8 ideas
| 10196 | The Axiom of Choice needs a criterion of choice [Black] |
| Full Idea: Some mathematicians seem to think that talk of an Axiom of Choice allows them to choose a single member of a collection when there is no criterion of choice. | |
| From: Max Black (The Identity of Indiscernibles [1952], p.68) |
| 9935 | Mathematical truth is always compromising between ordinary language and sensible epistemology [Benacerraf] |
| Full Idea: Most accounts of the concept of mathematical truth can be identified with serving one or another of either semantic theory (matching it to ordinary language), or with epistemology (meshing with a reasonable view) - always at the expense of the other. | |
| From: Paul Benacerraf (Mathematical Truth [1973], Intro) | |
| A reaction: The gist is that language pulls you towards platonism, and epistemology pulls you towards empiricism. He argues that the semantics must give ground. He's right. |
| 10245 | One geometry cannot be more true than another [Poincaré] |
| Full Idea: One geometry cannot be more true than another; it can only be more convenient. | |
| From: Henri Poincaré (Science and Method [1908], p.65), quoted by Stewart Shapiro - Philosophy of Mathematics | |
| A reaction: This is the culminating view after new geometries were developed by tinkering with Euclid's parallels postulate. |
| 17927 | Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Benacerraf, by Colyvan] |
| Full Idea: Benacerraf argues that realists about mathematical objects have a nice normal semantic but no epistemology, and anti-realists have a good epistemology but an unorthodox semantics. | |
| From: report of Paul Benacerraf (Mathematical Truth [1973]) by Mark Colyvan - Introduction to the Philosophy of Mathematics 1.2 |
| 9936 | The platonist view of mathematics doesn't fit our epistemology very well [Benacerraf] |
| Full Idea: The principle defect of the standard (platonist) account of mathematical truth is that it appears to violate the requirement that our account be susceptible to integration into our over-all account of knowledge. | |
| From: Paul Benacerraf (Mathematical Truth [1973], III) | |
| A reaction: Unfortunately he goes on to defend a causal theory of justification (fashionable at that time, but implausible now). Nevertheless, his general point is well made. Your theory of what mathematics is had better make it knowable. |
| 10194 | Two things can only be distinguished by a distinct property or a distinct relation [Black] |
| Full Idea: The only way we can discover that two things exist is by finding out that one has a quality not possessed by the other, or else that one has a relational characteristic that the other hasn't. | |
| From: Max Black (The Identity of Indiscernibles [1952], p.67) | |
| A reaction: At least this doesn't conflate relations with properties. Note that this idea is clearly epistemological, and in no way rules out the separateness of two objects which none of us can ever discern. Maybe the Earth has two Suns, which imperceptibly swap. |
| 10193 | The 'property' of self-identity is uselessly tautological [Black] |
| Full Idea: Saying that 'a has the property of being identical with a' is a roundabout way of saying nothing - a useless tautology - and means not more than 'a is a' | |
| From: Max Black (The Identity of Indiscernibles [1952], p.66) | |
| A reaction: This matter resembles the problem of the number zero, and the empty set, which seem to be crucial entities for logicians, but of no interest to a common sense view of the world. So much the worse for logic, I am inclined to say. |
| 10195 | If the universe just held two indiscernibles spheres, that refutes the Identity of Indiscernibles [Black] |
| Full Idea: Isn't it logically possible that the universe should have contained nothing but two exactly similar spheres? ...So two things would have all their properties in common, and this would refute the Principle of the Identity of Indiscernibles. | |
| From: Max Black (The Identity of Indiscernibles [1952], p.67) | |
| A reaction: [Black is the originator of this famous example] It also appears to be naturally possible. An observer at an instant of viewing will discern a relational difference relative to themselves. Most people take Black's objection to be decisive. |