16 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) |
| 8203 | All the arithmetical entities can be reduced to classes of integers, and hence to sets [Quine] |
| Full Idea: The arithmetic of ratios and irrational and imaginary numbers can all be reduced by definition to the theory of classes of positive integers, and this can in turn be reduced to pure set theory. | |
| From: Willard Quine (Vagaries of Definition [1972], p.53) | |
| A reaction: This summarises Quine's ontology of mathematics, which tries to eliminate virtually everything, but has to affirm the existence of sets. Can you count sets and their members, if the sets are used to define the numbers? |
| 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. |
| 18076 | Most theories are continually falsified [Kuhn, by Kitcher] |
| Full Idea: Kuhn contends that almost all theories are falsified at almost all times. | |
| From: report of Thomas S. Kuhn (Structure of Scientific Revolutions (2nd ed) [1962]) by Philip Kitcher - The Nature of Mathematical Knowledge 07.1 | |
| A reaction: This is obviously meant to demolish Karl Popper. |
| 22191 | Kuhn's scientists don't aim to falsifying their paradigm, because that is what they rely on [Kuhn, by Gorham] |
| Full Idea: In Kuhn's view scientists are decidedly not interested in falsifying their paradigm, because without a paradigm there is no systematic inquiry at all. | |
| From: report of Thomas S. Kuhn (Structure of Scientific Revolutions (2nd ed) [1962]) by Geoffrey Gorham - Philosophy of Science 3 | |
| A reaction: This seems to be one of the stronger aspects of Kuhn's account. You'd be leaving the big house, to go out on the road with a tent. |
| 22183 | Switching scientific paradigms is a conversion experience [Kuhn] |
| Full Idea: The transfer of allegiance from paradigm to paradigm is a conversion experience which cannot be forced. | |
| From: Thomas S. Kuhn (Structure of Scientific Revolutions (2nd ed) [1962]), quoted by Samir Okasha - Philosophy of Science: Very Short Intro (2nd ed) 5 | |
| A reaction: This is the controversial part of Kuhn, which says that the most important decisions are not really rational. Anyone who thought the interpretation of a bunch of evidence is logical needed their head examined. But it IS rational. |
| 6162 | Kuhn has a description theory of reference, so the reference of 'electron' changes with the descriptions [Rowlands on Kuhn] |
| Full Idea: Kuhn and Feyerabend adopt a description theory of reference; the term 'electron' refers to whatever satisfies the descriptions associated with electrons, and since these descriptions vary between theories, so too must the reference. | |
| From: comment on Thomas S. Kuhn (Structure of Scientific Revolutions (2nd ed) [1962]) by Mark Rowlands - Externalism Ch.3 | |
| A reaction: This is a key idea in modern philosophy, showing why all of reality and science were at stake when Kripke and others introduced a causal theory of reference. All the current debates about externalism and essentialism grow from this problem. |
| 22184 | Incommensurability assumes concepts get their meaning from within the theory [Kuhn, by Okasha] |
| Full Idea: The doctrine of incommensurability stems from Kuhn's belief that scientific concepts derive their meaning from the theory in which they play a role. | |
| From: report of Thomas S. Kuhn (Structure of Scientific Revolutions (2nd ed) [1962]) by Samir Okasha - Philosophy of Science: Very Short Intro (2nd ed) 5 | |
| A reaction: Quine was the source of this. Kripke's direct reference theory was meant to be the answer. |
| 7619 | Galileo's notions can't be 'incommensurable' if we can fully describe them [Putnam on Kuhn] |
| Full Idea: To tell us that Galileo had 'incommensurable' notions and then go on to describe them at length is totally incoherent. | |
| From: comment on Thomas S. Kuhn (Structure of Scientific Revolutions (2nd ed) [1962]) by Hilary Putnam - Reason, Truth and History Ch.5 | |
| A reaction: How refreshingly sensible. Incommensurability is the sort of nonsense you slide into if you take an instrumental view of science. But scientists are continually aim to pin down what is actually there. Translation between theories is very difficult! |
| 8202 | Meaning is essence divorced from things and wedded to words [Quine] |
| Full Idea: Meaning is essence divorced from the thing and wedded to the word. | |
| From: Willard Quine (Vagaries of Definition [1972], p.51) | |
| A reaction: Quine's strategy is that a demolition of essences will be a definition of meaning. Personally I would like to defend essences, though I admit to finding meaning tricky. That is because essences are external, but meanings are in minds. |
| 8201 | The distinction between meaning and further information is as vague as the essence/accident distinction [Quine] |
| Full Idea: The distinction between what belongs to the meaning of a word and what counts as further information is scarcely clearer than the distinction between the essence of a thing and its accidents. | |
| From: Willard Quine (Vagaries of Definition [1972], p.51) | |
| A reaction: In lots of cases the distinction between essence and accident strikes me as totally clear. Tricky borderline cases don't destroy a distinction. That bachelors are married is clearly not 'further information'. |