11 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. |
| 10779 | A comprehension axiom is 'predicative' if the formula has no bound second-order variables [Linnebo] |
| Full Idea: If φ contains no bound second-order variables, the corresponding comprehension axiom is said to be 'predicative'; otherwise it is 'impredicative'. | |
| From: Øystein Linnebo (Plural Quantification Exposed [2003], §1) | |
| A reaction: ['Predicative' roughly means that a new predicate is created, and 'impredicative' means that it just uses existing predicates] |
| 10781 | A 'pure logic' must be ontologically innocent, universal, and without presuppositions [Linnebo] |
| Full Idea: I offer these three claims as a partial analysis of 'pure logic': ontological innocence (no new entities are introduced), universal applicability (to any realm of discourse), and cognitive primacy (no extra-logical ideas are presupposed). | |
| From: Øystein Linnebo (Plural Quantification Exposed [2003], §1) |
| 10783 | Plural quantification depends too heavily on combinatorial and set-theoretic considerations [Linnebo] |
| Full Idea: If my arguments are correct, the theory of plural quantification has no right to the title 'logic'. ...The impredicative plural comprehension axioms depend too heavily on combinatorial and set-theoretic considerations. | |
| From: Øystein Linnebo (Plural Quantification Exposed [2003], §4) |
| 10778 | Can second-order logic be ontologically first-order, with all the benefits of second-order? [Linnebo] |
| Full Idea: According to its supporters, second-order logic allow us to pay the ontological price of a mere first-order theory and get the corresponding monadic second-order theory for free. | |
| From: Øystein Linnebo (Plural Quantification Exposed [2003], §0) |
| 13472 | Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD] |
| Full Idea: One of Hilbert's aims in 'The Foundations of Geometry' was to eliminate number [as measure of lengths and angles] from geometry. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by William D. Hart - The Evolution of Logic 2 | |
| A reaction: Presumably this would particularly have to include the elimination of ratios (rather than actual specific lengths). |
| 9546 | Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara] |
| Full Idea: Hilbert's geometrical axioms were existential in character, asserting the existence of certain geometrical objects (points and lines). Euclid's postulates do not assert the existence of anything; they assert the possibility of certain constructions. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Charles Chihara - A Structural Account of Mathematics 01.1 | |
| A reaction: Chihara says geometry was originally understood modally, but came to be understood existentially. It seems extraordinary to me that philosophers of mathematics can have become more platonist over the centuries. |
| 18742 | Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew] |
| Full Idea: In his formal investigation of Euclidean geometry, Hilbert uncovered congruence axioms that implicitly played a role in Euclid's proofs but were not explicitly recognised. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Horsten,L/Pettigrew,R - Mathematical Methods in Philosophy 2 | |
| A reaction: The writers are offering this as a good example of the benefits of a precise and formal approach to foundational questions. It's hard to disagree, but dispiriting if you need a PhD in maths before you can start doing philosophy. |
| 18217 | Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H] |
| Full Idea: Hilbert's formulation of the Euclidean theory is of special interest because (besides being rigorously axiomatised) it does not employ the real numbers in the axioms. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Hartry Field - Science without Numbers 3 | |
| A reaction: Notice that this job was done by Hilbert, and not by the fictionalist Hartry Field. |
| 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. |
| 10782 | The modern concept of an object is rooted in quantificational logic [Linnebo] |
| Full Idea: Our modern general concept of an object is given content only in connection with modern quantificational logic. | |
| From: Øystein Linnebo (Plural Quantification Exposed [2003], §2) | |
| A reaction: [He mentions Frege, Carnap, Quine and Dummett] This is the first thing to tell beginners in modern analytical metaphysics. The word 'object' is very confusing. I think I prefer 'entity'. |