11 ideas
| 21916 | Philosophers can't be religious, and don't need to be; philosophy is perilous but free [Schopenhauer] |
| Full Idea: No one who is religious attains to philosophy; he does not need it. No one who really philosophizes is religious; he walks without leading-strings, perilously but free. | |
| From: Arthur Schopenhauer (Manuscript remains [1855], II p.241-3), quoted by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: This is a direct reply to the opposite view expressed by Schleiermacher (and quoted by Lewis). I would say that to be a philosopher one must give priority to the philosophy, ahead of any religious beliefs. Thinking must be free. |
| 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. |
| 16674 | The quantity is just the matter, in that it has extended parts and is diffuse [Charleton] |
| Full Idea: The extension or quantity of a thing is merely modus materiae, or (rather) the matter itself composing that thing; insomuch as it consists not in a point, but has parts posited without parts, in respect whereof it is diffuse. | |
| From: Walter Charleton (Physiologia [1654], III.10.1.4), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 14.2 |
| 21924 | As the subject of willing I am wretched, but absorption in knowledge is bliss [Schopenhauer] |
| Full Idea: As the subject of willing I am an exceedingly wretched being, and all our suffering consistd in willing, ...but as soon as I am absorbed in knowledge, I am blissfully happy and nothing can assail me. | |
| From: Arthur Schopenhauer (Manuscript remains [1855], I p.137), quoted by Peter B. Lewis - Schopenhauer 4 | |
| A reaction: So the source of his pessimism is subjection to his own will. However, since becoming absorbed in knowledge is an easy task for a scholar, he has little to grumble about. Nietzsche mocked the great pessimist for playing the flute every day. |
| 21915 | To deduce morality from reason is blasphemy, because it is holy, and far above reason [Schopenhauer] |
| Full Idea: To deduce from reason [Vernunft] the moral element in conduct is blasphemy. In this element there is expressed the better consciousness which lies far above all reason. expresses itself in conduct as holiness, and is the true salvation of the world. | |
| From: Arthur Schopenhauer (Manuscript remains [1855], I p.47), quoted by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: Aimed at Kant. Only Plato could inspire a non-religious person to write about morality is such terms. Maybe also the stoic ideal of beautiful deeds (given the supreme value Schopenhauer placed on the arts). |