59 ideas
| 9641 | Definitions should be replaceable by primitives, and should not be creative [Brown,JR] |
| 17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
| 17765 | Propositional language can only relate statements as the same or as different [Walicki] |
| 17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
| 17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
| 17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
| 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] |
| 17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
| 9642 | A flock of birds is not a set, because a set cannot go anywhere [Brown,JR] |
| 17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
| 9605 | If a proposition is false, then its negation is true [Brown,JR] |
| 17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
| 17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
| 9649 | Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR] |
| 17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
| 17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
| 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] |
| 17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
| 17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
| 17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
| 17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
| 17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
| 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] |
| 17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
| 17754 | Inductive proof depends on the choice of the ordering [Walicki] |
| 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] |
| 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] |
| 17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
| 21833 | Research suggest that we overrate conscious experience [Flanagan] |
| 21834 | Sensations may be identical to brain events, but complex mental events don't seem to be [Flanagan] |
| 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] |
| 21837 | Morality is normative because it identifies best practices among the normal practices [Flanagan] |
| 21830 | For Darwinians, altruism is either contracts or genetics [Flanagan] |
| 21835 | We need Eudaimonics - the empirical study of how we should flourish [Flanagan] |
| 21831 | Alienation is not finding what one wants, or being unable to achieve it [Flanagan] |
| 9635 | Given atomism at one end, and a finite universe at the other, there are no physical infinities [Brown,JR] |
| 21832 | Buddhists reject God and the self, and accept suffering as key, and liberation through wisdom [Flanagan] |