From Pythagoreans to Hegel, and beyond, this book gives a brief overview of the history of the notion of graphs and introduces the main concepts of graph theory in order to apply them to philosophy. In addition, this book presents how philosophers can use various mathematical notions of order. Throughout the book, philosophical operations and concepts are defined through examining questions relating the two kinds of known infinities – discrete and continuous – and how Woodin's approach can influence elements of philosophy. We also examine how mathematics can help a philosopher to discover the elements of stability which will help to build an image of the world, even if various approaches (for example, negative theology) generally cannot be valid. Finally, we briefly consider the possibilities of weakening formal thought represented by fuzziness and neutrosophic graphs. In a nutshell, this book expresses the importance of graphs when representing ideas and communicating them clearly with others.

Intro -- Contents -- Prolegomena by Giuliano di Bernardo -- Preface -- The Scope and the Structure of this Project -- Acknowledgments -- Chapter 1 -- Philosophy, Science, and The Dialectic of Rational Dynamicity -- 1.1. The Meaning of Philosophy and Preliminary Concepts -- 1.2. The Abstract Study of a Being -- 1.2.1. Epistemological Presuppositions -- 1.2.2. The Significance and the Presence of a Being -- 1.2.3. The Knowledge of a Being -- Structuralism in Physics -- Newton's Three Laws of Kinematics -- Newton's Law of Universal Gravitation -- Conservation of Mass and Energy -- Laws of Thermodynamics -- Electrostatic Laws -- Quantum Mechanics -- Structuralism in Biology -- Structuralism in Linguistics -- Philosophical Structuralism and Hermeneutics -- 1.2.4. The Modes of Being -- 1.3. The Dialectic of Rational Dynamicity -- 1.3.1. Dynamized Time -- 1.3.2. Dynamized Space and the Problem of the Extension of the Quantum Formalism -- 1.3.3. Consciousness, the World, and the Dialectic of Rational Dynamicity -- 1.3.4. Matter, Life, and Consciousness -- Chapter 2 -- Foundations of Mathematical Analysis and Analytic Geometry -- 2.1. Sets, Relations, and Groups -- 2.1.2. Basic Operations on Sets -- Applications of Set Theory to Probability Theory -- 2.1.3. Relations -- 2.1.4. Groups -- 2.2. Number Systems, Algebra, and Geometry -- 2.2.1. Axiomatic Number Theory -- The System of Natural Numbers -- Principle of Mathematical Induction -- Recursion -- Properties of the System of Natural Numbers -- Enumeration -- Order in N and Ordinal Numbers -- Division -- 2.2.2. The Set of Integral Numbers -- 2.2.3. The Set of Rational Numbers -- 2.2.4. The Set of Real Numbers -- Dedekind Algebra -- R as a Field -- The Absolute Value of a Real Number -- Exponentiation and Logarithm -- Properties of the System of the Real Numbers.

1. Philosophy of Mathematics and Mathematical Practice in the Early Seventeenth Century p. 8 1.1 The Quaestio de Certitudine Mathematicarum p. 10 1.2 The Quaestio in the Seventeenth Century p. 15 1.3 The Quaestio and Mathematical Practice p. 24 2. Cavalieri's Geometry of Indivisibles and Guldin's Centers of Gravity p. 34 2.1 Magnitudes, Ratios, and the Method of Exhaustion p. 35 2.2 Cavalieri's Two Methods of Indivisibles p. 38 2.3 Guldin's Objections to Cavalieri's Geometry of Indivisibles p. 50 2.4 Guldin's Centrobaryca and Cavalieri's Objections p. 56 3. Descartes' Geometrie p. 65 3.1 Descartes' Geometrie p. 65 3.2 The Algebraization of Mathematics p. 84 4. The Problem of Continuity p. 92 4.1 Motion and Genetic Definitions p. 94 4.2 The "Causal" Theories in Arnauld and Bolzano p. 100 4.3 Proofs by Contradiction from Kant to the Present p. 105 5. Paradoxes of the Infinite p. 118 5.1 Indivisibles and Infinitely Small Quantities p. 119 5.2 The Infinitely Large p. 129 6. Leibniz's Differential Calculus and Its Opponents p. 150 6.1 Leibniz's Nova Methodus and L'Hopital's Analyse des Infiniment Petits p. 151 6.2 Early Debates with Cluver and Nieuwentijt p. 156 6.3 The Foundational Debate in the Paris Academy of Sciences p. 165 Appendix Giuseppe Biancani's De Mathematicarum Natura p. 178 Notes p. 213 References p. 249 Index p. 267.

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.

First published in 1974. Despite the tendency of contemporary analytic philosophy to put logic and mathematics at a central position, the author argues it failed to appreciate or account for their rich content. Through discussions of such mathematical concepts as number, the continuum, set, proof and mechanical procedure, the author provides an introduction to the philosophy of mathematics and an internal criticism of the then current academic philosophy. The material presented is also an illustration of a new, more general method of approach called substantial factualism which the author asserts allows for the development of a more comprehensive philosophical position by not trivialising or distorting substantial facts of human knowledge.

A sophisticated, original introduction to the philosophy of mathematics from one of its leading thinkers Mathematics is a model of precision and objectivity, but it appears distinct from the empirical sciences because it seems to deliver nonexperiential knowledge of a nonphysical reality of numbers, sets, and functions. How can these two aspects of mathematics be reconciled? This concise book provides a systematic, accessible introduction to the field that is trying to answer that question: the philosophy of mathematics. Øystein Linnebo, one of the world's leading scholars on the subject, introduces all of the classical approaches to the field as well as more specialized issues, including mathematical intuition, potential infinity, and the search for new mathematical axioms. Sophisticated but clear and approachable, this is an essential book for all students and teachers of philosophy and of mathematics.