10 ideas
| 21554 | Sets always exceed terms, so all the sets must exceed all the sets [Lackey] |
| Full Idea: Cantor proved that the number of sets in a collection of terms is larger than the number of terms. Hence Cantor's Paradox says the number of sets in the collection of all sets must be larger than the number of sets in the collection of all sets. | |
| From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
| A reaction: The sets must count as terms in the next iteration, but that is a normal application of the Power Set axiom. |
| 21553 | It seems that the ordinal number of all the ordinals must be bigger than itself [Lackey] |
| Full Idea: The ordinal series is well-ordered and thus has an ordinal number, and a series of ordinals to a given ordinal exceeds that ordinal by 1. So the series of all ordinals has an ordinal number that exceeds its own ordinal number by 1. | |
| From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
| A reaction: Formulated by Burali-Forti in 1897. |
| 18086 | Weierstrass eliminated talk of infinitesimals [Weierstrass, by Kitcher] |
| Full Idea: Weierstrass effectively eliminated the infinitesimalist language of his predecessors. | |
| From: report of Karl Weierstrass (works [1855]) by Philip Kitcher - The Nature of Mathematical Knowledge 10.6 |
| 18092 | Weierstrass made limits central, but the existence of limits still needed to be proved [Weierstrass, by Bostock] |
| Full Idea: After Weierstrass had stressed the importance of limits, one now needed to be able to prove the existence of such limits. | |
| From: report of Karl Weierstrass (works [1855]) by David Bostock - Philosophy of Mathematics 4.4 | |
| A reaction: The solution to this is found in work on series (going back to Cauchy), and on Dedekind's cuts. |
| 16078 | Clay is intrinsically and atomically the same as statue (and that lacks 'modal properties') [Rudder Baker] |
| Full Idea: Arguments for statue being the clay are: that the clay is intrinsically like the statue, that the clay has the same atoms as the statue', that objects don't have modal properties such as being necessarily F, and the reference of 'property' changes. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], II) | |
| A reaction: [my summary of the arguments she identifies - see text for details] Rudder Baker attempts to refute all four of these arguments, in defence of constitution as different from identity. |
| 16077 | The clay is not a statue - it borrows that property from the statue it constitutes [Rudder Baker] |
| Full Idea: I argue that a lump of clay borrows the property of being a statue from the statue. The lump is a statue because, and only because, there is something that the lump constitutes that is a statue. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], n9) | |
| A reaction: It is skating on very thin metaphysical ice to introduce the concept of 'borrowing' a property. I've spent the last ten minutes trying to 'borrow' some properties, but without luck. |
| 16080 | Is it possible for two things that are identical to become two separate things? [Rudder Baker] |
| Full Idea: A strong intuition shared by many philosophers is that some things that are in fact identical might not have been identical. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], IV) | |
| A reaction: This flies in the face of the Kripkean view that if Hesperus=Phosphorus then the identity is necessary. I don't think I have an intuition that some given thing might have been two things - indeed the thought seems totally weird. Amoeba? Statue/clay? |
| 16076 | Constitution is not identity, as consideration of essential predicates shows [Rudder Baker] |
| Full Idea: I want to resuscitate an essentialist argument against the view that constitution is identity, of the form 'x is essentially F, y is not essentially F, so x is not y'. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], Intro) | |
| A reaction: The point is that x might be essentially F and y only accidentally F. Thus a statue is essentially so, but a lump if clay is not essentially a statue. Another case where 'necessary' would do instead of 'essentially'. |
| 16081 | The constitution view gives a unified account of the relation of persons/bodies, statues/bronze etc [Rudder Baker] |
| Full Idea: Constitution-without-identity is superior to constitution-as-identity in that it provides a unified view of the relation between persons and bodies, statues and pieces of bronze, and so on. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], IV) | |
| A reaction: I have a problem with the intrinsic dualism of this whole picture. Clay needs shape, statues need matter - there aren't two 'things' here which have a 'relation'. |
| 16082 | Statues essentially have relational properties lacked by lumps [Rudder Baker] |
| Full Idea: The statue has relational properties which the lump of clay does not have essentially. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], V) | |
| A reaction: She has in mind relations to the community of artistic life. I don't think this is convincing. Is something only a statue if it is validated by an artistic community? That sounds like relative identity, which she doesn't like. |