19 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. |
| 23634 | Accepting the existence of anything presupposes the notion of existence [Reid] |
| Full Idea: The belief of the existence of anything seems to suppose a notion of existence - a notion too abstract, perhaps, to enter into the mind of an infant. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 05) | |
| A reaction: But even a small infant has to cope with the experience of waking up from a dream. I don't see how existence can be anything other than a primitive concept in any system of ontology. |
| 23635 | Truths are self-evident to sensible persons who understand them clearly without prejudice [Reid] |
| Full Idea: Self-evident propositions are those which appear evident to every man of sound understanding who apprehends the meaning of them distinctly, and attends to them without prejudice. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 10) | |
| A reaction: I suspect that there are some truths which are self-evident to dogs. There are also truths which are self-evident to experts, but not to ordinary persons of good understanding. Self-evidence is somewhat contextual. Self-evidence can be empirical. |
| 7631 | Sensation is not committed to any external object, but perception is [Reid] |
| Full Idea: Sensation, by itself, implies neither the conception nor belief of any external object. ...Perception implies a conviction and belief of something external. ...Things so different in their nature ought to be distinguished. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], II.16), quoted by Barry Maund - Perception | |
| A reaction: Maund sees this as the origin of the two-stage view of perception, followed by Chisholm, Evans, Dretske and Lowe. It implies that 'looks', 'tastes', 'sounds' etc. are ambiguous words, having either phenomenal or realist meanings. I like it. |
| 23637 | Primary qualities are the object of mathematics [Reid] |
| Full Idea: The primary qualities are the object of the mathematical sciences. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 17) | |
| A reaction: He spells out this crucial point, which is not so obvious in Locke. The sciences totally rely on the primary qualities, so it is ridiculous to reject the distinction (which Reid accepts). |
| 23638 | Secondary qualities conjure up, and are confused with, the sensations which produce them [Reid] |
| Full Idea: The thought of a secondary quality always carries us back to the sensation which it produces.We give the same name to both, and are apt to confound them. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 17) | |
| A reaction: 'Redness', for example. Reid puts the point very nicely. Secondary qualities are not entirely mental; they pick out features of the world, but are much harder to understand than the primary qualities. The qualia question lurks. |
| 23639 | It is unclear whether a toothache is in the mind or in the tooth, but the word has a single meaning [Reid] |
| Full Idea: If it be made a question whether the toothache be in the mind that feels it, or in tooth that is affected, much might be said on both sides, while it is not observed that the word has two meanings. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 18) | |
| A reaction: I'm glad Reid was struck by the weird phenomenon of the brain apparently 'projecting' a pain into a tooth. Presumably before the brain's role was known, people were unaware of this puzzle. There certainly are not two distinct experiences. |
| 6492 | Reid is seen as the main direct realist of the eighteenth century [Reid, by Robinson,H] |
| Full Idea: Reid is often represented by modern opponents of the empiricists as the outstanding protagonist of direct or naïve realism and common sense in the eighteenth century. | |
| From: report of Thomas Reid (Essays on Intellectual Powers 2: Senses [1785]) by Howard Robinson - Perception 1.6 | |
| A reaction: Robinson does not deny that this is Reid's view. Keith Lehrer is a great fan of Reid. Personally I think direct realism is quite clearly false, so I find myself losing interest in Reid's so-called 'common sense'. |
| 23641 | People dislike believing without evidence, and try to avoid it [Reid] |
| Full Idea: To believe without evidence is a weakness which every man is concerned to avoid, and which every man wishes to avoid. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 20) | |
| A reaction: It seems to be very common, though, for people to believe things on incredibly flimsy evidence, if they find the belief appealing. This is close to Clifford's Principle, but not quite as dogmatic. |
| 23642 | If non-rational evidence reaches us, it is reason which then makes use of it [Reid] |
| Full Idea: If Nature gives us information of things that concern us, by other means that by reasoning, reason itself will direct us to receive that information with thankfulness, and to make the best use of it. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 20) | |
| A reaction: This is more of a claim than an argument, but it is hard to see how anything could even be seen as evidence if some sort of rational judgement has not been made. The clever detective sees which facts are evidence. |
| 23640 | Only mature minds can distinguish the qualities of a body [Reid] |
| Full Idea: I think it requires some ripeness of understanding to distinguish the qualities of a body from the body; perhaps this distinction is not made by brutes, or by infants. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 19) | |
| A reaction: I'm glad the brutes get a mention in his assessment of these questions. I take such thinking to arise from what can be labelled the faculty of abstraction, which presumably only appears in a mature brain. It is second-level thinking. |
| 18552 | Forget about beauty; just concentrate on the virtues of delicacy and discernment admired in critics [Hume, by Scruton] |
| Full Idea: Hume suggest we get away from the fruitless discussion of beauty, and simply concentrate on the qualities we admire, and ought to admire, in a critic - qualities such as delicacy and discernment. | |
| From: report of David Hume (Of the standard of taste [1757]) by Roger Scruton - Beauty: a very short introduction 6 | |
| A reaction: We might wonder how you can admire 'discernment' without some view of the thing being discern, which is in danger of being beauty. How do you judge delicacy and discernment without judging the objects of the critic's taste? Mere authority? |
| 6608 | Strong sense, delicate sentiment, practice, comparisons, and lack of prejudice, are all needed for good taste [Hume] |
| Full Idea: Strong sense, united to delicate sentiment, improved by practice, perfected by comparison, and cleared of all prejudice, can alone entitle critics to the valuable character of having 'taste'. | |
| From: David Hume (Of the standard of taste [1757]), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.6 | |
| A reaction: I agree entirely with this, but then I am a very politically incorrect elitist when it comes to taste. It just seems screamingly obvious that professional wine-tasters have a better appreciation of wine than me, and so on for the rest of the arts. |