60 ideas
19735 | Wisdom has a higher value than understanding, which has a higher value than knowledge [Greco] |
9641 | Definitions should be replaceable by primitives, and should not be creative [Brown,JR] |
9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
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] |
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] |
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] |
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] |
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] |
9629 | For nomalists there are no numbers, only numerals [Brown,JR] |
9639 | Does some mathematics depend entirely on notation? [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] |
17979 | Research shows perceptual discrimination is sharper at category boundaries [Murphy] |
19734 | If value is practical, knowledge is no better than true opinion [Greco] |
19733 | Externalist theories don't explain why knowledge has value [Greco] |
18690 | Induction is said to just compare properties of categories, but the type of property also matters [Murphy] |
17980 | The main theories of concepts are exemplar, prototype and knowledge [Murphy] |
17969 | The classical definitional approach cannot distinguish typical and atypical category members [Murphy] |
17973 | The theoretical and practical definitions for the classical view are very hard to find [Murphy] |
17970 | Classical concepts follow classical logic, but concepts in real life don't work that way [Murphy] |
17971 | Classical concepts are transitive hierarchies, but actual categories may be intransitive [Murphy] |
17972 | The classical core is meant to be the real concept, but actually seems unimportant [Murphy] |
17975 | There is no 'ideal' bird or dog, and prototypes give no information about variability [Murphy] |
17976 | Prototypes are unified representations of the entire category (rather than of members) [Murphy] |
18691 | The prototype theory uses observed features, but can't include their construction [Murphy] |
17983 | The prototype theory handles hierarchical categories and combinations of concepts well [Murphy] |
17985 | Prototypes theory of concepts is best, as a full description with weighted typical features [Murphy] |
17986 | Learning concepts is forming prototypes with a knowledge structure [Murphy] |
17974 | The most popular theories of concepts are based on prototypes or exemplars [Murphy] |
17977 | The exemplar view of concepts says 'dogs' is the set of dogs I remember [Murphy] |
17987 | The concept of birds from exemplars must also be used in inductions about birds [Murphy] |
17982 | Exemplar theory struggles with hierarchical classification and with induction [Murphy] |
17981 | Children using knowing and essentialist categories doesn't fit the exemplar view [Murphy] |
17984 | Conceptual combination must be compositional, and can't be built up from exemplars [Murphy] |
17978 | We do not learn concepts in isolation, but as an integrated part of broader knowledge [Murphy] |
18687 | Concepts with familiar contents are easier to learn [Murphy] |
18688 | Some knowledge is involved in instant use of categories, other knowledge in explanations [Murphy] |
18689 | People categorise things consistent with their knowledge, even rejecting some good evidence [Murphy] |
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] |