43 ideas
| 18290 | But what is the reasoning of the body, that it requires the wisdom you seek? [Nietzsche] |
| Full Idea: There is more reason in your body than in your best wisdom. For who knows for what purpose your body requires precisely your best wisdom? | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.05) | |
| A reaction: Lovely question. For years I've paid lip-service to wisdom as the rough aim of all philosophy. Not quite knowing what wisdom is doesn't bother me, but knowing why I want wisdom certainly does, especially after this idea. |
| 18303 | Reject wisdom that lacks laughter [Nietzsche] |
| Full Idea: Let that wisdom be false to us that brought no laughter with it! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 3.12.23) |
| 18305 | To love truth, you must know how to lie [Nietzsche] |
| Full Idea: Inability to lie is far from being love of truth. ....He who cannot lie does not know what truth is. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 4.13.9) |
| 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. |
| 21500 | We rely on memory for empirical beliefs because they mutually support one another [Lewis,CI] |
| Full Idea: When the whole range of empirical beliefs is taken into account, all of them more or less dependent on memorial knowledge, we find that those which are most credible can be assured by their mutual support, or 'congruence'. | |
| From: C.I. Lewis (An Analysis of Knowledge and Valuation [1946], 334), quoted by Erik J. Olsson - Against Coherence 3.1 | |
| A reaction: Lewis may be over-confident about this, and is duly attacked by Olson, but it seems to me roughly correct. How do you assess whether some unusual element in your memory was a dream or a real experience? |
| 21501 | If we doubt memories we cannot assess our doubt, or what is being doubted [Lewis,CI] |
| Full Idea: To doubt our sense of past experience as founded in actuality, would be to lose any criterion by which either the doubt itself or what is doubted could be corroborated. | |
| From: C.I. Lewis (An Analysis of Knowledge and Valuation [1946], 358), quoted by Erik J. Olsson - Against Coherence 3.3.1 | |
| A reaction: Obviously scepticism about memory can come in degrees, but total rejection of short-term and clear memories looks like a non-starter. What could you put in its place? Hyper-rationalism? Even maths needs memory. |
| 6556 | If anything is to be probable, then something must be certain [Lewis,CI] |
| Full Idea: If anything is to be probable, then something must be certain. | |
| From: C.I. Lewis (An Analysis of Knowledge and Valuation [1946], 186), quoted by Robert Fogelin - Walking the Tightrope of Reason Intro | |
| A reaction: Lewis makes this comment when facing infinite regress problems. It is a very nice slogan for foundationalism, which embodies the slippery slope view. Personally I feel the emotional pull of foundations, but acknowledge the very strong doubts about them. |
| 21498 | Congruents assertions increase the probability of each individual assertion in the set [Lewis,CI] |
| Full Idea: A set of statements, or a set of supposed facts asserted, will be said to be congruent if and only if they are so related that the antecedent probability of any one of them will be increased if the remainder of the set can be assumed as given premises. | |
| From: C.I. Lewis (An Analysis of Knowledge and Valuation [1946], 338), quoted by Erik J. Olsson - Against Coherence 2.2 | |
| A reaction: This thesis is vigorously attacked by Erik Olson, who works through the probability calculations. There seems an obvious problem without that. How else do you assess 'congruence', other than by evidence of mutual strengthening? |
| 20757 | The powerful self behind your thoughts and feelings is your body [Nietzsche] |
| Full Idea: Behind your thoughts and feelings stands a powerful commander, an unknown wise man - he is called a self. He lives in your body; he is your body. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], I.4), quoted by Kevin Aho - Existentialism: an introduction 5 'Creature' | |
| A reaction: I find Nietzsche's view of the self very congenial, though I tend to see the self as certain central functions of the brain. The brain is enmeshed in the body (as in the location of pains). |
| 18289 | Forget the word 'I'; 'I' is performed by the intelligence of your body [Nietzsche] |
| Full Idea: You say 'I' and you are proud of this word. But greater than this - although you will not believe in it - is your body and its great intelligence, which does not say 'I' but performs 'I'. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.05) | |
| A reaction: I'm not sure if I understand this, but I offer it as a candidate for the most profound idea ever articulated about personal identity. |
| 5828 | Extension is the class of things, intension is the correct definition of the thing, and intension determines extension [Lewis,CI] |
| Full Idea: "The denotation or extension of a term is the class of all actual or existent things which the term correctly applies to or names; the connotation or intension of a term is delimited by any correct definition of it." ..And intension determines extension. | |
| From: C.I. Lewis (An Analysis of Knowledge and Valuation [1946]), quoted by Stephen P. Schwartz - Intro to Naming,Necessity and Natural Kinds §II | |
| A reaction: The last part is one of the big ideas in philosophy of language, which was rejected by Putnam and co. If you were to reverse the slogan, though, (to extension determines intension) how would you identify the members of the extension? |
| 18299 | The will is constantly frustrated by the past [Nietzsche] |
| Full Idea: Powerless against that which has been done, the will is an angry spectator of all things past. The will cannot will backwards; that it cannot break time and time's desire - that is the will's most lonely affliction. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 2.20) |
| 18297 | We created meanings, to maintain ourselves [Nietzsche] |
