Axiomatic Method And Category Theory

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Axiomatic Method and Category Theory

Andrei Rodin
Axiomatic Method and Category Theory Book PDF

Author : Andrei Rodin
Publisher : Springer Science & Business Media
Page : 285 pages
File Size : 54,8 Mb
Release : 2013-10-14
Category : Philosophy
ISBN : 9783319004044

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Axiomatic Method and Category Theory by Andrei Rodin Pdf

This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.

Entropy and Diversity

Tom Leinster
Entropy and Diversity Book PDF

Author : Tom Leinster
Publisher : Cambridge University Press
Page : 457 pages
File Size : 40,9 Mb
Release : 2021-04-22
Category : Language Arts & Disciplines
ISBN : 9781108832700

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Entropy and Diversity by Tom Leinster Pdf

Discover the mathematical riches of 'what is diversity?' in a book that adds mathematical rigour to a vital ecological debate.

Intuition and the Axiomatic Method

Emily Carson,Renate Huber
Intuition and the Axiomatic Method Book PDF

Author : Emily Carson,Renate Huber
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 46,7 Mb
Release : 2006-07-02
Category : Philosophy
ISBN : 9781402040405

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Intuition and the Axiomatic Method by Emily Carson,Renate Huber Pdf

Following developments in modern geometry, logic and physics, many scientists and philosophers in the modern era considered Kant’s theory of intuition to be obsolete. But this only represents one side of the story concerning Kant, intuition and twentieth century science. Several prominent mathematicians and physicists were convinced that the formal tools of modern logic, set theory and the axiomatic method are not sufficient for providing mathematics and physics with satisfactory foundations. All of Hilbert, Gödel, Poincaré, Weyl and Bohr thought that intuition was an indispensable element in describing the foundations of science. They had very different reasons for thinking this, and they had very different accounts of what they called intuition. But they had in common that their views of mathematics and physics were significantly influenced by their readings of Kant. In the present volume, various views of intuition and the axiomatic method are explored, beginning with Kant’s own approach. By way of these investigations, we hope to understand better the rationale behind Kant’s theory of intuition, as well as to grasp many facets of the relations between theories of intuition and the axiomatic method, dealing with both their strengths and limitations; in short, the volume covers logical and non-logical, historical and systematic issues in both mathematics and physics.

Category Theory in Context

Emily Riehl
Category Theory in Context Book PDF

Author : Emily Riehl
Publisher : Courier Dover Publications
Page : 272 pages
File Size : 44,7 Mb
Release : 2017-03-09
Category : Mathematics
ISBN : 9780486820804

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Category Theory in Context by Emily Riehl Pdf

Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.

Basic Category Theory

Tom Leinster
Basic Category Theory Book PDF

Author : Tom Leinster
Publisher : Cambridge University Press
Page : 193 pages
File Size : 50,6 Mb
Release : 2014-07-24
Category : Mathematics
ISBN : 9781107044241

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Basic Category Theory by Tom Leinster Pdf

A short introduction ideal for students learning category theory for the first time.

The Logical Foundations of Scientific Theories

Decio Krause,Jonas R.B. Arenhart
The Logical Foundations of Scientific Theories Book PDF

Author : Decio Krause,Jonas R.B. Arenhart
Publisher : Routledge
Page : 161 pages
File Size : 54,6 Mb
Release : 2016-07-15
Category : Mathematics
ISBN : 9781315535197

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The Logical Foundations of Scientific Theories by Decio Krause,Jonas R.B. Arenhart Pdf

This book addresses the logical aspects of the foundations of scientific theories. Even though the relevance of formal methods in the study of scientific theories is now widely recognized and regaining prominence, the issues covered here are still not generally discussed in philosophy of science. The authors focus mainly on the role played by the underlying formal apparatuses employed in the construction of the models of scientific theories, relating the discussion with the so-called semantic approach to scientific theories. The book describes the role played by this metamathematical framework in three main aspects: considerations of formal languages employed to axiomatize scientific theories, the role of the axiomatic method itself, and the way set-theoretical structures, which play the role of the models of theories, are developed. The authors also discuss the differences and philosophical relevance of the two basic ways of aximoatizing a scientific theory, namely Patrick Suppes’ set theoretical predicates and the "da Costa and Chuaqui" approach. This book engages with important discussions of the nature of scientific theories and will be a useful resource for researchers and upper-level students working in philosophy of science.

