27 ideas
| 15527 | Defining terms either enables elimination, or shows that they don't require elimination [Lewis] |
| 15717 | Using Choice, you can cut up a small ball and make an enormous one from the pieces [Kaplan/Kaplan] |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| 15712 | 1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan] |
| 15711 | The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan] |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| 15714 | 'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan] |
| 15715 | 'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan] |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| 15530 | A logically determinate name names the same thing in every possible world [Lewis] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 15528 | A Ramsey sentence just asserts that a theory can be realised, without saying by what [Lewis] |
| 15526 | There is a method for defining new scientific terms just using the terms we already understand [Lewis] |
| 15529 | It is better to have one realisation of a theory than many - but it may not always be possible [Lewis] |
| 15531 | The Ramsey sentence of a theory says that it has at least one realisation [Lewis] |
| 15713 | The first million numbers confirm that no number is greater than a million [Kaplan/Kaplan] |