29 ideas
| 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. |
| 2602 | What experience could prove 'If a=c and b=c then a=b'? [Descartes] |
| Full Idea: Please tell me what the corporeal motion is that is capable of forming some common notion to the effect that 'things which are equal to a third thing are equal to each other'. | |
| From: René Descartes (Comments on a Certain Broadsheet [1644], p.366) |
| 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. |
| 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. |
| 2600 | The mind's innate ideas are part of its capacity for thought [Descartes] |
| Full Idea: I have never written or taken the view that the mind requires innate ideas which are something distinct from its own faculty of thinking. | |
| From: René Descartes (Comments on a Certain Broadsheet [1644], p.365) |
| 2601 | Qualia must be innate, because physical motions do not contain them [Descartes] |
| Full Idea: The ideas of pains, colours, sounds etc. must be all the more innate if, on the occasion of certain corporeal motions, our mind is to be capable of representing them to itself, for there is no similarity between these ideas and the corporeal motions. | |
| From: René Descartes (Comments on a Certain Broadsheet [1644], p.365) | |
| A reaction: Simple and brilliant! We know perfectly well that there is no redness zooming through the air from a tomato (or the air would be pink!). Redness occurs when the light arrives, so we add the redness, so it is innate. |
| 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'. |
| 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. |