*Chris Heunen and Jamie Vicary*

- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780198739623
- eISBN:
- 9780191802584
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198739623.001.0001
- Subject:
- Mathematics, Mathematical Physics, Applied Mathematics

Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to ...
More

Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to understand many high-level quantum phenomena. Here, we lay the foundations for this categorical quantum mechanics, with an emphasis on the graphical calculus that makes computation intuitive. We describe superposition and entanglement using biproducts and dual objects, and show how quantum teleportation can be studied abstractly using these structures. We investigate monoids, Frobenius structures and Hopf algebras, showing how they can be used to model classical information and complementary observables. We describe the CP construction, a categorical tool to describe probabilistic quantum systems. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Previous knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text, we point out links with many other areas, such as representation theory, topology, quantum algebra, knot theory and probability theory, and present nonstandard models including sets and relations. All results are stated rigorously and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.Less

Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to understand many high-level quantum phenomena. Here, we lay the foundations for this categorical quantum mechanics, with an emphasis on the graphical calculus that makes computation intuitive. We describe superposition and entanglement using biproducts and dual objects, and show how quantum teleportation can be studied abstractly using these structures. We investigate monoids, Frobenius structures and Hopf algebras, showing how they can be used to model classical information and complementary observables. We describe the CP construction, a categorical tool to describe probabilistic quantum systems. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Previous knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text, we point out links with many other areas, such as representation theory, topology, quantum algebra, knot theory and probability theory, and present nonstandard models including sets and relations. All results are stated rigorously and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.

*Steven Phillips and William H. Wilson*

- Published in print:
- 2014
- Published Online:
- September 2014
- ISBN:
- 9780262027236
- eISBN:
- 9780262322461
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262027236.003.0009
- Subject:
- Philosophy, Philosophy of Mind

A theory of cognitive architecture should explain why systematicity is a necessary, not just possible consequence of the theory's core principles and base assumptions, without relying on arbitrary ...
More

A theory of cognitive architecture should explain why systematicity is a necessary, not just possible consequence of the theory's core principles and base assumptions, without relying on arbitrary (ad hoc) modifications to close explanatory gaps. Cognitive capacities are systematically distributed around common structures. Category theory, a mathematical theory of structure, provides a framework for a theory of systematicity that explains the nature of these structures. Underlying each collection of systematically related capacities is a categorical universal construction: each capacity is uniquely composed of a common (universal) component and capacity-specific component; constructions not admitting this form of compositionality are not categorical universal constructions. Hence, our category theoretical explanation transcends problems with other (classical/symbolic, connectionist) approaches, which admit both systematic and non-systematic architectures.Less

A theory of cognitive architecture should explain why systematicity is a necessary, not just possible consequence of the theory's core principles and base assumptions, without relying on arbitrary (ad hoc) modifications to close explanatory gaps. Cognitive capacities are systematically distributed around common structures. Category theory, a mathematical theory of structure, provides a framework for a theory of systematicity that explains the nature of these structures. Underlying each collection of systematically related capacities is a categorical universal construction: each capacity is uniquely composed of a common (universal) component and capacity-specific component; constructions not admitting this form of compositionality are not categorical universal constructions. Hence, our category theoretical explanation transcends problems with other (classical/symbolic, connectionist) approaches, which admit both systematic and non-systematic architectures.

*Anthony Chemero*

- Published in print:
- 2014
- Published Online:
- September 2014
- ISBN:
- 9780262027236
- eISBN:
- 9780262322461
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262027236.003.0014
- Subject:
- Philosophy, Philosophy of Mind

In its classical form, the systematicity of cognition is constitutively tied to the compositionality of the vehicles of cognition. A burgeoning research program in the cognitive sciences suggests ...
More

In its classical form, the systematicity of cognition is constitutively tied to the compositionality of the vehicles of cognition. A burgeoning research program in the cognitive sciences suggests that, in many cases, cognitive systems are interaction dominant. I will describe several examples of research showing that cognitive systems are interaction dominant, and argue that interaction dominance is inconsistent with the compositionally of the vehicles of cognition. As goes compositionality, so goes classical systematicity. So, to whatever extent cognitive systems are genuinely interaction dominant, cognition is not classically systematic.Less

In its classical form, the systematicity of cognition is constitutively tied to the compositionality of the vehicles of cognition. A burgeoning research program in the cognitive sciences suggests that, in many cases, cognitive systems are interaction dominant. I will describe several examples of research showing that cognitive systems are interaction dominant, and argue that interaction dominance is inconsistent with the compositionally of the vehicles of cognition. As goes compositionality, so goes classical systematicity. So, to whatever extent cognitive systems are genuinely interaction dominant, cognition is not classically systematic.

*Roberto G. de Almeida*

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198791492
- eISBN:
- 9780191868573
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198791492.003.0012
- Subject:
- Philosophy, Philosophy of Language

If there is a line between semantics and pragmatics, where is it drawn? In this essay I propose that appreciating a sentence is subject to two sets of processes: linguistic (viz., syntactic, ...
More

If there is a line between semantics and pragmatics, where is it drawn? In this essay I propose that appreciating a sentence is subject to two sets of processes: linguistic (viz., syntactic, semantic) driving the composition of shallow propositions, and unbounded pragmatic (viz., thinking). In section 1, I discuss some guiding assumptions on cognitive architecture, which constrain the nature of linguistic and cognitive representations and processes—and by implication, the conception of the semantics/pragmatics divide I have to offer. The phenomena I examine in section 2, relying on linguistic arguments and experimental evidence, suggest that for certain constructions there is an early “literal” process of interpretation followed by a period of uncertainty, indicating that the early linguistic computations produce a “shallow” semantic representation, not a fully enriched one. The cases I discuss, culminating in metaphors and so-called indeterminate sentences, challenge the prowess of linguistic computations for resolving—even suggesting—interpretations.Less

If there is a line between semantics and pragmatics, where is it drawn? In this essay I propose that appreciating a sentence is subject to two sets of processes: linguistic (viz., syntactic, semantic) driving the composition of shallow propositions, and unbounded pragmatic (viz., thinking). In section 1, I discuss some guiding assumptions on cognitive architecture, which constrain the nature of linguistic and cognitive representations and processes—and by implication, the conception of the semantics/pragmatics divide I have to offer. The phenomena I examine in section 2, relying on linguistic arguments and experimental evidence, suggest that for certain constructions there is an early “literal” process of interpretation followed by a period of uncertainty, indicating that the early linguistic computations produce a “shallow” semantic representation, not a fully enriched one. The cases I discuss, culminating in metaphors and so-called indeterminate sentences, challenge the prowess of linguistic computations for resolving—even suggesting—interpretations.