display all the ideas for this combination of texts
7 ideas
| 8297 | Numbers are universals, being sets whose instances are sets of appropriate cardinality [Lowe] |
| Full Idea: My view is that numbers are universals, beings kinds of sets (that is, kinds whose particular instances are individual sets of appropriate cardinality). | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10) | |
| A reaction: [That is, 12 is the set of all sets which have 12 members] This would mean, I take it, that if the number of objects in existence was reduced to 11, 12 would cease to exist, which sounds wrong. Or are we allowed imagined instances? |
| 8266 | Simple counting is more basic than spotting that one-to-one correlation makes sets equinumerous [Lowe] |
| Full Idea: That one-to-one correlated sets of objects are equinumerous is a more sophisticated achievement than the simple ability to count sets of objects. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 2.9) | |
| A reaction: This is an objection to Frege's way of defining numbers, in terms of equinumerous sets. I take pattern-recognition to be the foundation of number, and so spotting a pattern would have to precede spotting that two patterns were identical. |
| 8302 | Fs and Gs are identical in number if they one-to-one correlate with one another [Lowe] |
| Full Idea: What is now known as Hume's Principle says the number of Fs is identical with the number of Gs if and only if the Fs and the Gs are one-to-one correlated with one another. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.3) | |
| A reaction: This seems popular as a tool in attempts to get the concept of number off the ground. Although correlations don't seem to require numbers ('find yourself a partner'), at some point you have to count the correlations. Sets come first, to identify the Fs. |
| 8298 | Sets are instances of numbers (rather than 'collections'); numbers explain sets, not vice versa [Lowe] |
| Full Idea: I favour an account of sets which sees them as being instances of numbers, thereby avoiding the unhelpful metaphor which speaks of a set as being a 'collection' of things. This reverses the normal view, which explains numbers in terms of sets. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10) | |
| A reaction: Cf. Idea 8297. Either a set is basic, or a number is. We might graft onto Lowe's view an account of numbers in terms of patterns, which would give an empirical basis to the picture, and give us numbers which could be used to explain sets. |
| 8311 | If 2 is a particular, then adding particulars to themselves does nothing, and 2+2=2 [Lowe] |
| Full Idea: If 2 is a particular, 'adding' it to itself can, it would seem, only leave us with 2, not another number. (If 'Socrates + Socrates' denotes anything, it most plausibly just denotes Socrates). | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.7) | |
| A reaction: This suggest Kant's claim that arithmetical sums are synthetic (Idea 5558). It is a nice question why, when you put two 2s together, they come up with something new. Addition is movement. Among patterns, or along abstract sequences. |
| 8310 | Does the existence of numbers matter, in the way space, time and persons do? [Lowe] |
| Full Idea: Does it really matter whether the numbers actually exist - in anything like the way in which it matters that space and time or persons actually exist? | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.6) | |
| A reaction: Nice question! It might matter a lot. I take the question of numbers to be a key test case, popular with philosophers because they are the simplest and commonest candidates for abstract existence. The ontological status of values is the real issue. |
| 1635 | Mathematics reduces to set theory (which is a bit vague and unobvious), but not to logic proper [Quine] |
| Full Idea: Mathematics reduces only to set theory, and not to logic proper… but set theory cannot claim the same firmness and obviousness as logic. | |
| From: Willard Quine (Epistemology Naturalized [1968], p.69-70) |