34 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. |
| 14592 | Some abstract things have a beginning and end, so may exist in time (though not space) [Swoyer] |
| Full Idea: Many things that seem to be abstract also seem to have a beginning (and ending) in time, such as a language like Urdu. It may be tempting to say that such things exist in time but not in space, but where exactly? | |
| From: Chris Swoyer (Abstract Entities [2008], 1.1) | |
| A reaction: A few distinctions might be needed. Urdu-speaking is an ability of certain people. We abstract from that their 'language'. There is nothing there apart from that ability. It has no more abstract existence than the 'weather'. |
| 14594 | Ontologists seek existence and identity conditions, and modal and epistemic status for a thing [Swoyer] |
| Full Idea: Four things philosophers often want to know about a given sort of entity are: its existence conditions, its identity conditions, its modal status, and its epistemic status. | |
| From: Chris Swoyer (Abstract Entities [2008], 3) | |
| A reaction: I prefer 'modal profile' to 'modal status'. The 'existence conditions' sound rather epistemic. Why does the existence of anything require 'conditions' other than just existing? I suspect identity is irrelevant if humans aren't around. |
| 14595 | Can properties exemplify other properties? [Swoyer] |
| Full Idea: Can properties themselves exemplify properties? | |
| From: Chris Swoyer (Abstract Entities [2008], 3) | |
| A reaction: Since I espouse a rather strict causal view of true properties, and lump the rest into the category of 'predicates', I am inclined to answer 'no' to this. Most people would disagree. 'Bright red' seems to be an example. But it isn't. |
| 14593 | Quantum field theory suggests that there are, fundamentally, no individual things [Swoyer] |
| Full Idea: Quantum field theory strongly suggests that there are (at the fundamental level) no individual, particular things. | |
| From: Chris Swoyer (Abstract Entities [2008], 2.1) | |
| A reaction: When people introduce quantum theory into ontological discussions I reach for my shotgun, but it does rather look as if things turn to mush at the bottom level. |
| 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. |
| 22673 | Wherever there is a small community, the association of the people is natural [Tocqueville] |
| Full Idea: The village or township is the only association which is so perfectly natural that, wherever a number of men are collected, it seems to constitute itself. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.04) | |
| A reaction: Seems like a chicken and egg issue. I would have thought that association precedes the development of a village. |
| 22676 | The people are just individuals, and only present themselves as united to foreigners [Tocqueville] |
| Full Idea: The people in themselves are only individuals; and the special reason why they need to be united under one government is that they may appear to advantage before foreigners. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: I take this to be an observation on 1830s America, rather than a universal truth. It fits modern western societies rather well though. |
| 22679 | Vast empires are bad for well-being and freedom, though they may promote glory [Tocqueville] |
| Full Idea: Nothing is more opposed to the well-being and the freedom of men than vast empires. …But there is a love of glory in those who regard the applause of a great people as a worthy reward. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: Presumably the main the problem is the central dominance over distant colonies. There may also be some freedom in being distant from the centres, especially in 1830. The Wild West. |
| 22680 | People would be much happier and freer in small nations [Tocqueville] |
| Full Idea: If none but small nations existed, I do not doubt that mankind would be more happy and more free. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: In modern times many small states have appeared in Europe (in the Balkans and on the Baltic), and it looks to me a good thing. The prospect of Scottish independence may currently be looming, and De Tocqueville would approve. |
| 22675 | In American judges rule according to the Constitution, not the law [Tocqueville] |
| Full Idea: The Americans have acknowledged the right of judges to found their decisions on the Constitution, rather than on the laws. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.05) | |
| A reaction: Obviously the Constitution is one short document, so the details must be enshrined in the laws (which presumably defer to the Constitution). |
| 22677 | A monarchical family is always deeply concerned with the interests of the state [Tocqueville] |
| Full Idea: The advantages of a monarchy are that the private interests of a family are connected with the interests of the state, …and at least there is always someone available to conduct the affairs of a monarchy. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: The second one is not much of a reason! The same defence can be given for the dominance of the Mafia. His defences are deliberately feeble, I suspect. England had plenty of monarchs who showed limited interest. |
| 22683 | Despots like to see their own regulations ignored, by themselves and their agents [Tocqueville] |
| Full Idea: In despotic states the sovereign is so much attached to his power that he dislikes the constraints even of his own regulations, and likes to see his agents acting irregularly. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.11) | |
| A reaction: A nice observation. What would Machiavelli say? At least the citizens can see where the real power resides. |
| 22669 | Aristocracy is constituted by inherited landed property [Tocqueville] |
| Full Idea: Land is the basis of an aristocracy; …it is by landed property handed down from generation to generation that an aristocracy is constituted. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.01) | |
| A reaction: Presumably there can be aristocrats by mere royal patronage, who have perhaps gambled away their land. They need protection by the other aristocrats. |
| 22674 | In Europe it is thought that local government is best handled centrally [Tocqueville] |
| Full Idea: The partisans of centralisation in Europe are wont to maintain that the government can administer the affairs of each locality better than the citizens can do it for themselves. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.04) | |
| A reaction: In the modern UK we have lots of local government, which is thoroughly starved of funds by the central government. He is contrasting it with the strong local system in the U.S. |
| 22678 | An election, and its lead up time, are always a national crisis [Tocqueville] |
| Full Idea: The period which immediately precedes an election, and that during which the election is taking place, must always be considered as a national crisis. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: Rousseau said something similar. Election day in modern Britain is very peaceful and civilised, but it used to be chaotic. The weeks preceding it are invariably nasty. |
| 22682 | Universal suffrage is no guarantee of wise choices [Tocqueville] |
| Full Idea: Universal suffrage is by no means a guarantee of the wisdom of the popular choice. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.11) | |
| A reaction: This was precisely Plato's fear about democracy. There seems no way at all of preventing the people from electing representatives on superficial grounds of personality. |
| 22670 | Slavery undermines the morals and energy of a society [Tocqueville] |
| Full Idea: Slavery dishonours labour; it introduces idleness into society, and with idleness, ignorance and pride, luxury and distress. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.01) | |
| A reaction: A pretty feeble reason (in the 1830s) for disliking slavery. He seems only concerned with the adverse effects on the slave-owning society, and shows no interest in the slaves themselves. |
| 22681 | The liberty of the press is more valuable for what it prevents than what it promotes [Tocqueville] |
| Full Idea: I approve of the liberty of the press from a consideration more of the evils it prevents than of the advantages it ensures. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.10) | |
| A reaction: He accepts the freedom of the press as inevitable in a democracy, but he found U.S. newspapers to be nearly as bad then as they are now. |
| 22672 | It is admirable to elevate the humble to the level of the great, but the opposite is depraved [Tocqueville] |
| Full Idea: One manly and lawful passion for equality elevates the humble to the rank of the great. But there exists also a depraved taste for equality, which impels the weak to attempt to lower the powerful to their own level. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.02) | |
| A reaction: There is a distinction in modern political rhetoric between 'levelling down' and 'levelling up'. Since levelling down is just destructive, and levelling up is unaffordable, it seems obvious that true equality needs to be a compromise. |
| 22671 | Equality can only be established by equal rights for all (or no rights for anyone) [Tocqueville] |
| Full Idea: I know of only two methods of establishing equality in the political world; rights must be given to every citizen, or none at all to anyone. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.02) | |
| A reaction: We may have a vague concept of 'natural' rights, but primarily they are a tool of social engineering. You could grant equal rights on inheritance, for example, which turn out in practice to hugely favour the rich. |