31 ideas
18951 | For scientific purposes there is a precise concept of 'true-in-L', using set theory [Putnam] |
18953 | Modern notation frees us from Aristotle's restriction of only using two class-names in premises [Putnam] |
18949 | The universal syllogism is now expressed as the transitivity of subclasses [Putnam] |
18952 | '⊃' ('if...then') is used with the definition 'Px ⊃ Qx' is short for '¬(Px & ¬Qx)' [Putnam] |
18958 | In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type [Putnam] |
18954 | Before the late 19th century logic was trivialised by not dealing with relations [Putnam] |
18956 | Asserting first-order validity implicitly involves second-order reference to classes [Putnam] |
18962 | Unfashionably, I think logic has an empirical foundation [Putnam] |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
18961 | We can identify functions with certain sets - or identify sets with certain functions [Putnam] |
18955 | Having a valid form doesn't ensure truth, as it may be meaningless [Putnam] |
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] |
18959 | Sets larger than the continuum should be studied in an 'if-then' spirit [Putnam] |
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] |
18957 | Nominalism only makes sense if it is materialist [Putnam] |
18950 | Physics is full of non-physical entities, such as space-vectors [Putnam] |
5052 | When Gentiles follow the law, they must have the law written in their hearts [Paul] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
18960 | Most predictions are uninteresting, and are only sought in order to confirm a theory [Putnam] |
7572 | Power is ordained by God, so anyone who resists power resists God, and will be damned [Paul] |