93 ideas
| 18330 | Judging by the positive forces, the Renaissance was the last great age [Nietzsche] |
| Full Idea: Ages are to be assessed by their positive forces - and by this assessment the age of the Renaissance, so prodigal and so fateful, appears as the last great age. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 8.37) | |
| A reaction: I suspect that Nietzsche places art very high among the positive forces. Science and technology showed barely a glimmer during the Renaissance. Mathematics moved very little, Copernicus was ignored, and logic was static. |
| 2900 | I revere Heraclitus [Nietzsche] |
| Full Idea: I set apart with high reverence the name of Heraclitus. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 2.2) |
| 2913 | Thucydides was the perfect anti-platonist sophist [Nietzsche] |
| Full Idea: My recreation, my preference, my cure from all Platonism has always been Thucydides. …Sophist culture, by which I mean realist culture, attains in him its perfect expression. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 9.2) |
| 2909 | Thinking has to be learned in the way dancing has to be learned [Nietzsche] |
| Full Idea: Thinking has to be learned in the way dancing has to be learned. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 7.7) | |
| A reaction: Nice. At its deepest level thinking isn't a rational process? |
| 2892 | Wanting a system in philosophy is a lack of integrity [Nietzsche] |
| Full Idea: I mistrust all systematizers and avoid them. The will to system is a lack of integrity. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], Maxim 26) |
| 2896 | I want to understand the Socratic idea that 'reason equals virtue equals happiness' [Nietzsche] |
| Full Idea: I seek to understand out of what idiosyncrasy that Socratic equation 'reason equals virtue equals happiness' derives. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 1.04) |
| 2897 | With dialectics the rabble gets on top [Nietzsche] |
| Full Idea: With dialectics the rabble gets on top. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 1.05) |
| 2898 | Anything which must first be proved is of little value [Nietzsche] |
| Full Idea: What has first to have itself proved is of little value. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 1.05) |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 18317 | The 'real being' of things is a nothingness constructed from contradictions in the actual world [Nietzsche] |
| Full Idea: The characteristics which have been assigned to the 'real being' of things are the characteristics of non-being, of nothingness - the 'real world has been constructed out of the contradiction of the actual world. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 2.6) | |
| A reaction: I take this to be a critique of Hegel, in particular. Could we describe the metaphysics of Nietzsche as 'constructivist'? I certainly think he is underrated as a metaphysician, because the ideas are so fragmentary. |
| 18315 | We get the concept of 'being' from the concept of the 'ego' [Nietzsche] |
| Full Idea: Being is everywhere thought in, foisted on, as cause; it is only from the conception 'ego' that there follows, derivatively, the concept 'being'. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 2.5) | |
| A reaction: 'Being' is such a remote abstraction that I doubt whether we can say anything at all meaningful about where it 'comes from'. |
| 18316 | The grounds for an assertion that the world is only apparent actually establish its reality [Nietzsche] |
| Full Idea: The grounds upon which 'this' world has been designated as apparent establish rather its reality - another kind of reality is absolutely undemonstrable. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 2.6) |
| 18314 | In language we treat 'ego' as a substance, and it is thus that we create the concept 'thing' [Nietzsche] |
| Full Idea: It is the metaphysics of language (that is, of reason) ....which believes in the 'ego', in the ego as being, in the ego as substance, and which projects its belief in the ego-substance on to all things - only thus does it create the concept 'thing'. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 2.5) |
| 18309 | The evidence of the senses is falsified by reason [Nietzsche] |
| Full Idea: 'Reason' is the cause of our falsification of the evidence of the senses. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 2.1) | |
| A reaction: One for McDowell. |
| 18323 | Any explanation will be accepted as true if it gives pleasure and a feeling of power [Nietzsche] |
| Full Idea: To trace something unknown back to something known is alleviating, soothing, gratifying and gives moreover a feeling of power. ...First principle: any explanation is better than none. ...Proof by pleasure ('by potency') as criterion of truth. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 5.5) | |
| A reaction: By 'proof by pleasure' he means that we find an explanation so satisfying that we cling to it. I assume it is a criterion of rationality (an epistemic virtue) to reject the principle 'any explanation is better than none'. Negative capability. |
| 18310 | The 'highest' concepts are the most general and empty concepts [Nietzsche] |
| Full Idea: The 'highest concepts' ...are the most general, the emptiest concepts, the last fumes of evaporating reality. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 2.4) | |
| A reaction: This could be seen as an attack on the aspirations of all of philosophy, which seeks general truths out of the chaos of experience. Should we shut up, then, and just be and do? |
| 3238 | 'Dead person' isn't a contradiction, so 'person' is somewhat vague [Williams,B] |
| Full Idea: If we say (in opposition to a physical view of identity) that when Jones dies 'Jones ceases to exist' but 'Jones' body does not cease to exist', this shouldn't be pressed too hard, because it would make 'dead person' a contradiction. | |
| From: Bernard Williams (Are Persons Bodies? [1970], p.74) | |
| A reaction: A good point, which nicely challenges the distinction between a 'human' and a 'person', but the problem case is much more the one where Jones gets advanced Alzheimer's, rather than dies. A dead body ceases as a mechanism, as well as as a personality. |
| 3239 | You can only really love a person as a token, not as a type [Williams,B] |
| Full Idea: If you love a person as a type instead of as a token (i.e. a "person", instead of a physical body) you might prefer a run-down copy of them to no person at all, but at this point our idea of loving a person begins to crack. | |
| From: Bernard Williams (Are Persons Bodies? [1970], p.81) | |
| A reaction: Very persuasive. If you love a person you can cope with them getting old. If you own an original watercolour, you can accept that it fades, but you would replace a reproduction of it if that faded. But what, then, is it that you love? |
| 20368 | There are no 'individual' persons; we are each the sum of humanity up to this moment [Nietzsche] |
| Full Idea: The 'individual' ...is an error: he does not constitute a separate entity, an atom, a 'link in the chain', something merely inherited from the past - he constitutes the entire single line 'man' up to and including himself. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 8.33) | |
| A reaction: I'm not sure I understand this, but you can sort of imagine yourself as a culmination of something, rather than as an isolated entity. I'm not sure how that is supposed to affect my behaviour. |
| 2899 | The fanatical rationality of Greek philosophy shows that they were in a state of emergency [Nietzsche] |
| Full Idea: The fanaticism with which the whole of Greek thought throws itself at rationality betrays itself as a state of emergency: one was in peril, one had only one choice: either to perish or- be absurdly rational. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 1.10) |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| Full Idea: Cantor (in his exploration of infinities) pushed the bounds of conceivability further than anyone before him. To discover what is conceivable, we have to enquire into the concept. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics 6.5 | |
| A reaction: This remark comes during a discussion of Husserl's phenomenology. Intuitionists challenge Cantor's claim, and restrict what is conceivable to what is provable. Does possibility depend on conceivability? |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| Full Idea: Cantor thought that we abstract a number as something common to all and only those sets any one of which has as many members as any other. ...However one wants to see the logic of the inference. The irony is that set theory lays out this logic. | |
| From: comment on George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: The logic Hart has in mind is the notion of an equivalence relation between sets. This idea sums up the older and more modern concepts of abstraction, the first as psychological, the second as logical (or trying very hard to be!). Cf Idea 9145. |
| 18313 | The big error is to think the will is a faculty producing effects; in fact, it is just a word [Nietzsche] |
| Full Idea: At the beginning stands the great fateful error that the will is something which produces an effect - that will is a faculty.... Today we know it is merely a word. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 2.5) | |
| A reaction: This is despite Nietzsche's insistence that 'will to power' is the central fact of active existence. The misreading of Nietzsche is to think that this refers to the conscious exercising of a mental faculty. |
| 20133 | The 'motive' is superficial, and may even hide the antecedents of a deed [Nietzsche] |
| Full Idea: The so-called 'motive' is another error. Merely a surface phenomenon of consciousness - something alongside the deed which is more likely to cover up the antecedents of the deed than to represent them. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 6.3) | |
| A reaction: [Leiter gives 'VI.3', but I can't find it] As far as you can get from intellectualism about action, and is more in accord with the picture found in modern neuro-science. No one knows why they are 'interested' in something, and that's the start of it. |
| 18326 | The beautiful never stands alone; it derives from man's pleasure in man [Nietzsche] |
| Full Idea: Anyone who tried to divorce the beautiful from man's pleasure in man would at once feel the ground give way beneath him. The 'beautiful in itself' is not even a concept, merely a phrase. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 8.19) | |
| A reaction: I love the insult 'not even a concept'! It's like Pauli's 'not even wrong'! |
| 20101 | Without music life would be a mistake [Nietzsche] |
| Full Idea: Without music life would be a mistake. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], Maxim 33) | |
| A reaction: Cf Schopenhauer in Idea 21469. If you, dear reader, don't love music, then I sincerely hope that there is something in your life which can match it. |
