2007 | Introducing the Philosophy of Mathematics |
p.113 | 8710 | The powerset of all the cardinal numbers is required to be greater than itself |
p.128 | 8713 | In classical/realist logic the connectives are defined by truth-tables |
1.4 | p.12 | 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction |
1.4 | p.13 | 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega |
1.4 | p.13 | 8663 | Raising omega to successive powers of omega reveal an infinity of infinities |
1.5 | p.14 | 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant |
1.5 | p.15 | 8665 | A 'proper subset' of A contains only members of A, but not all of them |
1.5 | p.15 | 8666 | Infinite sets correspond one-to-one with a subset |
1.5 | p.16 | 8667 | The 'integers' are the positive and negative natural numbers, plus zero |
1.5 | p.17 | 8668 | The 'rational' numbers are those representable as fractions |
1.5 | p.17 | 8669 | Between any two rational numbers there is an infinite number of rational numbers |
1.5 | p.19 | 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps |
1.5 | p.19 | 8670 | A number is 'irrational' if it cannot be represented as a fraction |
1.5 | p.21 | 8672 | A 'powerset' is all the subsets of a set |
2.3 | p.26 | 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal |
2.3 | p.27 | 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' |
2.3 | p.29 | 8676 | Is mathematics based on sets, types, categories, models or topology? |
2.3 | p.32 | 8677 | Set theory makes a minimum ontological claim, that the empty set exists |
2.3 | p.33 | 8678 | Most mathematical theories can be translated into the language of set theory |
2.3 | p.34 | 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction |
2.4 | p.36 | 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects |
2.5 | p.36 | 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? |
2.6 | p.42 | 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity |
3.1 | p.51 | 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them |
3.4 | p.64 | 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) |
3.7 | p.77 | 8694 | Free logic was developed for fictional or non-existent objects |
4.1 | p.82 | 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves |
4.1 | p.82 | 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential |
4.4 | p.90 | 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? |
4.4 | p.91 | 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting |
4.4 | p.93 | 8701 | The number 8 in isolation from the other numbers is of no interest |
4.4 | p.93 | 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent |
4.5 | p.97 | 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' |
5.1 | p.104 | 8705 | Anti-realist see truth as our servant, and epistemically contrained |
5.1 | p.106 | 8706 | Constructivism rejects too much mathematics |
5.2 | p.106 | 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle |
5.2 | p.107 | 8708 | Double negation elimination is not valid in intuitionist logic |
5.2 | p.108 | 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false |
5.5 | p.122 | 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." |
6.1 | p.128 | 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory |
6.5 | p.145 | 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability |
6.6 | p.149 | 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research |
Glossary | p.172 | 8721 | An 'impredicative' definition seems circular, because it uses the term being defined |