An Aristotelian Realist Philosophy of Mathematics : Mathematics As the Science of Quantity and Structure

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge.background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. "Cover " -- "Half-Title" -- "Title " -- "Copyright " -- "Contents" -- "List of Figures" -- "List of Tables" -- "Introduction " -- "Part I The Science of Quantity and Structure " -- "1 The Aristotelian Realist Point of View" -- "2 Uninstantiated Universals and â#x80;Semi-Platonistâ#x80;#x99; Aristotelianism" -- "3 Elementary Mathematics: The Science of Quantity" -- "4 Higher Mathematics: Science of the Purely Structural" -- " 5 Necessary Truths about Reality" -- "6 The Formal Sciences Discover the Philosophersâ#x80;#x99; Stone" -- "7 Comparisons and Objections" -- "8 Infinity" -- "9 Geometry: Mathematics or Empirical Science?" -- "Part II Knowing Mathematical Reality " -- "10 Knowing Mathematics: Pattern Recognition and Perception of Quantity and Structure" -- "11 Knowing Mathematics: Visualization and Understanding" -- "12 Knowing Mathematics: Proof and Certainty" -- "13 Explanation in Mathematics" -- "14 Idealization: An Aristotelian View" -- "15 Non-Deductive Logic in Mathematics" -- "Epilogue: Mathematics, Last Bastion of Reason " -- "Notes " -- "Select Bibliography " Front Matter....Pages i-x Introduction....Pages 1-7 Front Matter....Pages 9-9 The Aristotelian Realist Point of View....Pages 11-20 Uninstantiated Universals and ‘Semi-Platonist’ Aristotelianism....Pages 21-30 Elementary Mathematics: The Science of Quantity....Pages 31-47 Higher Mathematics: Science of the Purely Structural....Pages 48-66 Necessary Truths about Reality....Pages 67-81 The Formal Sciences Discover the Philosophers’ Stone....Pages 82-100 Comparisons and Objections....Pages 101-128 Infinity....Pages 129-140 Geometry: Mathematics or Empirical Science?....Pages 141-162 Front Matter....Pages 163-163 Knowing Mathematics: Pattern Recognition and Perception of Quantity and Structure....Pages 165-179 Knowing Mathematics: Visualization and Understanding....Pages 180-191 Knowing Mathematics: Proof and Certainty....Pages 192-206 Explanation in Mathematics....Pages 207-221 Idealization: An Aristotelian View....Pages 222-240 Non-Deductive Logic in Mathematics....Pages 241-259 Epilogue: Mathematics, Last Bastion of Reason....Pages 260-262 Back Matter....Pages 263-308 "An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge."--Page 4 of cover Is quantity a kind of structure? 5 Necessary Truths about Reality; Examples of necessity; Objections and replies; 6 The Formal Sciences Discover the Philosophers' Stone; A brief survey of the formal or mathematical sciences; The formal sciences search for a place in the sun; Real certainty: program verification; Real certainty: the other formal sciences; Experiment in the formal sciences; 7 Comparisons and Objections; Frege's limited options; The Platonist/nominalist false dichotomy; Nominalism; Constructions in set theory; Avoiding the question: what are sets?; Overemphasis on the infinite Measurement and the applicability of mathematicsThe indispensability argument; Modal and Platonist structuralism; Epistemology and 'access'; Naturalism: non-Platonist realisms; 8 Infinity; Infinity, who needs it?; Paradoxes of infinity?; 'Potential' infinity?; Knowing the infinite; 9 Geometry: Mathematics or Empirical Science?; What is geometry? Plan A: multidimensional quantities; What is geometry? Plan B: the shapes of possible spaces; The grit-or-gunk controversy: does space consist of points?; Real non-spatial 'spaces' with geometric structure; The space of colours; Spaces of vectors The real space we live inNon-Euclidean geometry: the 'loss of certainty' in mathematics?; Part II Knowing Mathematical Reality; 10 Knowing Mathematics: Pattern Recognition and Perception of Quantity and Structure; The registering of mathematical properties by measurement devices and artificial intelligence; Babies and animals: the simplest mathematical perception; Animal and infant knowledge of quantity; Perceptual knowledge of pattern and structure; 11 Knowing Mathematics: Visualization and Understanding; Imagination and the uninstantiated; Visualization for understanding structure 3 Elementary Mathematics: The Science of QuantityTwo realist theories of mathematics: quantity versus structure; Continuous quantity and ratios; Discrete quantity and numbers; Discrete quantity and sets; Discrete and continuous quantity compared; Defining 'quantity'; 4 Higher Mathematics: Science of the Purely Structural; The rise of structure in mathematics; Structuralism in recent philosophy of mathematics; Abstract algebra, groups, and modern pure mathematics; Structural commonality in applied mathematics; Defining 'structure'; The sufficiency of mereology and logic Cover; Half-Title; Title; Copyright; Contents; List of Figures; List of Tables; Introduction; Part I The Science of Quantity and Structure; 1 The Aristotelian Realist Point of View; The reality of universals; Platonism and nominalism; The reality of relations and structure; 'Unit-making' properties and sets; Causality; Aristotelian epistemology; 2 Uninstantiated Universals and 'Semi-Platonist' Aristotelianism; Determinables and determinates; Uninstantiated shades of blue and huge numbers; Possibles by recombination?; Semi-Platonist Aristotelianism 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). The Book Develops A Complete Philosophy Of Mathematics That Contrasts With The Usual Platonist And Nominalist Options. By James Franklinches