75 ideas
| 2930 | The main aim of philosophy must be to determine the order of rank among values [Nietzsche] |
| Full Idea: The future task of the philosophers is the solution of the problem of value, the determination of the order of rank among values. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], I.§17 note) | |
| A reaction: 'Determine' is presumably either a power struggle, or needs criteria by which to do the judging. |
| 20143 | Scientific knowledge is nothing without a prior philosophical 'faith' [Nietzsche] |
| Full Idea: Strictly speaking there is no knowledge [science] without presuppositions; a philosophy, a 'faith', must always be there first of all, for knowledge to win from it a direction, a meaning, a limit, a method, a right to exist. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], III.§24) | |
| A reaction: He sees philosophers as the creators of this faith, and laughs at anyone who tries to set philosophy on a scientific basis. |
| 23722 | Objectivity is not disinterestedness (impossible), but the ability to switch perspectives [Nietzsche] |
| Full Idea: 'Objectivity' should be understood not as 'contemplation without interest' (a non-concept and an absurdity), but as having in our power the ability to engage and disengage our 'pros' and 'cons'; we can use the difference in perspectives for knowledge. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], III.§12) | |
| A reaction: Note that he will use perspectives to achieve knowledge. The idea that Perspectivalism is mere relativism is labelled as 'extreme' in Idea 4486. He is right that objectivity is a mental capacity and achievement of individuals. |
| 8797 | The negation of all my beliefs about my current headache would be fully coherent [Sosa] |
| Full Idea: If I have a headache, I could have a set of beliefs that I do not have a headache, that I am not in pain, that no one is in pain, and so on. The resulting system of beliefs would cohere as fully as does my actual system of beliefs. | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §9) | |
| A reaction: I think this is a misunderstanding of coherentism. Beliefs are not to be formulated through a process of coherence, but are evaluated that way. A belief that I have headache just arrives; I then see that its denial is incoherent, so I accept it. |
| 4417 | Only that which has no history is definable [Nietzsche] |
| Full Idea: Only that which has no history is definable. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], II.§13) | |
| A reaction: Too subtle to evaluate! It sounds as if it could be right, that some things are definable, but when the accretions of human history are interwoven into an identity, we can forget it. |
| 23715 | Psychologists should be brave and proud, and prefer truth to desires, even when it is ugly [Nietzsche] |
| Full Idea: I hope [psychologists] are actually brave, generous, proud animals, who know how to control their own pleasure and pain and are taught to sacrifice desirability to truth, even a bitter, ugly, unchristian, immoral truth - Because there are such truths. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], I.§01) | |
| A reaction: A nice expression of Nietzsche's values, which makes truth central, contrary to the widespread modern view that he was the high priest of relativism. If you think that, read him more carefully. |
| 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. |
| 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 |
| 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? |
| 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'. |
| 4421 | Philosophers have never asked why there is a will to truth in the first place [Nietzsche] |
| Full Idea: Both the earliest and most recent philosophers are all oblivious of how much the will to truth itself first requires justification: here there is a gap in every philosophy - how did this come about? | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], III.§24) | |
| A reaction: This seems to me a meta-philosophical question which will lead off into (quite interesting) cultural studies and (trite) evolutionary theory. Truth isn't a value, it is the biological function of brains. |
| 8794 | There are very few really obvious truths, and not much can be proved from them [Sosa] |
| Full Idea: Radical foundationalism suffers from two weaknesses: there are not so many perfectly obvious truths as Descartes thought; and if we restrict ourselves to what it truly obvious, very little supposed common sense knowledge can be proved. | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §3) | |
| A reaction: It is striking how few examples can ever be found of self-evident a priori truths. However, if there are self-evident truths about direct experience (pace Descartes), that would give us more than enough. |
| 23719 | Forgetfulness is a strong positive ability, not mental laziness [Nietzsche] |
| Full Idea: Forgetfulness is not just a vis inertiae, as superficial people believe, but is rather an active ability to suppress, positive in the strongest sense of the word. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], II.§01) | |
| A reaction: It is unimpressive when people remember small slights and grievances for a long time - and even being owed small sums - so the ability to forget such things is admirable. But wilfully forgetting some things is obviously shameful. |
| 8796 | A single belief can trail two regresses, one terminating and one not [Sosa] |
| Full Idea: A single belief can trail at once regresses of both sorts: one terminating and one not. | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §6) | |
| A reaction: This makes foundationalism possible, while admitting the existence of regresses. It is a good point, and triumphalist anti-foundationalists can't just point out a regress and then smugly troop off to the pub. |
| 8799 | If mental states are not propositional, they are logically dumb, and cannot be foundations [Sosa] |
| Full Idea: If a mental state is not propositional, then how can it possibly serve as a foundation for belief? How can one infer or justify anything on the basis of a state that, having no propositional content, must be logically dumb? | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §11) | |
