19 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. |
| 16435 | Plantinga proposes necessary existent essences as surrogates for the nonexistent things [Plantinga, by Stalnaker] |
| Full Idea: Plantinga proposes surrogates for nonexistent things - individual essences that are themselves necessary existents and that correspond one-to-one with all the 'things' that might exist. | |
| From: report of Alvin Plantinga (World and Essence [1970]) by Robert C. Stalnaker - Mere Possibilities 1 | |
| A reaction: There are an awful lot of competing concepts of essence flying around these days. This one seems to require some abstract 'third realm' (or worse) in which these essences can exist, awaiting the arrival of thinkers. Not for me. |
| 14655 | The 'identity criteria' of a name are a group of essential and established facts [Plantinga] |
| Full Idea: What we might call 'identity criteria' associated with a name such as 'Aristotle' are what the users of the name regard as essential and established facts about him. | |
| From: Alvin Plantinga (World and Essence [1970], I) | |
| A reaction: The problem here is that identifying something is superficial, whereas essences run deep. Plantinga is, in fact, talking about Lockean 'nominal essence' (and seems unaware of the fact, and never mentions the Lockean real/nominal distinction). |
| 14658 | 'Being Socrates' and 'being identical with Socrates' characterise Socrates, so they are among his properties [Plantinga] |
| Full Idea: Surely it is true of Socrates that he is Socrates and he is identical with Socrates. If these are true of him, then 'being Socrates' and 'being identical with Socrates' characterize him; they are among his properties or attributes. | |
| From: Alvin Plantinga (World and Essence [1970], II) | |
| A reaction: As far as I can see (if you insist on accepting self-identity as meaningful) the most you get here is that these are predicates that can attach to Socrates. If you identify predicates with properties you are in deep metaphysical trouble. |
| 14656 | Does Socrates have essential properties, plus a unique essence (or 'haecceity') which entails them? [Plantinga] |
| Full Idea: Does Socrates have, in addition to his essential properties, an 'essence' or 'haecceity' - a property essential to him that entails each of his essential properties and that nothing distinct from him has in the world? | |
| From: Alvin Plantinga (World and Essence [1970], II) | |
| A reaction: Plantinga says yes, and offers 'Socrateity' (borrowed from Boethius) as his candidate. This is a very odd use of the word 'essence'. I take an essence to be a complex set of fundamental properties. I am also puzzled by his use of the word 'entails'. |
| 14653 | X is essentially P if it is P in every world, or in every X-world, or in the actual world (and not ¬P elsewhere) [Plantinga] |
| Full Idea: Socrates has P essentially if he has P in every world, or has it in every world in which he exists, or - most plausible of all - has P in the actual world and has its complement [non-P] in no world. | |
| From: Alvin Plantinga (World and Essence [1970], Intro) | |
| A reaction: These strike me as mere necessary properties, which are not the same thing at all. Essences give rise to the other properties, but Plantinga offers nothing to do the job (and especially not 'Socrateity'!). Essences must explain, say I! |
| 14660 | If a property is ever essential, can it only ever be an essential property? [Plantinga] |
| Full Idea: Is it the case that any property had essentially by anything is had essentially by everything that has it? | |
| From: Alvin Plantinga (World and Essence [1970], III) | |
| A reaction: Plantinga says it is not true, but the only example he can give is Socrates having the property of 'being Socrates or Greek'. I take it to be universally false. There are not two types of property here. Properties sometimes play an essential role. |
| 14661 | Essences are instantiated, and are what entails a thing's properties and lack of properties [Plantinga] |
| Full Idea: E is an essence if and only if (a) 'has E essentially' is instantiated in some world or other, and (b) for any world W and property P, E entails 'has P in W' or 'does not have P in W'. | |
| From: Alvin Plantinga (World and Essence [1970], IV) | |
| A reaction: 'Entail' strikes me as a very odd word when you are talking about the structure of the physical world (or are we??). Why would a unique self-identity (his candidate for essence) do the necessary entailing? |
| 14654 | Properties are 'trivially essential' if they are instantiated by every object in every possible world [Plantinga] |
| Full Idea: Let us call properties that enjoy the distinction of being instantiated by every object in every possible world 'trivially essential properties'. | |
| From: Alvin Plantinga (World and Essence [1970], I) | |
| A reaction: These would appear to be trivially 'necessary' rather than 'essential'. This continual need for the qualifier 'trivial' shows that they are not talking about proper essences. |
| 14657 | Does 'being identical with Socrates' name a property? I can think of no objections to it [Plantinga] |
| Full Idea: Is there any reason to suppose that 'being identical with Socrates' names a property? Well, is there any reason to suppose that it does not? I cannot think of any, nor have I heard any that are at all impressive. | |
| From: Alvin Plantinga (World and Essence [1970], II) | |
| A reaction: Is there any reason to think that a planet somewhere is entirely under the control of white mice? Extraordinary. No wonder Plantinga believes in God and the Ontological Argument, as well as the existence of 'Socrateity' etc. |
| 14652 | 'De re' modality is as clear as 'de dicto' modality, because they are logically equivalent [Plantinga] |
| Full Idea: The idea of modality 'de re' is no more (although no less) obscure that the idea of modality 'de dicto'; for I think we can see that any statement of the former type is logically equivalent to some statement of the latter. | |
| From: Alvin Plantinga (World and Essence [1970], Intro) | |
| A reaction: If two things are logically equivalent, that doesn't ensure that they are equally clear! Personally I am on the side of de re modality. |
| 14659 | We can imagine being beetles or alligators, so it is possible we might have such bodies [Plantinga] |
| Full Idea: We easily understand Kafka's story about the man who wakes up to discover that he now has the body of a beetle; and in fact the state of affairs depicted is entirely possible. I can imagine being an alligator, so Socrates could have had an alligator body. | |
| From: Alvin Plantinga (World and Essence [1970], III) | |
| A reaction: This really is going the whole hog with accepting whatever is conceivable as being possible. I take this to be shocking nonsense, and it greatly reduces Plantinga in my esteem, despite his displays of intelligence and erudition. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |