7 ideas
| 9148 | I think of variables as objects rather than as signs [Fine,K] |
| Full Idea: It is natural nowadays to think of variables as a certain kind of sign, but I wish to think of them as a certain kind of object. | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §2) | |
| A reaction: Fine has a theory based on 'arbitrary objects', which is a rather charming idea. The cell of a spreadsheet is a kind of object, I suppose. A variable might be analogous to a point in space, where objects can locate themselves. |
| 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. |
| 9152 | If green is abstracted from a thing, it is only seen as a type if it is common to many things [Fine,K] |
| Full Idea: In traditional abstraction, the colour green merely has the intrinsic property of being green, other properties of things being abstracted away. But why should that be regarded as a type? It must be because the property is common to the instances. | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §5) | |
| A reaction: A nice question which shows that the much-derided single act of abstraction is not sufficient to arrive at a concept, so that abstraction is a more complex matter (perhaps even a rational one) than simple empiricists believe. |
| 9149 | To obtain the number 2 by abstraction, we only want to abstract the distinctness of a pair of objects [Fine,K] |
| Full Idea: In abstracting from the elements of a doubleton to obtain 2, we do not wish to abstract away from all features of the objects. We wish to take account of the fact that the two objects are distinct; this alone should be preserved under abstraction. | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §3) | |
| A reaction: This is Fine's strategy for meeting Frege's objection to abstraction, summarised in Idea 9146. It seems to use the common sense idea that abstraction is not all-or-nothing. Abstraction has degrees (and levels). |
| 9150 | We should define abstraction in general, with number abstraction taken as a special case [Fine,K] |
| Full Idea: Number abstraction can be taken to be a special case of abstraction in general, which can then be defined without recourse to the concept of number. | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §3) | |
| A reaction: At last, a mathematical logician recognising that they don't have a monopoly on abstraction. It is perfectly obvious that abstractions of simple daily concepts must be chronologically and logically prior to number abstraction. Number of what? |
| 9146 | After abstraction all numbers seem identical, so only 0 and 1 will exist! [Fine,K] |
| Full Idea: In Cantor's abstractionist account there can only be two numbers, 0 and 1. For abs(Socrates) = abs(Plato), since their numbers are the same. So the number of {Socrates,Plato} is {abs(Soc),abs(Plato)}, which is the same number as {Socrates}! | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §1) | |
| A reaction: Fine tries to answer this objection, which arises from §45 of Frege's Grundlagen. Fine summarises that "indistinguishability without identity appears to be impossible". Maybe we should drop talk of numbers in terms of sets. |
| 7607 | Nagarjuna and others pronounced the world of experience to be an illusion [Nagarjuna, by Armstrong,K] |
| Full Idea: Many later Buddhists (after Nagarjuna, c.120 CE) developed a belief that everything we experience is an illusion: in the West we would call them idealists. | |
| From: report of Nagarjuna (teachings [c.120]) by Karen Armstrong - A History of God Ch.3 | |
| A reaction: This is just one step beyond Plato (who at least hung onto the immediate world as an inferior reality), and is presumably intended to motivate meditators to break out of the misery of existence into a higher realm. Personally I am against it. |