11 ideas
| 17818 | How many? must first partition an aggregate into sets, and then logic fixes its number [Yourgrau] |
| Full Idea: We want to know How many what? You must first partition an aggregate into parts relevant to the question, where no partition is privileged. How the partitioned set is to be numbered is bound up with its unique members, and follows from logic alone. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'New Problem') | |
| A reaction: [Compressed wording of Yourgrau's summary of Frege's 'relativity argument'] Concepts do the partitioning. Yourgau says this fails, because the same argument applies to the sets themselves, as well as to the original aggregates. |
| 17822 | Nothing is 'intrinsically' numbered [Yourgrau] |
| Full Idea: Nothing at all is 'intrinsically' numbered. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'What the') | |
| A reaction: Once you are faced with distinct 'objects' of some sort, they can play the role of 'unit' in counting, so his challenge is that nothing is 'intrinsically' an object, which is the nihilism explored by Unger, Van Inwagen and Merricks. Aristotle disagrees... |
| 12154 | Are 'word token' and 'word type' different sorts of countable objects, or two ways of counting? [Geach, by Perry] |
| Full Idea: If we list the words 'bull', 'bull' and 'cow', it is often said that there are three 'word tokens' but only two 'word types', but Geach says there are not two kinds of object to be counted, but two different ways of counting the same object. | |
| From: report of Peter Geach (Reference and Generality (3rd ed) [1980]) by John Perry - The Same F II | |
| A reaction: Insofar as the notion that a 'word type' is an 'object', my sympathies are entirely with Geach, to my surprise. Geach's point is that 'bull' and 'bull' are the same meaning, but different actual words. Identity is relative to a concept. |
| 17817 | Defining 'three' as the principle of collection or property of threes explains set theory definitions [Yourgrau] |
| Full Idea: The Frege-Maddy definition of number (as the 'property' of being-three) explains why the definitions of Von Neumann, Zermelo and others work, by giving the 'principle of collection' that ties together all threes. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'A Fregean') | |
| A reaction: [compressed two or three sentences] I am strongly in favour of the best definition being the one which explains the target, rather than just pinning it down. I take this to be Aristotle's view. |
| 17815 | We can't use sets as foundations for mathematics if we must await results from the upper reaches [Yourgrau] |
| Full Idea: Sets could hardly serve as a foundation for number theory if we had to await detailed results in the upper reaches of the edifice before we could make our first move. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'Two') |
| 17821 | You can ask all sorts of numerical questions about any one given set [Yourgrau] |
| Full Idea: We can address a set with any question at all that admits of a numerical reply. Thus we can ask of {Carter, Reagan} 'How many feet do the members have?'. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'On Numbering') | |
| A reaction: This is his objection to the Fregean idea that once you have fixed the members of a set, you have thereby fixed the unique number that belongs with the set. |
| 8969 | We should abandon absolute identity, confining it to within some category [Geach, by Hawthorne] |
| Full Idea: Geach argued that the notion of absolute identity should be abandoned. ..We can only grasp the meaning of a count noun when we associate it with a criterion of identity, expressed by a particular relative identity sortal. | |
| From: report of Peter Geach (Reference and Generality (3rd ed) [1980]) by John Hawthorne - Identity | |
| A reaction: In other words, identity needs categorisation. Hawthorne concludes that Geach is wrong. Geach clearly has much common usage on his side. 'What's that?' usually invites a categorisation. Sameness of objects seems to need a 'respect'. |
| 16075 | Denial of absolute identity has drastic implications for logic, semantics and set theory [Wasserman on Geach] |
| Full Idea: Geach's denial of absolute identity has drastic implications for logic, semantics and set theory. He must deny the axiom of extensionality in set theory, for example. | |
| From: comment on Peter Geach (Reference and Generality (3rd ed) [1980]) by Ryan Wasserman - Material Constitution 6 | |
| A reaction: I'm beginning to think we have two entirely different concepts here - the logicians' and mathematicians' notion of when two things are identical, and the ordinary language concept of two things being 'the same'. 'We like the same music'. |
| 12152 | Identity is relative. One must not say things are 'the same', but 'the same A as' [Geach] |
| Full Idea: Identity is relative. When one says 'x is identical with y' this is an incomplete expression. It is short for 'x is the same A as y', where 'A' represents some count noun understood from the context of utterance. | |
| From: Peter Geach (Reference and Generality (3rd ed) [1980], p.39), quoted by John Perry - The Same F I | |
| A reaction: Perry notes that Geach's view is in conscious opposition to Frege, who had a pure notion of identity. We say 'they are the same insofar as they are animals', but not 'they are the same animal'. Perfect identity involves all possible A's. |
| 16073 | Leibniz's Law is incomplete, since it includes a non-relativized identity predicate [Geach, by Wasserman] |
| Full Idea: Geach rejects the standard formulation of Leibniz's Law as incomplete, since it includes a non-relativized identity predicate. | |
| From: report of Peter Geach (Reference and Generality (3rd ed) [1980]) by Ryan Wasserman - Material Constitution 6 | |
| A reaction: Not many people accept Geach's premiss that identity is a relative matter. I agree with Wiggins on this, that identity is an absolute (and possibly indefinable). The problem with the Law is what you mean by a 'property'. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |