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. |
| 14329 | Some dispositional properties (such as mental ones) may have no categorical base [Price,HH] |
| Full Idea: There is no a priori necessity for supposing that all disposition properties must have a 'categorical base'. In particular, there may be some mental dispositions which are ultimate. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.XI) | |
| A reaction: I take the notion that mental dispositions could be ultimate as rather old-fashioned, but I agree with the notion that dispositions might be more fundamental that categorical (actual) properties. Personally I like 'powers'. |
| 9032 | Before we can abstract from an instance of violet, we must first recognise it [Price,HH] |
| Full Idea: Abstraction is preceded by an earlier stage, in which we learn to recognize instances; before I can conceive of the colour violet in abstracto, I must learn to recognize instances of this colour when I see them. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.II) | |
| A reaction: The problem here might be one of circularity. If you are actually going to identify something as violet, you seem to need the abstract concept of 'violet' in advance. See Idea 9034 for Price's attempt to deal with the problem. |
| 9035 | If judgement of a characteristic is possible, that part of abstraction must be complete [Price,HH] |
| Full Idea: If we are to 'judge' - rightly or not - that this object has a specific characteristic, it would seem that so far as the characteristic is concerned the process of abstraction must already be completed. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.III) | |
| A reaction: Personally I think Price is right, despite the vicious attack from Geach that looms. We all know the experiences of familiarity, recognition, and identification that go on when see a person or picture. 'What animal is that, in the distance?' |
| 9034 | There may be degrees of abstraction which allow recognition by signs, without full concepts [Price,HH] |
| Full Idea: If abstraction is a matter of degree, and the first faint beginnings of it are already present as soon as anything has begun to feel familiar to us, then recognition by means of signs can occur long before the process of abstraction has been completed. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.III) | |
| A reaction: I like this, even though it is unscientific introspective psychology, for which no proper evidence can be adduced - because it is right. Neuroscience confirms that hardly any mental life has an all-or-nothing form. |
| 9036 | There is pre-verbal sign-based abstraction, as when ice actually looks cold [Price,HH] |
| Full Idea: We must still insist that some degree of abstraction, and even a very considerable degree of it, is present in sign-cognition, pre-verbal as it is. ...To us, who are familiar with northern winters, the ice actually looks cold. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.IV) | |
| A reaction: Price may be in the weak position of doing armchair psychology, but something like his proposal strikes me as correct. I'm much happier with accounts of thought that talk of 'degrees' of an activity, than with all-or-nothing cut-and-dried pictures. |
| 9037 | Intelligent behaviour, even in animals, has something abstract about it [Price,HH] |
| Full Idea: Though it may sound odd to say so, intelligent behaviour has something abstract about it no less than intelligent cognition; and indeed at the animal level it is unrealistic to separate the two. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.IV) | |
| A reaction: This elusive thought strikes me as being a key one for understanding human existence. To think is to abstract. Brains are abstraction machines. Resemblance and recognition require abstaction. |
| 9033 | Recognition must precede the acquisition of basic concepts, so it is the fundamental intellectual process [Price,HH] |
| Full Idea: Recognition is the first stage towards the acquisition of a primary or basic concept. It is, therefore, the most fundamental of all intellectual processes. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.II) | |
| A reaction: An interesting question is whether it is an 'intellectual' process. Animals evidently recognise things, though it is a moot point whether slugs 'recognise' tasty leaves. |
| 9030 | Abstractions can be interpreted dispositionally, as the ability to recognise or imagine an item [Price,HH] |
| Full Idea: An abstract idea may have a dispositional as well as an occurrent interpretation. ..A man who possesses the concept Dog, when he is actually perceiving a dog can recognize that it is one, and can think about dogs when he is not perceiving any dog. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.IX) | |
| A reaction: Ryle had just popularised the 'dispositional' account of mental events. Price is obviously right. The man may also be able to use the word 'dog' in sentences, but presumably dogs recognise dogs, and probably dream about dogs too. |
| 9029 | If ideas have to be images, then abstract ideas become a paradoxical problem [Price,HH] |
| Full Idea: There used to be a 'problem of Abstract Ideas' because it was assumed that an idea ought, somehow, to be a mental image; if some of our ideas appeared not to be images, this was a paradox and some solution must be found. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.VIII) | |
| A reaction: Berkeley in particular seems to be struck by the fact that we are incapable of thinking of a general triangle, simply because there is no image related to it. Most conversations go too fast for images to form even of very visual things. |
| 9031 | The basic concepts of conceptual cognition are acquired by direct abstraction from instances [Price,HH] |
| Full Idea: Basic concepts are acquired by direct abstraction from instances; unless there were some concepts acquired in this way by direct abstraction, there would be no conceptual cognition at all. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.II) | |
| A reaction: This seems to me to be correct. A key point is that not only will I acquire the concept of 'dog' in this direct way, from instances, but also the concept of 'my dog Spot' - that is I can acquire the abstract concept of an instance from an instance. |
| 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 |
| 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. |