14 ideas
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
| A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
| A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics. |
| 7746 | We don't normally think of names as having senses (e.g. we don't give definitions of them) [Searle] |
| Full Idea: If Tully=Cicero is synthetic, the names must have different senses, which seems implausible, for we don't normally think of proper names as having senses in the way that predicates do (we do not, e.g., give definitions of proper names). | |
| From: John Searle (Proper Names [1958], p.89) | |
| A reaction: It is probably necessary to prize apart the question of whether Tully 'has' (intrinsically) a sense, from whether we think of Tully in that way. Stacks of books have appeared about this one, since Kripke. |
| 7747 | How can a proper name be correlated with its object if it hasn't got a sense? [Searle] |
| Full Idea: It seems that a proper name could not have a reference unless it did have a sense, for how, unless the name has a sense, is it to be correlated with the object? | |
| From: John Searle (Proper Names [1958], p.91) | |
| A reaction: This might (just) be the most important question ever asked in modern philosophy, since it provoked Kripke into answering it, by giving a social, causal, externalist account of how names (and hence lots of language) actually work. But Searle has a point. |
| 7748 | 'Aristotle' means more than just 'an object that was christened "Aristotle"' [Searle] |
| Full Idea: Aristotle being identical with an object that was originally christened will not suffice, for the force of "Aristotle" is greater than the force of 'identical with an object named "Aristotle"', for not just any object named "Aristotle" will do. | |
| From: John Searle (Proper Names [1958], p.93) | |
| A reaction: This anticipates Kripke's proposal to base reference on baptism. I remain unsure about how rigid a designation of Aristotle could be, in a possible world where his father died young, and he became an illiterate soldier who hates philosophy. |
| 7749 | Reference for proper names presupposes a set of uniquely referring descriptions [Searle] |
| Full Idea: To use a proper name referringly is to presuppose the truth of certain uniquely referring descriptive statements. ...Names are pegs on which to hang descriptions. | |
| From: John Searle (Proper Names [1958], p.94) | |
| A reaction: This 'cluster' view of Searle's has become notorious, but I think one could at least try to mount a defence. The objection to Searle is that none of the descriptions are necessary, unlike just being the named object. |
| 7750 | Proper names are logically connected with their characteristics, in a loose way [Searle] |
| Full Idea: If asked whether or not proper names are logically connected with characteristics of the object to which they refer, the answer is 'yes, in a loose sort of way'. | |
| From: John Searle (Proper Names [1958], p.96) | |
| A reaction: It seems to be inviting trouble to assert that a connection is both 'logical' and 'loose'. Clearly Searle has been reading too much later Wittgenstein. This is probably the weakest point in Searle's proposal, which brought a landslide of criticism. |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth. | |
| From: Penelope Maddy (Defending the Axioms [2011], 5.3ii) |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
| Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40) |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.4) | |
| A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics. |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.3) | |
| A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor. |
| 23812 | Force is what turns man into a thing, and ultimately into a corpse [Weil] |
| Full Idea: To define 'force' - it is that x that turns anybody who is subjected to it into a thing. Exercised to the limit, it turns man into a thing in the most literal sense: it makes a corpse out of him. | |
| From: Simone Weil (The Iliad or the Poem of Force [1940], p.183) | |
| A reaction: She celebrates The Iliad as the great examination of force in human affairs. I have felt that sense of reduction to a thing whenever anyone above me in the hierarchy has arbitrarily exerted their power over me. |
| 23813 | Only people who understand force, and don't respect it, are capable of justice [Weil] |
| Full Idea: Only he who has measured the dominion of force, and knows how not to respect it, is capable of love and justice. | |
| From: Simone Weil (The Iliad or the Poem of Force [1940], p.212) | |
| A reaction: There are, of course, occasions when we are grateful to people who exercise appropriate force on our behalf. I think she was concerned with what is inappropriate. |