50 ideas
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| Full Idea: Second-order set theory is just like first-order set-theory, except that we use the version of Replacement with a universal second-order quantifier over functions from set to sets. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VII.4) |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| Full Idea: A member m of M is an 'upper bound' of a subset N of M if m is not less than any member of N. A member m of M is a 'least upper bound' of N if m is an upper bound of N such that if l is any other upper bound of N, then m is less than l. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: [if you don't follow that, you'll have to keep rereading it till you do] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| Full Idea: Since combinatorial collections are enumerated, some multiplicities may be too large to be gathered into combinatorial collections. But the size of a multiplicity seems quite irrelevant to whether it forms a logical connection. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| Full Idea: Many of those who are skeptical about the existence of infinite combinatorial collections would want to doubt or deny the Axiom of Choice. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| Full Idea: The Power Set is just he codification of the fact that the collection of functions from a mathematical collection to a mathematical collection is itself a mathematical collection that can serve as a domain of mathematical study. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| Full Idea: The Axiom of Replacement (of Skolem and Fraenkel) was remarkable for its universal acceptance, though it seemed to have no consequences except for the properties of the higher reaches of the Cantorian infinite. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| Full Idea: The Axiom of Foundation (Zermelo 1930) says 'Every (descending) chain in which each element is a member of the previous one is of finite length'. ..This forbids circles of membership, or ungrounded sets. ..The iterative conception gives this centre stage. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.4) |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| Full Idea: The controversy was not about Choice per se, but about the correct notion of function - between advocates of taking mathematics to be about arbitrary functions and advocates of taking it to be about functions given by rules. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| Full Idea: Combinatorial collections (defined just by the members) obviously obey the Axiom of Choice, while it is at best dubious whether logical connections (defined by a rule) do. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| Full Idea: The Peano-Russell notion of class is the 'logical' notion, where each collection is associated with some kind of definition or rule that characterises the members of the collection. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.1) |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| Full Idea: The iterative conception of set was not so much as suggested, let alone advocated by anyone, until 1947. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| Full Idea: The iterative conception of sets does not tell us how far to iterate, and so we must start with an Axiom of Infinity. It also presupposes the notion of 'transfinite iteration'. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| Full Idea: The iterative conception does not provide a conception that unifies the axioms of set theory, ...and it has had very little impact on what theorems can be proved. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) | |
| A reaction: He says he would like to reject the iterative conception, but it may turn out that Foundation enables new proofs in mathematics (though it hasn't so far). |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| Full Idea: Limitation of Size has it that if a collection is the same size as a set, then it is a set. The Axiom of Replacement is characteristic of limitation of size. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| Full Idea: A collection M is 'well-ordered' by a relation < if < linearly orders M with a least element, and every subset of M that has an upper bound not in it has an immediate successor. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3) | |
| A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'. |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
| A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku. |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| Full Idea: Mathematics is today thought of as the study of abstract structure, not the study of quantity. That point of view arose directly out of the development of the set-theoretic notion of abstract structure. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.2) | |
| A reaction: It sounds as if Structuralism, which is a controversial view in philosophy, is a fait accompli among mathematicians. |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| Full Idea: One reason to introduce the rational numbers is that it simplifes the theory of division, since every rational number is divisible by every nonzero rational number, while the analogous statement is false for the natural numbers. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.3) | |
| A reaction: That is, with rations every division operation has an answer. |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| Full Idea: The chief importance of the Continuum Hypothesis for Cantor (I believe) was that it would show that the real numbers form a set, and hence that they were encompassed by his theory. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| Full Idea: The Cauchy convergence criterion for a sequence: the sequence S0,S1,... has a limit if |S(n+r) - S(n)| is less than any given quantity for every value of r and sufficiently large values of n. He proved this necessary, but not sufficient. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], 2.5) |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| Full Idea: Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], II.6) | |
| A reaction: Thus there is the problem of whether the contents of the gap are one unique thing, or many. |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| Full Idea: Counting a set produces a well-ordering of it. Conversely, if one has a well-ordering of a set, one can count it by following the well-ordering. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: Cantor didn't mean that you could literally count the set, only in principle. |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| Full Idea: My proposal is that the concept of the infinite began with an extrapolation from the experience of indefinitely large size. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) | |
