59 research outputs found
Canonical Maps
Categorical foundations and set-theoretical foundations are sometimes presented as alternative foundational schemes. So far, the literature has mostly focused on the weaknesses of the categorical foundations. We want here to concentrate on what we take to be one of its strengths: the explicit identification of so-called canonical maps and their role in mathematics. Canonical maps play a central role in contemporary mathematics and although some are easily defined by set-theoretical tools, they all appear systematically in a categorical framework. The key element here is the systematic nature of these maps in a categorical framework and I suggest that, from that point of view, one can see an architectonic of mathematics emerging clearly. Moreover, they force us to reconsider the nature of mathematical knowledge itself. Thus, to understand certain fundamental aspects of mathematics, category theory is necessary (at least, in the present state of mathematics)
An Outline of Reality
This paper aims to provide a basic explanation of existence, fundamental aspects of reality, and consciousness. Existence in its most general sense is identified with the principle of logical consistency: to exist means to be logically consistent. The essence of the principle of logical consistency is that every thing is what it is and is not what it is not. From this principle follows the existence of intrinsic, indescribable identities of things and relations between them. There are three fundamental, logically necessary relations: similarity, composition and instantiation. Set theory, mathematics, logic and science are presented as relational descriptions of reality. Qualities of consciousness (qualia) are identified with intrinsic identities of things or at least a certain subset of them, especially in the context of a dynamic form of organized complexity
Category Theory and Model-Driven Engineering: From Formal Semantics to Design Patterns and Beyond
There is a hidden intrigue in the title. CT is one of the most abstract
mathematical disciplines, sometimes nicknamed "abstract nonsense". MDE is a
recent trend in software development, industrially supported by standards,
tools, and the status of a new "silver bullet". Surprisingly, categorical
patterns turn out to be directly applicable to mathematical modeling of
structures appearing in everyday MDE practice. Model merging, transformation,
synchronization, and other important model management scenarios can be seen as
executions of categorical specifications.
Moreover, the paper aims to elucidate a claim that relationships between CT
and MDE are more complex and richer than is normally assumed for "applied
mathematics". CT provides a toolbox of design patterns and structural
principles of real practical value for MDE. We will present examples of how an
elementary categorical arrangement of a model management scenario reveals
deficiencies in the architecture of modern tools automating the scenario.Comment: In Proceedings ACCAT 2012, arXiv:1208.430
In defense of mechanism
In Life Itself and in Essays on Life Itself, Robert Rosen (1991, 2000) argued that machines were, in principle, incapable of modeling the defining feature of living systems, which he claimed to be the existence of closed causal loops. Rosen's argument has been used to support critiques of computational models in ecological psychology. This article shows that Rosen's attack on mechanism is fundamentally misconceived. It is, in fact, of the essence of a mechanical system that it contains closed causal loops. Moreover, Rosen's epistemology is based on a strong form of indirect realism and his arguments, if correct, would call into question some of the fundamental principles of ecological psychology
Newton vs. Leibniz: Intransparency vs. Inconsistency
We investigate the structure common to causal theories that attempt to
explain a (part of) the world. Causality implies conservation of identity,
itself a far from simple notion. It imposes strong demands on the
universalizing power of the theories concerned. These demands are often met by
the introduction of a metalevel which encompasses the notions of 'system' and
'lawful behaviour'. In classical mechanics, the division between universal and
particular leaves its traces in the separate treatment of cinematics and
dynamics. This analysis is applied to the mechanical theories of Newton and
Leibniz, with some surprising results
LNCS
Extensionality axioms are common when reasoning about data collections, such as arrays and functions in program analysis, or sets in mathematics. An extensionality axiom asserts that two collections are equal if they consist of the same elements at the same indices. Using extensionality is often required to show that two collections are equal. A typical example is the set theory theorem (∀x)(∀y)x∪y = y ∪x. Interestingly, while humans have no problem with proving such set identities using extensionality, they are very hard for superposition theorem provers because of the calculi they use. In this paper we show how addition of a new inference rule, called extensionality resolution, allows first-order theorem provers to easily solve problems no modern first-order theorem prover can solve. We illustrate this by running the VAMPIRE theorem prover with extensionality resolution on a number of set theory and array problems. Extensionality resolution helps VAMPIRE to solve problems from the TPTP library of first-order problems that were never solved before by any prover
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