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Paul Lorenzen -- Mathematician and Logician

Author: Gerhard Heinzmann

Publisher: Springer Nature

Published: 2021-08-17

Total Pages: 268

ISBN-13: 3030658244

This open access book examines the many contributions of Paul Lorenzen, an outstanding philosopher from the latter half of the 20th century. It features papers focused on integrating Lorenzen's original approach into the history of logic and mathematics. The papers also explore how practitioners can implement Lorenzen’s systematical ideas in today’s debates on proof-theoretic semantics, databank management, and stochastics. Coverage details key contributions of Lorenzen to constructive mathematics, Lorenzen’s work on lattice-groups and divisibility theory, and modern set theory and Lorenzen’s critique of actual infinity. The contributors also look at the main problem of Grundlagenforschung and Lorenzen’s consistency proof and Hilbert’s larger program. In addition, the papers offer a constructive examination of a Russell-style Ramified Type Theory and a way out of the circularity puzzle within the operative justification of logic and mathematics. Paul Lorenzen's name is associated with the Erlangen School of Methodical Constructivism, of which the approach in linguistic philosophy and philosophy of science determined philosophical discussions especially in Germany in the 1960s and 1970s. This volume features 10 papers from a meeting that took place at the University of Konstanz.

A Beautiful Math

Author: Tom Siegfried

Publisher: National Academies Press

Published: 2006-09-21

Total Pages: 272

ISBN-13: 0309133807

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Millions have seen the movie and thousands have read the book but few have fully appreciated the mathematics developed by John Nash's beautiful mind. Today Nash's beautiful math has become a universal language for research in the social sciences and has infiltrated the realms of evolutionary biology, neuroscience, and even quantum physics. John Nash won the 1994 Nobel Prize in economics for pioneering research published in the 1950s on a new branch of mathematics known as game theory. At the time of Nash's early work, game theory was briefly popular among some mathematicians and Cold War analysts. But it remained obscure until the 1970s when evolutionary biologists began applying it to their work. In the 1980s economists began to embrace game theory. Since then it has found an ever expanding repertoire of applications among a wide range of scientific disciplines. Today neuroscientists peer into game players' brains, anthropologists play games with people from primitive cultures, biologists use games to explain the evolution of human language, and mathematicians exploit games to better understand social networks. A common thread connecting much of this research is its relevance to the ancient quest for a science of human social behavior, or a Code of Nature, in the spirit of the fictional science of psychohistory described in the famous Foundation novels by the late Isaac Asimov. In A Beautiful Math, acclaimed science writer Tom Siegfried describes how game theory links the life sciences, social sciences, and physical sciences in a way that may bring Asimov's dream closer to reality.

A Computational Logic

Author: Robert S. Boyer

Publisher: Academic Press

Published: 2014-06-25

Total Pages: 414

ISBN-13: 1483277887

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ACM Monograph Series: A Computational Logic focuses on the use of induction in proving theorems, including the use of lemmas and axioms, free variables, equalities, and generalization. The publication first elaborates on a sketch of the theory and two simple examples, a precise definition of the theory, and correctness of a tautology-checker. Topics include mechanical proofs, informal development, formal specification of the problem, well-founded relations, natural numbers, and literal atoms. The book then examines the use of type information to simplify formulas, use of axioms and lemmas as rewrite rules, and the use of definitions. Topics include nonrecursive functions, computing values, free variables in hypothesis, infinite backwards chaining, infinite looping, computing type sets, and type prescriptions. The manuscript takes a look at rewriting terms and simplifying clauses, eliminating destructors and irrelevance, using equalities, and generalization. Concerns include reasons for eliminating isolated hypotheses, precise statement of the generalization heuristic, restricting generalizations, precise use of equalities, and multiple destructors and infinite looping. The publication is a vital source of data for researchers interested in computational logic.

Language in Action

Author: Johan van Benthem

Publisher: MIT Press

Published: 1995

Total Pages: 390

ISBN-13: 9780262720243

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Language in Action demonstrates the viability of mathematical research into the foundations of categorial grammar, a topic at the border between logic and linguistics. Since its initial publication it has become the classic work in the foundations of categorial grammar. A new introduction to this paperback edition updates the open research problems and records relevant results through pointers to the literature. Van Benthem presents the categorial processing of syntax and semantics as a central component in a more general dynamic logic of information flow, in tune with computational developments in artificial intelligence and cognitive science. Using the paradigm of categorial grammar, he describes the substructural logics driving the dynamics of natural language syntax and semantics. This is a general type-theoretic approach that lends itself easily to proof-theoretic and semantic studies in tandem with standard logic. The emphasis is on a broad landscape of substructural categorial logics and their proof-theoretical and semantic peculiarities. This provides a systematic theory for natural language understanding, admitting of significant mathematical results. Moreover, the theory makes possible dynamic interpretations that view natural languages as programming formalisms for various cognitive activities.

