display all the ideas for this combination of philosophers
7 ideas
| 10979 | Although second-order arithmetic is incomplete, it can fully model normal arithmetic [Read] |
| Full Idea: Second-order arithmetic is categorical - indeed, there is a single formula of second-order logic whose only model is the standard model ω, consisting of just the natural numbers, with all of arithmetic following. It is nevertheless incomplete. | |
| From: Stephen Read (Thinking About Logic [1995], Ch.2) | |
| A reaction: This is the main reason why second-order logic has a big fan club, despite the logic being incomplete (as well as the arithmetic). |
| 10980 | Second-order arithmetic covers all properties, ensuring categoricity [Read] |
| Full Idea: Second-order arithmetic can rule out the non-standard models (with non-standard numbers). Its induction axiom crucially refers to 'any' property, which gives the needed categoricity for the models. | |
| From: Stephen Read (Thinking About Logic [1995], Ch.2) |
| 10554 | Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal [Dummett] |
| Full Idea: In the intuitionist view, the notion of an intuitive proof cannot be expected to coincide with that of a proof in a formal system, and Gödel's incompleteness theorem is thus unsurprising from an intuitionist point of view. | |
| From: Michael Dummett (Frege Philosophy of Language (2nd ed) [1973], Ch.14) |
| 10997 | Von Neumann numbers are helpful, but don't correctly describe numbers [Read] |
| Full Idea: The Von Neumann numbers have a structural isomorphism to the natural numbers - each number is the set of all its predecessors, so 2 is the set of 0 and 1. This helps proofs, but is unacceptable. 2 is not a set with two members, or a member of 3. | |
| From: Stephen Read (Thinking About Logic [1995], Ch.4) |
| 9829 | The identity of a number may be fixed by something outside structure - by counting [Dummett] |
| Full Idea: The identity of a mathematical object may sometimes be fixed by its relation to what lies outside the structure to which it belongs. It is more fundamental to '3' that if certain objects are counted, there are three of them. | |
| From: Michael Dummett (Frege philosophy of mathematics [1991], Ch. 5) | |
| A reaction: This strikes me as Dummett being pushed (by his dislike of the purely abstract picture given by structuralism) back to a rather empiricist and physical view of numbers, though he would totally deny that. |
| 9828 | Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1 [Dummett] |
| Full Idea: The number 0 is not differentiated from 1 by its position in a progression, otherwise there would be no difference between starting with 0 and starting with 1. That is enough to show that numbers are not identifiable just as positions in structures. | |
| From: Michael Dummett (Frege philosophy of mathematics [1991], Ch. 5) | |
| A reaction: This sounds conclusive, but doesn't feel right. If numbers are a structure, then where you 'start' seems unimportant. Where do you 'start' in St Paul's Cathedral? Starting sounds like a constructivist concept for number theory. |
| 9192 | The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett] |
| Full Idea: The number 4 cannot be characterized solely by its position in a system, because it has different positions in the system of natural numbers and that of the positive whole numbers, whereas these systems have the very same structure. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 6.1) | |
| A reaction: Dummett seems to think this is fairly decisive against structuralism. There is also the structure of the real numbers. We will solve this by saying that the wholes are abstracted from the naturals, which are abstracted from the reals. Job done. |