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) |
15938 | Platonists ruin infinity, which is precisely a growing structure which is never completed [Dummett] |
Full Idea: The platonist destroys the whole essence of infinity, which lies in the conception of a structure which is always in growth, precisely because the process of construction is never completed. | |
From: Michael Dummett (Elements of Intuitionism [1977], p.57), quoted by Thomas J. McKay - Plural Predication | |
A reaction: I don't warm to intuitionism, but I warm to this conception of infinity. Completed infinities are convenient reifications for mathematicians. |
15939 | For intuitionists it is constructed proofs (which take time) which make statements true [Dummett] |
Full Idea: For an intuitionist a mathematical statement is rendered true or false by a proof or disproof, that is, by a construction, and constructions are effected in time. | |
From: Michael Dummett (Elements of Intuitionism [1977], p.336), quoted by Shaughan Lavine - Understanding the Infinite VI.2 | |
A reaction: Lavine is quoting this to draw attention to the difficulties of thinking of it as all taking place 'in time', especially when dealing with infinities. |
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. |
9111 | God is not wise, but more-than-wise; God is not good, but more-than-good [William of Ockham] |
Full Idea: God is not wise, but more-than-wise; God is not good, but more-than-good. | |
From: William of Ockham (Reportatio [1330], III Q viii) | |
A reaction: [He is quoting 'Damascene'] I quote this for interest, but I very much doubt whether Damascene or William knew what it meant, and I certainly don't. There seems to have been a politically correct desire to invent super-powers for God. |
9112 | We could never form a concept of God's wisdom if we couldn't abstract it from creatures [William of Ockham] |
Full Idea: What we abstract is said to belong to perfection in so far as it can be predicated of God and can stand for Him. For if such a concept could not be abstracted from a creature, then in this life we could not arrive at a cognition of God's wisdom. | |
From: William of Ockham (Reportatio [1330], III Q viii) | |
A reaction: This seems to be the germ of an important argument. Without the ability to abstract from what is experienced, we would not be able to apply general concepts to things which are beyond experience. It is a key idea for empiricism. |