12 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. |
| 2614 | Modern phenomenalism holds that objects are logical constructions out of sense-data [Ayer] |
| Full Idea: Nowadays phenomenalism is held to be a theory of perception which says that physical objects are logical constructions out of sense-data. | |
| From: A.J. Ayer (Phenomenalism [1947], §1) |
| 2615 | The concept of sense-data allows us to discuss appearances without worrying about reality [Ayer] |
| Full Idea: The introduction of the term 'sense-datum' is a means of referring to appearances without prejudging the question of what it is, if anything, that they are appearances of. | |
| From: A.J. Ayer (Phenomenalism [1947], §1) |
| 15119 | Aristotelian explanation by essence may need to draw on knowledge of other essences [Aristotle, by Koslicki] |
| Full Idea: From Aristotle's biology we learn that a successful scientific explanation of the necessary (but non-essential) features of one type of phenomenon (e.g. camels) my require appeal to facts about the essences of other types of phenomena (stomachs). | |
| From: report of Aristotle (The History of Animals [c.344 BCE]) by Kathrin Koslicki - Essence, Necessity and Explanation 13.4 |
| 23302 | Plants have far less life than animals, but more life than other corporeal entities [Aristotle] |
| Full Idea: The genus of plants, whilst it is devoid of life compared with an animal, is endowed with life as compared with other corporeal entities. In the sea there are certain objects which one would be at a loss to determine whether they be animal or vegetable. | |
| From: Aristotle (The History of Animals [c.344 BCE], 588b09) | |
| A reaction: It seems that Aristotle takes life to come in degrees, assessed by the amount of physical vitality observed. This seems to make lambs more alive than sheep, which isn't very plausible. This is part of his 'gradualist' view of nature. |
| 23301 | There is a gradual proceeding from the inanimate to animals, with no clear borderlines [Aristotle] |
| Full Idea: Nature proceeds little by little from things lifeless to animal life so that it is impossible to determine the exact line of demarcation, nor on which side an intermediate form should lie. ...In plants there is a continuous ascent towards the animal. | |
| From: Aristotle (The History of Animals [c.344 BCE], 588b04) | |
| A reaction: This in itself should have alerted medieval Christians to the problematic nature of the idea that animal species were divinely created. |