29,894 research outputs found
The logic and topology of Kant's temporal continuum
In this article we provide a mathematical model of Kant?s temporal continuum that satisfies the (not obviously consistent) synthetic a priori principles for time that Kant lists in the Critique of pure Reason (CPR), the Metaphysical Foundations of Natural Science (MFNS), the Opus Postumum and the notes and frag- ments published after his death. The continuum so obtained has some affinities with the Brouwerian continuum, but it also has ‘infinitesimal intervals’ consisting of nilpotent infinitesimals, which capture Kant’s theory of rest and motion in MFNS. While constructing the model, we establish a concordance between the informal notions of Kant?s theory of the temporal continuum, and formal correlates to these notions in the mathematical theory. Our mathematical reconstruction of Kant?s theory of time allows us to understand what ?faculties and functions? must be in place for time to satisfy all the synthetic a priori principles for time mentioned. We have presented here a mathematically precise account of Kant?s transcendental argument for time in the CPR and of the rela- tion between the categories, the synthetic a priori principles for time, and the unity of apperception; the most precise account of this relation to date. We focus our exposition on a mathematical analysis of Kant’s informal terminology, but for reasons of space, most theorems are explained but not formally proven; formal proofs are available in (Pinosio, 2017).
The analysis presented in this paper is related to the more general project of developing a formalization of Kant’s critical philosophy (Achourioti & van Lambalgen, 2011). A formal approach can shed light on the most controversial concepts of Kant’s theoretical philosophy, and is a valuable exegetical tool in its own right. However, we wish to make clear that mathematical formalization cannot displace traditional exegetical methods, but that it is rather an exegetical tool in its own right, which works best when it is coupled with a keen awareness of the subtleties involved in understanding the philosophical issues at hand. In this case, a virtuous ?hermeneutic circle? between mathematical formalization and philosophical discourse arises
A Formalization of the Theorem of Existence of First-Order Most General Unifiers
This work presents a formalization of the theorem of existence of most
general unifiers in first-order signatures in the higher-order proof assistant
PVS. The distinguishing feature of this formalization is that it remains close
to the textbook proofs that are based on proving the correctness of the
well-known Robinson's first-order unification algorithm. The formalization was
applied inside a PVS development for term rewriting systems that provides a
complete formalization of the Knuth-Bendix Critical Pair theorem, among other
relevant theorems of the theory of rewriting. In addition, the formalization
methodology has been proved of practical use in order to verify the correctness
of unification algorithms in the style of the original Robinson's unification
algorithm.Comment: In Proceedings LSFA 2011, arXiv:1203.542
Formalization of Complex Vectors in Higher-Order Logic
Complex vector analysis is widely used to analyze continuous systems in many
disciplines, including physics and engineering. In this paper, we present a
higher-order-logic formalization of the complex vector space to facilitate
conducting this analysis within the sound core of a theorem prover: HOL Light.
Our definition of complex vector builds upon the definitions of complex numbers
and real vectors. This extension allows us to extensively benefit from the
already verified theorems based on complex analysis and real vector analysis.
To show the practical usefulness of our library we adopt it to formalize
electromagnetic fields and to prove the law of reflection for the planar waves.Comment: 15 pages, 1 figur
Computing Persistent Homology within Coq/SSReflect
Persistent homology is one of the most active branches of Computational
Algebraic Topology with applications in several contexts such as optical
character recognition or analysis of point cloud data. In this paper, we report
on the formal development of certified programs to compute persistent Betti
numbers, an instrumental tool of persistent homology, using the Coq proof
assistant together with the SSReflect extension. To this aim it has been
necessary to formalize the underlying mathematical theory of these algorithms.
This is another example showing that interactive theorem provers have reached a
point where they are mature enough to tackle the formalization of nontrivial
mathematical theories
Logical Conceptualization of Knowledge on the Notion of Language Communication
The main objective of the paper is to provide a conceptual apparatus
of a general logical theory of language communication. The aim of the paper is
to outline a formal-logical theory of language in which the concepts of the phenomenon
of language communication and language communication in general
are defined and some conditions for their adequacy are formulated. The theory
explicates the key notions of contemporary syntax, semantics, and pragmatics.
The theory is formalized on two levels: token-level and type-level. As such, it
takes into account the dual – token and type – ontological character of linguistic
entities. The basic notions of the theory: language communication, meaning
and interpretation are introduced on the second, type-level of formalization, and
their required prior formalization of some of the notions introduced on the first,
token-level; among others, the notion of an act of communication. Owing to the
theory, it is possible to address the problems of adequacy of both empirical acts
of communication and of language communication in general. All the conditions
of adequacy of communication discussed in the presented paper, are valid for
one-way communication (sender-recipient); nevertheless, they can also apply to
the reverse direction of language communication (recipient-sender). Therefore,
they concern the problem of two-way understanding in language communication
Actions and Events in Concurrent Systems Design
In this work, having in mind the construction of concurrent systems from
components, we discuss the difference between actions and events. For this
discussion, we propose an(other) architecture description language in which
actions and events are made explicit in the description of a component and a
system. Our work builds from the ideas set forth by the categorical approach to
the construction of software based systems from components advocated by Goguen
and Burstall, in the context of institutions, and by Fiadeiro and Maibaum, in
the context of temporal logic. In this context, we formalize a notion of a
component as an element of an indexed category and we elicit a notion of a
morphism between components as morphisms of this category. Moreover, we
elaborate on how this formalization captures, in a convenient manner, the
underlying structure of a component and the basic interaction mechanisms for
putting components together. Further, we advance some ideas on how certain
matters related to the openness and the compositionality of a component/system
may be described in terms of classes of morphisms, thus potentially supporting
a compositional rely/guarantee reasoning.Comment: In Proceedings LAFM 2013, arXiv:1401.056
Computational Soundness of Formal Encryption in Coq
We formalize Abadi and Rogaway's computational soundness result in the
Coq interactive theorem prover. This requires to model notions of provable
cryptography like indistinguishability between ensembles of
probability distributions, PPT reductions, and security notions for
encryption schemes.
Our formalization is the first computational soundness result to be
mechanized, and it shows the feasibility of rigorous reasoning of
computational cryptography inside a generic interactive theorem prover
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