Venus Homotopically

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Venus Homotopically

IfCoLoG Journal of Logics and their Applications. 2017. Vol. 4. No. 4. P. 1427–1446.

Rodin A.

The identity concept developed in the Homotopy Type theory (HoTT) supports an analysis of Frege's famous Venus example, which explains how empirical evidences justify judgements about identities. In the context of this analysis we consider the traditional distinction between the extension and the intension of concepts as it appears in HoTT, discuss an ontological signicance of this distinction and, nally, provide a homotopical reconstruction of a basic kinematic scheme, which is used in the Classical Mechanics, and discuss its relevance in the Quantum Mechanics.

Language: English

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Keywords: Homotopy Type theory

Models of HoTT and the Constructive View of Theories

Rodin A., , in: Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts.: Springer, 2019. Ch. 9 P. 191–219.

Homotopy Type theory and its Model theory provide a novel formal semantic framework for representing scientific theories. This framework supports a constructive view of theories according to which a theory is essentially characterised by its methods. The constructive view of theories was earlier defended by Ernest Nagel and a number of other philosophers of the past but available logical means ...

Added: October 30, 2019

Extra-Logical Proof-Theoretic Semantics in Homotopy Type Theory

Rodin A., , in: Одиннадцатые Смирновские чтения по логике: материалы Международной научной конференции, 19 – 21 июня 2019, г. Москва.: М.: Современные тетради, 2019. P. 42–43.

Kant famously argued that elementary geometrical statements such as Euclid's Triangle Angle Sum theorem cannot be deduced from the rst principles by purely logical means because their proofs require extra-logical geometrical constructions [1, A719/B747]. The discovery of non-Euclidean geometries in the 19-th century made Kant's analysis of geometrical reasoning untenable in its original form, and throughout the following 20-th century ...

Added: June 30, 2019

Model structures on categories of models of type theories

Valery Isaev, Mathematical Structures in Computer Science 2018 Vol. 28 No. 10 P. 1695–1722

Models of dependent type theories are contextual categories with some additional structure. We prove that if a theory T has enough structure, then the category T-Mod of its models carries the structure of a model category. We also show that if T has Σ-types, then weak equivalences can be characterized in terms of homotopy categories ...

Added: November 6, 2018

Univalence and Constructive Identity

Rodin A., , in: Philosophy, Mathematics, Linguistics: Aspects of Interaction (PhML 2012).: St. Petersburg: ВВМ, 2012. P. 170–174.

The non-standard identity concept developed in the Homotopy Type theory allows for an alternative analysis of Frege’s famous Venus example, which explains how empirical evidences justify judgements about identities and accounts for the constructive aspect of such judgements. ...

Added: June 6, 2018

Constructive Identities for Physics

Rodin A., , in: Frontiers of Fundamental Physics 14Vol. 224: EPISTEMOLOGY AND PHILOSOPHY.: [б.и.], 2014.

Homotopy Type theory instantiates a new form of axiomatic approach, which is more friendly to physics than the standard axiomatic approach stemming from Hilbert. This new axiomatic approach combines logical and geometrical methods in a new way and brings about a non-trivial constructive concept of identity applicable in various physical contexts including Quantum Mechanics and General Relativity. ...

Added: June 6, 2018

Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts.

Springer, 2019.

Homotopy Type theory and its Model theory provide a novel formal semantic framework for representing scientic theories. This framework supports a constructive view of theories according to which a theory is essentially characterised by its methods. The constructive view of theories was earlier defended by Ernest Nagel and a number of other philosophers of the past but available logical ...

Added: June 5, 2018

On the Constructive Axiomatic Method

Rodin A., Logique et Analyse 2018 Vol. 242 No. 2 P. 201–231

The received notion of axiomatic method stemming from Hilbert is not fully adequate to the recent successful practice of axiomatizing mathematical theories. The axiomatic architecture of Homotopy type theory (HoTT) does not ft the pattern of formal axiomatic theory in the standard sense of the word. However this theory falls under a more general and ...

Added: May 26, 2018