18 ideas
| 15957 | Essential definitions show the differences that discriminate things, and make them what they are [Boyle] |
| Full Idea: Essential definitions are such as are taken from the essential differences of things, which constitute them in such a sort of natural bodies, and discriminate them from all those of any other sort. | |
| From: Robert Boyle (The Origin of Forms and Qualities [1666], p.41?), quoted by Peter Alexander - Ideas, Qualities and Corpuscles | |
| A reaction: I don't think this goes as far as the aim Aristotle had in definitions, which was more than merely to 'discriminate' each thing. A full definition explains the thing as well. |
| 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. |
| 8732 | It is spooky the way mathematics anticipates physics [Weinberg] |
| Full Idea: It is positively spooky how the physicist finds the mathematician has been there before him or her. | |
| From: Steven Weinberg (Lecture on Applicability of Mathematics [1986], p.725), quoted by Stewart Shapiro - Thinking About Mathematics 2.3 | |
| A reaction: This suggests that mathematics might be the study of possibilities or hypotheticals, like mental rehearsals for physics. See Hellman's modal structuralism. Maybe mathematicians are reading the mind of God, but I doubt that. |
| 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. |
| 15965 | Boyle attacked a contemporary belief that powers were occult things [Boyle, by Alexander,P] |
| Full Idea: Boyle attacks an idea of powers, held by some modern schoolmen and chemists, that makes powers occult. | |
| From: report of Robert Boyle (The Origin of Forms and Qualities [1666]) by Peter Alexander - Ideas, Qualities and Corpuscles 03.3 | |
| A reaction: [This involves Boyle's famous example of a key having the power to turn a lock] On p.86 Alexander says the 'occult' belief is in affinities, antipathies, attractions and repulsions. How did Boyle explain magnetism? |
| 16735 | In the 17th century, 'disposition' usually just means the spatial arrangement of parts [Boyle, by Pasnau] |
| Full Idea: In Locke and Boyle, 'disposition' and its various cognates are standardly used to refer to the corpuscular structure of a body - the spatial arrangement of its parts - without reflecting any commitment to a dispositional property. | |
| From: report of Robert Boyle (The Origin of Forms and Qualities [1666]) by Robert Pasnau - Metaphysical Themes 1274-1671 23.2 | |
| A reaction: Here as a warning against enthusiasts for dispositional properties misreadigmg 17th century texts to their supposed advantage. Pasnau says none of them believe in dispositional properties or real powers. |
| 16034 | Form is not a separate substance, but just the manner, modification or 'stamp' of matter [Boyle] |
| Full Idea: I understand the word 'form' to mean, not a real substance distinct from matter, but only the matter itself of a natural body, with its peculiar manner of existence [corpuscular structure], which may be called its 'essential modification' or 'stamp'. | |
| From: Robert Boyle (The Origin of Forms and Qualities [1666], p.324), quoted by Jan-Erik Jones - Real Essence §3 | |
| A reaction: I don't think Aristotle ever thought that a form was separate from its matter, let alone qualifying as a substance. On the whole, Boyle attacks scholastic philosophy, rather than Aristotle. |
| 15953 | To cite a substantial form tells us what produced the effect, but not how it did it [Boyle] |
| Full Idea: If it be demanded why rhubarb purges choler, snow dazzles the eyes rather than grass etc., that these effects are performed by substantial forms of the respective bodies is at best but to tell me what is the agent, not how the effect is wrought. | |
| From: Robert Boyle (The Origin of Forms and Qualities [1666], p.47?), quoted by Peter Alexander - Ideas, Qualities and Corpuscles 01.2 | |
| A reaction: This is the problem of the 'virtus dormitiva' of opium (which at least tells you it was the opium what done it). I take Aristotle to have aspired to a lot more than this. He wanted a full definition, which would contain lots of information about the form. |
| 15962 | Boyle's term 'texture' is not something you feel, but is unobservable structures of particles [Boyle, by Alexander,P] |
| Full Idea: Perhaps Boyle's most important technical terms is 'texture'. ...It must not be confused with the way we feel the texture of a surface like sandpaper or velvet; it is rather a structure of unobservable particles and so it is not directly observable. | |
| From: report of Robert Boyle (The Origin of Forms and Qualities [1666]) by Peter Alexander - Ideas, Qualities and Corpuscles 03.2 | |
| A reaction: This is the basis for Alexander's reassessment of what Boyle and Locke meant by a 'secondary quality', which, he says, is a physical feature of objects, not a mental experience. |
| 15964 | Boyle's secondary qualities are not illusory, or 'in the mind' [Boyle, by Alexander,P] |
| Full Idea: There is no suggestion in Boyle that secondary qualities are, unlike primary qualities, somehow illusory, subjective or 'in the mind'. | |
| From: report of Robert Boyle (The Origin of Forms and Qualities [1666]) by Peter Alexander - Ideas, Qualities and Corpuscles 03.3 | |
| A reaction: [Alexander goes on to say that his also applied to Locke] |
| 15960 | Explanation is deducing a phenomenon from some nature better known to us [Boyle] |
| Full Idea: Explicating a phenomenon is to deduce it from something else in nature more known to us than the thing to be explained by it. | |
| From: Robert Boyle (The Origin of Forms and Qualities [1666], p.46?), quoted by Peter Alexander - Ideas, Qualities and Corpuscles | |
| A reaction: Interesting that the word 'deduce' is here, beloved of the 'covering law' view. But this may be deduced from the behaviour of other substances, as the iron filing behaviour may be explained by the magnet itself (or perhaps 'laws' of magnetism). |
| 15952 | The corpuscles just have shape, size and motion, which explains things without 'sympathies' or 'forces' [Boyle, by Alexander,P] |
| Full Idea: In Boyle's corpuscular philosophy, all material substances are composed of minute particles or corpuscles, with ordinary properties such as shape, size and motion. There was no need for occult relations between them, such as sympathies, or even forces. | |
| From: report of Robert Boyle (The Origin of Forms and Qualities [1666]) by Peter Alexander - Ideas, Qualities and Corpuscles 01.1 |
| 15972 | The corpuscular theory allows motion, but does not include forces between the particles [Boyle, by Alexander,P] |
| Full Idea: Though there is motion, the corpuscles will not be dynamic because the idea of forces between the particles or groups of them does not figure in the theory. | |
| From: report of Robert Boyle (The Origin of Forms and Qualities [1666]) by Peter Alexander - Ideas, Qualities and Corpuscles 5.2 | |
| A reaction: This is the view of Locke, as well as of Boyle. I quote this because I take to it be a particular target of Leibniz's disagreement. |