10 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. |
| 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. |
| 2596 | Maybe mind and body are parallel, like two good clocks [Leibniz] |
| Full Idea: Two clocks in perfect agreement must be by natural influence, or the control of a craftsman, or their perfect construction at the beginning. Only the third way (of "preestablished harmony" by God) is possible. | |
| From: Gottfried Leibniz (The Nature and Communication of Substance [1690], p.121) | |
| A reaction: Presumably 'natural influence' would cover the possibility that (unnoticed by you, apparently) one clock is attached to the other clock at the relevant points. If they are unconnected, presumably they are quite unaware of one another's existence. |
| 22457 | If the aim is good outcomes, why are killings worse than deaths? [Scheffler, by Foot] |
| Full Idea: It is not clear why, in the measurement of the goodness of states of affairs or total outcomes, killings for instance should count so much more heavily than deaths. | |
| From: report of Samuel Scheffler (The Rejection of Consequentialism [1982], pp.108-12) by Philippa Foot - Utilitarianism and the Virtues p.61 | |
| A reaction: Or drunken drivers worse than careless drivers. Or stolen bracelets than lost bracelets. The point is that morality is about the behaviour of people, and not about consequences. |
| 2595 | If the universe is a perfect agreement of uncommunicating substances, there must be a common source [Leibniz] |
| Full Idea: The perfect agreement of so many substances which have no communication whatever with each other can only come from a common source. | |
| From: Gottfried Leibniz (The Nature and Communication of Substance [1690], p.120) |