10 ideas
18832 | Mathematical statements and entities that result from an infinite process must lack a truth-value [Dummett] |
Full Idea: On an intuitionistic view, neither the truth-value of a statement nor any other mathematical entity can be given as the final result of an infinite process, since an infinite process is precisely one that does not have a final result. | |
From: Michael Dummett (Elements of Intuitionism (2nd ed) [2000], p.41), quoted by Ian Rumfitt - The Boundary Stones of Thought 7.3 | |
A reaction: This is rather a persuasive reason to sympathise with intuitionism. Mathematical tricks about 'limits' have lured us into believing in completed infinities, but actually that idea is incoherent. |
10304 | Very few things in set theory remain valid in intuitionist mathematics [Bernays] |
Full Idea: Very few things in set theory remain valid in intuitionist mathematics. | |
From: Paul Bernays (On Platonism in Mathematics [1934]) |
10303 | Restricted Platonism is just an ideal projection of a domain of thought [Bernays] |
Full Idea: A restricted Platonism does not claim to be more than, so to speak, an ideal projection of a domain of thought. | |
From: Paul Bernays (On Platonism in Mathematics [1934], p.261) | |
A reaction: I have always found Platonism to be congenial when it talks of 'ideals', and ridiculous when it talks of a special form of 'existence'. Ideals only 'exist' because we idealise things. I may declare myself, after all, to be a Restricted Platonist. |
10306 | Mathematical abstraction just goes in a different direction from logic [Bernays] |
Full Idea: Mathematical abstraction does not have a lesser degree than logical abstraction, but rather another direction. | |
From: Paul Bernays (On Platonism in Mathematics [1934], p.268) | |
A reaction: His point is that the logicists seem to think that if you increasingly abstract from mathematics, you end up with pure logic. |
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. |