23,354 research outputs found
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
Semi-simplicial Types in Logic-enriched Homotopy Type Theory
The problem of defining Semi-Simplicial Types (SSTs) in Homotopy Type Theory
(HoTT) has been recognized as important during the Year of Univalent
Foundations at the Institute of Advanced Study. According to the interpretation
of HoTT in Quillen model categories, SSTs are type-theoretic versions of Reedy
fibrant semi-simplicial objects in a model category and simplicial and
semi-simplicial objects play a crucial role in many constructions in homotopy
theory and higher category theory. Attempts to define SSTs in HoTT lead to some
difficulties such as the need of infinitary assumptions which are beyond HoTT
with only non-strict equality types.
Voevodsky proposed a definition of SSTs in Homotopy Type System (HTS), an
extension of HoTT with non-fibrant types, including an extensional strict
equality type. However, HTS does not have the desirable computational
properties such as decidability of type checking and strong normalization. In
this paper, we study a logic-enriched homotopy type theory, an alternative
extension of HoTT with equational logic based on the idea of logic-enriched
type theories. In contrast to Voevodskys HTS, all types in our system are
fibrant and it can be implemented in existing proof assistants. We show how
SSTs can be defined in our system and outline an implementation in the proof
assistant Plastic
Univalent Foundations as a Foundation for Mathematical Practice
I prove that invoking the univalence axiom is equivalent to arguing 'without loss of generality' (WLOG) within Propositional Univalent Foundations (PropUF), the fragment of Univalent Foundations (UF) in which all homotopy types are mere propositions. As a consequence, I argue that practicing mathematicians, in accepting WLOG as a valid form of argument, implicitly accept the univalence axiom and that UF rightly serves as a Foundation for Mathematical Practice. By contrast, ZFC is inconsistent with WLOG as it is applied, and therefore cannot serve as a foundation for practice
Quantum Structure in Cognition, Origins, Developments, Successes and Expectations
We provide an overview of the results we have attained in the last decade on
the identification of quantum structures in cognition and, more specifically,
in the formalization and representation of natural concepts. We firstly discuss
the quantum foundational reasons that led us to investigate the mechanisms of
formation and combination of concepts in human reasoning, starting from the
empirically observed deviations from classical logical and probabilistic
structures. We then develop our quantum-theoretic perspective in Fock space
which allows successful modeling of various sets of cognitive experiments
collected by different scientists, including ourselves. In addition, we
formulate a unified explanatory hypothesis for the presence of quantum
structures in cognitive processes, and discuss our recent discovery of further
quantum aspects in concept combinations, namely, 'entanglement' and
'indistinguishability'. We finally illustrate perspectives for future research.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1412.870
Mathematics and language
This essay considers the special character of mathematical reasoning, and
draws on observations from interactive theorem proving and the history of
mathematics to clarify the nature of formal and informal mathematical language.
It proposes that we view mathematics as a system of conventions and norms that
is designed to help us make sense of the world and reason efficiently. Like any
designed system, it can perform well or poorly, and the philosophy of
mathematics has a role to play in helping us understand the general principles
by which it serves its purposes well
Institutionalising Ontology-Based Semantic Integration
We address what is still a scarcity of general mathematical foundations for ontology-based semantic integration underlying current knowledge engineering methodologies in decentralised and distributed environments. After recalling the first-order ontology-based approach to semantic integration and a formalisation of ontological commitment, we propose a general theory that uses a syntax-and interpretation-independent formulation of language, ontology, and ontological commitment in terms of institutions. We claim that our formalisation generalises the intuitive notion of ontology-based semantic integration while retaining its basic insight, and we apply it for eliciting and hence comparing various increasingly complex notions of semantic integration and ontological commitment based on differing understandings of semantics
On Constructive Axiomatic Method
In this last version of the paper one may find a critical overview of some
recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
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