Social Constructivism as a Philosophy of Mathematics

Author: Paul Ernest

Publisher: SUNY Press

Published: 1998-01-01

Total Pages: 336

ISBN-13: 9780791435878

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Extends the ideas of social constructivism to the philosophy of mathematics, developing a powerful critique of traditional absolutist conceptions of mathematics, and proposing a reconceptualization of the philosophy of mathematics.

Constructive Measure Theory

Author: Errett Bishop

Publisher: American Mathematical Soc.

Published: 1972

Total Pages: 85

ISBN-13: 0821818163

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Handbook of Analysis and Its Foundations

Author: Eric Schechter

Publisher: Academic Press

Published: 1996-10-24

Total Pages: 907

ISBN-13: 0080532993

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Handbook of Analysis and Its Foundations is a self-contained and unified handbook on mathematical analysis and its foundations. Intended as a self-study guide for advanced undergraduates and beginning graduatestudents in mathematics and a reference for more advanced mathematicians, this highly readable book provides broader coverage than competing texts in the area. Handbook of Analysis and Its Foundations provides an introduction to a wide range of topics, including: algebra; topology; normed spaces; integration theory; topological vector spaces; and differential equations. The author effectively demonstrates the relationships between these topics and includes a few chapters on set theory and logic to explain the lack of examples for classical pathological objects whose existence proofs are not constructive. More complete than any other book on the subject, students will find this to be an invaluable handbook. Covers some hard-to-find results including: Bessagas and Meyers converses of the Contraction Fixed Point Theorem Redefinition of subnets by Aarnes and Andenaes Ghermans characterization of topological convergences Neumanns nonlinear Closed Graph Theorem van Maarens geometry-free version of Sperners Lemma Includes a few advanced topics in functional analysis Features all areas of the foundations of analysis except geometry Combines material usually found in many different sources, making this unified treatment more convenient for the user Has its own webpage: http://math.vanderbilt.edu/

Meaning, Logic and Ludics

Author: Alain Lecomte

Publisher: World Scientific

Published: 2011

Total Pages: 420

ISBN-13: 1848164564

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7. Grammatical reasoning. 7.1. Motivations. 7.2. Modal preliminary. 7.3. Residuation and modalities. 7.4. Linguistic applications. 7.5. Back to quantification. 7.6. Kripke semantics. 7.7. Concluding remarks and observations. 8. A type-theoretical version of minimalist grammars. 8.1. Inserting chains. 8.2. Head movement. 8.3. Adjoining and scrambling. 8.4. Semantics without cooper storage. 8.5. Concluding remarks : Some tracks to explore. 9. Grammars in deductive forms. 9.1. Introduction. 9.2. Convergent grammars. 9.3. Labelled linear grammars. 9.4. Binding in LLG. 9.5. On phases. 9.6. Comparing CVG and LLG. 9.7. Concluding remarks. 10. Continuations and contexts. 10.1. The use of continuations in semantics. 10.2. Symmetric calculi. 10.3. Concluding remarks and further works. 11. Proofs as meanings. 11.1. From intuitionistic logic to constructive type theory. 11.2. Formalizing Montague grammar in constructive type theory. 11.3. Dynamical interpretation and anaphoric expressions. 11.4. From sentences to dialogue -- pt. IV. Ludics. 12. Interaction and dialogue. 12.1. Dialogue and games. 12.2. Ludics. 12.3. Behaviours. 13. The future in conclusion

Second Philosophy

Author: Penelope Maddy

Publisher: Oxford University Press

Published: 2007-04-19

Total Pages: 461

ISBN-13: 0199273669

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Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In this book, Penelope Maddy describes and practises a particularly austere form of naturalism called 'Second Philosophy'. Without a definitive criterion for what counts as 'science' and what doesn't, Second Philosophy can't be specified directly - 'trust only the methods of science!' or some such thing - so Maddy proceeds instead by illustratingthe behaviours of an idealized inquirer she calls the 'Second Philosopher'. This Second Philosopher begins from perceptual common sense and progresses from there to systematic observation, active experimentation, theory formation and testing, working all the while to assess, correct and improve hermethods as she goes. Second Philosophy is then the result of the Second Philosopher's investigations.Maddy delineates the Second Philosopher's approach by tracing her reactions to various familiar skeptical and transcendental views (Descartes, Kant, Carnap, late Putnam, van Fraassen), comparing her methods to those of other self-described naturalists (especially Quine), and examining a prominent contemporary debate (between disquotationalists and correspondence theorists in the theory of truth) to extract a properly second-philosophical line of thought. She then undertakes to practise SecondPhilosophy in her reflections on the ground of logical truth, the methodology, ontology and epistemology of mathematics, and the general prospects for metaphysics naturalized.

Logic, Epistemology, and the Unity of Science

Author: Shahid Rahman

Publisher: Springer Science & Business Media

Published: 2009-03-15

Total Pages: 618

ISBN-13: 1402028083

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The first volume in this new series explores, through extensive co-operation, new ways of achieving the integration of science in all its diversity. The book offers essays from important and influential philosophers in contemporary philosophy, discussing a range of topics from philosophy of science to epistemology, philosophy of logic and game theoretical approaches. It will be of interest to philosophers, computer scientists and all others interested in the scientific rationality.