14 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. |
| 14637 | Only individuals have essences, so numbers (as a higher type based on classes) lack them [McMichael] |
| Full Idea: Essentialism is not verified by the observation that numbers have interesting essential properties, since they are properties of classes and so are entities of a higher logical type than individuals. | |
| From: Alan McMichael (The Epistemology of Essentialist Claims [1986], Intro) | |
| A reaction: This relies on a particular view of number (which might be challenged), but is interesting when it comes to abstract entities having essences. Only ur-elements in set theory could have essences, it seems. Why? Rising in type destroys essence? |
| 14636 | Essences are the interesting necessary properties resulting from a thing's own peculiar nature [McMichael] |
| Full Idea: Essentialism says some individuals have certain 'interesting' necessary properties. If it exists, it has that property. The properties are 'interesting' as had in virtue of their own peculiar natures, rather than as general necessary truths. | |
| From: Alan McMichael (The Epistemology of Essentialist Claims [1986], Intro) | |
| A reaction: [compressed] This is a modern commentator caught between two views. The idea that essence is the non-trivial-necessary properties is standard, but adding their 'peculiar natures' connects him to Aristotle, and Kit Fine's later papers. Good! |
| 14640 | Maybe essential properties have to be intrinsic, as well as necessary? [McMichael] |
| Full Idea: There is a tendency to think of essential properties as having some characteristic in addition to their necessity, such as intrinsicality. | |
| From: Alan McMichael (The Epistemology of Essentialist Claims [1986], VIII) | |
| A reaction: Personally I am inclined to take this view of all properties, and not just the 'essential' ones. General necessities, relations, categorisations, disjunctions etc. should not be called 'properties', even if they are 'predicates'. Huge confusion results. |
| 14638 | Essentialism is false, because it implies the existence of necessary singular propositions [McMichael] |
| Full Idea: Essentialism entails the existence of necessary singular propositions that are not instances of necessary generalizations. Therefore, since there are no such propositions, essentialism is false. | |
| From: Alan McMichael (The Epistemology of Essentialist Claims [1986], I) | |
| A reaction: This summarises the attack which McMichael wishes to deal with. I am wickedly tempted to say that essences actually have a contingent existence (or a merely hypothetical dependent necessity), and this objection might be grist for my mill. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 14639 | Individuals enter into laws only through their general qualities and relations [McMichael] |
| Full Idea: Individuals appear to enter into laws only through their general qualities and relations. | |
| From: Alan McMichael (The Epistemology of Essentialist Claims [1986], VIII) | |
| A reaction: This is a very significant chicken-or-egg issue. The remark seems to offer the vision of pre-existing general laws, which individuals then join (like joining a club). But surely the laws are derived from the individuals? Where else could they come from? |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |