14 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. |
| 8124 | The highest feelings of mankind can only be transmitted by art [Tolstoy] |
| Full Idea: The highest feelings to which mankind has attained can only be transmitted from man to man by art. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.17) | |
| A reaction: We are much more nervous these days of talking about 'highest' feelings. Tolstoy obviously considers religion to be an ingredient of the highest feelings, but that prevents us from judging them purely as feelings. Music is the place to rank feelings. |
| 8121 | Art is when one man uses external signs to hand on his feelings to another man [Tolstoy] |
| Full Idea: Art is a human activity in which one man consciously by means of external signs, hands on to others feelings he has lived through, and other are infected by those feelings, and also experience them. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.5) | |
| A reaction: Such definitions always work better for some art forms than for others. This may fit 'Anna Karenin' quite well, but probably not Bach's 'Art of Fugue'. Writing obscenities on someone's front door would fit this definition. |
| 8122 | True works of art transmit completely new feelings [Tolstoy] |
| Full Idea: Only that is a true work of art which transmits fresh feelings not previously experienced by man. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.9) | |
| A reaction: I think a great composer will probably not have any new feelings at all, but will discover new expressions which contain feelings by which even they are surprised (e.g. the Tristan chord). |
| 8123 | The purpose of art is to help mankind to evolve better, more socially beneficial feelings [Tolstoy] |
| Full Idea: The evolution of feeling proceeds by means of art - feelings less kind and less necessary for the well-being of mankind being replaced by others kinder and more needful for that end. That is the purpose of art. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.16) | |
| A reaction: Underneath his superficially expressivist view of art, Tolstoy is really an old-fashioned moralist about it, like Dr Johnson. This is the moralism of the great age of the nineteenth century novel (which was, er, the greatest age of the novel!). |
| 8064 | We separate the concept of beauty from goodness, unlike the ancients [Tolstoy] |
| Full Idea: The ancients had not that conception of beauty separated from goodness which forms the basis and aim of aesthetics in our time. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.3) | |
| A reaction: This is written at around the time of the Aesthetic Movement, but Tolstoy's own novels are intensely moral. This separation makes abstract painting possible. |
| 22710 | People estimate art according to their moral values [Tolstoy] |
| Full Idea: The estimation of the value of art …depends on men's perception of the meaning of life; depends on what they hold to be the good and evil of life. | |
| From: Leo Tolstoy (What is Art? [1898]), quoted by Iris Murdoch - The Sublime and the Good p.206 | |
| A reaction: [No ref given] This is put to the test by the insightful depiction of wickedness. We condemn the wickedness and admire the insight. Every reading of a novel is a moral journey, though I'm not sure how the true psychopath reads a novel. |
| 8125 | The upper classes put beauty first, and thus freed themselves from morality [Tolstoy] |
| Full Idea: The people of the upper class, more and more frequently encountering the contradictions between beauty and goodness, put the ideal of beauty first, thus freeing themselves from the demands of morality. | |
| From: Leo Tolstoy (What is Art? [1898], Ch.17) | |
| A reaction: The rich are a great deal freer to pursue the demands of beauty than are the poor. They also have a tradition of 'immorality' (such as duels and adultery) which was in place long before they discovered art. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |