12 ideas
| 6782 | Realism is the only philosophy of science that doesn't make the success of science a miracle [Putnam] |
| Full Idea: Realism….is the only philosophy science which does not make the success of science a miracle. | |
| From: Hilary Putnam (works [1980]), quoted by Alexander Bird - Philosophy of Science Ch.4 | |
| A reaction: This was from his earlier work; he became more pragmatist and anti-realist later. Personally I approve of the remark. The philosophy of science must certainly offer an explanation for its success. Truth seems the obvious explanation. |
| 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. |
| 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. |
| 22181 | Putnam says anti-realism is a bad explanation of accurate predictions [Putnam, by Okasha] |
| Full Idea: Putnam's 'no miracle' argument says that being an anti-realist is akin to believing in miracles (because of the accurate predictons). …It is a plausibility argument - an inference to the best explanation. | |
| From: report of Hilary Putnam (works [1980]) by Samir Okasha - Philosophy of Science: Very Short Intro (2nd ed) 4 | |
| A reaction: [not sure of ref] Putnam later backs off from this argument, but my personal realism rests on best explanation. Does anyone want to prefer an inferior explanation? The objection is that successful theories can turn out to be false. Phlogiston, ether. |
| 2602 | What experience could prove 'If a=c and b=c then a=b'? [Descartes] |
| Full Idea: Please tell me what the corporeal motion is that is capable of forming some common notion to the effect that 'things which are equal to a third thing are equal to each other'. | |
| From: René Descartes (Comments on a Certain Broadsheet [1644], p.366) |
| 2600 | The mind's innate ideas are part of its capacity for thought [Descartes] |
| Full Idea: I have never written or taken the view that the mind requires innate ideas which are something distinct from its own faculty of thinking. | |
| From: René Descartes (Comments on a Certain Broadsheet [1644], p.365) |
| 2601 | Qualia must be innate, because physical motions do not contain them [Descartes] |
| Full Idea: The ideas of pains, colours, sounds etc. must be all the more innate if, on the occasion of certain corporeal motions, our mind is to be capable of representing them to itself, for there is no similarity between these ideas and the corporeal motions. | |
| From: René Descartes (Comments on a Certain Broadsheet [1644], p.365) | |
| A reaction: Simple and brilliant! We know perfectly well that there is no redness zooming through the air from a tomato (or the air would be pink!). Redness occurs when the light arrives, so we add the redness, so it is innate. |