12 ideas
| 22115 | Wise people should contemplate and discuss the truth, and fight against falsehood [Aquinas] |
| Full Idea: The role of the wise person is to meditate on the truth, especially the truth regarding the first principle, and to discuss it with others, but also to fight against the falsity that is its contrary. | |
| From: Thomas Aquinas (Summa Contra Gentiles [1268], I.1.6), quoted by Kretzmann/Stump - Aquinas, Thomas 14 | |
| A reaction: So nice to hear someone (from no matter how long ago) saying that wisdom is concerned with truth. If you lose your grip on truth (which many thinkers seem to have done) you must also abandon wisdom. Then fools rule. |
| 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. |
| 3900 | Maybe experience is not essential to perception, but only to the causing of beliefs [Armstrong, by Scruton] |
| Full Idea: Armstrong has argued that experience, as normally understood, is not necessary to perception. To perceive is to acquire beliefs, through a causal process. | |
| From: report of David M. Armstrong (Belief Truth and Knowledge [1973]) by Roger Scruton - Modern Philosophy:introduction and survey 23.4 |
| 4253 | Externalism says knowledge involves a natural relation between the belief state and what makes it true [Armstrong] |
| Full Idea: Externalist accounts of non-inferential knowledge say what makes a true non-inferential belief a case of knowledge is some natural relation which holds between the belief state and the situation which makes the belief true. | |
| From: David M. Armstrong (Belief Truth and Knowledge [1973], 11.III.6) | |
| A reaction: Armstrong's concept is presumably a response to Quine's desire to 'naturalise epistemology'. Bad move, I suspect. It probably reduces knowledge to mere true belief, and hence a redundant concept. |
| 20700 | Without God's influence every operation would stop, so God causes everything [Aquinas] |
| Full Idea: If God's divine influence stopped, every operation would stop. Every operation, therefore, of everything is traced back to him as cause. | |
| From: Thomas Aquinas (Summa Contra Gentiles [1268], III.67), quoted by Brian Davies - Introduction to the Philosophy of Religion 3 'Freedom' | |
| A reaction: If the systematic interraction of mind and body counts as an 'operation', then this seems to imply Occasionalism. |
| 15202 | Eternity coexists with passing time, as the centre of a circle coexists with its circumference [Aquinas] |
| Full Idea: The centre of a circle is directly opposite any designated point on the circumference. In this way, whatever is in any part of time coexists with what is eternal as being present to it even though past or future with respect to another part of time. | |
| From: Thomas Aquinas (Summa Contra Gentiles [1268], I.66), quoted by Robin Le Poidevin - Past, Present and Future of Debate about Tense 2 c | |
| A reaction: A nice example of a really cool analogy which almost gets you to accept something which is actually completely incomprehensible. |