28 ideas
| 18851 | Pairing (with Extensionality) guarantees an infinity of sets, just from a single element [Rosen] |
| 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] |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| 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] |
| 18852 | A Meinongian principle might say that there is an object for any modest class of properties [Rosen] |
| 18849 | Metaphysical necessity is absolute and universal; metaphysical possibility is very tolerant [Rosen] |
| 18850 | 'Metaphysical' modality is the one that makes the necessity or contingency of laws of nature interesting [Rosen] |
| 18858 | Sets, universals and aggregates may be metaphysically necessary in one sense, but not another [Rosen] |
| 18857 | Standard Metaphysical Necessity: P holds wherever the actual form of the world holds [Rosen] |
| 18856 | Non-Standard Metaphysical Necessity: when ¬P is incompatible with the nature of things [Rosen] |
| 18848 | Something may be necessary because of logic, but is that therefore a special sort of necessity? [Rosen] |
| 18855 | Combinatorial theories of possibility assume the principles of combination don't change across worlds [Rosen] |
| 18853 | A proposition is 'correctly' conceivable if an ominiscient being could conceive it [Rosen] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 12165 | Romantics say music expresses ideas, or the Will, or intuitions, or feelings [Scruton] |
| 12164 | Expressing melancholy is a good thing, but arousing it is a bad thing [Scruton] |
| 18854 | The MRL view says laws are the theorems of the simplest and strongest account of the world [Rosen] |