| A reaction: This may be the best objection to foundationalism. McDowell tries to argue that conceptual content is inherent in perception, thus giving the beginnings of inbuilt propositional content. But an organism awash with bare experiences knows nothing. |
| 8795 | Mental states cannot be foundational if they are not immune to error [Sosa] |
| Full Idea: If a mental state provides no guarantee against error, then it cannot serve as a foundation for knowledge. | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §4) | |
| A reaction: That assumes that knowledge entails certainty, which I am sure it should not. On a fallibilist account, a foundation could be incredibly secure, despite a barely imaginable scenario in which it turned out to be false. |
| 8798 | Vision causes and justifies beliefs; but to some extent the cause is the justification [Sosa] |
| Full Idea: Visual experience is recognized as both the cause and the justification of our visual beliefs. But these are not wholly independent. Presumably the justification that something is red derives partly from the fact that it originates in visual experience. | |
| From: Ernest Sosa (The Raft and the Pyramid [1980], §10) | |
| A reaction: Yes, but the fact that certain visual experiences originate in dreams is taken as grounds for denying their truth, not affirming it. So why do we distinguish them? I am thinking that only in the 'space of reasons' can a cause become a justification. |
| 4420 | There is only 'perspective' seeing and knowing, and so the best objectivity is multiple points of view [Nietzsche] |
| Full Idea: There is only a perspective seeing, only a perspective "knowing", and the more different eyes we can use to observe one thing, the more complete will our "concept" of this thing, our "objectivity", be. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], III.§12) | |
| A reaction: A very perceptive statement of the most plausible and sophisticated version of relativism. It is hard to see how we could distinguish multiple viewpoints from pure objectivity. |
| 4414 | Philosophers invented "free will" so that our virtues would be permanently interesting to the gods [Nietzsche] |
| Full Idea: The philosophers invented "free will" - absolute human spontaneity in good and evil - to furnish a right to the idea that the interest of the gods in man, in human virtue, could never be exhausted. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], II.§07) | |
| A reaction: Wonderfully outrageous suggestion! If we had true metaphysical 'absolute' free will, we would be much more interesting, and have a much higher status in the cosmos. Nietzsche is probably right. |
| 4419 | People who think in words are orators rather than thinkers, and think about facts instead of thinking facts [Nietzsche] |
| Full Idea: Whoever thinks in words thinks as an orator and not as a thinker (it shows that he does not think facts, but only in relation to facts). | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], III.§08) | |
| A reaction: Good. It is certainly not true that we have to think in words, or else animals wouldn't think. Good thinking should focus on reality, and be too fast for words to keep up. |
| 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. |
| 4411 | It is a delusion to separate the man from the deed, like the flash from the lightning [Nietzsche] |
| Full Idea: Just as the popular mind separates the lightning from its flash and takes the latter for a 'action', so they separate strength from expressions of strength, but there is no such substratum; the deed is everything. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], I.§13) | |
| A reaction: Of course, there is no reason why an analysis should not separate the doer and the deed (to explain, for example, a well-meaning fool), but it is a blunder to think of a human action as a merely physical event. |
| 3793 | We must question the very value of moral values [Nietzsche] |
| Full Idea: We need a critique of moral values; the value of these values themselves must just be called in question. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], Pre f§3) | |
| A reaction: But we must start somewhere with values, to avoid an infinite regress. |
| 4408 | The concept of 'good' was created by aristocrats to describe their own actions [Nietzsche] |
| Full Idea: The judgement 'good' did not first originate with those to whom goodness was shown! Rather it was the 'good' themselves, that is to say the noble, powerful, high-stationed and high-minded who established themselves and their action as good. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], I.§02) | |
| A reaction: This may be right, but not very profound. Virtually all concepts are created by the most educated classes. The first recipient of charity may not have had the concept, but they would have been gobsmacked by the novelty. |
| 23716 | A strong rounded person soon forgets enemies, misfortunes, and even misdeeds [Nietzsche] |
| Full Idea: To be unable to take his enemies, his misfortunes and even his misdeeds seriously for long - that is the sign of strong, rounded natures with a superabundance of power. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], I.§10) | |
| A reaction: An aspect of the 'higher man' that I don't recall being mentioned elsewhere. I basically approve of this, if it means not holding grudges, and living for the future rather than for the past. |
| 20129 | All animals strive for the ideal conditions to express their power, and hate any hindrances [Nietzsche] |
| Full Idea: Every animal instinctively strives for an optimum of favourable conditions under which it can expend all its strength and achieve its maximal feeling of power; every animal abhors ...every hindrance that obstructs this path to the optimum. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], III.§07) | |
| A reaction: This became the lynchpin of Nietzsche's account of the source of values. It is a highly naturalistic view, fitting it into evolutionary theory (thought running deeper than that), so I have a lot of sympathy with the view. |
| 4409 | Only the decline of aristocratic morality led to concerns about "egoism" [Nietzsche] |
| Full Idea: It was only when aristocratic value judgements declined that the whole antithesis of "egoistic" and "unegoistic" obtruded itself more and more on the human conscience. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], I.§02) | |
