43 ideas
10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
5745 | Quine says quantified modal logic creates nonsense, bad ontology, and false essentialism [Melia on Quine] |
10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
8789 | Various strategies try to deal with the ontological commitments of second-order logic [Hale/Wright on Quine] |
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] |
16966 | Philosophers tend to distinguish broad 'being' from narrower 'existence' - but I reject that [Quine] |
16965 | All we have of general existence is what existential quantifiers express [Quine] |
16963 | Existence is implied by the quantifiers, not by the constants [Quine] |
16964 | Theories are committed to objects of which some of its predicates must be true [Quine] |
4216 | Express a theory in first-order predicate logic; its ontology is the types of bound variable needed for truth [Quine, by Lowe] |
18966 | Ontological commitment of theories only arise if they are classically quantified [Quine] |
14490 | You can be implicitly committed to something without quantifying over it [Thomasson on Quine] |
16961 | In formal terms, a category is the range of some style of variables [Quine] |
10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |