10 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. |
| 19691 | Unlike knowledge, you can achieve understanding through luck [Grimm] |
| Full Idea: It may be that understanding is compatible with luck, in a way that knowledge is not. | |
| From: Stephen R. Grimm (Understanding [2011], 3) | |
| A reaction: [He cites Kvanvig and Prichard] If so, then we cannot say that knowledge is a lesser type of understanding. If you ask a trusted person how a mechanism works, and they have a wild guess that is luckily right, you would then understand it. |
| 19690 | 'Grasping' a structure seems to be modal, because we must anticipate its behaviour [Grimm] |
| Full Idea: 'Graspng' a structure would seem to bring into play something like a modal sense or ability, not just to register how things are, but also to anticipate how certain elements of the system would behave. | |
| From: Stephen R. Grimm (Understanding [2011], 2) | |
| A reaction: In the case of the chronology of some historical events, talking of 'grasping' or 'understanding' seems wrong because the facts are static and invariant. That seems to support the present idea. But you might 'understand' a pattern if you can reproduce it. |
| 19692 | You may have 'weak' understanding, if by luck you can answer a set of 'why questions' [Grimm] |
| Full Idea: There may be a 'weak' sense of understanding, where all you need to do is to be able to answer 'why questions' successfully, where one might have come by this ability in a lucky way. | |
| From: Stephen R. Grimm (Understanding [2011], 3) | |
| A reaction: We can see this point (in Idea 19691), but the idea that one could come by true complex understanding of something by purely lucky means is a bit absurd. Surely you would get one or two why questions wrong? 100%, just by luck? |
| 1590 | The just man does not harm his enemies, but benefits everyone [Plato] |
| Full Idea: First, Socrates, you told me justice is harming your enemies and helping your friends. But later it seemed that the just man, since everything he does is for someone's benefit, never harms anyone. | |
| From: Plato (Clitophon [c.372 BCE], 410b) | |
| A reaction: Socrates certainly didn't subscribe to the first view, which is the traditional consensus in Greek culture. In general Socrates agreed with the views later promoted by Jesus. |