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Logical Monism: The Global Identity of Applicable Logic
 Advanced Studies in Mathematics and Logic
, 2005
"... Abstract. ‘One universe, one logic ’ takes the world as it is and leads to adjointness as the global logic of anything. The alternative approach to find a unification of known logics requires assumptions and is therefore consistent with the same conclusion for a universal logic has to be universally ..."
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Abstract. ‘One universe, one logic ’ takes the world as it is and leads to adjointness as the global logic of anything. The alternative approach to find a unification of known logics requires assumptions and is therefore consistent with the same conclusion for a universal logic has to be universally applicable. The universal characteristic of adjointness is that it has a natural construction from the concept of the arrow. The application to the test sentence, ‘John said that Mary believed he did not love her’, demonstrates adjointness as the logic of the postmodern world. 1 Unity of Applicable Logic There is one ultimate logic: it is a simple ontological but pragmatic argument of ‘one universe, one logic’. If more, how can we know unless there is a logic to compare them? If logic is a family of varying strength, what logic compares the variance? Only some ultimate logic. How do we even know this? It must still be the same logic that tells us this. And that logic must tell us about itself − − tell us that it has some recursive selfclosure. The same pragmatic cogency leads us into the world of physics and beyond into the humanities. The world must fit together according to this same ultimate logic. It is therefore an applicable logic. Universal logic means universally applicable logic. This study arises from the investigation of fundamentals in two large applied areas: one is schema design in interoperable databases, the other is in legal reasoning; both studies relate logic to realworld facts. Until we are able to identify the ultimate logic of the universe, it is not surprising that goals like unified field theory within a ”theory of everything ” are so elusive. Applicable logic is needed in new ways in biology, medicine, economics, legal science, natural computing, modern physics, etc. This means it has to be a logic which can manage the advances made in the twentieth century, many of
HILBERT’S 6TH PROBLEM: EXACT AND APPROXIMATE HYDRODYNAMIC MANIFOLDS FOR KINETIC EQUATIONS
"... Abstract. The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the ..."
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Abstract. The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the result of summation of the Chapman–Enskog asymptotic expansion and by the direct solution of the invariance equation. These model cases, comprising Grad’s moment systems, both linear and nonlinear, are studied in depth in order to gain understanding of what can be expected for the Boltzmann equation. Particularly, the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the shortwave limit and the viscosity modification at high divergence of the flow velocity are indicated as severe obstacles to the resolution of Hilbert’s 6th Problem. Furthermore, we review the derivation of the approximate hydrodynamic manifold for the Boltzmann equation using Newton’s iteration and avoiding smallness parameters, and compare this to the exact solutions. Additionally, we discuss the problem of projection of the
Locality, Weak or Strong Anticipation and Quantum Computing. I. Nonlocality in Quantum Theory
"... Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Categ ..."
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Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Category theory provides the necessary coordinatefree mathematical language which is both constructive and nonlocal to subsume the various interpretations of quantum theory in one pullback/pushout Dolittle diagram. This diagram can be used to test and classify physical devices and proposed algorithms for weak or strong anticipation. Quantum Information Science is more than a merger of ChurchTuring and quantum theories. It has constructively to bridge the nonlocal chasm between the weak anticipation of mathematics and the strong anticipation of physics.