Introduction to Mathematical Philosophy

Aggregates, 12 Alephs, 83, 92, 97, 125 Aliorelatives, 32 All, 158 ff. Analysis, 4 Ancestors, 25, 33 Argument of a function, 47, 108 Arithmetising of mathematics, 4 Associative law, 58, 94 Axioms, 1 Between, 38 ff., 58 Bolzano, 138 n. Boots and socks, 126 Boundary, 70, 98, 99 Cantor, Georg, 77, 79, 85 n., 86, 89, 95, 102, 136 Classes, 12, 137, 181 ff.; reflexive, 80, 127, 138; similar, 15, 16 Clifford, W. K., 76 Collections, infinite, 13 Commutative law, 58, 94 Conjunction, 147 Consecutiveness, 37, 38, 81 Constants, 202 Construction, method of, 73 Continuity, 86, 97 ff.; Cantorian, 102 ff.; Dedekindian, 101 ff.; in philosophy, 105; of functions, 106 ff. Contradictions, 135 ff. Convergence, 115 Converse, 16, 32, 49 Correlators, 54 Counterparts, objective, 61 Counting, 14, 16 Dedekind, 69, 99, 138 n. Deduction, 144 ff. Definition, 3; extensional and intensional, 12 Derivatives, 100 Descriptions, 139, 144 Descriptions, 167 Dimensions, 29 Disjunction, 147 Distributive law, 58, 94 Diversity, 87 Domain, 16, 32, 49 Equivalence, 183 Euclid, 67 Existence, 164, 171, 177 Exponentiation, 94, 120 Extension of a relation, 60 Fictions, logical, 14 n., 45, 137 Field of a relation, 32, 53 Finite, 27 Flux, 105 Form, 198 Fractions, 37, 64 Frege, 7, 10, 25 n., 77, 95, 146 n. Functions, 46; descriptive, 46, 180; intensional and extensional, 186; predicative, 189; propositional, 46, 144; propositional, 155; Gap, Dedekindian, 70 ff., 99 Generalisation, 156 Geometry, 29, 59, 67, 74, 100, 145; analytical, 4, 86 Greater and less, 65, 90 Hegel, 107 Hereditary properties, 21 Implication, 146, 153; formal, 163 Incommensurables, 4, 66 Incompatibility, 147 ff., 200 Incomplete symbols, 182 Indiscernibles, 192 Individuals, 132, 141, 173 Induction, mathematical, 20 ff., 87, 93, 185 Inductive properties, 21 Inference, 148 Infinite, 28; of rationals, 65; Cantorian, 65; of cardinals, 77 ff.; and series and ordinals, 89 ff. Infinity, axiom of, 66 n., 77, 131 ff., 202 Instances, 156 Integers, positive and negative, 64 Intervals, 115 Intuition, 145 Irrationals, 66, 72 [Pg 207] Kant, 145 Leibniz, 80, 107, 192 Lewis, C. I., 153, 154 Likeness, 52 Limit, 29, 69 ff., 97 ff.; of functions, 106 ff. Limiting points, 99 Logic, 159, 65, 194 ff.; mathematical, v, 201, 206 Logicising of mathematics, 7 Maps, 52, 60 ff., 80 Mathematics, 194 ff. Maximum, 70, 98 Median class, 104 Meinong, 169 Method, vi Minimum, 70, 98 Modality, 165 Multiplication, 118 ff. Multiplicative axiom, 92, 117 ff. Names, 173, 182 Necessity, 165 Neighbourhood, 109 Nicod, 148, 149, 151 Null-class, 23, 132 Number, cardinal, 10 ff., 56, 77 ff., 95; complex, 74 ff.; finite, 20 ff.; inductive, 27, 78, 131; infinite, 77 ff.; irrational, 66, 72; maximum? 135; multipliable, 130; natural, 2 ff., 22; non-inductive, 88, 127; real, 66, 72, 84; reflexive, 80, 127; relation, 56, 94; serial, 57 Occam, 184 Occurrences, primary and secondary, 179 Ontological proof, 203 Order 29ff.; cyclic, 40 Oscillation, ultimate, 111 Parmenides, 138 Particulars, 140 ff., 173 Peano, 5 ff., 23, 24, 78, 81, 131, 163 Peirce, 32 n. Permutations, 50 Philosophy, mathematical, v, 1 Plato, 138 Plurality, 10 Poincaré, 27 Points, 59 Posterity, 22 ff., 32; proper, 36 Postulates, 71, 73 Precedent, 98 Premisses of arithmetic, 5 Primitive ideas and propositions, 5, 202 Progressions, 8, 81 ff. Propositions, 155; analytic, 204; elementary, 161 Pythagoras, 4, 67 Quantity, 97, 195 Ratios, 64, 71, 84, 133 Reducibility, axiom of, 191 Referent, 48 Relation numbers, 56 ff. Relations, asymmetrical 31, 42; connected, 32; many-one, 15; one-many, 15, 45; one-one, 15, 47, 79; reflexive, 16; serial, 34; similar, 52; squares of, 32; symmetrical, 16, 44; transitive, 16, 32 Relatum, 48 Representatives, 120 Rigour, 144 Royce, 80 Section, Dedekindian, 69 ff.; ultimate, 111 Segments, 72, 98 Selections, 117 Sequent, 98 Series, 29 ff.; closed, 103; compact, 66, 93, 100; condensed in itself, 102; Dedekindian, 71, 73, 101; generation of, 41; infinite, 89; perfect, 102, 103; well-ordered, 92, 123 Sheffer, 148 Similarity, of classes, 15 ff.; of relations, 83; of relations, 52 Some, 158 ff. Space, 61, 86, 140 Structure, 60 ff. Sub-classes, 84 ff. Subjects, 142 Subtraction, 87 Successor of a number, 23, 35 Syllogism, 197 Tautology, 203, 205 The, 167, 172 ff. Time, 61, 86, 140 Truth-function, 147 Truth-value, 146 Types, logical, 53, 135 ff., 185, 188 Unreality, 168 Value of a function, 47, 108 Variables, 10, 161, 199 Veblen, 58 Verbs, 141 Weierstrass, 97, 107 Wells, H. G., 114 Whitehead, 64, 76, 107, 119 Wittgenstein, 205 n. Zermelo, 123, 129 Zero, 65