In Pi ( ) in Nature, Art, and Culture Marcel Danesi revisits the importance of as a pattern in the structure of reality, fitting in with the Pythagorean view of Order. Pi has cropped up in formulas that describe natural and physical structures which, on the surface, seem to have nothing to do with a circle, but might harbor the archetype of circularity as a principle. Through , this book thus revisits the implicit ancient Greek view that geometry was a 'hermeneutic science,' a discipline aiming to investigate the connectivity among numbers, shapes, and natural phenomena. It also examines its manifestations in aesthetic, symbolic and cultural structures, which point to an abiding fascination with the circle as an unconscious archetype. Hermeneutic geometry is ultimately about the exploration of the meanings of geometric-mathematical notions to science and human life.

This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects.

Agenda Relevance is the first volume in the authors' omnibus investigation of
the logic of practical reasoning, under the collective title, A Practical Logic
of Cognitive Systems. In this highly original approach, practical reasoning is
identified as reasoning performed with comparatively few cognitive assets,
including resources such as information, time and computational capacity. Unlike
what is proposed in optimization models of human cognition, a practical reasoner
lacks perfect information, boundless time and unconstrained access to
computational complexity. The practical reasoner is therefore obliged to be a
cognitive economizer and to achieve his cognitive ends with considerable
efficiency. Accordingly, the practical reasoner avails himself of various
scarce-resource compensation strategies. He also possesses neurocognitive
traits that abet him in his reasoning tasks. Prominent among these is the
practical agent's striking (though not perfect) adeptness at evading irrelevant
information and staying on task. On the approach taken here, irrelevancies are
impediments to the attainment of cognitive ends. Thus, in its most basic sense,
relevant information is cognitively helpful information. Information can then be
said to be relevant for a practical reasoner to the extent that it advances or
closes some cognitive agenda of his. The book explores this idea with a
conceptual detail and nuance not seen the standard semantic, probabilistic and
pragmatic approaches to relevance; but wherever possible, the authors seek to
integrate alternative conceptions rather than reject them outright. A further
attraction of the agenda-relevance approach is the extent to which its principal
conceptual findings lend themselves to technically sophisticated re-expression
in formal models that marshal the resources of time and action logics and
label led deductive systems.

Agenda Relevance is necessary reading for researchers in logic, belief
dynamics, computer science, AI, psychology and neuroscience, linguistics,
argumentation theory, and legal reasoning and forensic science, and will repay
study by graduate students and senior undergraduates in these same fields.

Key features:

relevance
action and agendas
practical reasoning
belief dynamics
non-classical logics
labelled deductive systems

"

Chemistry, physics and biology are by their nature genuinely difficult. Mathematics, however, is man-made, and therefore not as complicated. Two ideas form the basis for this book: 1) to use ordinary mathematics to describe the simplicity in the structure of mathematics and 2) to develop new branches of mathematics to describe natural sciences.

Mathematics can be described as the addition, subtraction or multiplication of planes. Using the exponential scale the authors show that the addition of planes gives the polyhedra, or any solid. The substraction of planes gives saddles. The multiplication of planes gives the general saddle equations and the multispirals. The equation of symmetry is derived, which contains the exponential scale with its functions for solids, the complex exponentials with the nodal surfaces, and the GD (Gauss Distribution) mathematics with finite periodicity.

Piece by piece, the authors have found mathematical functions for the geometrical descriptions of chemical structures and the structure building operations. Using the mathematics for dilatation; twins, trillings, fourlings and sixlings are made, and using GD mathematics these are made periodic. This description of a structure is the nature of mathematics itself. Crystal structures and 3D mathematics are synonyms. Mathematics are used to describe rod packings, Olympic rings and defects in solids. Giant molecules such as cubosomes, the DNA double helix, and certain building blocks in protein structures are also described mathematically.

Frege's Theorem collects eleven essays by Richard G Heck, Jr, one of the world's leading authorities on Frege's philosophy. The Theorem is the central contribution of Gottlob Frege's formal work on arithmetic. It tells us that the axioms of arithmetic can be derived, purely logically, from a single principle: the number of these things is the same as the number of those things just in case these can be matched up one-to-one with those. But that principle seems so utterly fundamental to thought about number that it might almost count as a definition of number. If so, Frege's Theorem shows that arithmetic follows, purely logically, from a near definition. As Crispin Wright was the first to make clear, that means that Frege's logicism, long thought dead, might yet be viable.
Heck probes the philosophical significance of the Theorem, using it to launch and then guide a wide-ranging exploration of historical, philosophical, and technical issues in the philosophy of mathematics and logic, and of their connections with metaphysics, epistemology, the philosophy of language and mind, and even developmental psychology. The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues. There are also new postscripts to five of the essays, which discuss changes of mind, respond to published criticisms, and advance the discussion yet further.

