10 ideas
23367 | Even pointing a finger should only be done for a reason [Epictetus] |
Full Idea: Philosophy says it is not right even to stretch out a finger without some reason. | |
From: Epictetus (fragments/reports [c.57], 15) | |
A reaction: The key point here is that philosophy concerns action, an idea on which Epictetus is very keen. He rather despise theory. This idea perfectly sums up the concept of the wholly rational life (which no rational person would actually want to live!). |
13750 | Analysis aims at the structure of facts, which are needed to give a rationale to analysis [Urmson, by Schaffer,J] |
Full Idea: Urmson explains the direction of analysis as 'towards a structure...more nearly similar to the structure of the fact', adding that this metaphysical picture is needed as a 'rationale of the practice of analysis'. | |
From: report of J.O. Urmson (Philosophical Analysis [1956], p.24-5) by Jonathan Schaffer - On What Grounds What n30 | |
A reaction: In other words, only realists can be truly motivated to keep going with analysis. Merely analysing language-games is doable, but hardly exciting. |
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] |