29 ideas
19370 | 'Blind thought' is reasoning without recognition of the ingredients of the reasoning [Leibniz, by Arthur,R] |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
19391 | We can assign a characteristic number to every single object [Leibniz] |
17453 | The meaning of a number isn't just the numerals leading up to it [Heck] |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
17457 | A basic grasp of cardinal numbers needs an understanding of equinumerosity [Heck] |
18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
17448 | In counting, numerals are used, not mentioned (as objects that have to correlated) [Heck] |
17456 | Counting is the assignment of successively larger cardinal numbers to collections [Heck] |
17455 | Is counting basically mindless, and independent of the cardinality involved? [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] |
19390 | Everything is subsumed under number, which is a metaphysical statics of the universe, revealing powers [Leibniz] |
8764 | Categories are the best foundation for mathematics [Shapiro] |
17459 | Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck] |
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] |
17454 | Children can use numbers, without a concept of them as countable objects [Heck] |
8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
17449 | We can understand cardinality without the idea of one-one correspondence [Heck] |
17458 | Equinumerosity is not the same concept as one-one correspondence [Heck] |
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] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |