10 ideas
21959 | Metaphysics is the most general attempt to make sense of things [Moore,AW] |
Full Idea: Metaphysics is the most general attempt to make sense of things. | |
From: A.W. Moore (The Evolution of Modern Metaphysics [2012], Intro) | |
A reaction: This is the first sentence of Moore's book, and a touchstone idea all the way through. It stands up well, because it says enough without committing to too much. I have to agree with it. It implies explanation as the key. I like generality too. |
9193 | ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett] |
Full Idea: ZF set theory is a first-order axiomatization. Variables range over sets, there are no second-order variables, and primitive predicates are just 'equals' and 'member of'. The axiom of extensionality says sets with the same members are identical. | |
From: Michael Dummett (The Philosophy of Mathematics [1998], 7) | |
A reaction: If the eleven members of the cricket team are the same as the eleven members of the hockey team, is the cricket team the same as the hockey team? Our cricket team is better than our hockey team, so different predicates apply to them. |
9194 | The main alternative to ZF is one which includes looser classes as well as sets [Dummett] |
Full Idea: The main alternative to ZF is two-sorted theories, with some variables ranging over classes. Classes have more generous existence assumptions: there is a universal class, containing all sets, and a class containing all ordinals. Classes are not members. | |
From: Michael Dummett (The Philosophy of Mathematics [1998], 7.1.1) | |
A reaction: My intuition is to prefer strict systems when it comes to logical theories. The whole point is precision. Otherwise we could just think about things, and skip all this difficult symbolic stuff. |
13547 | Limitation of Size is weak (Fs only collect is something the same size does) or strong (fewer Fs than objects) [Boolos, by Potter] |
Full Idea: Weak Limitation of Size: If there are no more Fs than Gs and the Gs form a collection, then Fs form a collection. Strong Limitation of Size: A property F fails to be collectivising iff there are as many Fs as there are objects. | |
From: report of George Boolos (Iteration Again [1989]) by Michael Potter - Set Theory and Its Philosophy 13.5 |
9195 | Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett] |
Full Idea: It must not be concluded from the rejection of excluded middle that intuitionistic logic operates with three values: true, false, and neither true nor false. It does not make use of true and false, but only with a construction being a proof. | |
From: Michael Dummett (The Philosophy of Mathematics [1998], 8.1) | |
A reaction: This just sounds like verificationism to me, with all its problems. It seems to make speculative statements meaningless, which can't be right. Realism has lots of propositions which are assumed to be true or false, but also unknowable. |
9186 | First-order logic concerns objects; second-order adds properties, kinds, relations and functions [Dummett] |
Full Idea: First-order logic is distinguished by generalizations (quantification) only over objects: second-order logic admits generalizations or quantification over properties or kinds of objects, and over relations between them, and functions defined over them. | |
From: Michael Dummett (The Philosophy of Mathematics [1998], 3.1) | |
A reaction: Second-order logic was introduced by Frege, but is (interestingly) rejected by Quine, because of the ontological commitments involved. I remain unconvinced that quantification entails ontological commitment, so I'm happy. |
9187 | Logical truths and inference are characterized either syntactically or semantically [Dummett] |
Full Idea: There are two ways of characterizing logical truths and correct inference. Proof-theoretic or syntactic characterizations, if the formalization admits of proof or derivation; and model-theoretic or semantic versions, being true in all interpretations. | |
From: Michael Dummett (The Philosophy of Mathematics [1998], 3.1) | |
A reaction: Dummett calls this distinction 'fundamental'. The second one involves truth, and hence meaning, where the first one just responds to rules. ..But how can you have a notion of correctly following a rule, without a notion of truth? |
9191 | Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett] |
Full Idea: It can be argued that the notion of ordinal numbers is more fundamental than that of cardinals. To count objects, we must count them in sequence. ..The theory of ordinals forms the substratum of Cantor's theory of cardinals. | |
From: Michael Dummett (The Philosophy of Mathematics [1998], 5) | |
A reaction: Depends what you mean by 'fundamental'. I would take cardinality to be psychologically prior ('that is a lot of sheep'). You can't order people by height without first acquiring some people with differing heights. I vote for cardinals. |
9192 | The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett] |
Full Idea: The number 4 cannot be characterized solely by its position in a system, because it has different positions in the system of natural numbers and that of the positive whole numbers, whereas these systems have the very same structure. | |
From: Michael Dummett (The Philosophy of Mathematics [1998], 6.1) | |
A reaction: Dummett seems to think this is fairly decisive against structuralism. There is also the structure of the real numbers. We will solve this by saying that the wholes are abstracted from the naturals, which are abstracted from the reals. Job done. |
21958 | Appearances are nothing beyond representations, which is transcendental ideality [Moore,AW] |
Full Idea: Appearances in general are nothing outside our representations, which is just what we mean by transcendental ideality. | |
From: A.W. Moore (The Evolution of Modern Metaphysics [2012], B535/A507) |