Not all scientific explanations work by describing causal connections between events or the world's overall causal structure. Some mathematical proofs explain why the theorems being proved hold. In this book, Marc Lange proposes philosophical accounts of many kinds of non-causal explanations in science and mathematics. These topics have been unjustly neglected in the philosophy of science and mathematics. One important kind of non-causal scientific explanation is termed explanation by constraint. These explanations work by providing information about what makes certain facts especially inevitable - more necessary than the ordinary laws of nature connecting causes to their effects. Facts explained in this way transcend the hurly-burly of cause and effect. Many physicists have regarded the laws of kinematics, the great conservation laws, the coordinate transformations, and the parallelogram of forces as having explanations by constraint. This book presents an original account of explanations by constraint, concentrating on a variety of examples from classical physics and special relativity. This book also offers original accounts of several other varieties of non-causal scientific explanation. Dimensional explanations work by showing how some law of nature arises merely from the dimensional relations among the quantities involved. Really statistical explanations include explanations that appeal to regression toward the mean and other canonical manifestations of chance. Lange provides an original account of what makes certain mathematical proofs but not others explain what they prove. Mathematical explanation connects to a host of other important mathematical ideas, including coincidences in mathematics, the significance of giving multiple proofs of the same result, and natural properties in mathematics. Introducing many examples drawn from actual science and mathematics, with extended discussions of examples from Lagrange, Desargues, Thomson, Sylvester, Maxwell, Rayleigh, Einstein, and Feynman, Because Without Cause's proposals and examples should set the agenda for future work on non-causal explanation.

Contents: Introduction; I. ONTOLOGY; 1. Existence (1987); 2. Nonexistence (1998); 3. Mythical Objects (2002); II. NECESSITY; 4. Modal Logic Kalish-and-Montague Style (1994); 5. Impossible Worlds (1984); 6. An Empire of Thin Air (1988); 7. The Logic of What Might Have Been (1989); III. IDENTITY; 8. The fact that x=y (1987); 9. This Side of Paradox (1993); 10. Identity Facts (2003); 11. Personal Identity: What's the Problem? (1995); IV. PHILOSOPHY OF MATHEMATICS; 12. Wholes, Parts, and Numbers (1997); 13. The Limits of Human Mathematics (2001); V. THEORY OF MEANING AND REFERENCE; 14. On Content (1992); 15. On Designating (1997); 16. A Problem in the Frege-Church Theory of Sense and Denotation (1993); 17. The Very Possibility of Language (2001); 18. Tense and Intension (2003); 19. Pronouns as Variables (2005)

According to Grothendieck, the notion of topos is "the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures". It is what he had "conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an "essence" which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things". The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision by building on the notion of classifying topos educed by categorical logicians. Mathematical theories (formalized within first-order logic) give rise to geometric objects called sites; the passage from sites to their associated toposes embodies the passage from the logical presentation of theories to their mathematical content, i.e. from syntax to semantics. The essential ambiguity given by the fact that any topos is associated in general with an infinite number of theories or different sites allows to study the relations between different theories, and hence the theories themselves, by using toposes as 'bridges' between these different presentations. The expression or calculation of invariants of toposes in terms of the theories associated with them or their sites of definition generates a great number of results and notions varying according to the different types of presentation, giving rise to a veritable mathematical morphogenesis.

This book is a specialized monograph on interpolation and definability, a notion central in pure logic and with significant meaning and applicability in all areas where logic is applied, especially computer science, artificial intelligence, logic programming, philosophy of science and natural language. Suitable for researchers and graduate students in mathematics, computer science and philosophy, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (second edition), J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2, and David J. Pym and Eike Ritter's Reductive Logic and Proof Search: Proof theory, semantics and control.

This book gathers the best presentations from the Topic Study Group 30: Mathematics Competitions at ICME-13 in Hamburg, and some from related groups, focusing on the field of working with gifted students. Each of the chapters includes not only original ideas, but also original mathematical problems and their solutions. The book is a valuable resource for researchers in mathematics education, secondary and college mathematics teachers around the globe as well as their gifted students.

