46 ideas
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] |
17453 | The meaning of a number isn't just the numerals leading up to it [Heck] |
9622 | 'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR] |
17457 | A basic grasp of cardinal numbers needs an understanding of equinumerosity [Heck] |
9648 | π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR] |
17448 | In counting, numerals are used, not mentioned (as objects that have to correlated) [Heck] |
17455 | Is counting basically mindless, and independent of the cardinality involved? [Heck] |
17456 | Counting is the assignment of successively larger cardinal numbers to collections [Heck] |
17450 | Understanding 'just as many' needn't involve grasping one-one correspondence [Heck] |
17451 | We can know 'just as many' without the concepts of equinumerosity or numbers [Heck] |
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] |
17459 | Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck] |
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] |
17454 | Children can use numbers, without a concept of them as countable objects [Heck] |
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] |
17458 | Equinumerosity is not the same concept as one-one correspondence [Heck] |
17449 | We can understand cardinality without the idea of one-one correspondence [Heck] |
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] |
8406 | Not all explanations are causal, but if a thing can be explained at all, it can be explained causally [Sanford] |
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] |
8407 | A totality of conditions necessary for an occurrence is usually held to be jointly sufficient for it [Sanford] |