8 ideas
| 9944 | We understand some statements about all sets [Putnam] |
| Full Idea: We seem to understand some statements about all sets (e.g. 'for every set x and every set y, there is a set z which is the union of x and y'). | |
| From: Hilary Putnam (Mathematics without Foundations [1967], p.308) | |
| A reaction: His example is the Axiom of Choice. Presumably this is why the collection of all sets must be referred to as a 'class', since we can talk about it, but cannot define it. |
| 9937 | I do not believe mathematics either has or needs 'foundations' [Putnam] |
| Full Idea: I do not believe mathematics either has or needs 'foundations'. | |
| From: Hilary Putnam (Mathematics without Foundations [1967]) | |
| A reaction: Agreed that mathematics can function well without foundations (given that the enterprise got started with no thought for such things), the ontology of the subject still strikes me as a major question, though maybe not for mathematicians. |
| 9939 | It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam] |
| Full Idea: I believe that under certain circumstances revisions in the axioms of arithmetic, or even of the propositional calculus (e.g. the adoption of a modular logic as a way out of the difficulties in quantum mechanics), is fully conceivable. | |
| From: Hilary Putnam (Mathematics without Foundations [1967], p.303) | |
| A reaction: One can change the axioms of a system without necessarily changing the system (by swapping an axiom and a theorem). Especially if platonism is true, since the eternal objects reside calmly above our attempts to axiomatise them! |
| 9940 | Maybe mathematics is empirical in that we could try to change it [Putnam] |
| Full Idea: Mathematics might be 'empirical' in the sense that one is allowed to try to put alternatives into the field. | |
| From: Hilary Putnam (Mathematics without Foundations [1967], p.303) | |
| A reaction: He admits that change is highly unlikely. It take hardcore Millian arithmetic to be only changeable if pebbles start behaving very differently with regard to their quantities, which appears to be almost inconceivable. |
| 9941 | Science requires more than consistency of mathematics [Putnam] |
| Full Idea: Science demands much more of a mathematical theory than that it should merely be consistent, as the example of the various alternative systems of geometry dramatizes. | |
| From: Hilary Putnam (Mathematics without Foundations [1967]) | |
| A reaction: Well said. I don't agree with Putnam's Indispensability claims, but if an apparent system of numbers or lines has no application to the world then I don't consider it to be mathematics. It is a new game, like chess. |
| 9943 | You can't deny a hypothesis a truth-value simply because we may never know it! [Putnam] |
| Full Idea: Surely the mere fact that we may never know whether the continuum hypothesis is true or false is by itself just no reason to think that it doesn't have a truth value! | |
| From: Hilary Putnam (Mathematics without Foundations [1967]) | |
| A reaction: This is Putnam in 1967. Things changed later. Personally I am with the younger man all they way, but I reserve the right to totally change my mind. |
| 16614 | Matter and form give true unity; subject and accident is just unity 'per accidens' [Duns Scotus] |
| Full Idea: From matter and form comes one thing per se. This is not so for subject and accident. Matter and form are instrinsic causes of a composite being, but whiteness and a human being are not. Humans can exist without whiteness, so it is one thing per accidens. | |
| From: John Duns Scotus (Oxford Commentary on Sentences [1301], II.12.1.14), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 | |
| A reaction: This isn't much of a theory, but at least it is focusing on an interesting question, and the distinction between genuinely unified, and unified by chance. Compare a loving couple with siblings who hate each other. |
| 5655 | Happiness is not satisfaction of desires, but fulfilment of values [Bradley, by Scruton] |
| Full Idea: For Bradley, the happiness of the individual is not to be understood in terms of his desires and needs, but rather in terms of his values - which is to say, in terms of those of his desires which he incorporates into his self. | |
| From: report of F.H. Bradley (Ethical Studies [1876]) by Roger Scruton - Short History of Modern Philosophy Ch.16 | |
| A reaction: Good. Bentham will reduce the values to a further set of desires, so that a value is a complex (second-level?) desire. I prefer to think of values as judgements, but I like Scruton's phrase of 'incorporating into his self'. Kant take note (Idea 1452). |