display all the ideas for this combination of texts
5 ideas
10061 | The If-thenist view only seems to work for the axiomatised portions of mathematics [Musgrave] |
Full Idea: The If-thenist view seems to apply straightforwardly only to the axiomatised portions of mathematics. | |
From: Alan Musgrave (Logicism Revisited [1977], §5) | |
A reaction: He cites Lakatos to show that cutting-edge mathematics is never axiomatised. One might reply that if the new mathematics is any good then it ought to be axiomatis-able (barring Gödelian problems). |
10065 | Perhaps If-thenism survives in mathematics if we stick to first-order logic [Musgrave] |
Full Idea: If we identify logic with first-order logic, and mathematics with the collection of first-order theories, then maybe we can continue to maintain the If-thenist position. | |
From: Alan Musgrave (Logicism Revisited [1977], §5) | |
A reaction: The problem is that If-thenism must rely on rules of inference. That seems to mean that what is needed is Soundness, rather than Completeness. That is, inference by the rules must work properly. |
10049 | Logical truths may contain non-logical notions, as in 'all men are men' [Musgrave] |
Full Idea: Containing only logical notions is not a necessary condition for being a logical truth, since a logical truth such as 'all men are men' may contain non-logical notions such as 'men'. | |
From: Alan Musgrave (Logicism Revisited [1977], §3) | |
A reaction: [He attributes this point to Russell] Maybe it is only a logical truth in its general form, as ∀x(x=x). Of course not all 'banks' are banks. |
10050 | A statement is logically true if it comes out true in all interpretations in all (non-empty) domains [Musgrave] |
Full Idea: The standard modern view of logical truth is that a statement is logically true if it comes out true in all interpretations in all (non-empty) domains. | |
From: Alan Musgrave (Logicism Revisited [1977], §3) |
11261 | Puzzles arise when reasoning seems equal on both sides [Aristotle] |
Full Idea: The equality of opposite reasonings is the cause of aporia; for it is when we reason on both [sides of a question] and it appears to us that everything can come about either way, that we are in a state of aporia about which of the two ways to take up. | |
From: Aristotle (Topics [c.331 BCE], 145b17), quoted by Vassilis Politis - Aristotle and the Metaphysics 3.1 | |
A reaction: Other philosophers give up on the subject in this situation, but I love Aristotle because he takes this to be the place where philosophy begins. |