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How can Nature help us compute
 SOFSEM 2006: Theory and Practice of Computer Science – 32nd Conference on Current Trends in Theory and Practice of Computer Science, Merin, Czech Republic, January 21–27
, 2006
"... Abstract. Ever since Alan Turing gave us a machine model of algorithmic computation, there have been questions about how widely it is applicable (some asked by Turing himself). Although the computer on our desk can be viewed in isolation as a Universal Turing Machine, there are many examples in natu ..."
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Abstract. Ever since Alan Turing gave us a machine model of algorithmic computation, there have been questions about how widely it is applicable (some asked by Turing himself). Although the computer on our desk can be viewed in isolation as a Universal Turing Machine, there are many examples in nature of what looks like computation, but for which there is no wellunderstood model. In many areas, we have to come to terms with emergence not being clearly algorithmic. The positive side of this is the growth of new computational paradigms based on metaphors for natural phenomena, and the devising of very informative computer simulations got from copying nature. This talk is concerned with general questions such as: • Can natural computation, in its various forms, provide us with genuinely new ways of computing? • To what extent can natural processes be captured computationally? • Is there a universal model underlying these new paradigms?
I.: Finitary Čechde Rham cohomology: much ado without C ∞  smoothness
 International Journal of Theoretical Physics
, 2001
"... Cordially dedicated to Jim Lambek, teacher, colleague and friend. The present paper comes to continue [40] and study the curved finitary spacetime sheaves of incidence algebras presented therein from a Čech cohomological perspective. In particular, we entertain the possibility of constructing a non ..."
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Cited by 7 (5 self)
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Cordially dedicated to Jim Lambek, teacher, colleague and friend. The present paper comes to continue [40] and study the curved finitary spacetime sheaves of incidence algebras presented therein from a Čech cohomological perspective. In particular, we entertain the possibility of constructing a nontrivial de Rham complex on these finite dimensional algebra sheaves along the lines of the first author’s axiomatic approach to differential geometry via the theory of vector and algebra sheaves [35, 36]. The upshot of this study is that important ‘classical ’ differential geometric constructions and results usually thought of as being intimately associated with C∞smooth manifolds carry through, virtually unaltered, to the finitaryalgebraic regime with the help of some quite universal, because abstract, ideas taken mainly from sheafcohomology as developed in [35, 36]. At the end of the paper, and due to the fact that the incidence algebras involved have been interpreted as quantum causal sets [47, 40], we discuss how these ideas may be used in certain aspects of current research on discrete Lorentzian quantum gravity. 1 The general question motivating our quest
FinitaryAlgebraic ‘Resolution ’ of the Inner Schwarzschild Singularity
, 2004
"... A ‘resolution ’ of the interior singularity of the spherically symmetric Schwarzschild solution of the Einstein equations for the gravitational field of a pointparticle is carried out entirely and solely by finitistic and algebraic means. To this end, the background differential spacetime manifold ..."
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Cited by 6 (3 self)
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A ‘resolution ’ of the interior singularity of the spherically symmetric Schwarzschild solution of the Einstein equations for the gravitational field of a pointparticle is carried out entirely and solely by finitistic and algebraic means. To this end, the background differential spacetime manifold and, in extenso, Calculusfree purely algebraic (:sheaftheoretic) conceptual and technical machinery of Abstract Differential Geometry (ADG) is employed via Sorkin’s finitary (:locally finite) poset substitutes of continuous manifolds in their Gel’fanddual picture in terms of discrete differential incidence algebras and the finitary spacetime sheaves thereof. It is shown that the Einstein equations hold not only at the finitary poset level of ‘discrete events’—as it were, when only finitely many ‘degrees of freedom ’ of the gravitational field are involved, so that no infinity or uncontrollable divergence of the latter arises at all in our inherently finitisticalgebraic scenario, let alone that the law of gravity—still modelled in ADG by a differential equation proper—breaks down in any (differential geometric) sense in the vicinity of the locus of the pointmass as it is currently maintained in the usual manifold based analysis of spacetime singularities in General Relativity (GR), but also that
When champions meet: Rethinking the Bohr–Einstein debate
, 2006
"... Einstein’s philosophy of physics (as clarified by Fine and Howard) was predicated on his Trennungsprinzip, a combination of separability and locality, without which he believed “physical thought ” and “physical laws ” to be impossible. Bohr’s philosophy (as elucidated by Hooker, Scheibe, Folse, Howa ..."
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Einstein’s philosophy of physics (as clarified by Fine and Howard) was predicated on his Trennungsprinzip, a combination of separability and locality, without which he believed “physical thought ” and “physical laws ” to be impossible. Bohr’s philosophy (as elucidated by Hooker, Scheibe, Folse, Howard, and others), on the other hand, was grounded in a seemingly different doctrine about the possibility of objective knowledge, namely the necessity of classical concepts. In fact, it follows from Raggio’s Theorem in algebraic quantum theory that within a suitable class of physical theories Einstein’s doctrine is mathematically equivalent to Bohr’s, so that quantum mechanics accommodates Einstein’s Trennungsprinzip if and only if it is interpreted à la Bohr through classical physics. Unfortunately, the protagonists themselves failed to discuss their differences in a constructive way, since in its early phase their debate was blurred by an undue emphasis on the uncertainty relations, whereas in its second stage it was dominated by Einstein’s flawed attempts to establish the “incompleteness ” of quantum mechanics. These two aspects of their debate may still be understood and appreciated, however, as reflecting a much deeper and insurmountable disagreement between Bohr and Einstein on the knowability of Nature. Using the theological controversy on the knowability of God as a analogy, Einstein was a Spinozist, whereas Bohr could be said to be on the side of Maimonides. Thus Einstein’s offthecuff characterization of Bohr as a ‘Talmudic philosopher ’ was spoton.
