7 ideas
| 15943 | Limitation of Size is not self-evident, and seems too strong [Lavine on Neumann] |
| Full Idea: Von Neumann's Limitation of Size axiom is not self-evident, and he himself admitted that it seemed too strong. | |
| From: comment on John von Neumann (An Axiomatization of Set Theory [1925]) by Shaughan Lavine - Understanding the Infinite VII.1 |
| 13672 | All the axioms for mathematics presuppose set theory [Neumann] |
| Full Idea: There is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory. | |
| From: John von Neumann (An Axiomatization of Set Theory [1925]), quoted by Stewart Shapiro - Foundations without Foundationalism 8.2 | |
| A reaction: Von Neumann was doubting whether set theory could have axioms, and hence the whole project is doomed, and we face relativism about such things. His ally was Skolem in this. |
| 13768 | Validity can preserve certainty in mathematics, but conditionals about contingents are another matter [Edgington] |
| Full Idea: If your interest in logic is confined to applications to mathematics or other a priori matters, it is fine for validity to preserve certainty, ..but if you use conditionals when arguing about contingent matters, then great caution will be required. | |
| From: Dorothy Edgington (Conditionals [2001], 17.2.1) |
| 13770 | There are many different conditional mental states, and different conditional speech acts [Edgington] |
| Full Idea: As well as conditional beliefs, there are conditional desires, hopes, fears etc. As well as conditional statements, there are conditional commands, questions, offers, promises, bets etc. | |
| From: Dorothy Edgington (Conditionals [2001], 17.3.4) |
| 13764 | Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? [Edgington] |
| Full Idea: Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? Are they non-truth-functional, like 'because' or 'before'? Do the values of A and B, in some cases, leave open the value of 'If A,B'? | |
| From: Dorothy Edgington (Conditionals [2001], 17.1) | |
| A reaction: I would say they are not truth-functional, because the 'if' asserts some further dependency relation that goes beyond the truth or falsity of A and B. Logical ifs, causal ifs, psychological ifs... The material conditional ⊃ is truth-functional. |
| 13765 | 'If A,B' must entail ¬(A & ¬B); otherwise we could have A true, B false, and If A,B true, invalidating modus ponens [Edgington] |
| Full Idea: If it were possible to have A true, B false, and If A,B true, it would be unsafe to infer B from A and If A,B: modus ponens would thus be invalid. Hence 'If A,B' must entail ¬(A & ¬B). | |
| From: Dorothy Edgington (Conditionals [2001], 17.1) | |
| A reaction: This is a firm defence of part of the truth-functional view of conditionals, and seems unassailable. The other parts of the truth table are open to question, though, if A is false, or they are both true. |
| 21516 | We want certainty in order achieve secure results for action [Dewey] |
| Full Idea: The ultimate ground of the quest for cognitive certainty is the need for security in the results for action. | |
| From: John Dewey (The Quest for Certainty [1929], p.39), quoted by Erik J. Olsson - Against Coherence 10.5 | |
| A reaction: Just what a pragmatist should say. This may be true within an evolutionary account of human nature, but seems unlikely when doing a sudoku. The 'ground' of the quest may not be the same as its 'source'. |