| A reaction: I think it might be better to talk of an 'abstraction' than an 'extrapolition', since the latter is just more of the same, which doesn't get you to concept. Lavine spends 100 pages working out his proposal. |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| Full Idea: The indiscernibility of indefinitely large sizes will be a critical part of the theory of indefinitely large sizes. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| Full Idea: The intuitionist endorse the actual finite, but only the potential infinite. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| Full Idea: The symbol 'aleph-nought' denotes the cardinal number of the set of natural numbers. The symbol 'aleph-one' denotes the next larger cardinal number. 'Aleph-omega' denotes the omega-th cardinal number. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.3) |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| Full Idea: The ordinals are basic because the transfinite sets are those that can be counted, or (equivalently for Cantor), those that can be numbered by an ordinal or are well-ordered. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: Lavine observes (p.55) that for Cantor 'countable' meant 'countable by God'! |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| Full Idea: The paradox of the largest ordinal (the 'Burali-Forti') is that the class of all ordinal numbers is apparently well-ordered, and so it has an ordinal number as order type, which must be the largest ordinal - but all ordinals can be increased by one. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| Full Idea: The paradox of the largest cardinal ('Cantor's Paradox') says the diagonal argument shows there is no largest cardinal, but the class of all individuals (including the classes) must be the largest cardinal number. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| Full Idea: Every theorem of mathematics has a counterpart with set theory - ...but that theory cannot serve as a basis for the notion of proof. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3) |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| Full Idea: In modern mathematics virtually all work is only up to isomorphism and no one cares what the numbers or points and lines 'really are'. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
| A reaction: At least that leaves the field open for philosophers, because we do care what things really are. So should everybody else, but there is no persuading some people. |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| Full Idea: Intuitionism in philosophy of mathematics rejects set-theoretic foundations. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3 n33) |
| 20429 | Most of us are too close to our own motives to understand them [Fry] |
| Full Idea: The motives we actually experience are too close to us to enable us to feel them clearly. They are in a sense unintelligible. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.30) | |
| A reaction: Fry is defending the role of art in clarifying and highlighting such things, but I am not convinced by his claim. We can grasp most of our motives with a little introspection, and those we can't grasp are probably too subtle for art as well. |
| 20424 | Imaginative life requires no action, so new kinds of perception and values emerge in art [Fry] |
| Full Idea: In the imaginative life no action is necessary, so the whole consciousness may be focused upon the perceptive and the emotional aspects of the experience. Hence we get a different set of values, and a different kind of perception | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.24) | |
| A reaction: Good. A huge range of human activities are like scientific experiments, where you draw on our evolved faculties, but put them in controlled conditions, where the less convenient and stressful parts are absent. War and sport. Real and theatrical tragedy. |
| 20427 | Everyone reveals an aesthetic attitude, looking at something which only exists to be seen [Fry] |
| Full Idea: It is only when an object exists for no other purpose than to be seen that we really look at it, …and then even the most normal person adopts to some extent the artistic attitude of pure vision abstracted from necessity. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.29) | |
| A reaction: A painter of still life looks at things which exist for other purposes, with just the attitude which Fry attributes to the viewers of the paintings. We can encourage a child to look at a flower with just this attitude. |
| 20433 | 'Beauty' can either mean sensuous charm, or the aesthetic approval of art (which may be ugly) [Fry] |
| Full Idea: There is an apparent contradiction between two distinct uses of the word 'beauty', one for that which has sensuous charm, and one for the aesthetic approval of works of imaginative art where the objects presented to us are often of extreme ugliness. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.33) | |
| A reaction: The gouging of eyes in 'King Lear' was always the big problem case for aesthetics, just as nowadays it is Marcel Duchamp's wretched 'Fountain'. |
| 20430 | In life we neglect 'cosmic emotion', but it matters, and art brings it to the fore [Fry] |
| Full Idea: Those feelings unhappily named cosmic emotion find almost no place in life, but, since they seem to belong to certain very deep springs of our nature, do become of great importance in the arts. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.31) | |
| A reaction: Focus on the sublime was big in the romantic era, but Fry still sees its importance, and I don't think it ever goes away. Art styles which scorn the sublime are failing to perform their social duty, say I. |
| 20431 | Art needs a mixture of order and variety in its sensations [Fry] |
| Full Idea: The first quality that we demand in our [artistic] sensations will be order, without which our sensations will be troubled and perplexed, and the other will be variety, without which they will not be fully stimulated. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.32) | |
| A reaction: He makes good claims, but gives unconvincing reasons for them. Some of us rather like 'troubled and perplexed' sensations. And a very narrow range of sensations could still be highly stimulated. Is Fry a good aesthetician but a modest philosopher? |
