92 ideas
| 5486 | Essentialism says metaphysics can't be done by analysing unreliable language [Ellis] |
| 9641 | Definitions should be replaceable by primitives, and should not be creative [Brown,JR] |
| 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
| 9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
| 10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
| 9613 | Naïve set theory assumed that there is a set for every condition [Brown,JR] |
| 9615 | Nowadays conditions are only defined on existing sets [Brown,JR] |
| 9617 | The 'iterative' view says sets start with the empty set and build up [Brown,JR] |
| 9642 | A flock of birds is not a set, because a set cannot go anywhere [Brown,JR] |
| 9605 | If a proposition is false, then its negation is true [Brown,JR] |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| 9649 | Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR] |
| 9638 | Berry's Paradox finds a contradiction in the naming of huge numbers [Brown,JR] |
| 9604 | Mathematics is the only place where we are sure we are right [Brown,JR] |
| 9622 | 'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR] |
| 10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
| 9648 | π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR] |
| 9621 | Mathematics represents the world through structurally similar models. [Brown,JR] |
| 9646 | There is no limit to how many ways something can be proved in mathematics [Brown,JR] |
| 9647 | Computers played an essential role in proving the four-colour theorem of maps [Brown,JR] |
| 10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| 10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
| 9643 | Set theory may represent all of mathematics, without actually being mathematics [Brown,JR] |
| 9644 | When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR] |
| 10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
| 10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
| 10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
| 10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
| 10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
| 10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
| 10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
| 9625 | To see a structure in something, we must already have the idea of the structure [Brown,JR] |
| 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
| 10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
| 9628 | Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR] |
| 10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
| 9606 | The irrationality of root-2 was achieved by intellect, not experience [Brown,JR] |
| 9612 | There is an infinity of mathematical objects, so they can't be physical [Brown,JR] |
| 9610 | Numbers are not abstracted from particulars, because each number is a particular [Brown,JR] |
| 9620 | Empiricists base numbers on objects, Platonists base them on properties [Brown,JR] |
| 9639 | Does some mathematics depend entirely on notation? [Brown,JR] |
| 9629 | For nomalists there are no numbers, only numerals [Brown,JR] |
| 9630 | The most brilliant formalist was Hilbert [Brown,JR] |
| 9608 | There are no constructions for many highly desirable results in mathematics [Brown,JR] |
| 9645 | Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR] |
| 9619 | David's 'Napoleon' is about something concrete and something abstract [Brown,JR] |
| 5468 | Properties are 'dispositional', or 'categorical' (the latter as 'block' or 'intrinsic' structures) [Ellis, by PG] |
| 5469 | The passive view of nature says categorical properties are basic, but others say dispositions [Ellis] |
| 5456 | Redness is not a property as it is not mind-independent [Ellis] |
| 5481 | Properties have powers; they aren't just ways for logicians to classify objects [Ellis] |
| 5458 | Nearly all fundamental properties of physics are dispositional [Ellis] |
| 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
| 5443 | Kripke and others have made essentialism once again respectable [Ellis] |
| 5444 | 'Individual essences' fix a particular individual, and 'kind essences' fix the kind it belongs to [Ellis] |
| 5462 | Essential properties are usually quantitatively determinate [Ellis] |
| 5448 | 'Real essence' makes it what it is; 'nominal essence' makes us categorise it a certain way [Ellis] |
| 5477 | One thing can look like something else, without being the something else [Ellis] |
| 5479 | Scientific essentialists say science should define the limits of the possible [Ellis] |
| 5483 | Essentialists deny possible worlds, and say possibilities are what is compatible with the actual world [Ellis] |
| 5447 | Metaphysical necessities are true in virtue of the essences of things [Ellis] |
| 5476 | Essentialists say natural laws are in a new category: necessary a posteriori [Ellis] |
| 5478 | Imagination tests what is possible for all we know, not true possibility [Ellis] |
| 5482 | Possible worlds realism is only needed to give truth conditions for modals and conditionals [Ellis] |
| 5453 | Essentialists mostly accept the primary/secondary qualities distinction [Ellis] |
| 5466 | Primary qualities are number, figure, size, texture, motion, configuration, impenetrability and (?) mass [Ellis] |
| 5485 | Emeralds are naturally green, and only an external force could turn them blue [Ellis] |
| 5484 | Essentialists don't infer from some to all, but from essences to necessary behaviour [Ellis] |
| 9611 | 'Abstract' nowadays means outside space and time, not concrete, not physical [Brown,JR] |
| 9609 | The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars [Brown,JR] |
| 9640 | A term can have not only a sense and a reference, but also a 'computational role' [Brown,JR] |
| 5457 | Predicates assert properties, values, denials, relations, conventions, existence and fabrications [Ellis, by PG] |
| 5488 | Regularity theories of causation cannot give an account of human agency [Ellis] |
| 5489 | Humans have variable dispositions, and also power to change their dispositions [Ellis] |
| 5490 | Essentialism fits in with Darwinism, but not with extreme politics of left or right [Ellis] |
| 9635 | Given atomism at one end, and a finite universe at the other, there are no physical infinities [Brown,JR] |
| 5472 | Natural kinds are of objects/substances, or events/processes, or intrinsic natures [Ellis] |
| 5471 | Essentialism says natural kinds are fundamental to nature, and determine the laws [Ellis] |
| 5446 | For essentialists two members of a natural kind must be identical [Ellis] |
| 5480 | The whole of our world is a natural kind, so all worlds like it necessarily have the same laws [Ellis] |
| 5445 | Essentialists regard inanimate objects as genuine causal agents [Ellis] |
| 5463 | Essentialists believe causation is necessary, resulting from dispositions and circumstances [Ellis] |
| 5491 | A general theory of causation is only possible in an area if natural kinds are involved [Ellis] |
| 5442 | For 'passivists' behaviour is imposed on things from outside [Ellis] |
| 5473 | The laws of nature imitate the hierarchy of natural kinds [Ellis] |
| 5474 | Laws of nature tend to describe ideal things, or ideal circumstances [Ellis] |
| 5475 | We must explain the necessity, idealisation, ontology and structure of natural laws [Ellis] |
| 5460 | Causal relations cannot be reduced to regularities, as they could occur just once [Ellis] |
| 5459 | Essentialists say dispositions are basic, rather than supervenient on matter and natural laws [Ellis] |
| 5461 | The essence of uranium is its atomic number and its electron shell [Ellis] |
| 5464 | For essentialists, laws of nature are metaphysically necessary, being based on essences of natural kinds [Ellis] |
| 5487 | Essentialism requires a clear separation of semantics, epistemology and ontology [Ellis] |