25 ideas
4424 | A warlike philosopher challenges problems to single combat [Nietzsche] |
Full Idea: A warlike philosopher challenges problems to single combat. | |
From: Friedrich Nietzsche (Ecce Homo [1889], Wise §7) | |
A reaction: And what do pacifist philosophers do? It is a moot point whether philosophy is even possible without a streak of aggression. Otherwise you circle the problem, but don't confront it. |
21829 | Philosophy aims to understand how things (broadly understood) hang together (broadly understood) [Sellars] |
Full Idea: The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term. | |
From: Wilfrid Sellars (Philosophy and Scientific Image of Man [1962], p.3), quoted by Owen Flanagan - The Really Hard Problem 1 'Vocation' | |
A reaction: I'm happier with broad things than broad hanging together, but to me this sounds about right. |
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. |
6550 | Reduction requires that an object's properties consist of its constituents' properties and relations [Sellars] |
Full Idea: The 'Principle of Reducibility' says if an object is a system of objects, then every property of the object must consist in the fact that its constituents have such and such qualities and such and such relations | |
From: Wilfrid Sellars (Philosophy and Scientific Image of Man [1962], p.27), quoted by William Lycan - Consciousness | |
A reaction: This sounds to me a more promising attitude to reduction than all this talk of Ernest Nagel's 'Bridge Laws'. If we ask HOW a higher level property arises because of a lower level property, we can describe a mechanism rather than a law. |
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. |
2886 | The distinction between egoistic and non-egoistic acts is absurd [Nietzsche] |
Full Idea: There are neither egoistic nor unegoistic actions: both concepts are psychologically nonsense. | |
From: Friedrich Nietzsche (Ecce Homo [1889], 4.5) | |
A reaction: Not quite true, but I like this observation. The idea that you could divide everyone's actions into these two groups is certainly nonsense. But some people are more altruistic than others! |
4426 | A bad result distorts one's judgement about the virtue of what one has done [Nietzsche] |
Full Idea: I should prefer to exclude the bad result, the consequences, from the question of value as a matter of principle. Faced with a bad result, one loses all too easily the right perspective for what one has done. | |
From: Friedrich Nietzsche (Ecce Homo [1889], Clever §1) | |
A reaction: If the perspective is easily lost, we should make more effort, not ignore consequences. The question is whether you could have foreseen or controlled the consequences. |
4425 | The overcoming of pity I count among the noble virtues [Nietzsche] |
Full Idea: The overcoming of pity I count among the noble virtues. | |
From: Friedrich Nietzsche (Ecce Homo [1889], Wise §4) | |
A reaction: Hm. I can just about see that there might be more important things than compassion for suffering, but I can't see any human activity that makes it worthwhile to trample on pity. |
20132 | To become what you are you must have no self-awareness [Nietzsche] |
Full Idea: To become what one is, one must not have the faintest notion of what one is. | |
From: Friedrich Nietzsche (Ecce Homo [1889], II.9), quoted by Brian Leiter - Nietzsche On Morality 3 'fatalism' | |
A reaction: [Don't understand 'II.9'] Enigmatic but striking. As I understand it, Nietzsche thought that knowing what you are is virtually impossible, though he spent a lifetime studying himself. Would you recognise someone who had become what they are? |
20144 | Eternal recurrence is the highest attainable affirmation [Nietzsche] |
Full Idea: Eternal recurrence is the highest formula of affirmation that is at all attainable. | |
From: Friedrich Nietzsche (Ecce Homo [1889], Zar 1) | |
A reaction: Did Nietzsche have in mind an even higher formulation that was unattainable? The aim of eternal recurrence is to offer the highest possible ideal that remains rooted in the nature of ordinary life. It is a cut-down version of the Form of the Good. |
2889 | One repays a teacher badly if one remains only a pupil [Nietzsche] |
Full Idea: One repays a teacher badly if one remains only a pupil. | |
From: Friedrich Nietzsche (Ecce Homo [1889], Fore) |
2887 | I am not an atheist because of reasoning or evidence, but because of instinct [Nietzsche] |
Full Idea: I have absolutely no knowledge of atheism as an outcome of reasoning, still less an event: with me it is obvious by instinct. | |
From: Friedrich Nietzsche (Ecce Homo [1889], 3.1) |