58 ideas
| 9641 | Definitions should be replaceable by primitives, and should not be creative [Brown,JR] |
| 17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
| 17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
| 9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
| 9615 | Nowadays conditions are only defined on existing sets [Brown,JR] |
| 9613 | Naïve set theory assumed that there is a set for every condition [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] |
| 17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
| 17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
| 9649 | Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR] |
| 17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
| 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] |
| 17928 | Ordinal numbers represent order relations [Colyvan] |
| 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] |
| 17923 | Intuitionists only accept a few safe infinities [Colyvan] |
| 17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
| 17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
| 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] |
| 17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
| 17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
| 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] |
| 17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
| 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] |
| 17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
| 9606 | The irrationality of root-2 was achieved by intellect, not experience [Brown,JR] |
| 9610 | Numbers are not abstracted from particulars, because each number is a particular [Brown,JR] |
| 9612 | There is an infinity of mathematical objects, so they can't be physical [Brown,JR] |
| 9620 | Empiricists base numbers on objects, Platonists base them on properties [Brown,JR] |
| 9630 | The most brilliant formalist was Hilbert [Brown,JR] |
| 9629 | For nomalists there are no numbers, only numerals [Brown,JR] |
| 9639 | Does some mathematics depend entirely on notation? [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] |
| 16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
| 16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
| 9619 | David's 'Napoleon' is about something concrete and something abstract [Brown,JR] |
| 16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
| 16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
| 17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
| 17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
| 17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
| 17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
| 17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
| 17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
| 17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
| 17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
| 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] |
| 9635 | Given atomism at one end, and a finite universe at the other, there are no physical infinities [Brown,JR] |