Philosophy of Mathematics: An Introduction to the World of Proofs and PicturesPhilosophy of Mathematics is an excellent introductory text. This student friendly book discusses the great philosophers and the importance of mathematics to their thought. It includes the following topics: |
Contents
Introduction The Mathematical Image | 1 |
Platonism | 8 |
Some Recent Views | 9 |
What is Platonism? | 11 |
The Problem of Access | 15 |
The Problem of Certainty | 18 |
Platonism and its Rivals | 23 |
Pictureproofs and Platonism | 25 |
Lakatos | 107 |
Concluding Remarks | 112 |
Constructive Approaches | 113 |
From Kant to Brouwer | 114 |
Brouwers Intuitionism | 115 |
Bishops Constructivism | 117 |
Dummetts Antirealism | 118 |
Logic | 120 |
What Did Bolzano Do? | 28 |
Different Theorems Different Concepts? | 29 |
Inductive Mathematics | 30 |
Special and General Cases | 33 |
Instructive Examples | 34 |
Representation | 37 |
A Kantian Objection | 39 |
Three Analogies | 40 |
Are Pictures Explanatory? | 42 |
So Why Worry? | 43 |
What is Applied Mathematics? | 46 |
Representations | 47 |
Artifacts | 49 |
Bogus Applications | 51 |
Does Science Need Mathematics? | 52 |
Representation vs Description | 55 |
Structuralism | 57 |
Hilbert and Gödel | 62 |
Early Formalism | 63 |
Hilberts Formalism | 64 |
Hilberts Programme | 68 |
Small Problems | 70 |
Gödels Theorem | 71 |
Gödels Second Theorem | 75 |
The Upshot for Hilberts Programme | 77 |
Knots and Notation | 79 |
Knots | 81 |
The Dowker Notation | 83 |
The Conway Notation | 84 |
Polynomials | 86 |
Creation or Revelation? | 88 |
Sense Reference and Something Else | 92 |
What is a Definition? | 94 |
The FregeHilbert Debate | 95 |
Reductionism | 102 |
Graph Theory | 103 |
Problems | 122 |
Proofs Pictures and Procedures in Wittgenstein | 130 |
Following a Rule | 132 |
Platonism | 136 |
Algorithms | 138 |
Knowing Our Own Intentions | 139 |
Radical Conventionalism | 140 |
Bizarre Examples | 141 |
Naturalism | 142 |
The Sceptical Solution | 144 |
Modus Ponens or Modus Tollens? | 145 |
What is a Rule? | 146 |
Grasping a Sense | 147 |
Platonism vs Realism | 149 |
Surveyability | 151 |
The Sense of a Picture | 152 |
Computation Proof and Conjecture | 154 |
Fallibility | 155 |
Surveyability | 157 |
Inductive Mathematics | 158 |
Perfect Numbers | 160 |
Computation | 162 |
Is 𝝅 Normal? | 164 |
Fermats Last Theorem | 165 |
The Riemann Hypothesis | 166 |
Clusters of Conjectures | 167 |
Polya and Putnam | 168 |
Conjectures and Axioms | 170 |
Calling the Bluff | 172 |
Calling the Bluff | 179 |
A Report from the Front | 181 |
The Mathematical Image | 191 |
Notes | 193 |
| 199 | |
| 208 | |
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Common terms and phrases
abstract entities actually answer argument arithmetic assert axioms Bolzano Brouwer causal Chapter claim concept consequences consistency constructive constructivist counter-example course defined definition diagram distinct Dowker notation Dummett ematics evidence example existence fact fallibilism fallible false Fields Medal Figure finite number fractal Frege geometry Gödel Goldbach's conjecture graph theory grasp Hilbert ibid inductive inference infinitely intuition isomorphic Kantian knot theory Lakatos logic math mathematical entities mathematical objects mathematical truths mathematicians means Mersenne primes natural numbers notation notion number theory paradoxes particular perfect number perhaps philosophers philosophy of mathematics physical picture-proofs Platonism Platonistic polyhedron polynomial prime number principle priori problem proof properties proposition prove Putnam rational real numbers realism reason Reidemeister moves relation representation result Riemann hypothesis rigorous role rule sceptical sense sequence set theory simply statement structure surveyable symbols theorem things tion true unlabelled graph verbal/symbolic Wittgenstein