30 ideas
| 10702 | Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter] |
| Full Idea: Set theory has three roles: as a means of taming the infinite, as a supplier of the subject-matter of mathematics, and as a source of its modes of reasoning. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], Intro 1) | |
| A reaction: These all seem to be connected with mathematics, but there is also ontological interest in set theory. Potter emphasises that his second role does not entail a commitment to sets 'being' numbers. |
| 10713 | Usually the only reason given for accepting the empty set is convenience [Potter] |
| Full Idea: It is rare to find any direct reason given for believing that the empty set exists, except for variants of Dedekind's argument from convenience. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 04.3) |
| 13044 | Infinity: There is at least one limit level [Potter] |
| Full Idea: Axiom of Infinity: There is at least one limit level. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 04.9) | |
| A reaction: A 'limit ordinal' is one which has successors, but no predecessors. The axiom just says there is at least one infinity. |
| 10708 | Nowadays we derive our conception of collections from the dependence between them [Potter] |
| Full Idea: It is only quite recently that the idea has emerged of deriving our conception of collections from a relation of dependence between them. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.2) | |
| A reaction: This is the 'iterative' view of sets, which he traces back to Gödel's 'What is Cantor's Continuum Problem?' |
| 13546 | The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter] |
| Full Idea: We group under the heading 'limitation of size' those principles which classify properties as collectivizing or not according to how many objects there are with the property. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 13.5) | |
| A reaction: The idea was floated by Cantor, toyed with by Russell (1906), and advocated by von Neumann. The thought is simply that paradoxes start to appear when sets become enormous. |
| 10707 | Mereology elides the distinction between the cards in a pack and the suits [Potter] |
| Full Idea: Mereology tends to elide the distinction between the cards in a pack and the suits. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1) | |
| A reaction: The example is a favourite of Frege's. Potter is giving a reason why mathematicians opted for set theory. I'm not clear, though, why a pack cannot have either 4 parts or 52 parts. Parts can 'fall under a concept' (such as 'legs'). I'm puzzled. |
| 10704 | We can formalize second-order formation rules, but not inference rules [Potter] |
| Full Idea: In second-order logic only the formation rules are completely formalizable, not the inference rules. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 01.2) | |
| A reaction: He cites Gödel's First Incompleteness theorem for this. |
| 10703 | Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter] |
| Full Idea: A 'supposition' axiomatic theory is as concerned with truth as a 'realist' one (with undefined terms), but the truths are conditional. Satisfying the axioms is satisfying the theorem. This is if-thenism, or implicationism, or eliminative structuralism. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 01.1) | |
| A reaction: Aha! I had failed to make the connection between if-thenism and eliminative structuralism (of which I am rather fond). I think I am an if-thenist (not about all truth, but about provable truth). |
| 10712 | If set theory didn't found mathematics, it is still needed to count infinite sets [Potter] |
| Full Idea: Even if set theory's role as a foundation for mathematics turned out to be wholly illusory, it would earn its keep through the calculus it provides for counting infinite sets. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.8) |
| 17882 | It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter] |
| Full Idea: It is a remarkable fact that all the arithmetical properties of the natural numbers can be derived from such a small number of assumptions (as the Peano Axioms). | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 05.2) | |
| A reaction: If one were to defend essentialism about arithmetic, this would be grist to their mill. I'm just saying. |
| 13043 | A relation is a set consisting entirely of ordered pairs [Potter] |
| Full Idea: A set is called a 'relation' if every element of it is an ordered pair. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 04.7) | |
| A reaction: This is the modern extensional view of relations. For 'to the left of', you just list all the things that are to the left, with the things they are to the left of. But just listing the ordered pairs won't necessarily reveal how they are related. |
| 13042 | If dependence is well-founded, with no infinite backward chains, this implies substances [Potter] |
| Full Idea: The argument that the relation of dependence is well-founded ...is a version of the classical arguments for substance. ..Any conceptual scheme which genuinely represents a world cannot contain infinite backward chains of meaning. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3) | |
| A reaction: Thus the iterative conception of set may imply a notion of substance, and Barwise's radical attempt to ditch the Axiom of Foundation (Idea 13039) was a radical attempt to get rid of 'substances'. Potter cites Wittgenstein as a fan of substances here. |
| 13041 | Collections have fixed members, but fusions can be carved in innumerable ways [Potter] |
| Full Idea: A collection has a determinate number of members, whereas a fusion may be carved up into parts in various equally valid (although perhaps not equally interesting) ways. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1) | |
| A reaction: This seems to sum up both the attraction and the weakness of mereology. If you doubt the natural identity of so-called 'objects', then maybe classical mereology is the way to go. |
| 10709 | Priority is a modality, arising from collections and members [Potter] |
| Full Idea: We must conclude that priority is a modality distinct from that of time or necessity, a modality arising in some way out of the manner in which a collection is constituted from its members. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3) | |
| A reaction: He is referring to the 'iterative' view of sets, and cites Aristotle 'Metaphysics' 1019a1-4 as background. |
| 12157 | Kant gave form and status to aesthetics, and Hegel gave it content [Kant, by Scruton] |
| Full Idea: Kant gave form and status to aesthetics, and Hegel endowed it with content. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Roger Scruton - Recent Aesthetics in England and America p.3 |
| 20346 | The aesthetic attitude is a matter of disinterestedness [Kant, by Wollheim] |
| Full Idea: The aesthetic attitude is defined by Kant in terms of disinterestedness. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Richard Wollheim - Art and Its Objects 54 | |
| A reaction: This is presumably, mainly, to explain our enjoyment of the miseries of tragedy. We just give ourselves up to a merry jig by Haydn. |
| 18547 | Only rational beings can experience beauty [Kant, by Scruton] |
| Full Idea: Kant is surely right that the experience of beauty, like the judgements in which it issues, is the prerogative of rational beings. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Roger Scruton - Beauty: a very short introduction 1 | |
| A reaction: I'm not sure how Scruton can say that Kant is 'surely right'. It is an interesting speculation. Are we to dogmatically affirm that bees get no aesthetic thrill when they spot a promising flower? Something in their little brains attracts them. |
| 24172 | It is hard to see why we would have developed Kant's 'disinterested' aesthetic attitude [Cochrane on Kant] |
| Full Idea: The Kantian notion of disinterest isolated aesthetic value from the rest of our lives. It is hard to understand why we should have developed a tendency that is detached from our everyday practical purposes. | |
| From: comment on Immanuel Kant (Critique of Judgement I: Aesthetic [1790], §2) by Tom Cochrane - The Aesthetic Value of the World 1.4 | |
| A reaction: Cochrane always seeks an evolutionary framework for accounts of aesthetics, and I agree with him. That doesn't devalue them. The best things in life, like piano music, are obviously offshoots of things which evolved for other reasons. |
| 20408 | With respect to the senses, taste is an entirely personal matter [Kant] |
| Full Idea: With regard to the agreeable, the principle Everyone has his own taste (of the senses) is valid. | |
| From: Immanuel Kant (Critique of Judgement I: Aesthetic [1790], CUP 7 5:212), quoted by Elizabeth Schellekens - Immanuel Kant (aesthetics) 1 | |
| A reaction: This is a preliminary concession, and he goes on to defend more objective views of taste. |
| 20409 | When we judge beauty, it isn't just personal; we judge on behalf of everybody [Kant] |
| Full Idea: It is ridiculous if someone justifies his tast by saying 'this object is beautiful for me'. . .If he pronounces that something is beautiful, then he expects the very same satisfaction of others: he judges not merely for himself, but for everyone. | |
| From: Immanuel Kant (Critique of Judgement I: Aesthetic [1790], CUP 7 5:213), quoted by Elizabeth Schellekens - Immanuel Kant (aesthetics) 1 | |
| A reaction: For Kant this would also be the hallmark of rationality - that we expect, or hope for, a consensus when we express a rational judgement. But this expectation is far less in cases of beauty. We do not expect total agreement from very tasteful people. |
| 20411 | Saying everyone has their own taste destroys the very idea of taste [Kant] |
| Full Idea: To say thast 'Everyone has his special taste' would be to dismiss the very possibility of aesthetic taste, and to deny that there could be aesthetic judgement 'that could make a rightful claim to the assent of everyone'. | |
| From: Immanuel Kant (Critique of Judgement I: Aesthetic [1790], CUP 7 5:213), quoted by Elizabeth Schellekens - Immanuel Kant (aesthetics) 2.2 | |
| A reaction: I am a great believer in the objectivity of taste (within sensible reason). But the great evidence against it is the shifting standards of taste over the centuries. Nineteenth century collectors wasted fortunes on inferior works, it seems to us. |
| 24170 | Kant thinks beauty ignores its objects, because it is only 'form' engaging with mind [Cochrane on Kant] |
| Full Idea: Kant thinks that the ideal of beauty requires no concept of what the object is. Universality demands that appreciation be purely a matter of the way the form of the object fits one's cognitive machinery. | |
| From: comment on Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Tom Cochrane - The Aesthetic Value of the World 1.3 | |
