1981 | Sets and Numbers |
I | p.345 | 17823 | If mathematical objects exist, how can we know them, and which objects are they? |
II | p.347 | 17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative |
II | p.347 | 17824 | The master science is physical objects divided into sets |
III | p.347 | 17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology |
III | p.349 | 17828 | Numbers are properties of sets, just as lengths are properties of physical objects |
III | p.349 | 17827 | Sets exist where their elements are, but numbers are more like universals |
IV | p.350 | 17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first |
V | p.353 | 17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties |
1988 | Believing the Axioms I |
§0 | p.482 | 13011 | New axioms are being sought, to determine the size of the continuum |
§1.1 | p.484 | 13013 | The Axiom of Extensionality seems to be analytic |
§1.1 | p.484 | 13014 | Extensional sets are clearer, simpler, unique and expressive |
§1.3 | p.485 | 13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes |
§1.3 | p.485 | 13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances |
§1.5 | p.486 | 13022 | Infinite sets are essential for giving an account of the real numbers |
§1.5 | p.486 | 13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics |
§1.6 | p.486 | 13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum |
§1.7 | p.487 | 13024 | Efforts to prove the Axiom of Choice have failed |
§1.7 | p.488 | 13026 | A large array of theorems depend on the Axiom of Choice |
§1.7 | p.488 | 13025 | Modern views say the Choice set exists, even if it can't be constructed |
§1.8 | p.489 | 13028 | Replacement was added when some advanced theorems seemed to need it |
1990 | Realism in Mathematics |
p.191 | 17733 | We know mind-independent mathematical truths through sets, which rest on experience |
p.223 | 8755 | Maddy replaces pure sets with just objects and perceived sets of objects |
p.224 | 8756 | Intuition doesn't support much mathematics, and we should question its reliability |
3 §2 | p.19 | 10718 | A natural number is a property of sets |
1997 | Naturalism in Mathematics |
§107 | p.117 | 18182 | The extension of concepts is not important to me |
I Intro | p.1 | 18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms |
I.1 | p.5 | 18164 | Frege solves the Caesar problem by explicitly defining each number |
I.1 | p.7 | 18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number |
I.1 | p.9 | 18168 | 'Propositional functions' are propositions with a variable as subject or predicate |
I.1 | p.11 | 18169 | Axiom of Reducibility: propositional functions are extensionally predicative |
I.1 | p.16 | 18172 | Infinity has degrees, and large cardinals are the heart of set theory |
I.1 | p.17 | 18175 | For any cardinal there is always a larger one (so there is no set of all sets) |
I.1 | p.17 | 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size |
I.1 | p.17 | 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities |
I.1 n39 | p.15 | 18171 | Cantor and Dedekind brought completed infinities into mathematics |
I.2 | p.23 | 18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets |
I.2 | p.26 | 18183 | Set theory brings mathematics into one arena, where interrelations become clearer |
I.2 | p.26 | 18184 | Making set theory foundational to mathematics leads to very fruitful axioms |
I.2 | p.26 | 18185 | Unified set theory gives a final court of appeal for mathematics |
I.2 | p.26 | 18186 | Identifying geometric points with real numbers revealed the power of set theory |
I.2 | p.27 | 18188 | The line of rationals has gaps, but set theory provided an ordered continuum |
I.2 | p.27 | 18187 | Theorems about limits could only be proved once the real numbers were understood |
I.2 n8 | p.24 | 18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) |
I.2 n8 | p.24 | 18178 | For Zermelo the successor of n is {n} (rather than n U {n}) |
I.2 n9 | p.24 | 18180 | Von Neumann numbers are preferred, because they continue into the transfinite |
I.3 | p.51 | 18190 | Completed infinities resulted from giving foundations to calculus |
I.3 | p.52 | 18191 | Axiom of Infinity: completed infinite collections can be treated mathematically |
I.3 | p.60 | 18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy |
I.4 | p.66 | 18194 | 'Forcing' can produce new models of ZFC from old models |
I.5 | p.73 | 18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy |
I.5 | p.74 | 18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets |
II.5 | p.131 | 18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible |
II.6 | p.143 | 18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real |
II.6 | p.143 | 18206 | Science idealises the earth's surface, the oceans, continuities, and liquids |
II.6 | p.152 | 18207 | Maybe applications of continuum mathematics are all idealisations |
2007 | Second Philosophy |
III.8 n1 | p.299 | 10594 | Henkin semantics is more plausible for plural logic than for second-order logic |
2011 | Defending the Axioms |
1.3 | p.31 | 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization |
1.3 | p.35 | 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres |
2.3 | p.53 | 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics |
2.4 n40 | p.56 | 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals |
3.3 | p.99 | 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying |
3.4 | p.82 | 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness |
5.3ii | p.129 | 17625 | If two mathematical themes coincide, that suggest a single deep truth |