EARTHDANCE: Living Systems in Evolution
, 1999
"... Dancing is surely the most basic and relevant of all forms of expression. Nothing else can so effectively give outward form to an inner experience. Poetry and music exist in time. Painting and architecture are a part of space But only the dance lives at once in both space and time In it the creator ..."
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Dancing is surely the most basic and relevant of all forms of expression. Nothing else can so effectively give outward form to an inner experience. Poetry and music exist in time. Painting and architecture are a part of space But only the dance lives at once in both space and time In it the creator and the thing created, the artist and the expression, are one. Each participant is completely in the other. There could be no better metaphor for an understanding of the...cosmos. We begin to realize that our universe is in a sense brought into being by the participation of those involved in it. It is a dance, for participation is its organizing principle. This is the important new concept of quantum mechanics. It takes the place in our understanding of the old notion of observation, of watching without getting involved. Quantum theory says it can’t be done. That
Quantum fractals on nspheres
 http://arxiv.org/abs/quantph/0608117v2
"... Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on ndim ..."
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Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on ndimensional spheres, jumps that are induced by symmetric configurations of noncommuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (Möbius transformations) and realize iterated function systems (IFS) with fractal attractors located on ndimensional spheres. We also extend the formalism to mixed states, represented by “density matrices ” in the standard formalism, (the nballs), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (onedimensional sphere and pentagon), two–sphere (octahedron), and on threedimensional sphere (hypercubetesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we give computational details of the proofs of some of our results and discuss the Hamilton’s “icossian calculus ” and its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell. 1 1
The Incomputable Alan Turing
"... The last century saw dramatic challenges to the Laplacian predictability which had underpinned scientific research for around 300 years. Basic to this was Alan Turing’s 1936 discovery (along with Alonzo Church) of the existence of unsolvable problems. This paper focuses on incomputability as a power ..."
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Cited by 1 (1 self)
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The last century saw dramatic challenges to the Laplacian predictability which had underpinned scientific research for around 300 years. Basic to this was Alan Turing’s 1936 discovery (along with Alonzo Church) of the existence of unsolvable problems. This paper focuses on incomputability as a powerful theme in Turing’s work and personal life, and examines its role in his evolving concept of machine intelligence. It also traces some of the ways in which important new developments are anticipated by Turing’s ideas in logic.
Quantum fractals on n–spheres. Clifford Algebra approach.
, 2008
"... Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on ndim ..."
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Cited by 1 (1 self)
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Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on ndimensional spheres, jumps that are induced by symmetric configurations of noncommuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (Möbius transformations) and realize iterated function systems (IFS) with fractal attractors located on ndimensional spheres. We also extend the formalism to mixed states, represented by “density matrices ” in the standard formalism, (the nballs), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (onedimensional sphere and pentagon), two–sphere (octahedron), and on threedimensional sphere (hypercubetesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we calculate the RadonNikodym derivative of the SO(n+1) invariant measure on S n under SO(1, n + 1) transformations and discuss the Hamilton’s “icossian calculus ” as well as its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell. As a byproduct of this work we obtain several Clifford algebraic results, such as a characterization of positive elements in a Clifford algebra C(n + 1) as generalized Lorentz “spin–boosts”, and their action as Moebius transformation on nsphere, and a decomposition of any element of Spin + (1, n + 1) into a spin–boost and a spin–rotation, including the explicit formula for the pullback of the SO(n+1) invariant 1 Riemannian metric with respect to the associated Möbius transformation. 2 1
Incomputability, Emergence and the Turing Universe
"... Amongst the huge literature concerning emergence, reductionism and mechanism, there is a role for analysis of the underlying mathematical constraints. Much of the speculation, confusion, controversy and descriptive verbiage might be clarified via suitable modelling and theory. The key ingredients we ..."
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Amongst the huge literature concerning emergence, reductionism and mechanism, there is a role for analysis of the underlying mathematical constraints. Much of the speculation, confusion, controversy and descriptive verbiage might be clarified via suitable modelling and theory. The key ingredients we bring to this project are the mathematical notions of definability and invariance, a computability theoretic framework in a realworld context, and within that, the modelling of basic causal environments via Turing’s 1939 notion of interactive computation over a structure described in terms of reals. Useful outcomes are: a refinement of what one understands to be a causal relationship, including nonmechanistic, irreversible causal relationships; an appreciation of how the mathematically simple origins of incomputability in definable hierarchies are materialised in the real world; and an understanding of the powerful explanatory role of current computability theoretic developments. The theme of this article concerns the way in which mathematics can structure everyday discussions around a range of important issues — and can also reinforce intuitions about theoretical links between different aspects of the real world. This fits with the widespread sense of excitement and expectation felt in many fields — and of a corresponding confusion — and of a tension characteristic of a Kuhnian paradigm shift. What we have below can be seen as tentative steps towards the sort of mathematical modelling needed for such a shift to be completed. In section 1, we outline the decisive role mathematics played in the birth of modern science; and how, more recently, it has helped us towards a better understanding of the nature and limitations of the scientific enterprise. In section 2, we review how the mathematics brings out inherent contradictions in the Laplacian model of scientific activity. And we look at some of the approaches to dealing