| A reaction: This confirms further my increasingly negative view of Kant. Everything in him points to idealism (despite denials by his fans), and via Hegel we arrive at the idea that our values are all 'cultural constructs', rather than responses to reality. |
| 22711 | The beautiful is not conceptualised as moral, but it symbolises or resembles goodness [Kant, by Murdoch] |
| Full Idea: Kant insists that the beautiful must not be tainted with the good (that is, not conceptualised in any way which would bring it into the sphere of moral judgement) yet he says that the beautiful symbolises the good, it is an analogy of the good. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Iris Murdoch - The Sublime and the Good p.209 | |
| A reaction: Kant evidently wanted a very pure view of the aesthetic experience, drained of any overlapping feelings or beliefs. I'm not sure I understand how the beautiful can symbolise or be analogous to the good, while being devoid of it. |
| 4025 | Kant saw beauty as a sort of disinterested pleasure, which has become separate from the good [Kant, by Taylor,C] |
| Full Idea: Kant, in his third critique, defined beauty in terms of a certain kind of disinterested pleasure;….this is the basis for a declaration of independence of the beautiful relative to the good. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Charles Taylor - Sources of the Self §23.1 | |
| A reaction: This is a rebellion against the Greeks, especially Plato, and prepares the ground for the idea of 'art for art's sake'. Personally, I'm with Plato. |
| 20412 | Beauty is only judged in pure contemplation, and not with something else at stake [Kant] |
| Full Idea: If the question is whether something is beautiful, one does not want to know whether there is something that is or that could be at stake, for us or for someone else, in the existence of the thing, but rather how we judge it in mere contemplation. | |
| From: Immanuel Kant (Critique of Judgement I: Aesthetic [1790], CUP 2 5:204), quoted by Elizabeth Schellekens - Immanuel Kant (aesthetics) 2.3 | |
| A reaction: This evidently denies that function has anything to do with beauty, and seems to be a prelude to 'art for art's sake'. But a running cheetah cannot be separated from the sheer efficiencey and focus of the performance. |
| 22046 | The mathematical sublime is immeasurable greatness; the dynamical sublime is overpowering [Kant, by Pinkard] |
| Full Idea: Kant distinguished the 'mathematical' and 'dynamical' sublime. The former involves immeasurable greatness (or smallness) such that we cannot even present them to ourselves. The latter is of something large and overpowering, which we can morally resist. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Terry Pinkard - German Philosophy 1760-1860 13 | |
| A reaction: Presumably Cantor revealed the full extent of the mathematical sublime ('heaven', according to Hilbert). We await the comet that destroys the Earth to fully experience the other one. |
| 21458 | The sublime is a moral experience [Kant, by Gardner] |
| Full Idea: The sublime is understood by Kant as a moral experience. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790], 28-9) by Sebastian Gardner - Kant and the Critique of Pure Reason 09 'Judgment' | |
| A reaction: Gardner give the source in Kant. I can't accept that the initial experience of the sublime is moral in character. It could easily acquire a moral character after contemplation by someone who had such inclinations. |
| 5643 | Aesthetic values are not objectively valid, but we must treat them as if they are [Kant, by Scruton] |
| Full Idea: The 'Critique of Judgement' argues, then, not for the objective validity of aesthetic values, but for the fact that we must think of them as objectively valid. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Roger Scruton - Short History of Modern Philosophy §11.7 | |
| A reaction: The trouble with these transcendental arguments of Kant is that they render you powerless to discuss the question of whether values are actually objective. We are all trapped in presuppositions, instead of testing suppositions. |
| 20410 | The judgement of beauty is not cognitive, but relates, via imagination, to pleasurable feelings [Kant] |
| Full Idea: In order to understand whether or not something is beautiful, we do not relate the representation by means of understanding to the object for cognition, but relate it by means of the imagination ..to the subject and its feeling of pleasure or displeasure. | |
| From: Immanuel Kant (Critique of Judgement I: Aesthetic [1790], CUP 1 5:203), quoted by Elizabeth Schellekens - Immanuel Kant (aesthetics) 2.1 | |
| A reaction: This is to distinguish the particular type of judgement which counts as 'aesthetic' - the point being that it is not cognitive - it is not a matter of knowledge and facts, but a cool judgement made about a warm feeling of pleasure. I think. |
| 7825 | The politics of Leibniz was the reunification of Christianity [Stewart,M] |
| Full Idea: The politics of Leibniz may be summed up in one word: theocracy. The specific agenda motivating much of his work was to reunite the Protestant and Catholic churches | |
| From: Matthew Stewart (The Courtier and the Heretic [2007], Ch. 5) | |
| A reaction: This would be a typical project for a rationalist philosopher, who thinks that good reasoning will gradually converge on the one truth. |