their union defines a surface of CP2 of dimension dc = 4 − D 0 phases associated with real polynomials as approximations of real analytic functions would correspond to a hierarchy of inclusions of hyperfinite-factors of type II1 realized as physical systems and giving rise to finite cognition based on finite-D extensions of rationals and corresponding extensions of p-adic number fields. 3. The construction of 2-D periodic minimal surfaces inspires a construction of minimal surfaces with a temporal periodicity. For Λ > 0 this happens by gluing copies of minimal surface and its mirror image together and for Λ = 0 by using a periodic frame. A more general engineering construction using different basic pieces fitting together like legos gives rise to a model of logical thinking with thoughts as legos. This also allows an improved understanding of how M8 − H duality manages to be consistent with the Uncertainty Principle. 4. At the physical level, one gains a deeper understanding of the space-time correlates of particle massivation and of the TGD counterparts of twistor diagrams. Twistor lift predicts M4 K¨ahler action and its Chern-Simons implying CP breaking. This part is necessary in order to have particles with non-vanishing momentum in the Λ = 0 phase. (shrink)

The idea of human reason’s positivist self-limitation is used by Benedict XVI and others to characterize an aspect of the long-lasting intellectual situation of Western technological-scientific civilization. Liberating human reason from its positivist self-limitation requires, in general, an overcoming of the historical process of Natural Science’s drifting away from Philosophy. In the case of Physics, it requires a point where both physics and philosophy have to deal together with the same problem. This paper first identifies a problem caused by specific (...) reductionisms in Physics. These reductionisms cause certain deformations of physical knowledge, which in turn makes it desirable for physicists to dispose of an assessment of them. The paper then proposes specific steps in philosophically assessing these reductionisms. Such an assessment in turn is based on common experiential knowledge which is not restricted by any reductionisms. That excludes experiments and, thus, cannot be done with physical means, but only with philosophical ones. All this already constitutes a grain of sand of human reason’s liberation from its positivist self-limitation. It is not any imposition from outside physics, but a desideratum from inside physics. The second purpose of this paper is to briefly present the main ideas of that assessment. Implementing its consequences would bring about an epistemological mindset in Physics as a whole that is open to natural theology. Furthermore, it is suited to mitigate, or even eliminate, a certain quasi-contradiction in a physicist’s mind and professional work. To show that is the third purpose of this paper. Precisely these two issues offer a certain flanking aid to, though not a part of, the Evangelization. More specifically, that flanking aid consists in offering the epistemological mindset of Natural Realism and can be circumscribed by four aspects: the first is a general corroboration of the stance that Natural Realism is the true form of man’s relationship to reality; the second is a sort of contemplative mindset; the third is the elimination or the mitigating of the quasi contradiction referred to in the preceding paragraph, and which can be called ‘unity of life’. The fourth aspect is a ripe fruit of the three aforementioned: an uncommon quietness and serenity of the spirit. (shrink)