display all the ideas for this combination of texts
11 ideas
| 9846 | Defining 'direction' by parallelism doesn't tell you whether direction is a line [Dummett on Frege] |
| Full Idea: The stipulation that the direction of a line a is to be the same as that of a line b just in case a is parallel to b does not determine whether the direction of a line is itself a line or something quite different. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §60) by Michael Dummett - Frege philosophy of mathematics Ch.11 | |
| A reaction: Nice point. Maybe not being able to say exactly what something is is either a symptom of nonsense, and simply a symptom that we are dealing with an abstract concept. If abstractions don't exist, they don't need individuation criteria. |
| 9976 | Frege accepts abstraction to the concept of all sets equipollent to a given one [Tait on Frege] |
| Full Idea: Frege's own conception of abstraction (although he disapproves of the term) is in agreement with the view that abstracting from the particular nature of the elements of M would yield the concept under which fall all sets equipollent to M. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by William W. Tait - Frege versus Cantor and Dedekind III | |
| A reaction: Nice! This shows how difficult it is to slough off the concept of abstractionism and live with purely logical concepts of it. If we 'construct' a set, then there is a process of creation to be explained; we can't just think of platonic givens. |
| 10803 | Frege himself abstracts away from tone and color [Yablo on Frege] |
| Full Idea: Frege himself abstracts away from tone and color. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Stephen Yablo - Carving Content at the Joints §3 | |
| A reaction: Gotcha! I have been searching for instances where Frege perpetrates psychological abstraction right in the heart of his theory. No one can avoid it, if they are in the business of trying to formulate new concepts. Reference ignores sense, and vice versa. |
| 9988 | If we abstract 'from' two cats, the units are not black or white, or cats [Tait on Frege] |
| Full Idea: When from a set of two cats, one black and one white, we 'abstract' the number two as a set of pure units, the units are not black and white, respectively, and they are not cats. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §34) by William W. Tait - Frege versus Cantor and Dedekind XI | |
| A reaction: Well said. Frege is contemptuous of this approach, as if we were incapable of thinking of a black cat as anything other than as black or cat, when we can catch cats as 'food', or 'objects', or just plain 'countables'. |
| 9855 | Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects [Frege, by Dummett] |
| Full Idea: Like psychological abstractionism, Frege's method (which we can call 'logical abstraction') aims at isolating what is in common between the members of any equivalent sets of objects, by identifying the feature with the maximal set of such objects. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.14 | |
| A reaction: [compressed] So Frege's approach to abstraction is a branch of the view that properties are sets. This view, in addition to being vulnerable to Russell's paradox, ignores the causal role of properties, making them all categorical (which is daft). |
| 10802 | Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first [Yablo on Frege] |
| Full Idea: Frege's discussion of 'direction' borders on incoherent. He claims that the equivalence of lines a and b and their directions being equal have the same content, which leads to the concept of direction, but we grasp the equivalence before the equality. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Stephen Yablo - Carving Content at the Joints § 1 | |
| A reaction: [The Frege is in Grundlagen §64] Well said. The notion that we get the full concept of 'direction' from such paltry resources seems very weak. For a start, parallel lines exhibit two directions, not one. |
| 10525 | Frege put the idea of abstraction on a rigorous footing [Frege, by Fine,K] |
| Full Idea: It was Frege who first showed how the idea of abstraction could be put on a rigorous footing. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Kit Fine - Precis of 'Limits of Abstraction' p.305 | |
| A reaction: This refers to the crucial landmark in philosophical thought about abstraction. The question is whether Frege had to narrow the concept of abstraction and abstract entities too severely, in order to achieve his rigour. |
| 10526 | Fregean abstraction creates concepts which are equivalences between initial items [Frege, by Fine,K] |
| Full Idea: Fregean abstraction rests on initial items, taken to be related by an equivalence relation (e.g. parallelism, or equinumerosity), and then an operation for forming abstraction (e.g. direction or number), with identity related to their equivalence. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Kit Fine - Precis of 'Limits of Abstraction' p.305 | |
| A reaction: [compressed] This is the best summary I have found of the modern theory of abstraction, as opposed to the nature of the abstracta themselves. A minimum of two items is needed to implement the process. |
| 10556 | We create new abstract concepts by carving up the content in a different way [Frege] |
| Full Idea: (In creating the concept of direction..) We carve up the content in a way different from the original way, and this yields us a new concept. ...It is a matter of drawing boundary lines that were not previously given. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64) | |
| A reaction: [second half in §88] 'Recarving' is now the useful shorthand for Frege's way of creating abstract concepts (rather than the old psychological way of ignoring some features of an object). |
| 9882 | You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables [Dummett on Frege] |
| Full Idea: Frege's root confusion (over abstraction by identity, and other things) was to believe that he could simultaneously fix the truth-conditions of such statements and the domain over which the individual variables were to range. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64-68) by Michael Dummett - Frege philosophy of mathematics Ch.18 | |
| A reaction: This strikes me as a wonderfully penetrating criticism, but it also seems to me to threaten Dummett's whole programme of doing ontology through language. If a quantified sentences needs a domain, how do you first decide your domain? |
| 9881 | From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements [Frege, by Dummett] |
| Full Idea: Having rightly perceived that the fundamental class here was statements of identity between directions, Frege leapt to the conclusion that the basis for introducing new abstract terms consisted of determining the truth-conditions of identity-statements. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64-68) by Michael Dummett - Frege philosophy of mathematics Ch.18 | |
| A reaction: This seems to be the modern view - that abstraction consists of the assertion of an equivalence principle. Dummett criticises Frege here (see Idea 9882). There always seems to be a chicken/egg problem. Why would the identity be asserted? |