27 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. |
| 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. |
| 604 | Knowledge is mind and knowing 'cohabiting' [Lycophron, by Aristotle] |
| Full Idea: Lycophron has it that knowledge is the 'cohabitation' (rather than participation or synthesis) of knowing and the soul. | |
| From: report of Lycophron (fragments/reports [c.375 BCE]) by Aristotle - Metaphysics 1045b | |
| A reaction: This sounds like a rather passive and inert relationship. Presumably knowing something implies the possibility of acting on it. |
| 23874 | Armies and businesses create moralities in which their activity can do no wrong [Marx, by Weil] |
| Full Idea: Marx saw that social groups manufacture moralities for their own use, so their activity is placed outside the reach of evil. Thus the first articles of soldiers and businessmen is to deny that it is possible to do evil while waging war or doing business. | |
| From: report of Karl Marx (works [1860]) by Simone Weil - Fragments: London 1943 p.146 | |
| A reaction: This is especially true of the modern reverence for 'market forces'. It is a key debate in the ethics of warfare - compare Walzer and McMahon. A striking thought, obviously containing a lot of truth. |
| 18662 | Liberal freedom is the right to be separate, and ignores the union of man with man [Marx] |
| Full Idea: The liberal right of man to freedom is not based on the union of man with man, but on the separation of man from man; it is the right to this separation. | |
| From: Karl Marx (works [1860]), quoted by Will Kymlicka - Contemporary Political Philosophy (1st edn) 7.2.a | |
| A reaction: [quoted from an anthology] It is interesting that liberal freedom is the right NOT to be involved in politics, and even not to vote in elections. Home counties England (high hedges etc) is the embodiment of the freedom not to be involved in society. |
| 23372 | Liberals want the right to be separate, rather than for people to be united [Marx] |
| Full Idea: The [liberal] right of man to freedom is not based on the union of man with man, but on the separation of man from man. It is the right to this separation. | |
| From: Karl Marx (works [1860], p.53), quoted by Will Kymlicka - Contemporary Political Philosophy (2nd edn) 7 | |
| A reaction: [in collection ed.McLelland p.53] That nicely encapsulates the debate. Modern liberal thinkers regret the loss of community, but people in authoritarian communities yearn for separation. You can have too much 'union'! |
| 20576 | Early Marx anticipates communitarian objections to liberalism [Marx, by Oksala] |
| Full Idea: The early writings of Marx anticipate the communitarian critique of liberalism. | |
| From: report of Karl Marx (works [1860]) by Johanna Oksala - Political Philosophy: all that matters Ch.8 | |
| A reaction: [Oksala says modern writers seem to prefer this to the hardcore later Marx, which is presumably too 'scientific'. He says 'Capital Vol 1' is Marx's most important work] |
| 23862 | By saying the material dialectic of history aspires to the best, Marx agreed with capitalism [Weil on Marx] |
| Full Idea: When Marx inverted Hegel's dialectic of history, by substituting matter for mind as the motive, he attributed to matter the essence of mind, an unceasing aspiration towards the best - which was in keeping with the general current of capitalist thought. | |
| From: comment on Karl Marx (works [1860]) by Simone Weil - Reflections on Liberty and Social Oppression p.43 | |
| A reaction: [compressed] A rather nice debating point! Marx seems to share the universal nineteenth century belief in unremitting progress. Without that, it is impossible to believe that a revolution will necessarily improve anything. |
| 21999 | False consciousness results from concealment by the superstructure [Marx, by Singer] |
| Full Idea: False consciousness involves failing to see things as they really are. It comes about because a society's superstructure can conceal the real basis of the society. | |
| From: report of Karl Marx (works [1860]) by Peter Singer - Marx 9 | |
| A reaction: That seems a poor label, probably revealing a Hegelian background. It seems a matter of knowledge rather than consciousness. Can a whole mind be in a state of error? |
| 23875 | Marx says force is everything, and that the weak will become strong, while remaining the weak [Weil on Marx] |
| Full Idea: Marx posits on the one hand that force alone governs social relations to the exclusion of anything else, and on the other hand that one day the weak, while remaining weak, will nevertheless be stronger. He believed in miracles. | |
| From: comment on Karl Marx (works [1860]) by Simone Weil - Fragments: London 1943 p.149 | |
| A reaction: This is close to the obvious contradiction if the working classes despise the middle classes (the dreaded 'bourgeoisie') while their only aspiration is to be like them. It is hard to custom design a new class to which they could both aspire. |
| 18653 | Marx rejected equal rights because they never actually treat people as equals [Marx, by Kymlicka] |
| Full Idea: Marx rejected the idea of equal rights, not because he was not a friend to the idea of treating people as equals, but precisely because he thought rights failed to live up to that ideal. | |
| From: report of Karl Marx (works [1860]) by Will Kymlicka - Contemporary Political Philosophy (1st edn) 5.1 | |
| A reaction: Presumably because the power to award 'rights' goes to the highest bidder. If equality is to be enshrined in law, it is a bit difficult to see how else to manage it. |
| 20958 | Capitalism changes the world, by socialising the idea of a commodity [Marx, by Bowie] |
| Full Idea: In Marx's view the essential factor in capitalism is that the encroachment of the commodity form into society fundamentally changes the world. | |
| From: report of Karl Marx (works [1860]) by Andrew Bowie - Introduction to German Philosophy 6 'Historical' | |
| A reaction: The main point is that people and their labour become commodities. Haven't animals always been treated as commodities? Clearly slave were commodities, long before capitalism. Capitalism universalises it? |
| 23876 | The essence of capitalism is the subordination of people to things [Marx, by Weil] |
| Full Idea: Marx discovered a formula impossible to surpass when he said that the essence of capitalism lies in the subordination of subject to object, of man to thing. | |
| From: report of Karl Marx (works [1860]) by Simone Weil - Fragments: London 1943 p,155 | |
| A reaction: I find this rather too vague to be a penetrating observation. I would suggest the obliteration of cooperation and community, in favour of competition. Winners and losers. |
| 20960 | Marx thought capitalism was partly liberating, and could make labour and ownership more humane [Marx, by Bowie] |
| Full Idea: Marx did not disapprove per se of capitalism. New divisions of labour and forms of ownership could transform individuals in modern societies, creating a more humane world with the means capitalism had liberated from feudalism. | |
| From: report of Karl Marx (works [1860]) by Andrew Bowie - Introduction to German Philosophy 11 'Metaphysics' | |
| A reaction: I'm guessing this might be early Marx, which has less to say about the 'scientific' inevitably of deep change, and the necessity for revolution. Nowadays we tinker with humane changes at the poorer end, while the rich run rampant. |