65 ideas
| 8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
| 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
| 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| 8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
| 8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
| 8694 | Free logic was developed for fictional or non-existent objects [Friend] |
| 8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
| 8672 | A 'powerset' is all the subsets of a set [Friend] |
| 8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |
| 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
| 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
| 8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
| 8668 | The 'rational' numbers are those representable as fractions [Friend] |
| 8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
| 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
| 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| 8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
| 8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
| 8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
| 8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
| 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
| 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
| 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
| 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
| 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
| 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
| 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
| 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
| 8706 | Constructivism rejects too much mathematics [Friend] |
| 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
| 16657 | Substance, Quantity and Quality are real; other categories depend on those three [Henry of Ghent] |
| 16658 | The only reality in the category of Relation is things from another category [Henry of Ghent] |
| 16645 | Accidents are diminished beings, because they are dispositions of substance (unqualified being) [Henry of Ghent] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 22012 | Kant says things-in-themselves cause sensations, but then makes causation transcendental! [Henry of Ghent, by Pinkard] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 21181 | Relativity and Quantum theory give very different accounts of forces [Hesketh] |
| 21183 | Thermodynamics introduced work and entropy, to understand steam engine efficiency [Hesketh] |
| 21191 | Photons are B and W° bosons, linked by the Higgs mechanism [Hesketh] |
| 21199 | Spinning electric charge produces magnetism, so all fermions are magnets [Hesketh] |
| 21189 | Electrons may have smaller components, bound by a new force [Hesketh] |
| 21180 | Electrons are fundamental and are not made of anything; they are properties without size [Hesketh] |
| 21182 | Quantum mechanics is our only theory, and is very precise, and repeatedly confirmed [Hesketh] |
| 21184 | Physics was rewritten to explain stable electron orbits [Hesketh] |
| 21187 | Virtual particles can't be measured, and can ignore the laws of physics [Hesketh] |
| 21185 | Colour charge is positive or negative, and also has red, green or blue direction [Hesketh] |
| 21194 | The Standard Model omits gravity, because there are no particles involved [Hesketh] |
| 21195 | In Supersymmetry the Standard Model simplifies at high energies [Hesketh] |
| 21197 | Standard Model forces are one- two- and three-dimensional [Hesketh] |
| 21188 | Quarks and leptons have a weak charge, for the weak force [Hesketh] |
| 21186 | Quarks rush wildly around in protons, restrained by the gluons [Hesketh] |
| 21192 | Neutrinos only interact with the weak force, but decays produce them in huge numbers [Hesketh] |
| 21196 | To combine the forces, they must all be the same strength at some point [Hesketh] |
| 21190 | 'Space' in physics just means location [Hesketh] |
| 21193 | The universe is 68% dark energy, 27% dark matter, 5% regular matter [Hesketh] |
| 21198 | If a cosmic theory relies a great deal on fine-tuning basic values, it is probably wrong [Hesketh] |