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| 9978 | Analytic philosophy focuses too much on forms of expression, instead of what is actually said |
| Full Idea: The tendency to attack forms of expression rather than attempting to appreciate what is actually being said is one of the more unfortunate habits that analytic philosophy inherited from Frege. | |||
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], IV) | |||
| A reaction: The key to this, I say, is to acknowledge the existence of propositions (in brains). For example, this belief will make teachers more sympathetic to pupils who are struggling to express an idea, and verbal nit-picking becomes totally irrelevant. |
| 9986 | The null set was doubted, because numbering seemed to require 'units' |
| Full Idea: The conception that what can be numbered is some object (including flocks of sheep) relative to a partition - a choice of unit - survived even in the late nineteenth century in the form of the rejection of the null set (and difficulties with unit sets). | |||
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], IX) | |||
| A reaction: This old view can't be entirely wrong! Frege makes the point that if asked to count a pack of cards, you must decide whether to count cards, or suits, or pips. You may not need a 'unit', but you need a concept. 'Units' name concept-extensions nicely! |
| 9984 | We can have a series with identical members |
| Full Idea: Why can't we have a series (as opposed to a linearly ordered set) all of whose members are identical, such as (a, a, a...,a)? | |||
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], VII) | |||
| A reaction: The question is whether the items order themselves, which presumably the natural numbers are supposed to do, or whether we impose the order (and length) of the series. What decides how many a's there are? Do we order, or does nature? |
| 9981 | Abstraction is 'logical' if the sense and truth of the abstraction depend on the concrete |
| Full Idea: If the sense of a proposition about the abstract domain is given in terms of the corresponding proposition about the (relatively) concrete domain, ..and the truth of the former is founded upon the truth of the latter, then this is 'logical abstraction'. | |||
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], V) | |||
| A reaction: The 'relatively' in parentheses allows us to apply his idea to levels of abstraction, and not just to the simple jump up from the concrete. I think Tait's proposal is excellent, rather than purloining 'abstraction' for an internal concept within logic. |
| 9982 | Cantor and Dedekind use abstraction to fix grammar and objects, not to carry out proofs |
| Full Idea: Although (in Cantor and Dedekind) abstraction does not (as has often been observed) play any role in their proofs, but it does play a role, in that it fixes the grammar, the domain of meaningful propositions, and so determining the objects in the proofs. | |||
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], V) | |||
| A reaction: [compressed] This is part of a defence of abstractionism in Cantor and Dedekind (see K.Fine also on the subject). To know the members of a set, or size of a domain, you need to know the process or function which created the set. |
| 9985 | Abstraction may concern the individuation of the set itself, not its elements |
| Full Idea: A different reading of abstraction is that it concerns, not the individuating properties of the elements relative to one another, but rather the individuating properties of the set itself, for example the concept of what is its extension. | |||
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], VIII) | |||
| A reaction: If the set was 'objects in the room next door', we would not be able to abstract from the objects, but we might get to the idea of things being contain in things, or the concept of an object, or a room. Wrong. That's because they are objects... Hm. |
| 9972 | Why should abstraction from two equipollent sets lead to the same set of 'pure units'? |
| Full Idea: Why should abstraction from two equipollent sets lead to the same set of 'pure units'? | |||
| From: William W. Tait (Frege versus Cantor and Dedekind [1996]) | |||
| A reaction: [Tait is criticising Cantor] This expresses rather better than Frege or Dummett the central problem with the abstractionist view of how numbers are derived from matching groups of objects. |
| 9980 | If abstraction produces power sets, their identity should imply identity of the originals |
| Full Idea: If the power |A| is obtained by abstraction from set A, then if A is equipollent to set B, then |A| = |B|. But this does not imply that A = B. So |A| cannot just be A, taken in abstraction, unless that can identify distinct sets, ..or create new objects. | |||
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], V) | |||
| A reaction: An elegant piece of argument, which shows rather crucial facts about abstraction. We are then obliged to ask how abstraction can create an object or a set, if the central activity of abstraction is just ignoring certain features. |