Quantum gravity is the name given to a theory that unites general relativity - Einstein's theory of gravitation and spacetime - with quantum field theory, our framework for describing non-gravitational forces. The Structural Foundations of Quantum Gravity brings together philosophers and physicists to discuss a range of conceptual issues that surface in the effort to unite these theories, focusing in particular on the ontological nature of the spacetime that results. Although there has been a great deal written about quantum gravity from the perspective of physicists and mathematicians, very little attention has been paid to the philosophical aspects. This volume closes that gap, with essays written by some of the leading researchers in the field. Individual papers defend or attack a structuralist perspective on the fundamental ontologies of our physical theories, which offers the possibility of shedding new light on a number of foundational problems. It is a book that will be of interest not only to physicists and philosophers of physics but to anyone concerned with foundational issues and curious to explore new directions in our understanding of spacetime and quantum physics.

This book explores an important central thread that unifies Russell's thoughts on logic in two works previously considered at odds with each other, the Principles of Mathematics and the later Principia Mathematica. This thread is Russell's doctrine that logic is an absolutely general science and that any calculus for it must embrace wholly unrestricted variables. The heart of Landini's book is a careful analysis of Russell's largely unpublished "substitutional" theory. On Landini's showing, the substitutional theory reveals the unity of Russell's philosophy of logic and offers new avenues for a genuine solution of the paradoxes plaguing Logicism.

A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true.

This book collects research papers on the philosophical foundations of probability, causality, spacetime and quantum theory. The papers are related to talks presented in six subsequent workshops organized by The Budapest-Krakow Research Group on Probability, Causality and Determinism. Coverage consists of three parts. Part I focuses on the notion of probability from a general philosophical and formal epistemological perspective. Part II applies probabilistic considerations to address causal questions in the foundations of quantum mechanics. Part III investigates the question of indeterminism in spacetime theories. It also explores some related questions, such as decidability and observation. The contributing authors are all philosophers of science with a strong background in mathematics or physics. They believe that paying attention to the finer formal details often helps avoiding pitfalls that exacerbate the philosophical problems that are in the center of focus of contemporary research. The papers presented here help make explicit the mathematical-structural assumptions that underlie key philosophical argumentations. This formally rigorous and conceptually precise approach will appeal to researchers and philosophers as well as mathematicians and statisticians.

The nature of truth in mathematics is a problem which has exercised the minds of thinkers from at least the time of the ancient Greeks. The great advances in mathematics and philosophy in the twentieth century--and in particular the proof of Gödel's theorem and the development of the notion of independence in mathematics--have led to new viewpoints on his question. This book is the result of the interaction of a number of outstanding mathematicians and philosophers--including Yurii Manin, Vaughan Jones, and Per Martin-Löf--and their discussions of this problem. It provides an overview of the forefront of current thinking, and is a valuable introduction and reference for researchers in the area.

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl's phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of "naturalist" and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the "unreasonable" effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies.

This book presents 25 selected papers from the International Conference on "Developing Synergies between Islam & Science and Technology for Mankind's Benefit" held at the International Institute for Advanced Islamic Studies Malaysia, Kuala Lumpur, in October 2014. The papers cover a broad range of issues reflecting the main conference themes: Cosmology and the Universe, Philosophy of Science and the Emergence of Biological Systems, Principles and Applications of Tawhidic Science, Medical Applications of Tawhidic Science and Bioethics, and the History and Teaching of Science from an Islamic Perspective. Highlighting the relationships between the Islamic religious worldview and the physical sciences, the book challenges secularist paradigms on the study of Science and Technology. Integrating metaphysical perspectives of Science, topics include Islamic approaches to S&T such as an Islamic epistemology of the philosophy of science, a new quantum theory, environmental care, avoiding wasteful consumption using Islamic teachings, and emotional-blasting psychological therapy. Eminent contributing scholars include Osman Bakar, Mohammad Hashim Kamali, Mehdi Golshani, Mohd. Kamal Hassan, Adi Setia and Malik Badri. The book is essential reading for a broad group of academics and practitioners, from Islamic scholars and social scientists to (physical) scientists and engineers.

