7 ideas
| 9935 | Mathematical truth is always compromising between ordinary language and sensible epistemology [Benacerraf] |
| Full Idea: Most accounts of the concept of mathematical truth can be identified with serving one or another of either semantic theory (matching it to ordinary language), or with epistemology (meshing with a reasonable view) - always at the expense of the other. | |
| From: Paul Benacerraf (Mathematical Truth [1973], Intro) | |
| A reaction: The gist is that language pulls you towards platonism, and epistemology pulls you towards empiricism. He argues that the semantics must give ground. He's right. |
| 8203 | All the arithmetical entities can be reduced to classes of integers, and hence to sets [Quine] |
| Full Idea: The arithmetic of ratios and irrational and imaginary numbers can all be reduced by definition to the theory of classes of positive integers, and this can in turn be reduced to pure set theory. | |
| From: Willard Quine (Vagaries of Definition [1972], p.53) | |
| A reaction: This summarises Quine's ontology of mathematics, which tries to eliminate virtually everything, but has to affirm the existence of sets. Can you count sets and their members, if the sets are used to define the numbers? |
| 17927 | Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Benacerraf, by Colyvan] |
| Full Idea: Benacerraf argues that realists about mathematical objects have a nice normal semantic but no epistemology, and anti-realists have a good epistemology but an unorthodox semantics. | |
| From: report of Paul Benacerraf (Mathematical Truth [1973]) by Mark Colyvan - Introduction to the Philosophy of Mathematics 1.2 |
| 9936 | The platonist view of mathematics doesn't fit our epistemology very well [Benacerraf] |
| Full Idea: The principle defect of the standard (platonist) account of mathematical truth is that it appears to violate the requirement that our account be susceptible to integration into our over-all account of knowledge. | |
| From: Paul Benacerraf (Mathematical Truth [1973], III) | |
| A reaction: Unfortunately he goes on to defend a causal theory of justification (fashionable at that time, but implausible now). Nevertheless, his general point is well made. Your theory of what mathematics is had better make it knowable. |
| 8202 | Meaning is essence divorced from things and wedded to words [Quine] |
| Full Idea: Meaning is essence divorced from the thing and wedded to the word. | |
| From: Willard Quine (Vagaries of Definition [1972], p.51) | |
| A reaction: Quine's strategy is that a demolition of essences will be a definition of meaning. Personally I would like to defend essences, though I admit to finding meaning tricky. That is because essences are external, but meanings are in minds. |
| 8201 | The distinction between meaning and further information is as vague as the essence/accident distinction [Quine] |
| Full Idea: The distinction between what belongs to the meaning of a word and what counts as further information is scarcely clearer than the distinction between the essence of a thing and its accidents. | |
| From: Willard Quine (Vagaries of Definition [1972], p.51) | |
| A reaction: In lots of cases the distinction between essence and accident strikes me as totally clear. Tricky borderline cases don't destroy a distinction. That bachelors are married is clearly not 'further information'. |
| 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'). |