From a Geometrical Point of View

Jean-Pierre Marquis
From a Geometrical Point of View Book PDF

Author : Jean-Pierre Marquis
Publisher : Springer Science & Business Media
Page : 316 pages
File Size : 49,7 Mb
Release : 2008-11-20
Category : Science
ISBN : 9781402093845

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From a Geometrical Point of View by Jean-Pierre Marquis Pdf

From a Geometrical Point of View explores historical and philosophical aspects of category theory, trying therewith to expose its significance in the mathematical landscape. The main thesis is that Klein’s Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane’s work in the early 1940’s and follows the major developments of the theory from this perspective. Particular attention is paid to the philosophical elements involved in this development. The book ends with a presentation of categorical logic, some of its results and its significance in the foundations of mathematics. From a Geometrical Point of View aims to provide its readers with a conceptual perspective on category theory and categorical logic, in order to gain insight into their role and nature in contemporary mathematics. It should be of interest to mathematicians, logicians, philosophers of mathematics and science in general, historians of contemporary mathematics, physicists and computer scientists.

Axiomatic Projective Geometry

A. Heyting
Axiomatic Projective Geometry Book PDF

Author : A. Heyting
Publisher : Elsevier
Page : 160 pages
File Size : 43,6 Mb
Release : 2014-05-12
Category : Mathematics
ISBN : 9781483259314

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Axiomatic Projective Geometry by A. Heyting Pdf

Bibliotheca Mathematica: A Series of Monographs on Pure and Applied Mathematics, Volume V: Axiomatic Projective Geometry, Second Edition focuses on the principles, operations, and theorems in axiomatic projective geometry, including set theory, incidence propositions, collineations, axioms, and coordinates. The publication first elaborates on the axiomatic method, notions from set theory and algebra, analytic projective geometry, and incidence propositions and coordinates in the plane. Discussions focus on ternary fields attached to a given projective plane, homogeneous coordinates, ternary field and axiom system, projectivities between lines, Desargues' proposition, and collineations. The book takes a look at incidence propositions and coordinates in space. Topics include coordinates of a point, equation of a plane, geometry over a given division ring, trivial axioms and propositions, sixteen points proposition, and homogeneous coordinates. The text examines the fundamental proposition of projective geometry and order, including cyclic order of the projective line, order and coordinates, geometry over an ordered ternary field, cyclically ordered sets, and fundamental proposition. The manuscript is a valuable source of data for mathematicians and researchers interested in axiomatic projective geometry.

Elements of ?-Category Theory

Emily Riehl,Dominic Verity
Elements of Category Theory Book PDF

Author : Emily Riehl,Dominic Verity
Publisher : Cambridge University Press
Page : 781 pages
File Size : 45,8 Mb
Release : 2022-02-10
Category : Mathematics
ISBN : 9781108837989

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Elements of ?-Category Theory by Emily Riehl,Dominic Verity Pdf

This book develops the theory of infinite-dimensional categories by studying the universe, or ∞-cosmos, in which they live.

Model Theory and the Philosophy of Mathematical Practice

John T. Baldwin
Model Theory and the Philosophy of Mathematical Practice Book PDF

Author : John T. Baldwin
Publisher : Cambridge University Press
Page : 365 pages
File Size : 47,5 Mb
Release : 2018-01-25
Category : Mathematics
ISBN : 9781107189218

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Model Theory and the Philosophy of Mathematical Practice by John T. Baldwin Pdf

Recounts the modern transformation of model theory and its effects on the philosophy of mathematics and mathematical practice.

Iconicity and Abduction

Gianluca Caterina,Rocco Gangle
Iconicity and Abduction Book PDF

Author : Gianluca Caterina,Rocco Gangle
Publisher : Springer
Page : 180 pages
File Size : 54,7 Mb
Release : 2016-09-13
Category : Philosophy
ISBN : 9783319442457

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Iconicity and Abduction by Gianluca Caterina,Rocco Gangle Pdf

This book consolidates and extends the authors’ work on the connection between iconicity and abductive inference. It emphasizes a pragmatic, experimental and fallibilist view of knowledge without sacrificing formal rigor. Within this context, the book focuses particularly on scientific knowledge and its prevalent use of mathematics. To find an answer to the question “What kind of experimental activity is the scientific employment of mathematics?” the book addresses the problems involved in formalizing abductive cognition. For this, it implements the concept and method of iconicity, modeling this theoretical framework mathematically through category theory and topoi. Peirce's concept of iconic signs is treated in depth, and it is shown how Peirce's diagrammatic logical notation of Existential Graphs makes use of iconicity and how important features of this iconicity are representable within category theory. Alain Badiou’s set-theoretical model of truth procedures and his relational sheaf-based theory of phenomenology are then integrated within the Peircean logical context. Finally, the book opens the path towards a more naturalist interpretation of the abductive models developed in Peirce and Badiou through an analysis of several recent attempts to reformulate quantum mechanics with categorical methods. Overall, the book offers a comprehensive and rigorous overview of past approaches to iconic semiotics and abduction, and it encompasses new extensions of these methods towards an innovative naturalist interpretation of abductive reasoning.

Tool and Object

Ralph Krömer
Tool and Object Book PDF

Author : Ralph Krömer
Publisher : Springer Science & Business Media
Page : 400 pages
File Size : 43,6 Mb
Release : 2007-06-25
Category : Mathematics
ISBN : 9783764375249

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Tool and Object by Ralph Krömer Pdf

Category theory is a general mathematical theory of structures and of structures of structures. It occupied a central position in contemporary mathematics as well as computer science. This book describes the history of category theory whereby illuminating its symbiotic relationship to algebraic topology, homological algebra, algebraic geometry and mathematical logic and elaboratively develops the connections with the epistemological significance.

Categories for the Working Philosopher

Elaine M. Landry
Categories for the Working Philosopher Book PDF

Author : Elaine M. Landry
Publisher : Oxford University Press
Page : 486 pages
File Size : 52,5 Mb
Release : 2017
Category : Mathematics
ISBN : 9780198748991

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Categories for the Working Philosopher by Elaine M. Landry Pdf

This is the first book on category theory for a broad philosophical readership. There is no other discussion of category theory comparable in its scope. It is designed to show the interest and significant of category theory for philosophers working in a range of areas, including mathematics, proof theory, computer science, ontology, physics, biology, cognition, mathematical modelling, the structure of scientific theories, and the structure of the world. Moreover, it does this in a way that is accessible to non specialists. Each chapter is written by either a category-theorist or a philosopher working in one of the represented fields, in a way that builds on the concepts already familiar to philosophers working in these areas. The book is split into two halves. The 'pure' chapters focus on the use of category theory for mathematical, foundational, and logical purposes, while the 'applied' chapters consider the use of category theory for representational purposes, investigating category theory as a framework for theories of physics and biology, for mathematical modelling more generally, and for the structure of scientific theories. Book jacket.

Functional Analysis and Related Fields

F. E. Browder
Functional Analysis and Related Fields Book PDF

Author : F. E. Browder
Publisher : Springer Science & Business Media
Page : 248 pages
File Size : 50,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642482724

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Functional Analysis and Related Fields by F. E. Browder Pdf

On May 20-24. 1968, a Conference on Functional Analysis and Related Fields was held at the Center for Continuing Education of the University cl Chicago in honor of ProfessoLMARSHALL HARVEY STONE on the occasion of his retirement from active service at the University. The Conference received support from the Air Force Office of Scientific Research under the Grant AFOSR 68-1497. The Organizing committee for this Conference consisted of ALBERTO P. CALDERON, SAUNDERS MACLANE, ROBERT G. POHRER, and FELIX E. BROWDER (Chairman). The present volume contains some of the papers presented at the Conference. nther talks which were presented at the Conference for which papers are noLinduded hereare: K. CHANDRASEKHARAN, "Zeta functions of quadratic fields"; J. L. DooB, "An application of prob ability theory to the Choquet boundary" ; HALMOS, "Irreducible operators"; P. R. KADISON, "Strong continuity of operator functions"; L. NIRENBERG, "Intrinsic norms on complex manifolds"; D. SCOTT, "Some problems and recent results in Boolean algebras"; 1. M. SINGER, "A conjecture relating the Reidemeister torsion and the zeta function of the Laplacian". A banquet in honor of Professor STONE was held during the Con ference, with brief talks by S. S. CHERN, A. A. ALBERT, S. MACLANE, E. HEWITT, K. CHANDRASEKHARAN, and F. E. BROWDER (as Toast master), as weH as a response by Professor STONE.

The SAGE Encyclopedia of Theory in Science, Technology, Engineering, and Mathematics

James Mattingly
The SAGE Encyclopedia of Theory in Science Technology Engineering and Mathematics Book PDF

Author : James Mattingly
Publisher : SAGE Publications
Page : 1801 pages
File Size : 53,5 Mb
Release : 2022-10-28
Category : Reference
ISBN : 9781506353289

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The SAGE Encyclopedia of Theory in Science, Technology, Engineering, and Mathematics by James Mattingly Pdf

Project Description: Theories are part and parcel of every human activity that involves knowing about the world and our place in it. In all areas of inquiry from the most commonplace to the most scholarly and esoteric, theorizing plays a fundamental role. The SAGE Encyclopedia of Theory in Science, Technology, Engineering, and Mathematics focuses on the ways that various STEM disciplines theorize about their subject matter. How is thinking about the subject organized? What methods are used in moving a novice in given field into the position of a competent student of that subject? Within the pages of this landmark work, readers will learn about the complex decisions that are made when framing a theory, what goes into constructing a powerful theory, why some theories change or fail, how STEM theories reflect socio-historical moments in time and how – at their best – they form the foundations for exploring and unlocking the mysteries of the world around us. Featuring more than 200 authoritative articles written by experts in their respective fields, the encyclopedia includes a Reader’s Guide that organizes entries by broad themes; lists of Further Readings and cross-references that conclude each article; and a Resource Guide listing classic books in the field, leading journals, associations, and key websites.