49 ideas
| 9641 | Definitions should be replaceable by primitives, and should not be creative [Brown,JR] |
| 9921 | 'True' is only occasionally useful, as in 'everything Fermat believed was true' [Burgess/Rosen] |
| 9924 | Modal logic gives an account of metalogical possibility, not metaphysical possibility [Burgess/Rosen] |
| 9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
| 9933 | The paradoxes are only a problem for Frege; Cantor didn't assume every condition determines a set [Burgess/Rosen] |
| 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] |
| 9928 | Mereology implies that acceptance of entities entails acceptance of conglomerates [Burgess/Rosen] |
| 9605 | If a proposition is false, then its negation is true [Brown,JR] |
| 9926 | A relation is either a set of sets of sets, or a set of sets [Burgess/Rosen] |
| 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] |
| 9932 | The paradoxes no longer seem crucial in critiques of set theory [Burgess/Rosen] |
| 9604 | Mathematics is the only place where we are sure we are right [Brown,JR] |
| 9923 | We should talk about possible existence, rather than actual existence, of numbers [Burgess/Rosen] |
| 9622 | 'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR] |
| 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] |
| 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] |
| 9925 | Structuralism and nominalism are normally rivals, but might work together [Burgess/Rosen] |
| 9625 | To see a structure in something, we must already have the idea of the structure [Brown,JR] |
| 9628 | Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR] |
| 9606 | The irrationality of root-2 was achieved by intellect, not experience [Brown,JR] |
| 9934 | Number words became nouns around the time of Plato [Burgess/Rosen] |
| 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] |
| 9918 | Abstract/concrete is a distinction of kind, not degree [Burgess/Rosen] |
| 9929 | Much of what science says about concrete entities is 'abstraction-laden' [Burgess/Rosen] |
| 9619 | David's 'Napoleon' is about something concrete and something abstract [Brown,JR] |
| 9927 | Mathematics has ascended to higher and higher levels of abstraction [Burgess/Rosen] |
| 9930 | Abstraction is on a scale, of sets, to attributes, to type-formulas, to token-formulas [Burgess/Rosen] |
| 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] |
| 9919 | The old debate classified representations as abstract, not entities [Burgess/Rosen] |
| 9640 | A term can have not only a sense and a reference, but also a 'computational role' [Brown,JR] |
| 20239 | Unlike us, the early Greeks thought envy was a good thing, and hope a bad thing [Hesiod, by Nietzsche] |
| 9635 | Given atomism at one end, and a finite universe at the other, there are no physical infinities [Brown,JR] |
| 9922 | If space is really just a force-field, then it is a physical entity [Burgess/Rosen] |