25 ideas
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| 13733 | Frege considered definite descriptions to be genuine singular terms [Frege, by Fitting/Mendelsohn] |
| 9874 | Contradiction arises from Frege's substitutional account of second-order quantification [Dummett on Frege] |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| 18252 | Real numbers are ratios of quantities, such as lengths or masses [Frege] |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| 18271 | We can't prove everything, but we can spell out the unproved, so that foundations are clear [Frege] |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| 10623 | Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions [Frege, by Hale/Wright] |
| 9975 | Frege ignored Cantor's warning that a cardinal set is not just a concept-extension [Tait on Frege] |
| 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] |
| 18165 | My Basic Law V is a law of pure logic [Frege] |
| 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] |
| 16672 | Quantity is the quantified parts of a thing, plus location and coordination [Olivi] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 9190 | A concept is a function mapping objects onto truth-values, if they fall under the concept [Frege, by Dummett] |
| 13665 | Frege took the study of concepts to be part of logic [Frege, by Shapiro] |