29 ideas
| 24084 | Seeing with other eyes is more egoism, but exploring other perspectives leads to objectivity [Nietzsche] |
| Full Idea: Wanting to know things as they are - that alone is the good inclination: not seeing ..with other eyes; that would be merely a change of place of egoistic seeing. …Practise at seeing with other eyes, and without human relationships, hence objectively! | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 11[013]) | |
| A reaction: That Nietzsche thinks we should try to see things objectively will come as a bit of a shock to those who have him catalogued among the relativists. It's clear from other writings that he thinks (rightly) that perfect objectivity is unattainable. |
| 24092 | I tell the truth, even if it is repulsive [Nietzsche] |
| Full Idea: As a man I tell the truth, even the repulsive ones. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 12[86]) | |
| A reaction: I wonder if even Nietzsche had his limits. He is quite coy about sexual matters, for example, before Freud and various sexual revolutions. To ruthlessly tell difficult truths strikes me as a scientific approach to the world. |
| 2572 | Logical truth seems much less likely to 'correspond to the facts' than factual truth does [Haack] |
| Full Idea: It is surely less plausible to suppose that logical truth consists in correspondence to the facts than that 'factual' truth does. | |
| From: Susan Haack (Philosophy of Logics [1978], 7.6) |
| 2570 | The same sentence could be true in one language and meaningless in another, so truth is language-relative [Haack] |
| Full Idea: The definition of truth will have to be, Tarski argues, relative to a language, for one and the same sentence may be true in one language, and false or meaningless in another. | |
| From: Susan Haack (Philosophy of Logics [1978], 7.5) |
| 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. |
| 24089 | Essences are fictions needed for beings who represent things [Nietzsche] |
| Full Idea: The true essence of things is a fiction of representing being, without which being is unable to represent. 11[330] Thinking must assert substance and identity because a knowing of complete flux is impossible. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 11[329]) | |
| A reaction: I have defended (in my PhD) the thesis that the concept of essence is required for explanation. Do animals need the concept of essence in order to represent? I think people and animals ascribe essential natures to most things. |
| 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. |
| 24090 | Our inclinations would not conflict if we were a unity; we imagine unity for our multiplicity [Nietzsche] |
| Full Idea: How is it that we satisfy our stronger inclinations at the expense of our weaker inclinations? - In itself, if we were a unity, this split could not exist. In fact we are a multiplicity that has imagined a unity for itself. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 12[35]) | |
| A reaction: Plato had the same thought, but stopped at three parts, rather than a multiplicity. What Nietzsche fails to say, I think, is that this 'imagined' unity of the mind is not optional, and obviously has a real link to the one body and the one life. |
| 24087 | People who miss beauty seek the sublime, where even the ugly shows its 'beauty' [Nietzsche] |
| Full Idea: Whoever does not achieve the beautiful seeks the wildly sublime, because there even the ugly can show its 'beauty'. Likewise we seek the wildly sublime morality. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 11[049]) | |
| A reaction: Is the 'we' here Nietzsche, or the herd? The former, I guess, since some the values he likes seem rather ugly to me. He is a fan of war, for example. I'm guessing that massive destruction is sublime but ugly. |
| 24091 | The sublimity of nature which dwarfs us was a human creation [Nietzsche] |
| Full Idea: This beauty and sublimity of nature, before which every human being seems small, was first imposed on nature by us. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 12[38]) | |
| A reaction: I was struck when I was 10 with how indifferent to a landscape I was, when my mother told me it was 'beautiful'. Five years later I saw it differently. I assume nature is not intrinsically sublime. Dwarfed by our own concept is a bit odd. |
| 24093 | We can aspire to greatness by creating new functions for ourselves [Nietzsche] |
| Full Idea: To see the new greatness not above oneself, not outside oneself, but to make a new function from it. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 13[19]) | |
| A reaction: Thus we might combine the Aristotelian and the existentialist views! Do we discover our function or invent it? Anyone who acquires an expertise is creating a new function for themselves, presumably with a high value. |
| 24085 | For absolute morality a goal for mankind is needed [Nietzsche] |
| Full Idea: I deny absolute morality because I do not know an absolute goal of mankind. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 11[037]) | |
| A reaction: Christianity dreams of union of souls with God (clustering around God like goldfish to food, according to Dante). That is an absolute goal, justifying an absolute morality. If Aristotelians could identify human nature, its flourishing might be absolute. |
| 24094 | Humans are vividly aware of short-term effects, and almost ignorant of the long-term ones [Nietzsche] |
| Full Idea: How weakly human beings feel responsible for their indirect and distant effects! And how cruelly and exaggeratedly the closest effect that we exert pounces on us - the effect we see, for which our myopic vision is still just sharp enough! | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 15[11]) | |
| A reaction: This strikes me as both accurate and important, because consequentialist ethics is largely committed to judging by a very distorted image of their own objective. |
| 24086 | The goal is to settle human beings, like other animals, but humans are still changeable [Nietzsche] |
| Full Idea: Obviously the goal is to make human beings as steady and firm as most animal species; they have adapted to the conditions of the earth etc. and do not change essentially. The human being is still changeable - is still becoming. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 11[044]) | |
| A reaction: I favour an Aristotelian view, based on the flourishing of human nature, but this thought obviously challenges such a view. Great changes to a culture can make some difference to the apparent nature of people. |
| 24088 | See our present lives as eternal! Religions see it as fleeting, and aim at some different life [Nietzsche] |
| Full Idea: Let us press the image of eternity on our life! This thought contains more than all religions that despise this life as fleeting and taught us to look toward an unspecified different life. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 11[159]) | |
| A reaction: This is the best statement of the idea of eternal recurrence I have so far found. His ideal is to design a life for ourselves which we would be happy to see endlessly repeated. A lot of thought would have to go into that! |
| 24095 | Our growth is too subtle to perceive, and long events are too slow for us to grasp [Nietzsche] |
| Full Idea: The slowness of the events in the history of human beings is not suited to the human sense of time - and the subtlety and smallness of all growth defies human vision. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1881-82 [1882], 15[41]) | |
| A reaction: The only way we can study history is by 'periods'. It is as if English history has its slate wiped clean in 1066, 1485, 1603 and 1689. All historians know that the reality of it all is totally beyond our grasp. |