24 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. |
| 19855 | The purpose of society is to protect the rights of liberty, property, security and resistance [Mirabeau/committee] |
| Full Idea: The aim of all political associations is the conservation of the natural and imprescriptible rights of man. These rights are liberty, property, security, and resistance to oppression. | |
| From: Mirabeau and committee (Declaration of the Rights of Man [1789], 02) | |
| A reaction: Radical thinkers will obviously be doubtful about property being on the list, because that entrenches huge inequalities, between peasants and their landlords. Resistance to oppression will bother the likes of Edmund Burke. |
| 19856 | The law expresses the general will, and all citizens can participate [Mirabeau/committee] |
| Full Idea: The law is the expression of the general will. All citizens have the right to take part in person or through their representatives in its formulation. It must be the same for all, whether it protects or whether it punishes. | |
| From: Mirabeau and committee (Declaration of the Rights of Man [1789], 06) | |
| A reaction: Now you are wondering who qualifies as a 'citizen'. Rousseau would have been excited until he found that the citizens could send 'representatives', instead of voting themselves. Rousseau aimed at foundational laws, not all of the laws. |
| 19861 | There is only a constitution if rights are assured, and separation of powers defined [Mirabeau/committee] |
| Full Idea: Any society in which the guarantee of Rights is not assured, nor the separation of Power determined, has no Constitution. | |
| From: Mirabeau and committee (Declaration of the Rights of Man [1789], 16) | |
| A reaction: I wonder if they had Britain in mind with this one? The British latched onto Magna Carta in the early 19th century, because it offered some semblance of a constitution. |
| 19858 | No one should be molested for their opinions, if they do not disturb the established order [Mirabeau/committee] |
| Full Idea: No man is to be molested on account of his opinions, even his religious opinions, provided that their manifestation does not disturb the public order established by law. | |
| From: Mirabeau and committee (Declaration of the Rights of Man [1789], 10) | |
| A reaction: Virtually any opinion will 'disturb' the established order a little bit, so this gives the option of suppressing quite mild beliefs, on the grounds of their small disturbance. It is still a wonderful proposal, though. |
| 19859 | Free speech is very precious, and everyone may speak and write freely (but take responsibility for it) [Mirabeau/committee] |
| Full Idea: The free communication of thoughts and opinions is one of man's most precious rights. Every citizen may therefore speak, write, and publish freely; except that he shall be responsible for the abuse of that freedom in cases determined by law. | |
| From: Mirabeau and committee (Declaration of the Rights of Man [1789], 11) | |
| A reaction: Wonderful, and very nicely expressed. Tom Paine will have been a huge influence on this clause. |
| 20544 | Berlin distinguishes 'negative' and 'positive' liberty, and rejects the latter [Berlin, by Swift] |
| Full Idea: Isaiah Berlin draws a famous distinction between 'negative' and 'positive' concepts of liberty, and argues that the latter should be seen as a wrong turning (because totalitarian regimes have invoked it). | |
| From: report of Isaiah Berlin (Two Concepts of Liberty [1958]) by Adam Swift - Political Philosophy (3rd ed) 2 'Intro' | |
| A reaction: Swift argues against him, saying that positive liberty is not a single concept (it's three), and has aspects that should be defended. I think I'm with Swift on that. Is religious freedom a freedom 'from' something, or a freedom 'to do' something? |
| 19857 | All citizens are eligible for roles in the state, purely on the basis of merit [Mirabeau/committee] |
| Full Idea: All citizens being equal in the eyes of the law are equally eligible to all honours, offices, and public employments, according to their abilities and without other distinction than that of their virtues and talents. | |
| From: Mirabeau and committee (Declaration of the Rights of Man [1789], 06) | |
| A reaction: This proclamation of meritocracy must have rung bells around the cities of Europe, and was a reason why many people enjoyed being invaded by Napoleon. |
| 19862 | Property is a sacred right, breached only when essential, and with fair compensation [Mirabeau/committee] |
| Full Idea: Since property is an inviolable and sacred right, no man may be deprived of it except when public necessity, lawfully constituted, evidently requires it; and on condition that a just indemnity be paid in advance. | |
| From: Mirabeau and committee (Declaration of the Rights of Man [1789], 17) | |
| A reaction: This covers compulsory purchase orders. Is the ownership of slaves inviolable? Will aristocrats be compensated for the confiscation of their vast estates? |
| 19860 | Everyone must contribute to the state's power and administration, in just proportion [Mirabeau/committee] |
| Full Idea: For the maintenance of public force and the expenses of administration, a common contribution is indispensable. It must be equally apportioned among all citizens according to their abilities. | |
| From: Mirabeau and committee (Declaration of the Rights of Man [1789], 13) | |
| A reaction: Presumably this enshrines graduated income tax, an eighteenth century invention. Could you contribute just by your labour, or by fighting for the army? Those may be greater contributions than mere money. |