12 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. |
| 9295 | Not only substances have attributes; events, actions, states and qualities can have them [Teichmann] |
| Full Idea: It is not true that only substances have attributes; events, actions, states and qualities can all be characterized. | |
| From: Jenny Teichmann (The Mind and the Soul [1974], Ch.2) | |
| A reaction: This is why it is so important to distinguish the actual properties in nature from those that can be fancifully hypothesized by a linguistic being. Is there any limit to the possible number of levels of meta-properties? |
| 9293 | Body-spirit interaction ought to result in losses and increases of energy in the material world [Teichmann] |
| Full Idea: Since the interaction of bodies themselves involves energy-flow, it looks as if interaction between body and spirit ought to result in losses and increases of energy in the material world. | |
| From: Jenny Teichmann (The Mind and the Soul [1974], Ch.2) | |
| A reaction: A nice statement of an important argument. It forces the dualist to go the whole way, asserting that not only is the mind immaterial, but that it can be active without energy, and cover its traces in the physical world. Doesn't look good. |
| 23261 | A people, not government, creates a constitution, which is essential for legitimacy [Paine] |
| Full Idea: A constitution is not the act of a government, but of a people constituting a government, and a government without a constitution is power without right. | |
| From: Thomas Paine (Rights of Man [1792], Ch.7), quoted by A.C. Grayling - The Good State 5 | |
| A reaction: A constitution looks like the ultimate focus of a social contract (though Greeks had them long ago). It is hard to say why a government should consider itself to be sovereign if it hasn't got it in writing. |
| 9292 | The Soul has no particular capacity (in the way thinking belongs to the mind) [Teichmann] |
| Full Idea: On the whole, the Soul has no capacities which belong to it pre-eminently in the way that thinking 'belongs' to the mind. | |
| From: Jenny Teichmann (The Mind and the Soul [1974], Ch.1) | |
| A reaction: There are no phenomena which have to be saved by postulating a soul. It lacks a function within a human being, but it has a crucial function within a large theological picture. |
| 9294 | No individuating marks distinguish between Souls [Teichmann] |
| Full Idea: There are no individuating marks which could serve to differentiate one Soul from another. | |
| From: Jenny Teichmann (The Mind and the Soul [1974], Ch.2) | |
| A reaction: Presumably they could have at least much identity as two different electrons (if they are in space-time?). It is hard to see why anyone would be interested in their 'own' immortality, if loss of all individuality was a condition. |