55 ideas
15169 | Metaphysics is clarifying how we speak and think (and possibly improving it) [Sidelle] |
15164 | We seem to base necessities on thought experiments and imagination [Sidelle] |
10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
8764 | Categories are the best foundation for mathematics [Shapiro] |
10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
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] |
10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
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] |
15180 | There doesn't seem to be anything in the actual world that can determine modal facts [Sidelle] |
10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
15184 | Causal reference presupposes essentialism if it refers to modally extended entities [Sidelle] |
15172 | Clearly, essential predications express necessary properties [Sidelle] |
15181 | Being a deepest explanatory feature is an actual, not a modal property [Sidelle] |
15173 | That the essence of water is its microstructure is a convention, not a discovery [Sidelle] |
15185 | We aren't clear about 'same stuff as this', so a principle of individuation is needed to identify it [Sidelle] |
15175 | Evaluation of de dicto modalities does not depend on the identity of its objects [Sidelle] |
15032 | Necessary a posteriori is conventional for necessity and nonmodal for a posteriority [Sidelle, by Sider] |
15179 | To know empirical necessities, we need empirical facts, plus conventions about which are necessary [Sidelle] |
15171 | The necessary a posteriori is statements either of identity or of essence [Sidelle] |
15167 | Empiricism explores necessities and concept-limits by imagining negations of truths [Sidelle] |
15177 | Contradictoriness limits what is possible and what is imaginable [Sidelle] |
15176 | The individuals and kinds involved in modality are also a matter of convention [Sidelle] |
15174 | A thing doesn't need transworld identity prior to rigid reference - that could be a convention of the reference [Sidelle] |
15183 | 'Dthat' operates to make a singular term into a rigid term [Sidelle] |
15165 | A priori knowledge is entirely of analytic truths [Sidelle] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
15168 | That water is essentially H2O in some way concerns how we use 'water' [Sidelle] |
15166 | Causal reference seems to get directly at the object, thus leaving its nature open [Sidelle] |
15182 | Because some entities overlap, reference must have analytic individuation principles [Sidelle] |
15178 | Can anything in science reveal the necessity of what it discovers? [Sidelle] |