Univalent Foundations and the Equivalence Principle

Ahrens, Benedikt; North, Paige Randall

doi:10.1007/978-3-030-15655-8_6

Cited by 12 publications

(11 citation statements)

References 9 publications

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“…In Theorem 3.6, we prove that they are in fact homotopy equivalent. The same correspondence for graphs also arises for many other structures [1,2], for example, groups, and topological spaces. Proof.…”

Section: The Category Of Graphsmentioning

confidence: 62%

On Planarity of Graphs in Homotopy Type Theory

Prieto-Cubides¹,

Gylterud²

2021

Preprint

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In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces, and maps of graphs embedded in the sphere, in homotopy type theory. This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for homotopy type theory.CCS Concepts: • Theory of computation → Constructive mathematics; Type theory; • Mathematics of computing → Graphs and surfaces.

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Section: The Category Of Graphsmentioning

confidence: 62%

On Planarity of Graphs in Homotopy Type Theory

Prieto-Cubides¹,

Gylterud²

2021

Preprint

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“…axiomatic Set theory that comes with its internal identity relation provides a rigorous sense in which isomorphic structures can be said to be identical. We shall shortly see how this problem drives the category-theoretic approach in FOM and how it is settled in the Univalent FOM [1]. In Bourbaki's Elements the notion of self-standing mathematical structure independent of any set-theoretic background remains a wishful thinking or, on a more charitable interpretation, a regulative idea in the Kantian sense of the word.…”

Section: Foundations At Workmentioning

confidence: 99%

One Mathematic(s) or Many? Foundations of Mathematics in Today's Mathematical Practice

Rodin

2023

Preprint

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The received Hilbert-style axiomatic foundations of mathematics has been designed by Hilbert and his followers as a tool for meta-theoretical research. Foundations of mathematics of this type fail to satisfactory perform more basic and more practically-oriented functions of theoretical foundations such as verification of mathematical constructions and proofs. Using alternative foundations of mathematics such as the Univalent Foundations is compatible with using the received set-theoretic foundations for meta-mathematical purposes provided the two foundations are mutually interpretable.Changes in foundations of mathematics do not, generally, disqualify mathematical theories based on older foundations but allow for reconstruction of these theories on new foundations. Mathematics is one but its foundations are many.

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“…26 Only since the advance of the HoTT (circa 2014) we are finally in a position to discuss the alternative approaches to identity motivated by the general CT and the 20th century mathematical practice more formally. 27 B. Ahrens and P. R. North explore the issue of identity and sameness in [1] in the context of HoTT: 28 "What should it mean for two objects…”

Section: Assessing Identity and Extensionalitymentioning

confidence: 99%

“…Once one has reasons to employ second-order languages, i.e., languages that allow for quantification of predicate variables, to formalize a (philosophical or scientific) theory, Nominalists' attitude to formal ontology is primarily concerned with providing semantic strategies to avoid the ontologically committing reference to second-order entities, e.g., universals, "naively" intended to be the reference of quantified second-order variables. They deploy a peculiar substitutional semantics, one restricted to second-order quantification, 1 against the Realist approach that appeals to the so-called "full" or "referential" semantics of second-order languages [9]. 2 This semantics interprets quantified second-order formulas of the form "∃ X (Xa)", 3 over classes of first-order open formulas ϕ added to the original first-order model that are as "substituends" of the second-order quantificational apparatus.…”

Section: Introductionmentioning

confidence: 99%

An argument against nominalism

Ferrari

2022

Synthese

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Nominalism in formal ontology is still the thesis that the only acceptable domain of quantification is the first-order domain of particulars. Nominalists may assert that second-order well-formed formulas can be fully and completely interpreted within the first-order domain, thereby avoiding any ontological commitment to second-order entities, by means of an appropriate semantics called "substitutional". In this paper I argue that the success of this strategy depends on the ability of Nominalists to maintain that identity, and equivalence relations more in general, is first-order and invariant. Firstly, I explain why Nominalists are formally bound to this first-order concept of identity. Secondly, I show that the resources needed to justify identity, a certain conception of identity invariance, are out of the nominalist's reach.

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Univalent Foundations and the Equivalence Principle

Cited by 12 publications

References 9 publications

On Planarity of Graphs in Homotopy Type Theory

On Planarity of Graphs in Homotopy Type Theory

One Mathematic(s) or Many? Foundations of Mathematics in Today's Mathematical Practice

An argument against nominalism

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