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. |
| 23476 | Logical constants seem to be entities in propositions, but are actually pure form [Russell] |
| Full Idea: 'Logical constants', which might seem to be entities occurring in logical propositions, are really concerned with pure form, and are not actually constituents of the propositions in the verbal expressions of which their names occur. | |
| From: Bertrand Russell (The Theory of Knowledge [1913], 1.IX) | |
| A reaction: This seems to entirely deny the existence of logical constants, and yet he says that they are named. Russell was obviously under pressure here from Wittgenstein. |
| 23477 | We use logical notions, so they must be objects - but I don't know what they really are [Russell] |
| Full Idea: Such words as or, not, all, some, plainly involve logical notions; since we use these intelligently, we must be acquainted with the logical objects involved. But their isolation is difficult, and I do not know what the logical objects really are. | |
| From: Bertrand Russell (The Theory of Knowledge [1913], 1.IX) | |
| A reaction: See Idea 23476, from the previous page. Russell is struggling. Wittgenstein was telling him that the constants are rules (shown in truth tables), rather than objects. |
| 18273 | Logical truths are known by their extreme generality [Russell] |
| Full Idea: A touchstone by which logical propositions may be distinguished from all others is that they result from a process of generalisation which has been carried to its utmost limits. | |
| From: Bertrand Russell (The Theory of Knowledge [1913], p.129), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap 7 'What' |
| 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. |
| 22315 | There can't be a negative of a complex, which is negated by its non-existence [Potter on Russell] |
| Full Idea: On Russell's pre-war conception it is obvious that a complex cannot be negative. If a complex were true, what would make it false would be its non-existence, not the existence of some other complex. | |
| From: comment on Bertrand Russell (The Theory of Knowledge [1913]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 41 'Neg' | |
| A reaction: It might be false because it doesn't exist, but also 'made' false by a rival complex (such as Desdemona loving Othello). |
| 16764 | The soul conserves the body, as we see by its dissolution when the soul leaves [Toletus] |
| Full Idea: Every accident of a living thing, as well as all its organs and temperaments and its dispositions are conserved by the soul. We see this from experience, since when that soul recedes, all these dissolve and become corrupted. | |
| From: Franciscus Toletus (Commentary on 'De Anima' [1572], II.1.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.5 | |
| A reaction: A nice example of observing a phenemonon, but not being able to observe the dependence relation the right way round. Compare Descartes in Idea 16763. |