33 ideas
| 21959 | Metaphysics is the most general attempt to make sense of things [Moore,AW] |
| Full Idea: Metaphysics is the most general attempt to make sense of things. | |
| From: A.W. Moore (The Evolution of Modern Metaphysics [2012], Intro) | |
| A reaction: This is the first sentence of Moore's book, and a touchstone idea all the way through. It stands up well, because it says enough without committing to too much. I have to agree with it. It implies explanation as the key. I like generality too. |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 20339 | Classes rarely share properties with their members - unlike universals and types [Wollheim] |
| Full Idea: Classes can share properties with their members (e.g. the class of big things is big), but this is very rare. ....In the case of both universals and types, there will be shared properties. Red things can be exhilarating, and so can redness. | |
| From: Richard Wollheim (Art and Its Objects [1968], 92) | |
| A reaction: 'Exhilarating' is an extrinsic property, so not the best illustration. This is interesting, but would need checking with a wide range of examples. (Too busy for that right now) |
| 21958 | Appearances are nothing beyond representations, which is transcendental ideality [Moore,AW] |
| Full Idea: Appearances in general are nothing outside our representations, which is just what we mean by transcendental ideality. | |
| From: A.W. Moore (The Evolution of Modern Metaphysics [2012], B535/A507) |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 20338 | We often treat a type as if it were a sort of token [Wollheim] |
| Full Idea: Much of the time we think and talk of a type as though it were itself a kind of token. | |
| From: Richard Wollheim (Art and Its Objects [1968], 35) | |
| A reaction: A helpful way of connecting what I call 'objectification' to the more conventional modern philosophical vocabulary. Thus I might claim that beauty is superior to truth, as if they were two tokens. |
| 20342 | Interpretation is performance for some arts, and critical for all arts [Wollheim] |
| Full Idea: Performative interpretation occurs only with certain arts, but critical intepretation pertains to all. | |
| From: Richard Wollheim (Art and Its Objects [1968], 38) | |
| A reaction: Fairly obvious, but this is the first point to make about the concept of 'interpretation'. Does the word in fact have two meanings? Or do I perform a painting when I look carefully at it? |
| 20343 | A love of nature must precede a love of art [Wollheim] |
| Full Idea: We could not have a feeling for the beauties of art unless we had been correspondingly moved in front of nature. | |
| From: Richard Wollheim (Art and Its Objects [1968], 43) | |
| A reaction: Wollheim offers this in defence of Kant's view, without necessarily agreeing. Similarly one could hardly care for fictional characters, but not for real people. So the aesthetic attitude may arise from life, rather than from art. Is art hence unimportant? |
| 20348 | A criterion of identity for works of art would be easier than a definition [Wollheim] |
| Full Idea: Maybe, rather than defining art, it would be more fruitful, and more realistic, to seek a general method of identifying works of art. | |
| From: Richard Wollheim (Art and Its Objects [1968], 60) | |
| A reaction: The whole enterprise is ruined by Marcel Duchamp! I'm more interested in identifying or defining good art. |
| 20347 | If beauty needs organisation, then totally simple things can't be beautiful [Wollheim] |
| Full Idea: It is said that beauty cannot consist in organisation because, if it did, we would not be able to predicate beauty of totally simple objects. | |
| From: Richard Wollheim (Art and Its Objects [1968], 59) | |
| A reaction: [He says this idea originates in Plotinus] I'm struggling to think of an example of something which is 'totally' simple and beautiful. Maybe a patch of colour like the breast of a bullfinch? |
| 20345 | Some say art must have verbalisable expression, and others say the opposite! [Wollheim] |
| Full Idea: The view that a work of art expresses nothing if it can't be put into other words ...is reduced by the view that a work of art has no value if what it expresses or says can be put into (other) words. | |
| From: Richard Wollheim (Art and Its Objects [1968], 49) | |
| A reaction: I prefer the second view. Poetry is what is lost in translation. Good art actually seems to evoke emotions which one virtually never feels in ordinary life. But how could that be possible? What are those emotions doing there? |
| 20331 | It is claimed that the expressive properties of artworks are non-physical [Wollheim] |
| Full Idea: The argument that works of art have properties that physical objects could not have characteristically concentrates on the expressive properties of works of art. | |
