25 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. |
| 23029 | Knowledge is secured by the relations between its parts, through differences and identities [Green,TH, by Muirhead] |
| Full Idea: What gives reality and stability to our knowledge is the reality and stability of the relations established between its parts..…by the differences and identities with other things which have similarly achieved comparative fixity and substantiality. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State I | |
| A reaction: Although I don't sympathise with Green's idealist metaphysics, and nevertheless think that this internalist account of knowledge is correct. |
| 23223 | The word 'respect' ranges from mere non-interference to the highest levels of reverence [Blackburn] |
| Full Idea: The word 'respect' seems to span a spectrum from simply not interfering, passing by on the other side, through admiration, right up to reverence and deference. This makes it uniquely well placed for ideological purposes. | |
| From: Simon Blackburn (Religion and Respect [2005], p.2) | |
| A reaction: Most people understand the world perfectly well, but only when they fully understand the context. I've taken to distinguishing conditional from unconditional forms of respect. Everyone is entitled to the unconditional form, which has limits. |
| 23046 | States only have full authority if they heed the claims of human fellowship [Green,TH] |
| Full Idea: The claim of the state is only absolutely paramount on the supposition that in its commands and prohibitions it takes account of all the claims that arise out of human fellowship. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882], §146), quoted by John H. Muirhead - The Service of the State III | |
| A reaction: He rejects the idea of the general will in ordinary political activity, so it is not clear how this condition could ever be met in practice. Hideous governments just pay lip service to 'human fellowship'. How could you tell whether they believe it? |
| 23051 | Equality also implies liberty, because equality must be of opportunity as well as possessions [Green,TH] |
| Full Idea: Liberty was essential, not only as a means to equality, but as part of it. …because the opportunity which was to be equalised was not merely to have and to be happy, but to do and to realise. It was 'the right of man to make the best of himself'. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882]), quoted by John H. Muirhead - The Service of the State IV | |
| A reaction: This nicely identifies the core idea of civilised liberalism (as opposed to the crazy self-seeking kind). I think 'give people the right to make the best of themselves' makes a good slogan, because it implies ensuring that they have the means. |
| 23028 | The highest political efforts express our deeper social spirit [Green,TH, by Muirhead] |
| Full Idea: Political effort in all its highest forms is the expression of a belief in the reality of the social spirit as the deeper element in the individual. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State I | |
| A reaction: Although Green is rather literally spiritual, if we express it as a central aspect of human nature, this idea strikes me as correct. Writing in 2021, I am totally bewildered by the entire absence of any 'higher' forms of political expression. |
| 23054 | Communism is wrong because it restricts the freedom of individuals to contribute to the community [Green,TH, by Muirhead] |
| Full Idea: Green condemned pure communism, not in the name of any abstract rights of the individual, but of the right of the community itself to the best that individuals can contribute through the free and spontaneous exercise of their powers of self-expression. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State IV | |
| A reaction: Interesting. In a very authoritarian communist state it does seem that citizens are less able to contribute to the general good. But extreme liberty seems also to undermine the general good. Hm. |
| 23047 | Original common ownership is securing private property, not denying it [Green,TH, by Muirhead] |
| Full Idea: Common ownership in early societies is not the denial of a man's private property in the products of his own labour, but the only way under the circumstances of securing it. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882], §218) by John H. Muirhead - The Service of the State III | |
| A reaction: This is announced with some confidence, but it is very speculative. I think there is some truth in Locke's thought that putting work into a creation creates natural ownership. But who owns the raw materials? Why is work valued highly? |
| 23042 | National spirit only exists in the individuals who embody it [Green,TH, by Muirhead] |
| Full Idea: A national spirit cannot exist apart from the individuals who embody it. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State II | |
| A reaction: We see this in football supporters. They are thrilled by the glory of a great victory, but the reality is just the thrill of the players, and the exuberance in each supporter's mind. There is no further entity called the 'glory'. Green was a liberal. |
| 23048 | The ground of property ownership is not force but the power to use it for social ends [Green,TH, by Muirhead] |
| Full Idea: It is not the power of forcible tenure but the power of utilisation for social ends that is the ground of the permanent recognition that constitutes a right to property. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State III | |
| A reaction: Tell that to the aristocratic owners of British grouse moors! This just seems to be wishful thinking. Does that mean that I have no right to property if my ends are not 'social'? |
| 23049 | Property is needed by all citizens, to empower them to achieve social goods [Green,TH] |
| Full Idea: The rationale of property is that every one should be secured by society in the power of getting and keeping the means of realising a will which in possibility is a will directed to social good. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882], §220), quoted by John H. Muirhead - The Service of the State III | |
| A reaction: An interesting argument. If you want free citizens in a liberal society to be capable of achieving social good, you must allow them the right to acquire the means of doing so. |