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. |
7783 | Bodies, properties, relations, events, numbers, sets and propositions are 'things' if they exist [Lowe] |
Full Idea: Not only material bodies but also properties, relations, events, numbers, sets, and propositions are—if they are acknowledged as existing—to be accounted ‘things’. | |
From: E.J. Lowe (Things [1995]) | |
A reaction: There might be lots of borderline cases here. Is the sky a thing? Is air a thing? How is transparency a thing? Is minus-one a thing? Is an incomplete proposition a thing? Etc. |
7518 | If folk psychology gives a network of causal laws, that fits neatly with functionalism [Churchland,PM] |
Full Idea: The portrait of folk psychology as a network of causal laws dovetailed neatly with the emerging philosophy of mind called functionalism. | |
From: Paul M. Churchland (Folk Psychology [1996], II) | |
A reaction: And from the lower levels functionalism is supported by the notion that the brain is modular. Note the word 'laws'; this implies an underlying precision in folk psychology, which is then easily attacked. Maybe the network is too complex for simple laws. |
7519 | Many mental phenomena are totally unexplained by folk psychology [Churchland,PM] |
Full Idea: Folk psychology fails utterly to explain a considerable variety of central psychological phenomena: mental illness, sleep, creativity, memory, intelligence differences, and many forms of learning, to cite just a few. | |
From: Paul M. Churchland (Folk Psychology [1996], III) | |
A reaction: If folk psychology is a theory, it will have been developed to predict behaviour, rather than as a full-blown psychological map. The odd thing is that some people seem to be very bad at folk psychology. |
7520 | Folk psychology never makes any progress, and is marginalised by modern science [Churchland,PM] |
Full Idea: Folk psychology has not progressed significantly in the last 2500 years; if anything, it has been steadily in retreat during this period; it does not integrate with modern science, and its emerging wallflower status bodes ill for its future. | |
From: Paul M. Churchland (Folk Psychology [1996], III) | |
A reaction: [compressed] However, while shares in alchemy and astrology have totally collapsed, folk psychology shows not the slightest sign of going away, and it is unclear how it ever could. See Idea 3177. |