27 ideas
| 9307 | Modern Western culture suddenly appeared in Jena in the 1790s [Svendsen] |
| Full Idea: Foucault was right to say that Jena in the 1790s was the arena where the fundamental interests in modern Western culture suddenly had their breakthrough. | |
| From: Lars Svendsen (A Philosophy of Boredom [2005], Ch.2) | |
| A reaction: [Hölderlin, Novalis, Tieck, Schlegel, based on Kant and Fichte] Romanticism seems to have been born then. Is that the essence of modernism? Foucault and his pals are hoping to destroy the Enlightenment by ignoring it, but that is modern too. |
| 9297 | You can't understand love in terms of 'if and only if...' [Svendsen] |
| Full Idea: I once began reading a philosophical article on love. The following statement soon came up: 'Bob loves Kate if and only if...' At that point I stopped reading. Such a formalized approach was unsuitable, because the actual phenomenon would be lost. | |
| From: Lars Svendsen (A Philosophy of Boredom [2005], Pref) | |
| A reaction: It is hard to disagree! However, if your best friend comes to you and says, 'I can't decide whether I am really in love with Kate; what do you think?', how are you going to respond. You offer 'if and only if..', but in a warm and sympathetic way! |
| 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. |
| 9308 | If subjective and objective begin to merge, then so do primary and secondary qualities [Svendsen] |
| Full Idea: It is doubtful whether the traditional dichotomy between the strictly subjective and the strictly objective can still be maintained; if not, we must also revise the distinction between primary and secondary qualities. | |
| From: Lars Svendsen (A Philosophy of Boredom [2005], Ch.3) | |
| A reaction: Very perceptive. The reason why I am so keen to hang onto the primary/secondary distinction is because I want to preserve objectivity (and realism). I much prefer Locke to Hume, as empiricist spokesmen. |
| 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. |
| 9309 | Emotions have intentional objects, while a mood is objectless [Svendsen] |
| Full Idea: An emotion normally has an intentional object, while a mood is objectless. | |
| From: Lars Svendsen (A Philosophy of Boredom [2005], Ch.3) | |
| A reaction: It doesn't follow that the object of the emotion is clearly understood, or even that it is conscious. One may experience rising anger while struggling to see what its object is. Artistic symbolism seems to involve objects that create moods. |
| 22187 | Genetic behaviours that have enhanced human success include aggression, rape and xenophobia [Wilson,EO, by Okasha] |
| Full Idea: Wilson claimed that many human behaviours, including aggression, rape, and xenophobia, had a genetic basis, and were adaptations favoured by natural selection because they enhanced the reproductive success of our ancestors. | |
| From: report of Edmund O. Wilson (Sociobiology [1975]) by Samir Okasha - Philosophy of Science: Very Short Intro (2nd ed) 7 | |
| A reaction: This led to the Sociobiology Wars, when E.O. Wilson was attacked by Richard Lewontin and Stephen Jay Gould. |
| 9304 | Death appears to be more frightening the less one has lived [Svendsen] |
| Full Idea: Death appears to be more frightening the less one has lived. | |
| From: Lars Svendsen (A Philosophy of Boredom [2005], Ch.2) | |
| A reaction: [He credits Adorno with this] A good thought, which should be immediately emailed to Epicurus for comment. Which is worse - to die when you have barely started your great work (Ramsey), or dying in full flow (Schubert)? |
| 9298 | We can be unaware that we are bored [Svendsen] |
| Full Idea: It is perfectly possible to be bored without being aware of the fact. | |
| From: Lars Svendsen (A Philosophy of Boredom [2005], Ch.1) | |
| A reaction: True. Also, I sometimes mistake indecision for boredom. It becomes very hard to say for certain whether you are bored. I am certain that I am bored if I am forced to do something which has no interest for me. The big one is free-but-bored. |
| 9301 | Boredom is so radical that suicide could not overcome it; only never having existed would do it [Svendsen] |
| Full Idea: Boredom is so radical that it cannot even be overcome by suicide, only by something completely impossible - not to have existed at all. | |
| From: Lars Svendsen (A Philosophy of Boredom [2005], Ch.1) | |
| A reaction: [he cites Fernando Pessoa for this] The actor George Sanders left a suicide note saying that he was just bored. A cloud of boredom is left hanging in the air where he was. |
| 9302 | We are bored because everything comes to us fully encoded, and we want personal meaning [Svendsen] |
| Full Idea: Boredom results from a lack of personal meaning, which is due to the fact that all objects and actions come to us fully encoded, while we (as descendants of Romanticism) insist on a personal meaning. | |
| From: Lars Svendsen (A Philosophy of Boredom [2005], Ch.2) | |
| A reaction: This idea justifies me categorising Boredom under Existentialism. This is an excellent idea, and perfectly captures the experience of most teenagers, for whom it is impossible to impose a personal meaning on such a vast cultural reality. |
| 9310 | The profoundest boredom is boredom with boredom [Svendsen] |
| Full Idea: In the profound form of boredom, I am bored by boredom itself. | |
| From: Lars Svendsen (A Philosophy of Boredom [2005], Ch.3) | |
| A reaction: Boredom is boring, which is why I try to avoid it. Third-level boredom is a rather enchanting idea. It sounds remarkably similar to the Buddha experiencing enlightenment. |
| 9311 | We have achieved a sort of utopia, and it is boring, so that is the end of utopias [Svendsen] |
| Full Idea: There can hardly be any new utopias. To the extent that we can imagine a utopia, it must already have been realised. A utopia cannot, by definition, include boredom, but the 'utopia' we are living in is boring. | |
| From: Lars Svendsen (A Philosophy of Boredom [2005], Ch.4) | |
| A reaction: Compare Idea 8989. Lots of people (including me) think that we have achieved a kind of liberal, democratic, individualistic 'utopia', but the community needs of people are not being met, so we still have a way to go. |
| 9303 | The concept of 'alienation' seems no longer applicable [Svendsen] |
| Full Idea: I do not believe that the concept of 'alienation' is all that applicable any more. | |
| From: Lars Svendsen (A Philosophy of Boredom [2005], Ch.1) | |
| A reaction: Interesting but puzzling. If alienation is the key existential phenomenon of a capitalist society, why should it fade away if we remain capitalist? He is proposing that it has metamorphosed into boredom, which may be a different sort of alienation. |