| From: Richard Wollheim (Art and Its Objects [1968], 10) | |
| A reaction: Since the idea of an object having non-physical properties strikes me as ridiculous, this gets off to a bad start. If artworks are abstract objects, then all of their properties are non-physical. |
| 20336 | Style can't be seen directly within a work, but appreciation needs a grasp of style [Wollheim] |
| Full Idea: 'Style' would seem to be a concept that cannot be applied to a work solely on the basis of what is represented and yet it is also essential to a proper understanding or appreciation of a work. | |
| From: Richard Wollheim (Art and Its Objects [1968], 32) | |
| A reaction: Sounds right. One long held musical note creates an expectation which depends on the presumed style of the piece of music. A single bar from a piece may well not exhibit its characteristic style. |
| 20337 | The traditional view is that knowledge of its genre to essential to appreciating literature [Wollheim] |
| Full Idea: From Aristotle onwards it has been a tenet of the traditional rhetoric that the proper understanding of a literary work involves the location of it in the correct genre, that is, as drama, epic or lyric. | |
| From: Richard Wollheim (Art and Its Objects [1968], 32) | |
| A reaction: Walton argues this persuasively. I've seen the climax of a Jacobean tragedy ruined by laughter from the audience. Genre dictates appropriate responses, so it is a communal concept. |
| 20333 | If artworks are not physical objects, they are either ideal entities, or collections of phenomena [Wollheim] |
| Full Idea: In denying that works of art are physical objects, one theory (the 'ideal') withdraws them altogether from experience, and a second theory ('phenomenal') pins them too it inescapably and at all points. | |
| From: Richard Wollheim (Art and Its Objects [1968], 21) | |
| A reaction: I incline towards them being transient ideals, created by human minds. As with so much, we idealise and objectify them as 'works', and abstract their image from the instance(s) we encounter. |
| 20334 | The ideal theory says art is an intuition, shaped by a particular process, and presented in public [Wollheim] |
| Full Idea: The ideal theory of Croce and Collingwood says art is first an inner intuition or expression of the artist, resulting from a particular process of organisation and unification, which can be externalised in public form. | |
| From: Richard Wollheim (Art and Its Objects [1968], 22) | |
| A reaction: [compressed] As stated this doesn't sound very controversial or 'ideal'. I take it the theory is intended to be more platonist than this expression of it suggests. I think the idea that it is an 'expression' of the artist is wrong. |
| 20335 | The ideal theory of art neglects both the audience and the medium employed [Wollheim] |
| Full Idea: Because the ideal theory makes a work of art inner or mental, the link between the artist and the audience has been severed .....and it also totally ignores the significance of the medium. | |
| From: Richard Wollheim (Art and Its Objects [1968], 23) | |
| A reaction: Emily Dickinson had virtually no audience for her poetry. The medium used to perform Bach's 'Art of Fugue' seems unimportant. For paintings of painterly painters paint matters. For some visual art many different media will suffice. |
| 20340 | A musical performance has virtually the same features as the piece of music [Wollheim] |
| Full Idea: With the usual reservations, there is nothing that can be predicated of a performance of a piece of music that could not also be predicated of that piece of music itself. | |
| From: Richard Wollheim (Art and Its Objects [1968], 37) | |
| A reaction: He offers this as evidence that it fits the performance being a token, and music (and all other art) being a type. There are quite a few 'reservations'. Music too difficult to perform. Great music always badly performed. |
| 20341 | An interpretation adds further properties to the generic piece of music [Wollheim] |
| Full Idea: Interpretation may be regarded as the production of a token that has properties in excess of those of the type. | |
| From: Richard Wollheim (Art and Its Objects [1968], 37) | |
| A reaction: I suppose so. If you play accurately everything that is written in the score, then anything else has to be an addition. If you play less than the score, you aren't quite playing that piece of music. |
| 20332 | A drawing only represents Napoleon if the artist intended it to [Wollheim] |
| Full Idea: It is necessary, if a drawing is to represent Napoleon, that the draughtsman should intend it to be Napoleon. | |
| From: Richard Wollheim (Art and Its Objects [1968], 13) | |
| A reaction: Does a perfect and intended representation of a person also count as a representation of the person's identical twin? The families of both might well order copies. |