26 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. |
304 | Beautiful things must be different from beauty itself, but beauty itself must be present in each of them [Plato] |
Full Idea: Are fine things different from or identical to fineness? They are different from fineness itself, but fineness itself is in a sense present in each of them. | |
From: Plato (Euthydemus [c.379 BCE], 301a) |
16120 | Knowing how to achieve immortality is pointless without the knowledge how to use immortality [Plato] |
Full Idea: If there exists the knowledge of how to make men immortal, but without the knowledge of how to use this immortality, there seems to be no value in it. | |
From: Plato (Euthydemus [c.379 BCE], 289b) | |
A reaction: I take this to be not a gormless utilitarianism about knowledge, but a plea for holism, that knowledge only has value as part of some larger picture. The big view is the important view. He's wrong, though. Work out the use later. |
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. |
303 | Say how many teeth the other has, then count them. If you are right, we will trust your other claims [Plato] |
Full Idea: If each of you says how many teeth the other has, and when they are counted we find you do know, we will believe your other claims as well. | |
From: Plato (Euthydemus [c.379 BCE], 294c) | |
A reaction: This is the clairvoyant problem for reliabilism, if truth is delivered for no apparent reason. Useful, but hardly knowledge. HOW did you know the number of teeth? |
302 | What knowledge is required to live well? [Plato] |
Full Idea: What knowledge would enable us to live finely for the rest of our lives? | |
From: Plato (Euthydemus [c.379 BCE], 293a) | |
A reaction: A successful grasp of other people's points of view might lead to respect for them. Also a realisation that we are not isolated individuals. We really are all in it together. |
301 | Only knowledge of some sort is good [Plato] |
Full Idea: Nothing is good except knowledge of some sort. | |
From: Plato (Euthydemus [c.379 BCE], 292b) | |
A reaction: I've heard it suggested that truth is the only value. This is the Socratic idea that moral goodness is a matter of successful rational judgement. Not convinced, but interesting. |
305 | Something which lies midway between two evils is better than either of them [Plato] |
Full Idea: Something which is composed of two factors which are bad for different purposes and lies midway between them is better than either of the factors. | |
From: Plato (Euthydemus [c.379 BCE], 306a) |
23831 | The essence of power is illusory prestige [Weil] |
Full Idea: Prestige, which is an illusion, is the very essence of power. | |
From: Simone Weil (The Power of Words [1934], p.255) | |
A reaction: It is hard to maintain illusory prestige if there is no actual power behind it. |
23830 | A group is only dangerous if it endorses an abstract entity [Weil] |
Full Idea: Any group which has not secreted an abstract entity will probably not be dangerous. | |
From: Simone Weil (The Power of Words [1934], p.255) | |
A reaction: Written in the 1930s, the era of many political -isms. No group can be united if it lacks a clear label, and a few simple slogans. |
23829 | National leaders want to preserve necessary order - but always the existing order [Weil] |
Full Idea: Those in command see their duty as defending order, without which no social life can survive; and the only order they conceive is the existing one. | |
From: Simone Weil (The Power of Words [1934], p.249) | |
A reaction: She sympathises with them, because a new order is such an unknown. But it always struck me as weird that traditions are preserved because they are traditions, and not because they are good. (My old school, for example!). |
23828 | National prestige consists of behaving as if you could beat the others in a war [Weil] |
Full Idea: What is called national prestige consists in behaving always in such a way as to demoralise other nations by giving them the impression that, if it comes to war, one would certainly defeat them. | |
From: Simone Weil (The Power of Words [1934], p.244) | |
A reaction: It's true. No nation gains prestige because of the happy lives of its citizens, or the creativity of its culture. |
23827 | Modern wars are fought in the name of empty words which are given capital letters [Weil] |
Full Idea: For our contemporaries the role of Helen in the Trojan War is is played by words with capital letters. …When empty words are given capital letters, then, on the slightest pretext, men will begin shedding blood for them and piling up ruin in their name. | |
From: Simone Weil (The Power of Words [1934], p.241) | |
A reaction: This seems particularly true of the 1930s, where specific dogmatic ideologies seemed to grip and divide people. Simple aggressive nationalism seems to be the cause of current wars, now the fear of Communism has diminished. |