| Full Idea: Man first implanted values into things to maintain himself - he first created the meaning of things, a human meaning! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.16) | |
| A reaction: It is certainly hard to see anything resembling values or meaning in the cosmos, if you remove the human beings. We should expect an evolutionary grounding in their explanation. |
| 18293 | The noble man wants new virtues; the good man preserves what is old [Nietzsche] |
| Full Idea: The noble man wants to create new things and a new virtue. The good man wants the old things and that the old things shall be preserved. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.09) | |
| A reaction: There is a limit to how many plausible virtues the noble men can come up with. We may already have run out. Are we going to have to re-run the Iliad? |
| 18301 | We only really love children and work [Nietzsche] |
| Full Idea: One loves from the very heart only one's child and one's work. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 3.03) | |
| A reaction: Very Nietzchean (and masculine?) to cite one's work. Rachmaninov said he was 85% musician and 15% human being, so I guess he loved music from the very heart. |
| 18307 | I want my work, not happiness! [Nietzsche] |
| Full Idea: Do I aspire after happiness? I aspire after my work! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 4.20) | |
| A reaction: I empathise with aspiring to do something, rather than be something. But what do we wish for our children? Happiness first, then achievement? |
| 18291 | Virtues can destroy one another, through jealousy [Nietzsche] |
| Full Idea: Every virtue is jealous of the others, and jealousy is a terrible thing. Even virtues can be destroyed through jealousy. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.07) | |
| A reaction: How much more subtle and plausible than the picture of accumulating virtues, like medals! Zarathustra says it is best to have just one virtue. |
| 18287 | People now find both wealth and poverty too much of a burden [Nietzsche] |
| Full Idea: Nobody grows rich or poor any more: both are too much of a burden. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.01) | |
| A reaction: True. Most people I know are just puzzled by people who actually seem to want to be extremely wealthy. |
| 18295 | If you want friends, you must be a fighter [Nietzsche] |
| Full Idea: If you want a friend, you must be willing to wage war for him: and to wage war, you must be capable of being an enemy. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.15) |
| 18286 | The greatest experience possible is contempt for your own happiness, reason and virtue [Nietzsche] |
| Full Idea: What is the greatest thing you can experience? It is the hour of the great contempt. The hour in which even your happiness grows loathsome to you, and your reason and your virtue also. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.01) | |
| A reaction: This would be a transient state for Nietzsche, in which you realise the hollowness of those traditional ideas, and begin to seek something else. |
| 18296 | An enduring people needs its own individual values [Nietzsche] |
| Full Idea: No people could live without evaluating; but if it wishes to maintain itself it must not evaluate as its neighbour evaluates. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.16) | |
| A reaction: Political philosophers say plenty about a 'people', but little about what unifies them, or about what keeps one people distinct from another. Most people's are proud of their local values. |
| 18294 | The state coldly claims that it is the people, but that is a lie [Nietzsche] |
| Full Idea: The state is the coldest of all cold monsters. Coldly it lies, too; and this lie creeps from its mouth: 'I, the state, am the people'. It is a lie! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.12) | |
| A reaction: This strikes me as just as true even after everyone gets the vote. Rulers can't help gradually forgetting about the people. |
| 18304 | Saints want to live as they desire, or not to live at all [Nietzsche] |
| Full Idea: 'To live as I desire to live or not to live at all': that is what I want, that is what the most saintly man wants. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 4.09) | |
| A reaction: [spoken by Zarathustra] |
| 18300 | Whenever we have seen suffering, we have wanted the revenge of punishment [Nietzsche] |
| Full Idea: The spirit of revenge: my friends, that, up to now, has been mankind's chief concern; and where there was suffering, there was always supposed to be punishment. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 2.20) |
| 18302 | Man and woman are deeply strange to one another! [Nietzsche] |
| Full Idea: Who has fully conceived how strange man and woman are to one another! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 3.10.2) |
| 18292 | I can only believe in a God who can dance [Nietzsche] |
| Full Idea: I should believe only in a God who understood how to dance. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.08) |
| 18298 | Not being a god is insupportable, so there are no gods! [Nietzsche] |
| Full Idea: If there were gods, how could I endure not to be a god! Therefore there are no gods. ...For what would there to be create if gods - existed! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 2.02) | |
| A reaction: [Zarathustra says this, not Nietzsche!] |
| 18288 | Heaven was invented by the sick and the dying [Nietzsche] |
| Full Idea: It was the sick and dying who despised the body and the earth and invented the things of heaven and the redeeming drops of blood. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.04) |
| 18306 | We don't want heaven; now that we are men, we want the kingdom of earth [Nietzsche] |
| Full Idea: We certainly do not want to enter into the kingdom of heaven: we have become men, so we want the kingdom of earth. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 4.18.2) |