13 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. |
| 16659 | Relations do not add anything to reality, though they are real aspects of the world [Olivi] |
| Full Idea: It does not seem that a relation adds anything real to that on which it is founded, but only makes for another real aspect belonging to the same thing. It is real since an aspect exists in re, not solely in the intellect, but it is not another thing. | |
| From: Peter John Olivi (Summa quaestionum super Sententias [1290], II.54), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.4 |
| 16673 | Quantity just adds union and location to the extension of parts [Olivi] |
| Full Idea: Quantity or extension adds absolutely nothing really distinct to the quantified matter or to the extended and quantified form, except perhaps the union and location and position of those parts. | |
| From: Peter John Olivi (Summa quaestionum super Sententias [1290], II:58,II:440), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 14.1 | |
| A reaction: Other views seem to say that the Quantity provides the extension, but he seems to take that as given. |
| 7522 | A full neural account of qualia will give new epistemic access to them, beyond private experience [Churchlands] |
| Full Idea: When the hidden neurophysiological structure of qualia (if there is any) gets revealed by unfolding research, then we will automatically gain a new epistemic access to qualia, beyond each person's native and exclusive capacity for internal discrimination. | |
| From: Churchland / Churchland (Recent Work on Consciousness [1997]) | |
| A reaction: Carefully phrased and hard to deny, but something is impenetrable. What experience does an insect have when it encounters ultra-violet light? Nothing remotely interesting about their qualia is likely to emerge from the study of insect brains. |
| 7521 | It is question-begging to assume that qualia are totally simple, hence irreducible [Churchlands] |
| Full Idea: One of the crucial premises of the antireductionists - concerning the intrinsic, nonrelational, metaphysical simplicity of our sensory qualia - is a question-begging and unsupported assumption. | |
| From: Churchland / Churchland (Recent Work on Consciousness [1997]) | |
| A reaction: This is a key point for reductionists, with emphasis on the sheer numbers of connections involved in a simple quale (I estimate a billion involved in one small patch of red). |
| 7523 | The qualia Hard Problem is easy, in comparison with the co-ordination of mental states [Churchlands] |
| Full Idea: The so-called Hard Problem (of qualia) appears to be one of the easiest, in comparison with the problems of short-term memory, fluid and directable attention, the awake state vs sleep, and the unity of consciousness. | |
| From: Churchland / Churchland (Recent Work on Consciousness [1997]) | |
| A reaction: Most of their version of the Hard Problems centre on personal identity, and the centralised co-ordination of mental events. I am inclined to agree with them. Worriers about qualia should think more about the complexity of systems of neurons. |
| 16663 | Things are limited by the species to certain modes of being [Olivi] |
| Full Idea: A subject is limited by its species to certain modes of being. | |
| From: Peter John Olivi (Summa quaestionum super Sententias [1290], I:586-7), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 13.2 | |
| A reaction: I think this is so very the wrong way round. Species characteristics are generalisations about similar individual creatures. The 'species' doesn't do anything at all. It is a classification. See ring species, for example. |