10 ideas
| 9879 | NF has no models, but just blocks the comprehension axiom, to avoid contradictions [Quine, by Dummett] |
| Full Idea: Quine's New Foundations system of set theory, devised with no model in mind, but on the basis of a hunch that a purely formal restriction on the comprehension axiom would block all contradictions. | |
| From: report of Willard Quine (New Foundations for Mathematical Logic [1937]) by Michael Dummett - Frege philosophy of mathematics Ch.18 | |
| A reaction: The point is that Quine (who had an ontological preference for 'desert landscapes') attempted to do without an ontological commitment to objects (and their subsequent models), with a purely formal system. Quine's NF is not now highly regarded. |
| 8717 | Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Hilbert, by Friend] |
| Full Idea: Hilbert wanted to derive ideal mathematics from the secure, paradox-free, finite mathematics (known as 'Hilbert's Programme'). ...Note that for the realist consistency is not something we need to prove; it is a precondition of thought. | |
| From: report of David Hilbert (works [1900], 6.7) by Michčle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: I am an intuitive realist, though I am not so sure about that on cautious reflection. Compare the claims that there are reasons or causes for everything. Reality cannot contain contradicitions (can it?). Contradictions would be our fault. |
| 10113 | The grounding of mathematics is 'in the beginning was the sign' [Hilbert] |
| Full Idea: The solid philosophical attitude that I think is required for the grounding of pure mathematics is this: In the beginning was the sign. | |
| From: David Hilbert (works [1900]), quoted by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: Why did people invent those particular signs? Presumably they were meant to designate something, in the world or in our experience. |
| 10115 | Hilbert substituted a syntactic for a semantic account of consistency [Hilbert, by George/Velleman] |
| Full Idea: Hilbert replaced a semantic construal of inconsistency (that the theory entails a statement that is necessarily false) by a syntactic one (that the theory formally derives the statement (0 =1 ∧ 0 not-= 1). | |
| From: report of David Hilbert (works [1900]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: Finding one particular clash will pinpoint the notion of inconsistency, but it doesn't seem to define what it means, since the concept has very wide application. |
| 10116 | Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [Hilbert, by George/Velleman] |
| Full Idea: Hilbert's project was to establish the consistency of classical mathematics using just finitary means, to convince all parties that no contradictions will follow from employing the infinitary notions and reasoning. | |
| From: report of David Hilbert (works [1900]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: This is the project which was badly torpedoed by Gödel's Second Incompleteness Theorem. |
| 16065 | Constitution is identity (being in the same place), or it isn't (having different possibilities) [Wasserman] |
| Full Idea: Some insist that constitution is identity, on the grounds that distinct material objects cannot occupy the same place at the same time. Others argue that constitution is not identity, since the statue and its material differ in important respects. | |
| From: Ryan Wasserman (Material Constitution [2009], Intro) | |
| A reaction: The 'important respects' seem to concern possibilities rather than actualities, which is suspicious. It is misleading to think we are dealing with two things and their relation here. Objects must have constitutions; constitutions make objects. |
| 16067 | Constitution is not identity, because it is an asymmetric dependence relation [Wasserman] |
| Full Idea: For those for whom 'constitution is not identity' (the 'constitution view'), constitution is said to be an asymmetric relation, and also a dependence relation (unlike identity). | |
| From: Ryan Wasserman (Material Constitution [2009], 2) | |
| A reaction: It seems obvious that constitution is not identity, because there is more to a thing's identity than its mere constitution. But this idea makes it sound as if constitution has nothing to do with identity (chalk and cheese), and that can't be right. |
| 16069 | There are three main objections to seeing constitution as different from identity [Wasserman] |
| Full Idea: The three most common objections to the constitution view are the Impenetrability Objection (two things in one place?), the Extensionality Objection (mereology says wholes are just their parts), and the Grounding Objection (their ground is the same). | |
| From: Ryan Wasserman (Material Constitution [2009], 2) | |
| A reaction: [summary] He adds a fourth, that if two things can be in one place, why stop at two? [Among defenders of the Constitution View he lists Baker, Fine, Forbes, Koslicki, Kripke, Lowe, Oderberg, N.Salmon, Shoemaker, Simons and Yablo.] |
| 16068 | The weight of a wall is not the weight of its parts, since that would involve double-counting [Wasserman] |
| Full Idea: We do not calculate the weight of something by summing the weights of all its parts - weigh bricks and the molecules of a wall and you will get the wrong result, since you have weighed some parts more than once. | |
| From: Ryan Wasserman (Material Constitution [2009], 2) | |
| A reaction: In fact the complete inventory of the parts of a thing is irrelevant to almost anything we would like to know about the thing. The parts must be counted at some 'level' of division into parts. An element can belong to many different sets. |
| 16074 | Relative identity may reject transitivity, but that suggests that it isn't about 'identity' [Wasserman] |
| Full Idea: If the relative identity theorist denies transitivity (to deal with the Ship of Theseus, for example), this would make us suspect that relativised identity relations are not identity relations, since transitivity seems central to identity. | |
| From: Ryan Wasserman (Material Constitution [2009], 6) | |
| A reaction: The problem here, I think, focuses on the meaning of the word 'same'. One change of plank leaves you with the same ship, but that is not transitive. If 'identical' is too pure to give the meaning of 'the same' it's not much use in discussing the world. |