| 2902 | Healthy morality is dominated by an instinct for life [Nietzsche] |
| Full Idea: All naturalism in morality, that is all healthy morality, is dominated by an instinct for life. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 4.4) | |
| A reaction: Sounds right. There is no reasoning against a moral nihilist, because they seem to have no instinct in favour of life. It is the given of morality. |
| 18311 | Philosophers hate values having an origin, and want values to be self-sufficient [Nietzsche] |
| Full Idea: For philosophers, the higher must not be allowed to grow out of the lower, must not be allowed to have grown at all ...Moral: everything of the first rank must be causa sui. Origin in something else counts as an objection, as casting a doubt on value. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 2.4) | |
| A reaction: This is so deep and central that I wrote a paper on it, advocating that the theory of values should focus of value-makers. |
| 18324 | There are no moral facts, and moralists believe in realities which do not exist [Nietzsche] |
| Full Idea: An insight formulated by me: that there are no moral facts whatever. Moral judgement has this in common with religious judgement that it believes in realities which do not exist. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 6.1) | |
| A reaction: Not only a slogan for non-cognitivism, but also a clear statement of the error theory about morality, a century before John Mackie. |
| 2904 | The doctrine of free will has been invented essentially in order to blame and punish people [Nietzsche] |
| Full Idea: The doctrine of will has been invented essentially for the purpose of punishment, that is of finding guilty. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 5.7) | |
| A reaction: Michael Frede says free will was invented to feel wholly in charge of our own actions. I doubt whether punishment was the first motive. The will just gives a simple explanation of actions. |
| 18322 | When we establish values, that is life itself establishing them, through us [Nietzsche] |
| Full Idea: When we speak of values we do so under the inspiration and from the perspective of life: life itself evaluates through us when we establish values | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 4.5) | |
| A reaction: I love Nietzsche's ideas about the source of values, and his remarks about the value of life. Other thinkers sound so simplistic in comparison. |
| 2893 | In every age the wisest people have judged life to be worthless [Nietzsche] |
| Full Idea: In every age the wisest have passed the identical judgement on life: it is worthless. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 1.01) | |
| A reaction: I guess he was having a bad day. Since the whole universe is clearly 'worthless', this judgement must in some sense be correct. But I love my books. |
| 2894 | Value judgements about life can never be true [Nietzsche] |
| Full Idea: Judgements, value judgements concerning life, for or against it, can in the last resort never be true. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 1.02) | |
| A reaction: I suppose this is in the same spirit as judging whether celery tastes nice. Are you for or against the Moon? |
| 18321 | To evaluate life one must know it, but also be situated outside of it [Nietzsche] |
| Full Idea: One would have to be situated outside life ....[and yet know it thoroughly] ....to be permitted to touch on the problem of the value of life at all. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 4.5) | |
| A reaction: Can practising artists question the value of their art? The whole point of objectivity is that we can mentally step 'outside' of something, without actually withdrawing from it. |
| 18308 | A philosopher fails in wisdom if he thinks the value of life is a problem [Nietzsche] |
| Full Idea: For a philosopher to see a problem in the value of life thus even constitutes an objection to him, a question-mark as to his wisdom, a piece of unwisdom. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 1.02) | |
| A reaction: I take his point to be neither that life is unquestionably valuable nor that it is valueless, but that the very question is ridiculous. If we live, we value living. Sounds right. |
| 2895 | The value of life cannot be estimated [Nietzsche] |
| Full Idea: The value of life cannot be estimated. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 1.02) | |
| A reaction: Military leaders apparently judge that the death of one of their own soldiers is worth between 12 and 20 enemy deaths (so history suggests). How about ransom money? |
| 18319 | Love is the spiritualisation of sensuality [Nietzsche] |
| Full Idea: The spiritualization of sensuality is called 'love': it is a great triumph over Christianity. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 4.3) | |
| A reaction: I'm not quite clear what 'spiritualization' means, particularly when it comes from Nietzsche. |
| 2903 | A good human will be virtuous because they are happy [Nietzsche] |
| Full Idea: A well-constituted human being, a 'happy one', must perform certain actions and shrink from other actions. In a formula: his virtue is the consequence of his happiness. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 5.2) | |
| A reaction: A nice reversal of basic Aristotle, though Aristotle does say that the truly virtuous person is happy in their actions. Treat unhappy people with caution! |
| 2891 | Only the English actually strive after happiness [Nietzsche] |
