74 ideas
| 13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
| 13477 | The problems are the monuments of philosophy [Hart,WD] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
| 13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD] |
| 13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth [Hart,WD] |
| 13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD] |
| 13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD] |
| 13456 | Set theory articulates the concept of order (through relations) [Hart,WD] |
| 13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD] |
| 13443 | ∈ relates across layers, while ⊆ relates within layers [Hart,WD] |
| 13442 | Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD] |
| 13493 | In the modern view, foundation is the heart of the way to do set theory [Hart,WD] |
| 13495 | Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD] |
| 13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
| 13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
| 13516 | If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD] |
| 13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD] |
| 13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD] |
| 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
| 13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
| 13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
| 13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
| 13481 | Maybe sets should be rethought in terms of the even more basic categories [Hart,WD] |
| 13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
| 13513 | Models are ways the world might be from a first-order point of view [Hart,WD] |
| 13505 | Model theory studies how set theory can model sets of sentences [Hart,WD] |
| 13511 | Model theory is mostly confined to first-order theories [Hart,WD] |
| 13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
| 13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
| 13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD] |
| 13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD] |
| 13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD] |
| 13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
| 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
| 13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
| 13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
| 13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
| 13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
| 13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
| 13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
| 17405 | If a theory can be fudged, so can observations [Scerri] |
| 17397 | The periodic system is the big counterexample to Kuhn's theory of revolutionary science [Scerri] |
| 17393 | Scientists eventually seek underlying explanations for every pattern [Scerri] |
| 17403 | The periodic table suggests accommodation to facts rates above prediction [Scerri] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| 17394 | Natural kinds are what are differentiated by nature, and not just by us [Scerri] |
| 17421 | If elements are natural kinds, might the groups of the periodic table also be natural kinds? [Scerri] |
| 17396 | The colour of gold is best explained by relativistic effects due to fast-moving inner-shell electrons [Scerri] |
| 17420 | The stability of nuclei can be estimated through their binding energy [Scerri] |
| 17411 | If all elements are multiples of one (of hydrogen), that suggests once again that matter is unified [Scerri] |
| 17409 | Does radioactivity show that only physics can explain chemistry? [Scerri] |
| 17392 | How can poisonous elements survive in the nutritious compound they compose? [Scerri] |
| 17391 | Periodicity and bonding are the two big ideas in chemistry [Scerri] |
| 17404 | Chemistry does not work from general principles, but by careful induction from large amounts of data [Scerri] |
| 17407 | The electron is the main source of chemical properties [Scerri] |
| 17415 | A big chemistry idea is that covalent bonds are shared electrons, not transfer of electrons [Scerri] |
| 17418 | It is now thought that all the elements have literally evolved from hydrogen [Scerri] |
| 17398 | 19th C views said elements survived abstractly in compounds, but also as 'material ingredients' [Scerri] |
| 17395 | Elements were ordered by equivalent weight; later by atomic weight; finally by atomic number [Scerri] |
| 17406 | Moseley, using X-rays, showed that atomic number ordered better than atomic weight [Scerri] |
| 17408 | Some suggested basing the new periodic table on isotopes, not elements [Scerri] |
| 17413 | Elements in the table are grouped by having the same number of outer-shell electrons [Scerri] |
| 17416 | Orthodoxy says the periodic table is explained by quantum mechanics [Scerri] |
| 17417 | To explain the table, quantum mechanics still needs to explain order of shell filling [Scerri] |
| 17419 | Since 99.96% of the universe is hydrogen and helium, the periodic table hardly matters [Scerri] |
| 17414 | Pauli explained the electron shells, but not the lengths of the periods in the table [Scerri] |
| 17410 | Moseley showed the elements progress in units, and thereby clearly identified the gaps [Scerri] |
| 17412 | Elements are placed in the table by the number of positive charges - the atomic number [Scerri] |
| 17422 | The best classification needs the deepest and most general principles of the atoms [Scerri] |