30 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. |
| 15200 | How could change consist of a conjunction of changeless facts? [McTaggart, by Le Poidevin] |
| Full Idea: McTaggart objects, to Russell 1903, that change cannot consist of a conjunction of changeless facts. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by Robin Le Poidevin - Past, Present and Future of Debate about Tense 1 (b) | |
| A reaction: I agree with McTaggart. Logicians like to model processes with domains of timeless entities, but it just won't do. |
| 14761 | Change is not just having two different qualities at different points in some series [McTaggart] |
| Full Idea: The fact that it is hot at one point in a series and cold at other points cannot give change, if neither of these facts change. If two points on a line have different properties, this doesn't give change. | |
| From: J.M.E. McTaggart (The Nature of Existence vol.2 [1927], 33.315-6), quoted by Theodore Sider - Four Dimensionalism 6.2 | |
| A reaction: [The second half compresses an example about the Meridian] This objection is aimed at Russell's view, that change is just different properties at different times. I (unlike Sider) am wholly with McTaggart on this one. Change is 'dynamic'. |
| 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. |
| 22252 | Capitalism may actually be the best way to foster community [Conway,D] |
| Full Idea: Not only is there no good reason for supposing capitalism inimical to community, but there is reason to think it more conducive to community than the feasible alternatives to it. | |
| From: David Conway (Capitalism and Community [1996], I) | |
| A reaction: Conway is defending an obviously unorthodox view, while attacking the hopes of communitarians. |
| 22253 | Capitalism is just the market, with optional limited government, and perhaps democracy [Conway,D] |
| Full Idea: There are three types of capitalism: 1) the market - private ownership, labor contracts and profit, 2) limited government - the state provides goods the market cannot do, 3) limited government with democracy - with political freedom and elections. | |
| From: David Conway (Capitalism and Community [1996], II) | |
| A reaction: [compressed] I would have thought that capitalism is compatible with a fair degree of workplace democracy, which would make a fourth type. |
| 22256 | Capitalism prefers representative democracy, which avoids community decision-making [Conway,D] |
| Full Idea: By opting for representative rather than direct democracy, capitalism is said to preclude political community, for which the citizens of a state must possess a common will, which needs their direct participation in decisions. | |
| From: David Conway (Capitalism and Community [1996], V) | |
| A reaction: Conway does not accept this claim. I'm beginning to wonder whether the famous British electoral system is actually a capitalist conspiracy against the people. |
| 22255 | Capitalism breaks up extended families, and must then provide welfare for the lonely people [Conway,D] |
| Full Idea: It is said that capitalism encourages the breakup of extended families, which creates the need for extensive state welfare for those indigent members of society who can no longer rely on their own family to take care of them. | |
| From: David Conway (Capitalism and Community [1996], V) | |
| A reaction: Conway does not accept this claim. It seems to simplistic to say that capitalism is the sole culprit. Any rise of mechanisation in agriculture would break up rural extended families. |
| 22257 | Capitalism is anti-community, by only valuing individuals, and breaking up families [Conway,D] |
| Full Idea: Communitarns say capitalism is inimical to family community, because it encourages an individualistic mentality which only values self-fulfilment, and because it demands labour mobility which is disruptive of families. | |
| From: David Conway (Capitalism and Community [1996], VI) | |
| A reaction: Chicken-and-egg with the first one. Small entrepreneurs are individualists who seek their own gain. It is big capitalism that sucks in the others. Traditional community is based on labour-intensive agriculture. |
| 2608 | For McTaggart time is seen either as fixed, or as relative to events [McTaggart, by Ayer] |
| Full Idea: McTaggart says we can speak of events in time in two ways, as past, present or future, or as being before or after or simultaneous with one another. The first cannot be reduced to the second, as the second makes no provision for the passage of time. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927], II.329-) by A.J. Ayer - The Central Questions of Philosophy 1.D |
| 22936 | A-series time positions are contradictory, and yet all events occupy all of them! [McTaggart, by Le Poidevin] |
| Full Idea: McTaggart's proof of time's unreality: A-series positions (past, present and future) are mutually incompatible, so no event can exhibit more than one of them; but since A-series events change position, all events have all A-series posititions. Absurd! | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by Robin Le Poidevin - Travels in Four Dimensions 08 'McTaggart's' | |
| A reaction: I'm not convinced that this is any more contradictory than someone being married at one time and unmarried at another. No one is suggesting that an A-series event can be both past and future simultaneously. |
| 4231 | Time involves change, only the A-series explains change, but it involves contradictions, so time is unreal [McTaggart, by Lowe] |
| Full Idea: McTaggart argued that time involves change, only the A-series can explain change, the A-series involves contradictions (past, present and future), and hence time is unreal. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by E.J. Lowe - A Survey of Metaphysics p.313 | |
| A reaction: I doubt whether it is a logical contradiction to say Waterloo has been past, present and future, though it is odd. |
| 8591 | There could be no time if nothing changed [McTaggart] |
| Full Idea: It is universally admitted.... that there could be no time if nothing changed. | |
| From: J.M.E. McTaggart (The Nature of Existence vol.2 [1927], II p.11), quoted by Sydney Shoemaker - Time Without Change p.49 | |
| A reaction: This is set up alongside Aristotle (Idea 8590) to be attacked by Shoemaker. I think Shoemaker is right, and that the rejection of McTaggart's view is a key result in modern metaphysics. |
| 22935 | The B-series can be inferred from the A-series, but not the other way round [McTaggart, by Le Poidevin] |
| Full Idea: McTaggart says the A-series is more fundamental than the B-series. An objective being could not deduce the present moment of the A-series from the B-series, but the B-series can be deduced from the A-series. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by Robin Le Poidevin - Travels in Four Dimensions 08 'McTaggart's' | |
| A reaction: [summarised] This has no ontological importance for McTaggart, since he thinks time is unreal either way. But giving the A-series priority because it reveals the present moment seems to nullify the B-series as incomplete. |
| 7802 | A-series uses past, present and future; B-series uses 'before' and 'after' [McTaggart, by Girle] |
| Full Idea: The A-series puts events into past, present and future. The B-series puts events into a series based on relationships of 'before' and 'after'. McTaggart said the A-series was contradictory, and the B-series failed to cope with essential features of time. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by Rod Girle - Modal Logics and Philosophy 8.10 | |
| A reaction: The A-series is indexical. |
| 4230 | A-series expressions place things in time, and their truth varies; B-series is relative, and always true [McTaggart, by Lowe] |
| Full Idea: A-series expressions include words like 'today' and 'five weeks ago', and can be true at one time and false at another; B-series expressions are like 'simultaneously', and are always true, if true at all. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927]) by E.J. Lowe - A Survey of Metaphysics p.308 | |
| A reaction: A-series gives time separate existence, where B-series time is purely relational. Intuition favours the A-series, but how fast do events travel against this fixed background? |
| 15199 | The B-series must depend on the A-series, because change must be explained [McTaggart, by Le Poidevin] |
| Full Idea: McTaggart's argument is 1) B-series relations are temporal relations, 2) There cannot be temporal relations unless there is change, 3) There cannot be change unless there is real A-series ordering, so there can't be a B-series unless there is an A-series. | |
| From: report of J.M.E. McTaggart (The Nature of Existence vol.2 [1927], vol.ii) by Robin Le Poidevin - Past, Present and Future of Debate about Tense 1 a |