1985 | Sets, Aggregates and Numbers |
'A Fregean' | p.356 | 17817 | Defining 'three' as the principle of collection or property of threes explains set theory definitions |
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. |
'New Problem' | p.357 | 17818 | How many? must first partition an aggregate into sets, and then logic fixes its number |
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. |
'On Numbering' | p.358 | 17821 | You can ask all sorts of numerical questions about any one given set |
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. |
'Two' | p.356 | 17815 | We can't use sets as foundations for mathematics if we must await results from the upper reaches |
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') |
'What the' | p.359 | 17822 | Nothing is 'intrinsically' numbered |
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... |