17 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. |
| 22167 | Our images of bodies are not produced by the bodies, but by our own minds [Augustine, by Aquinas] |
| Full Idea: Augustine says bodies don't form images in our spirit; our spirit does that itself with amazing quickness. ...So the appearances under which mind knows things aren't drawn from the things themselves. | |
| From: report of Augustine (works [c.415]) by Thomas Aquinas - Quodlibeta 8.2.1 | |
| A reaction: This is Augustine's theory of 'illumination' - that God creates experience within us. His theory was soon discarded by the early scholastics. |
| 22117 | Our minds grasp reality by direct illumination (rather than abstraction from experience) [Augustine, by Matthews] |
| Full Idea: Instead of supposing that what we know can be abstracted from sensible particulars that instantiate such knowledge, Augustine insists that our mind is so constituted as to see 'intelligible realities' directly by inner illumination. | |
| From: report of Augustine (works [c.415]) by Gareth B. Matthews - Augustine p.74 | |
| A reaction: His 'theory of illumination'. This seems to be a sort of super-rationalism. This doesn't make clear the role of sensations. Surely he doesn't thing that we just bypass them? |
| 22118 | Augustine created the modern concept of the will [Augustine, by Matthews] |
| Full Idea: The modern concept of the will is often said to originate with Augustine. | |
| From: report of Augustine (works [c.415]) by Gareth B. Matthews - Augustine p.74 | |
| A reaction: I'm beginning to think that this is the source of the trouble. How can a thing be intrinsically free? Surely freedom is always a contextual concept? |
| 4348 | Love, and do what you will [Augustine] |
| Full Idea: Love, and do what you will. | |
| From: Augustine (works [c.415]) | |
| A reaction: This sounds libertarian, but Augustine had a stern concept of what love required. It nicely captures one of the essential ideas of virtue ethics. |
| 7821 | Pagans produced three hundred definitions of the highest good [Augustine, by Grayling] |
| Full Idea: Augustine claimed that the pagan schools between them had produced nearly three hundred different definitions of the highest good. | |
| From: report of Augustine (works [c.415]) by A.C. Grayling - What is Good? Ch.5 | |
| A reaction: I would expect the right definition to be in there somewhere, but no doubt Augustine's definition made it 301. Perhaps the biggest problem of human life is that (as with the Kennedy assassination) proliferating stories obscure the true story. |
| 22119 | Augustine said (unusually) that 'ought' does not imply 'can' [Augustine, by Matthews] |
| Full Idea: Augustine insisted that 'ought' does not, in any straightforward way, imply 'can' - which distinguishes him from most modern ethicists. | |
| From: report of Augustine (works [c.415]) by Gareth B. Matthews - Augustine p.74 | |
| A reaction: Not unreasonable. I ought to help my ailing friend who lives abroad, but I haven't the time or money to do it. We can experience impossibilities as duties. Impossibilities are just excuses. Augustine is opposing the Pelagian heresy. |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| Full Idea: Philosophers are the forefathers of heretics. | |
| From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
| 22116 | Augustine identified Donatism, Pelagianism and Manicheism as the main heresies [Augustine, by Matthews] |
| Full Idea: Augustine did the most to define Christian heresy. The three most prominent were Donatism, Pelagianism (that humans are perfectible), and Manicheism (that good and evil are equally basic metaphysical realities). | |
| From: report of Augustine (works [c.415]) by Gareth B. Matthews - Augustine p.73 | |
| A reaction: Manicheans had presumably been studying Empedocles. (I suppose it's too late to identify Christianity as a heresy?). |
| 6610 | I believe because it is absurd [Tertullian] |
| Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
| From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
| A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |
| 19338 | Augustine said evil does not really exist, and evil is a limitation in goodness [Augustine, by Perkins] |
| Full Idea: Augustine solution to the problem of evil was to say that, strictly speaking, evil does not exist. Human beings are not part evil and part good, but rather just a limited amount of goodness. | |
| From: report of Augustine (works [c.415]) by Franklin Perkins - Leibniz: Guide for the Perplexed 2.III | |
| A reaction: Augustine was rebelling against Manicheanism, which he espoused when young, which proposed a good and an evil force. An apathetic slob seems devoid of goodness, but is not evil. It takes extra effort to perform active evil. |