This volume contains thirteen papers that were presented at the 2017 Annual Meeting of the Canadian Society for History and Philosophy of Mathematics/Societe canadienne d'histoire et de philosophie des mathematiques, which was held at Ryerson University in Toronto. It showcases rigorously reviewed modern scholarship on an interesting variety of topics in the history and philosophy of mathematics from Ancient Greece to the twentieth century. A series of chapters all set in the eighteenth century consider topics such as John Marsh's techniques for the computation of decimal fractions, Euler's efforts to compute the surface area of scalene cones, a little-known work by John Playfair on the practical aspects of mathematics, and Monge's use of descriptive geometry. After a brief stop in the nineteenth century to consider the culture of research mathematics in 1860s Prussia, the book moves into the twentieth century with an examination of the historical context within which the Axiom of Choice was developed and a paper discussing Anatoly Vlasov's adaptation of the Boltzmann equation to ionized gases. The remaining chapters deal with the philosophy of twentieth-century mathematics through topics such as an historically informed discussion of finitism and its limits; a reexamination of Mary Leng's defenses of mathematical fictionalism through an alternative, anti-realist approach to mathematics; and a look at the reasons that mathematicians select specific problems to pursue. Written by leading scholars in the field, these papers are accessible to not only mathematicians and students of the history and philosophy of mathematics, but also anyone with a general interest in mathematics.

One hundred years ago, Russell and Whitehead published their epoch-making Principia Mathematica (PM), which was initially conceived as the second volume of Russell's Principles of Mathematics (PoM) that had appeared ten years before. No other works can be credited to have had such an impact on the development of logic and on philosophy in the twentieth century. However, until now, scholars only focused on the first parts of the books - that is, on Russell's and Whitehead's theory of logic, set-theory and arithmetic.
Sebastien Gandon aims at reversing the perspective. His goal is to give a picture of Russell's logicism based on a detailed reading of the developments dealing with advanced mathematics - namely projective geometry and the theory of quantity. This book is not only the first study ever made of the 'later' portions of PoM and PM, it also shows how this shift of perspective compels us to change our view of the logicist program taken as a whole.

Truth Through Proof defends an anti-platonist philosophy of mathematics derived from game formalism. Classic formalists claimed implausibly that mathematical utterances are truth-valueless moves in a game. Alan Weir aims to develop a more satisfactory successor to game formalism utilising a widely accepted, broadly neo-Fregean framework, in which the proposition expressed by an utterance is a function of both sense and background circumstance. This framework allows for sentences whose truth-conditions are not representational, which are made true or false by conditions residing in the circumstances of utterances but not transparently in the sense.
Applications to projectivism and fiction pave the way for the claim that mathematical utterances are made true or false by the existence of concrete proofs or refutations, though these truth-making conditions form no part of their sense or informational content.
The position is compared with rivals, an account of the applicability of mathematics developed, and a new account of the nature of idealisation proffered in which it is argued that the finitistic limitations Godel placed on proofs are without rational justification. Finally a non-classical logical system is provided in which excluded middle fails, yet enough logical power remains to recapture the results of standard mathematics.

This book examines mathematical discourse from the perspective of Michael Halliday's social semiotic theory. In this approach, mathematics is conceptualized as a multisemiotic discourse involving language, visual images and symbolism. The book discusses the evolution of the semiotics of mathematical discourse, and then, proceeds to examine the grammar of mathematical symbolism, the grammar of mathematical visual images, intersemiosis between language, visual images and symbolism and the subsequent ways in which mathematics orders reality. The focus of this investigation is written mathematical texts. The aims of the book are to understand the semantic realm of mathematics and to appreciate the metaphorical expansions and simultaneous limitations of meaning in mathematical discourse. The book is intended for linguists, semioticians, social scientists and those interested in mathematics and science education. In addition, the close study of the multisemiotic mature of mathematics has implications for other studies adopting a social semiotic approach to multimodality.

Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.

Mathematics is as much a science of the real world as biology is. It is the science of the world's quantitative aspects (such as ratio) and structural or patterned aspects (such as symmetry). This book develops a complete philosophy of mathematics that contrasts with the usual Platonist and nominalist options.

Church's Thesis (CT) was first published by Alonzo Church in 1935. CT is a proposition that identifies two notions: an intuitive notion of a effectively computable function defined in natural numbers with the notion of a recursive function. Despite of the many efforts of prominent scientists, Church's Thesis has never been falsified. There exists a vast literature concerning the thesis. The aim of the book is to provide one volume summary of the state of research on Church's Thesis. These include the following: different formulations of CT, CT and intuitionism, CT and intensional mathematics, CT and physics, the epistemic status of CT, CT and philosophy of mind, provability of CT and CT and functional programming.