| Full Idea: Man does not strive after happiness; only the Englishman does that. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], Maxim 12) | |
| A reaction: The Danes keeping being voted the happiest nation, so presumably that results from some sort of effort on their part. The easiest is happiness is to achieve security, then do nothing. |
| 18327 | A wholly altruistic morality, with no egoism, is a thoroughly bad thing [Nietzsche] |
| Full Idea: An 'altruistic' morality, a morality under which egoism languishes - is under all circumstances a bad sign. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 8.35) |
| 15606 | Military idea: what does not kill me makes me stronger [Nietzsche] |
| Full Idea: From the military school of life. - What does not kill me makes me stronger. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], Maxim 08) | |
| A reaction: The published version! Perhaps the most famous remark in all of Nietzsche, and no one realises it is ironic! It is a sarcastic remark about the battering ram mentality of the Prussian militarist! He had served in the army. |
| 18328 | Invalids are parasites [Nietzsche] |
| Full Idea: The invalid is a parasite on society. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 8.36) | |
| A reaction: I'll skip the rest, but you get the idea. The point (with which I sympathise) is that life is primarily about what healthy people do. Something has gone wrong if all we do is worry about the sick and the suffering. |
| 18331 | Democracy is organisational power in decline [Nietzsche] |
| Full Idea: Democracy has always been the declining form of the power to organise. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 8.39) | |
| A reaction: Even when Nietzsche is wrong (and who knows, here?) he always challenges you to think! |
| 18332 | The creation of institutions needs a determination which is necessarily anti-liberal [Nietzsche] |
| Full Idea: For institutions to exist there must exist the kind of will, instinct, imperative which is anti-liberal to the point of malice: the will to tradition, to authority, to centuries-long responsibility, to solidarity between succeeding generations. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 8.39) | |
| A reaction: This sounds like a lovely challenge to Popper, who seems to have been a liberal who pinned his faith on institutions. |
| 2911 | True justice is equality for equals and inequality for unequals [Nietzsche] |
| Full Idea: 'Equality for equals, inequality for unequals' - that would be the true voice of justice. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 8.48) |
| 18320 | To renounce war is to renounce the grand life [Nietzsche] |
| Full Idea: One has renounced grand life when one renounces war. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 4.3) | |
| A reaction: Nietzsche was a medical orderly in the 1870 Franco-Prussian war, so he had seen it at first hand. I think the machine gun and the heavy bomber would have changed his attitude to warfare. He sounds a bit silly now. Nostalgia for the Iliad. |
| 2908 | There is a need for educators who are themselves educated [Nietzsche] |
| Full Idea: There is a need for educators who are themselves educated. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 7.5) |
| 18329 | Sometimes it is an error to have been born - but we can rectify it [Nietzsche] |
| Full Idea: We have no power to prevent ourselves being born: but we can rectify this error - for sometimes it is an error. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 8.36) |
| 2905 | 'Purpose' is just a human fiction [Nietzsche] |
| Full Idea: We invented the concept 'purpose': in reality purpose is lacking. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 5.8) |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |
| 18312 | The supreme general but empty concepts must be compatible, and hence we get 'God' [Nietzsche] |
| Full Idea: The supreme concepts of philosophers cannot be incommensurate with one another, be incompatible with one another... Thus they acquired their stupendous concept 'God'.... The last, thinnest, emptiest is placed as the first, as cause in itself. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 2.4) |
| 2906 | By denying God we deny human accountability, and thus we redeem the world [Nietzsche] |
| Full Idea: We deny God; in denying God we deny accountability; only by doing that do we redeem the world. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 5.8) |
| 18325 | Christians believe that only God can know what is good for man [Nietzsche] |
| Full Idea: Christianity presupposes that man does not know, cannot know what is good for him and what evil: he believes in God, who alone knows. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 8.05) |
| 2901 | How could the Church intelligently fight against passion if it preferred poorness of spirit to intelligence? [Nietzsche] |
| Full Idea: The primitive church fought against the 'intelligent' in favour of the 'poor in spirit': how could one expect from it an intelligent war against passion? | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 4.1) |
| 18318 | People who disparage actual life avenge themselves by imagining a better one [Nietzsche] |
| Full Idea: If there is a strong instinct for slandering, disparaging and accusing life within us, then we revenge ourselves on life by means of the phantasmagoria of 'another', a 'better' life. | |
| From: Friedrich Nietzsche (Twilight of the Idols [1889], 2.6) |