10,477 research outputs found
Loops and Knots as Topoi of Substance. Spinoza Revisited
The relationship between modern philosophy and physics is discussed. It is
shown that the latter develops some need for a modernized metaphysics which
shows up as an ultima philosophia of considerable heuristic value, rather than
as the prima philosophia in the Aristotelian sense as it had been intended, in
the first place. It is shown then, that it is the philosophy of Spinoza in
fact, that can still serve as a paradigm for such an approach. In particular,
Spinoza's concept of infinite substance is compared with the philosophical
implications of the foundational aspects of modern physical theory. Various
connotations of sub-stance are discussed within pre-geometric theories,
especially with a view to the role of spin networks within quantum gravity. It
is found to be useful to intro-duce a separation into physics then, so as to
differ between foundational and empirical theories, respectively. This leads to
a straightforward connection bet-ween foundational theories and speculative
philosophy on the one hand, and between empirical theories and sceptical
philosophy on the other. This might help in the end, to clarify some recent
problems, such as the absence of time and causality at a fundamental level. It
is implied that recent results relating to topos theory might open the way
towards eventually deriving logic from physics, and also towards a possible
transition from logic to hermeneutic.Comment: 42 page
Computing with Coloured Tangles
We suggest a diagrammatic model of computation based on an axiom of
distributivity. A diagram of a decorated coloured tangle, similar to those that
appear in low dimensional topology, plays the role of a circuit diagram.
Equivalent diagrams represent bisimilar computations. We prove that our model
of computation is Turing complete, and that with bounded resources it can
moreover decide any language in complexity class IP, sometimes with better
performance parameters than corresponding classical protocols.Comment: 36 pages,; Introduction entirely rewritten, Section 4.3 adde
Columnar order and Ashkin-Teller criticality in mixtures of hard-squares and dimers
We show that critical exponents of the transition to columnar order in a {\em
mixture} of dimers and hard-squares on the square
lattice {\em depends on the composition of the mixture} in exactly the manner
predicted by the theory of Ashkin-Teller criticality, including in the
hard-square limit. This result settles the question regarding the nature of the
transition in the hard-square lattice gas. It also provides the first example
of a polydisperse system whose critical properties depend on composition. Our
ideas also lead to some interesting predictions for a class of frustrated
quantum magnets that exhibit columnar ordering of the bond-energies at low
temperature.Comment: 4pages, 2-column format + supplementary material; v2: published
version including supplemental materia
- …