| 20423 | If graphic arts only aim at imitation, their works are only trivial ingenious toys [Fry] |
| Full Idea: If imitation is the sole purpose of the graphic arts, it is surprising that the works of such arts are ever looked upon as more than curiosities, or ingenious toys, and are ever taken seriously by grown-up people. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.23) | |
| A reaction: But then you might say that same about fine wines. A mere nice taste is hardly worthy of grown ups, and yet lots of grown ups feeling quite passionately about it. What about Fabergé eggs? |
| 20428 | Popular opinion favours realism, yet most people never look closely at anything! [Fry] |
| Full Idea: Ordinary people have almost no idea of what things really look like, so that the one standard that popular criticism applies to painting (whether it is like nature or not) is the one which most people are prevented frm applying properly. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.29) | |
| A reaction: A nice remark, though there is a streak of Bloomsbury artistic snobbery running through Fry. Ordinary people recognise photographic realism, so they can study things closely either in the reality or the picture, should they so choose. |
| 20432 | When viewing art, rather than flowers, we are aware of purpose, and sympathy with its creator [Fry] |
| Full Idea: In our reaction to a work of art (rather than a flower) there is the consciousness of purpose, of a peculiar relation of sympathy with the man who made this thing in order to arouse precisely the sensations we experience. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.33) | |
| A reaction: I think this is entirely right. I like the mention of 'sympathy' as well as 'purpose'. |
| 20400 | Intentions either succeed or fail, so external evidence for them is always irrelevant [Wimsatt/Beardsley, by Davies,S] |
| Full Idea: Wimsatt and Beardsley claimed that either the intention succeeded, so one does not need to look outside the work for its meaning, or the intention failed, so external evidence does not help. | |
| From: report of W Wimsatt/W Beardsley (The Intentional Fallacy [1946]) by Stephen Davies - The Philosophy of Art (2nd ed) 5.3 | |
| A reaction: Actually, the external evidence may tell you much more clearly and accurately what the intention was than the work itself does. The best example may be the title of the work, which is presumably outside the work. |
| 7266 | The author's intentions are irrelevant to the judgement of a work's success [Wimsatt/Beardsley] |
| Full Idea: The design or intention of the author is neither available nor desirable as a standard for judging the success of a work of literary art. | |
| From: W Wimsatt/W Beardsley (The Intentional Fallacy [1946], §I) | |
| A reaction: This famous proposal may have been misunderstood. Note that it is a comment about judging the work, not about understanding it. The idea allows for a work being much more successful than the author's humble intentions (e.g. Pepys). |
| 7267 | Poetry, unlike messages, can be successful without communicating intentions [Wimsatt/Beardsley] |
| Full Idea: Poetry differs from practical messages, which are successful if and only if we correctly infer the intention. | |
| From: W Wimsatt/W Beardsley (The Intentional Fallacy [1946], §I) | |
| A reaction: I am not convinced by this claim. It is plausible that a work does much more than it intends (Astaire said he danced "to make a buck"), but it is rather odd to rate very highly a work of which you have missed the point. |
| 7268 | The thoughts of a poem should be imputed to the dramatic speaker, and hardly at all to the poet [Wimsatt/Beardsley] |
| Full Idea: We ought to impute the thoughts and attitudes of the poem immediately to the dramatic speaker, and if to the author at all, only by an act of biographical inference. | |
| From: W Wimsatt/W Beardsley (The Intentional Fallacy [1946], §I) | |
| A reaction: Wrong. If in Browning's "My Last Duchess" (say), we only inferred the mind of the speaker (and his Duchess), and took no interest in Browning's view of things, we would miss the point. We might end up respecting the Duke, which would be daft. |
| 7269 | The intentional fallacy is a romantic one [Wimsatt/Beardsley] |
| Full Idea: The intentional fallacy is a romantic one. | |
| From: W Wimsatt/W Beardsley (The Intentional Fallacy [1946], §II) | |
| A reaction: Wrong. Even with those most famous of anonymous artists, the architects and carvers of medieval cathedrals, without some discernment of the purpose you won't get it. The Taj Mahal is a love letter, not a potential ice cream parlour. |
| 7271 | Biography can reveal meanings and dramatic character, as well as possible intentions [Wimsatt/Beardsley] |
| Full Idea: The use of biographical evidence need not involve intentionalism, because while it may be evidence of what the author intended, it may also be evidence of the meaning of his words and the dramatic character of his utterance. | |
| From: W Wimsatt/W Beardsley (The Intentional Fallacy [1946], §IV) | |
| A reaction: I am very keen to penetrate the author's intentions, but I have always be doubtful about the use of biography as a means to achieve this. Most of the effort to infer intentions must come from a study of the work itself, not introductions, letters etc. |
| 20425 | In the cinema the emotions are weaker, but much clearer than in ordinary life [Fry] |
| Full Idea: One notices in the visions of the cinematograph that whatever emotions are aroused by them, though they are likely to be weaker than those of ordinary life, are presented more clearly to the conscious. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.25) | |
| A reaction: Fry had probably only seen very simple melodramas, but the general idea that artistic emotions are weaker than real life, but much clearer, is quite plausible. |
| 20426 | For pure moralists art must promote right action, and not just be harmless [Fry] |
| Full Idea: To the pure moralist, accepting nothing but ethical values, to be justified, the life of the imagination must be shown not only not to hinder but actually to forward right action, otherwise it is not only useless but, by absorbing energies, harmful. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.26) | |
| A reaction: I think this is the sort of attitude you find in Samuel Johnson. Puritans even reject light music, which seems pleasantly harmless to the rest of us. 'Absorbing energies' doesn't sound much of an objection, and may not be the actual objection. |