11 ideas
17824 | The master science is physical objects divided into sets [Maddy] |
Full Idea: The master science can be thought of as the theory of sets with the entire range of physical objects as ur-elements. | |
From: Penelope Maddy (Sets and Numbers [1981], II) | |
A reaction: This sounds like Quine's view, since we have to add sets to our naturalistic ontology of objects. It seems to involve unrestricted mereology to create normal objects. |
17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy] |
Full Idea: If you wonder why multiplication is commutative, you could prove it from the Peano postulates, but the proof offers little towards an answer. In set theory Cartesian products match 1-1, and n.m dots when turned on its side has m.n dots, which explains it. | |
From: Penelope Maddy (Sets and Numbers [1981], II) | |
A reaction: 'Turning on its side' sounds more fundamental than formal set theory. I'm a fan of explanation as taking you to the heart of the problem. I suspect the world, rather than set theory, explains the commutativity. |
17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy] |
Full Idea: The standard account of the relationship between numbers and sets is that numbers simply are certain sets. This has the advantage of ontological economy, and allows numbers to be brought within the epistemology of sets. | |
From: Penelope Maddy (Sets and Numbers [1981], III) | |
A reaction: Maddy votes for numbers being properties of sets, rather than the sets themselves. See Yourgrau's critique. |
17828 | Numbers are properties of sets, just as lengths are properties of physical objects [Maddy] |
Full Idea: I propose that ...numbers are properties of sets, analogous, for example, to lengths, which are properties of physical objects. | |
From: Penelope Maddy (Sets and Numbers [1981], III) | |
A reaction: Are lengths properties of physical objects? A hole in the ground can have a length. A gap can have a length. Pure space seems to contain lengths. A set seems much more abstract than its members. |
17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
Full Idea: A set of things is located where the aggregate of those things is located, ...but a number is simultaneously located at many different places (10 in my hand, and a baseball team) ...so numbers seem more like universals than particulars. | |
From: Penelope Maddy (Sets and Numbers [1981], III) | |
A reaction: My gut feeling is that Maddy's master idea (of naturalising sets by building them from ur-elements of natural objects) won't work. Sets can work fine in total abstraction from nature. |
17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
Full Idea: I am not suggesting a reduction of number theory to set theory ...There are only sets with number properties; number theory is part of the theory of finite sets. | |
From: Penelope Maddy (Sets and Numbers [1981], V) |
17823 | If mathematical objects exist, how can we know them, and which objects are they? [Maddy] |
Full Idea: The popular challenges to platonism in philosophy of mathematics are epistemological (how are we able to interact with these objects in appropriate ways) and ontological (if numbers are sets, which sets are they). | |
From: Penelope Maddy (Sets and Numbers [1981], I) | |
A reaction: These objections refer to Benacerraf's two famous papers - 1965 for the ontology, and 1973 for the epistemology. Though he relied too much on causal accounts of knowledge in 1973, I'm with him all the way. |
17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy] |
Full Idea: Number words are not like normal adjectives. For example, number words don't occur in 'is (are)...' contexts except artificially, and they must appear before all other adjectives, and so on. | |
From: Penelope Maddy (Sets and Numbers [1981], IV) | |
A reaction: [She is citing Benacerraf's arguments] |
21982 | I only wish I had such eyes as to see Nobody! It's as much as I can do to see real people. [Carroll,L] |
Full Idea: "I see nobody on the road," said Alice. - "I only wish I had such eyes," the King remarked. ..."To be able to see Nobody! ...Why, it's as much as I can do to see real people." | |
From: Lewis Carroll (C.Dodgson) (Through the Looking Glass [1886], p.189), quoted by A.W. Moore - The Evolution of Modern Metaphysics 07.7 | |
A reaction: [Moore quotes this, inevitably, in a chapter on Hegel] This may be a better candidate for the birth of philosophy of language than Frege's Groundwork. |
11945 | In addition to laws, God must also create appropriate natures for things [Leibniz] |
Full Idea: It isn't sufficient to say that God has made a general law, for in addition to the decree there has also to be a natural way of carrying it out. It is necessary, that is, that what happens should be explicable in terms of the God-given nature of things. | |
From: Gottfried Leibniz (Letter to the Editor about Bayle [1698], p.205) | |
A reaction: Thus Leibniz is an ancestor of scientific essentialism, but he was too frightened to take the next step, which is to see that once God has endowed the natures, he doesn't need to wield his laws as well. The natures will do the whole job. |
19348 | All that is real in motion is the force or power which produces change [Leibniz] |
Full Idea: As for motion, what is real in it is force or power; that is to say, what there is in the present state which carries with it a change in the future. The rest is only phenomena and relations. | |
From: Gottfried Leibniz (Letter to the Editor about Bayle [1698], §13) | |
A reaction: This presumably contradicts Newton's concept of inertia, which allows constant motion without force. I always like a reference to powers. What is 'kinetic energy' in this context? |