| A reaction: But Aristotle, who is no aristocrat, has a balanced and sensible view of 'egoism', where it isn't the patronising arrogance that Nietzsche seems to like, but a proper concern with one's own character. |
| 3259 | Nietzsche rejects impersonal morality, and asserts the idea of living well [Nietzsche, by Nagel] |
| Full Idea: Nietzsche's rejection of impersonal morality is an assertion of the dominance of the ideal of living well. | |
| From: report of Friedrich Nietzsche (On the Genealogy of Morals [1887], I) by Thomas Nagel - The View from Nowhere X.2 |
| 4416 | Basic justice is the negotiation of agreement among equals, and the imposition of agreement [Nietzsche] |
| Full Idea: Justice on the elementary level is good will among parties of approximately equal power to come to terms with one another, and to compel parties of lesser power to reach a settlement among themselves. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], II.§08) | |
| A reaction: This pinpoints a key problem with the social contract as a moral theory - that it requires equals, and recognises only terror of superiors, and indifference to useless inferiors who have nothing to offer (paraplegics and animals). |
| 4418 | A masterful and violent person need have nothing to do with contracts [Nietzsche] |
| Full Idea: He who can command, he who is "master", he who is violent in act and bearing - what has he to do with contracts! | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], II.§17) | |
| A reaction: The persistent problem with social contract theory is that those much stronger or much weaker seem to have no interest in morality at all, and yet they can all have standards of behaviour. |
| 4407 | Plato, Spinoza and Kant are very different, but united in their low estimation of pity [Nietzsche] |
| Full Idea: Plato, Spinoza, La Rochefoucauld, and Kant are four spirits very different from one another, but united in one thing: their low estimation of pity. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], Pref §5) | |
| A reaction: Plato is no surprise, as virtually no Greeks value pity. Spinoza and Kant are interesting. Presumably Kant's 'contractualism' places respect far above pity, and is theoretical neglect of animals would fit. Remember Nietzsche embraced a horse in Turin. |
| 4415 | Guilt and obligation originated in the relationship of buying and selling, credit and debt [Nietzsche] |
| Full Idea: The feeling of guilt, of personal obligation, had its origin in the oldest and most primitive personal relationship, that between buyer and seller, between creditor and debtor. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], II.§08) | |
| A reaction: In other words, lofty Kantian ideals started life in the grubby world of the Hobbesian social contract, and self-seeking has been disguised by idealism. Too harsh on Kant, who explains why contracts have force, not just convenience. |
| 23718 | If we say birds of prey could become lambs, that makes them responsible for being birds of prey [Nietzsche] |
| Full Idea: Scientists …do not defend any belief more strongly than that the strong are free to be weak, and the birds of prey are free to be lambs: - in this way, they gain the right to make the birds of prey responsible for being birds of prey. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], I.§13) | |
| A reaction: This is a flat rejection of the Sartrean idea that we can what sort of person we want to be. He cares about birds of prey, but also lambs can't become eagles. I would say that adolescents have a reasonable degree of choice about what they will become. |
| 23717 | Modern nihilism is now feeling tired of mankind [Nietzsche] |
| Full Idea: The sight of man now makes us tired - what is nihilism today if it is not that? …We are tired of man… | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], I.§12) | |
| A reaction: That is close to Hume's nihilist, who would destroy the world to protect his own finger from a scratch. The actor George Sanders committed suicide because he was bored. Don't ever think that Nietzsche was a nihilist, just because he mentions it a lot! |
| 23721 | Old tribes always felt an obligation to the earlier generations, and the founders [Nietzsche] |
| Full Idea: Within the original tribal association the living generation always acknowledged a legal obligation towards the earlier generation, and in particular towards the earliest. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], II.§19) | |
| A reaction: This is still a factor in modern politics, though the people remember are either military heroes or the great figures of a particular political movement. We remember the big artists and personalities, but don't feel obligated to them. |
| 20142 | The state begins with brutal conquest of a disorganised people, not with a 'contract' [Nietzsche] |
| Full Idea: Some pack of blond beasts of prey, on a war footing, unscrupulously lays its dreadful paws on a populace which is shapeless. In this way the 'state' began on earth. I think I have dispensed with the fantasy which has it begin with a 'contract'. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], II.§17) | |
| A reaction: [compressed] It is certainly likely that a tribe which got itself well organised and focused on some end would achieve total dominance over other tribes that just focus on food. |
| 23720 | Punishment makes people harder, more alienated, and hostile [Nietzsche] |
| Full Idea: On the whole, punishment makes men harder and colder, it concentrates, it sharpens the feeling of alienation; it strengthens the power to resist. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], II.§14) | |
| A reaction: If the school system involves routine harsh punishments, that means that the whole population ends up in that state. I would have thought that this was an obvious truth about punishment, but no one seems to want to face up to it. |
| 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'. |
| 4410 | The truly great haters in world history have always been priests [Nietzsche] |
| Full Idea: The truly great haters in world history have always been priests. | |
| From: Friedrich Nietzsche (On the Genealogy of Morals [1887], I.§07) | |
| A reaction: Wicked, but it has a lot of truth. Priests have a lot to defend, and a lot of reasons for feeling threatened. |