This work presents a unied pattern-based epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in terms of its objective calibration protocols - procedures that are implemented on top of the participating subjective knowledge-patches. They are the procedures of agreement and obedience that characterize the coherence of any type of interaction, and which are used here in order to formalize the concept of participator consciousness in terms of the inverse-direct limit duality of Category Theory.

Abstract. In Das Kontinuum, Weyl showed how a large body of classical mathematics could be developed on a purely predicative foundation. We present a logic-enriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definition of the cardinality of a set and several results from real analysis — has been formalised, using the proof assistant Plastic that implements a logical framework. This case study shows how type theory can be used to represent a non-constructive foundation for mathematics. Key words: logic-enriched type theory, predicativism, formalisation 1

In constructive mathematics it is of interest to consider a more general, but classically equivalent, notion of linear order, so called pseudoorder.

different logical foundations

Abstract. Information systems anticipate the real world. Classical databases store, organise and search collections of data of that real world but only as weak anticipatory information systems. This is because of the reductionism and normalisation needed to map the structuralism of natural data on to idealised machines with von Neumann architectures consisting of fixed instructions. Category theory developed as a formalism to explore the theoretical concept of naturality shows that methods like sketches arising from graph theory as only non-natural models of naturality cannot capture real-world structures for strong anticipatory information systems. Databases need a schema of the natural world. Natural computing databases need the schema itself to be also natural. Natural computing methods including neural computers, evolutionary automata, molecular and nanocomputing and quantum computation have the potential to be strong. At present they are mainly at the stage of weak anticipatory systems.

This article gives a brief introduction to a new discipline called the cognitive science of mathematics (Lakoff & Núñez, 2000), that is, the empirical and multidisciplinary study of mathematics (itself) as a scientific subject matter. The theoretical background of the arguments is based on embodied cognition, and on relatively recent findings in cognitive linguistics. The article discusses Mathematical Idea Analysis—the set of techniques for studying implicit (largely unconscious) conceptual structures in mathematics. Particular attention is paid to everyday cognitive mechanisms such as image schemas and conceptual metaphors, showing how they play a fundamental role in constituting the very fabric of mathematics. The analyses, illustrated with a discussion of some issues of set and hyperset theory, show that it is (human) meaning what makes mathematics what it is: Mathematics is not transcendentally objective, but it is not arbitrary either (not the result of pure social conventions). Some implications for mathematics education are suggested. Have you ever thought why (I mean, really why) the multiplication of two negative numbers yields a positive one? Or why the empty class is a subclass of all

Abstract. Is the universe computable? If yes, is it computationally a polynomial place? In standard quantum mechanics, which permits infinite parallelism and the infinitely precise specification of states, a negative answer to both questions is not ruled out. On the other hand, empirical evidence suggests that NP-complete problems are intractable in the physical world. Likewise, computational problems known to be algorithmically uncomputable do not seem to be computable by any physical algorithms on the one hand, and physical computers on the other, finds a natural explanation if the universe is assumed to be algorithmic; that is, that physical reality is the product of discrete subphysical information processing equivalent to the actions of a probabilistic Turing machine. This assumption can be reconciled with the observed exponentiality of quantum systems at microscopic scales, and the consequent possibility of implementing Shor's quantum polynomial time algorithm at that scale, provided the degree of superposition is intrinsically, finitely upper-bounded. If this bound is associated with the quantum-classical divide (the Heisenberg cut), a natural resolution to the quantum measurement problem arises. From this viewpoint, macroscopic classicality is an evidence that the universe is in BPP, and both questions raised above receive affirmative answers. A recently proposed computational model of quantum measurement, which relates the Heisenberg cut to the discreteness of Hilbert space, is briefly discussed. A connection to quantum gravity is noted. Our results are compatible with the philosophy that mathematical truths are independent of the laws of physics.

Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical

We construct a logic-enriched type theory LTTw that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTw, including Weyl’s definition of the cardinality of a set and several results from real analysis, using the proof assistant Plastic that implements the logical framework LF. This case study shows how type theory can be used to represent a non-constructive foundation for mathematics.

We identify specific properties of the complex plane that allow functions of a continuous n-dimensional (Hilbert) measure space to be transformed into a well ordered counting sequence. We discuss proof strategies for problems in number theory (Goldbach Conjecture) and topology (Poincare ´ Conjecture) that suggest correspondence between the physical principle of least action and the mathematical concept of well ordering. The result implies a deeply organic connection between physics and mathematics. 1.0 Counting & Measure 1.1 Counting is so fundamental that even Kindergartners have little trouble grasping the concept of ordering objects into discrete sets. The concept of a continuous measurement function has been formalized only in recent centuries, however. Newton’s calculus of the rate of change of the rate of change, e.g., is not intuitively obvious. 1.1.1 The growth of generalization in geometry (topology) in the last century, and the proliferation of sophisticated techniques in number theory (Wiles ’ proof of Fermat’s Last Theorem, e.g.) leave clues to subtle relations between continuous functions and discrete