display all the ideas for this combination of philosophers
5 ideas
| 10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
| Full Idea: Some realists argue that simple types can be apprehended through their tokens, via abstraction. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.2) | |
| A reaction: One might rephrase that to say that types are created by abstraction from tokens (and then preserved in language). |
| 9626 | A structure is an abstraction, focussing on relationships, and ignoring other features [Shapiro] |
| Full Idea: A structure is the abstract form of a system, focussing on the interrelationships among the objects, and ignoring any features of them that do not affect how they relate to other objects in the system. | |
| From: Stewart Shapiro (Structure and Ontology [1989], 146), quoted by James Robert Brown - Philosophy of Mathematics Ch.4 | |
| A reaction: I find this account very attractive, even though it appeals to supposedly outmoded psychological abstractionism. It seems pretty close to Aristotle's view of things. Shapiro's account must face up to Frege's worries about these matters. |
| 10217 | We can apprehend structures by focusing on or ignoring features of patterns [Shapiro] |
| Full Idea: One way to apprehend a particular structure is through a process of pattern recognition, or abstraction. One observes systems in a structure, and focuses attention on the relations among the objects - ignoring features irrelevant to their relations. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1) | |
| A reaction: A lovely statement of the classic Aristotelian abstractionist approach of focusing-and-ignoring. But this is made in 1997, long after Frege and Geach ridiculed it. It just won't go away - not if you want a full and unified account of what is going on. |
| 9554 | We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro] |
| Full Idea: One can observe a system and focus attention on the relations among the objects - ignoring those features of the objects not relevant to the system. For example, we can understand a baseball defense system by going to several games. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], p.74), quoted by Charles Chihara - A Structural Account of Mathematics | |
| A reaction: This is Shapiro perpetrating precisely the wicked abstractionism which Frege and Geach claim is ridiculous. Frege objects that abstract concepts then become private, but baseball defences are discussed in national newspapers. |
| 10231 | Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro] |
| Full Idea: Perhaps we can introduce abstract objects by abstraction over an equivalence relation on a base class of entities, just as Frege suggested that 'direction' be obtained from parallel lines. ..Properties must be equinumerous, but need not be individuated. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.5) | |
| A reaction: [He cites Hale and Wright as the originators of this} It is not entirely clear why this is 'abstraction', rather than just drawing attention to possible groupings of entities. |