13 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. |
| 8979 | Slow and continuous events (like balding or tree-growth) are called 'processes', not 'events' [Simons] |
| Full Idea: Some changes are slow and continuous and are called 'processes' rather than events; the growth of a tree or the greying of John's hair. | |
| From: Peter Simons (Events [2003], 3.2) | |
| A reaction: So making a loaf of bread is an event rather than a process, and World War I was a process rather than an event? If you slow down a dramatic event (on film), you see that it is really a process. I take 'process' to be a much more illuminating word. |
| 8981 | Maybe processes behave like stuff-nouns, and events like count-nouns [Simons] |
| Full Idea: There is arguably a parallel between the mass-count distinction among meanings of nouns and the process-event distinction among meanings of verbs. Processes, like stuff, do not connote criteria for counting, whereas events, like things, do. | |
| From: Peter Simons (Events [2003], 6.2) | |
| A reaction: Hm. You can have several processes, and a process can come to an end - but then you can have several ingredients of a cake, and you can run out of one of them. This may be quite a helpful distinction. |
| 8973 | Einstein's relativity brought events into ontology, as the terms of a simultaneity relationships [Simons] |
| Full Idea: The ontology of events rose in philosophy with the rise of relativity theory in physics. Einstein postulated the relativity of simultaneity to an observer's state of motion. The terms of the relation of simultaneity must be events or their parts. | |
| From: Peter Simons (Events [2003], 1.1.2) | |
| A reaction: Intriguing. Philosophers no doubt think they are way ahead of physicists in such a metaphysical area. Personally I regard the parentage of the concept as good grounds for scepticism about it. See Idea 7621 for my reason. |
| 19355 | The soul doesn't understand many of its own actions, if perceptions are confused and desires buried [Leibniz] |
| Full Idea: The soul does many things without knowing how it does them - when it does them by means of confused perceptions and unconscious inclinations or appetites. | |
| From: Gottfried Leibniz (On Note L to Bayle's 'Rorarius' [1705], [L]) | |
| A reaction: This increasingly strikes me as a wonderful and important insight for its time. He's really paid attention to his own mind, and given up the simplistic view that derives from Descartes. Are birds conscious? Yes or no! Silly. |
| 19350 | We should say that body is mechanism and soul is immaterial, asserting their independence [Leibniz] |
| Full Idea: I think we should keep both sides: we should be more Democritean and make all actions of bodies mechanical and independent of souls, and we should also be more than Platonic and hold that all actions of souls are immaterial and independent of mechanism. | |
| From: Gottfried Leibniz (On Note L to Bayle's 'Rorarius' [1705], [C]) | |
| A reaction: This is about as dualist as it is possible to get. It certainly looks as if many of Leibniz's doctrines are rebellions against Spinoza (in this case his 'dual aspect monism'). I take Leibniz to be utterly but heroically wrong. |
| 19356 | Minds unconsciously count vibration beats in music, and enjoy it when they coincide [Leibniz] |
| Full Idea: In music, the soul counts the beats of the vibrating object which makes the sound, and when these beats regularly coincide at short intervals, it finds them pleasing. Thus it counts without knowing it. | |
| From: Gottfried Leibniz (On Note L to Bayle's 'Rorarius' [1705], [L]) | |
| A reaction: Only a mathematician would see music this way! He is defending his account of the unconscious mind. The proposal that we unconsciously count sounds highly implausible. He needs to recognise the patterns that ground mathematics. |