77 ideas
| 23183 | Different abilities are needed for living in an incomplete and undogmatic system [Nietzsche] |
| Full Idea: There is an entirely different strength and mobility to maintaining oneself in an incomplete system, with free, open vistas, than in a dogmatic world. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[025]) | |
| A reaction: This is like Keats's 'negative capability' - the ability to live in a state of uncertainty. I'm a fan of attempts to create a philosophical system, but dogmatism would seem to be the death of such a project. How would you live with your system? Nice. |
| 23188 | Bad writers use shapeless floating splotches of concepts [Nietzsche] |
| Full Idea: Bad writers have only shapeless floating splotches of concepts in their heads. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[083]) | |
| A reaction: Under 'conceptual analyis' not because he analyses concepts, but because he recognises their foundation importance in philosophy. I get more irritated by unchallenged concepts than by drifting concepts. Writer must know and challenge their key concepts. |
| 23212 | A text has many interpretations, but no 'correct' one [Nietzsche] |
| Full Idea: The same text allows innumerable interpretations: there is no 'correct' interpretation. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 1[120]) | |
| A reaction: It is hard to defend a 'correct' interpretation, but I think it is obvious to students of literature that some interpretations are very silly, such as reading things allegorically when there was no such intention. |
| 23199 | What is the search for truth if it isn't moral? [Nietzsche] |
| Full Idea: What is searching for truth, truthfulness, honesty if not something moral? | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 35[05]) | |
| A reaction: Feels right to me. It might be an effect of the virtue of respect. If you respect a person you tell them the truth (assuming they want the truth). Lying to someone is a sort of contempt. |
| 23202 | Like all philosophers, I love truth [Nietzsche] |
| Full Idea: I, too, love truth, like all philosophers. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 37[02]) | |
| A reaction: Please pay attention to this remark! His perspectivalism is not a denial of truth. It is an epistemological phenomenon, not a metaphysical one. The perspectives are the nearest we can get to truth. Humanity therefore needs teamwork. |
| 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 |
| 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 |
| 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). |
| 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 |
| 17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
| Full Idea: Zermelo's paper sets out to show that the standard set-theoretic axioms (what he calls the 'constitutive axioms', thus the ZF axioms minus the axiom of infinity) have an unending sequence of different models, thus that they are non-categorical. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1209 | |
| A reaction: Hallett says later that Zermelo is working with second-order set theory. The addition of an Axiom of Infinity seems to have aimed at addressing the problem, and the complexities of that were pursued by Gödel. |
| 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. |
| 13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
| Full Idea: Zermelo included Replacement in 1930, after it was noticed that the sequence of power sets was needed, and Replacement gave the ordinal form of the well-ordering theorem, and justification for transfinite recursion. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Penelope Maddy - Believing the Axioms I §1.8 | |
| A reaction: Maddy says that this axiom suits the 'limitation of size' theorists very well, but is not so good for the 'iterative conception'. |
| 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. |
| 23196 | Logic is a fiction, which invents the view that one thought causes another [Nietzsche] |
| Full Idea: The model of a complete fiction is logic. Here a thinking is made up where a thought is posited as the cause of another thought. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[249]) | |
| A reaction: He could almost be referring to Frege's Third Realm. Most hard core analytic philosophers seem to think that propositions have tight logical relationships which are nothing to do with the people who think them. |
| 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] |
| 17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
| Full Idea: Two opposite tendencies of thought, the idea of creative advance and of collection and completion (underlying the Kantian 'antinomies') find their symbolic representation and their symbolic reconciliation in the transfinite numbers based on well-ordering. | |
| From: Ernst Zermelo (On boundary numbers and domains of sets [1930], §5) | |
| A reaction: [a bit compressed] It is this sort of idea, from one of the greatest set-theorists, that leads philosophers to think that the philosophy of mathematics may offer solutions to metaphysical problems. As an outsider, I am sceptical. |
| 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 |
| 23186 | Numbers enable us to manage the world - to the limits of counting [Nietzsche] |
| Full Idea: Numbers are our major means of making the world manageable. We comprehend as far as we can count, i.e. as far as a constancy can be perceived. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[058]) | |
| A reaction: I don't agree with 'major', but it is a nice thought. The intermediate concept is a 'unit', which means identifying something as a 'thing', which is how we seem to grasp the world. So to what extent do we comprehend the infinite. Enter Cantor |
| 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. |
| 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 |
| 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 |
| 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 |
| 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. |
| 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. |
| 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. |
| 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 |
| 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. |
| 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'. |
| 23211 | Events are just interpretations of groups of appearances [Nietzsche] |
| Full Idea: There is no event in itself. What happens is a group of appearances selected and summarised by an interpreting being. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 1[115]) | |
| A reaction: Since innumerable events are nested within one another, such as the events at a carnival, this is obviously true. A primitive 'Kim event' (an object changes a property) might have objective existence. Carnivals happen, though. |
| 23201 | The 'I' does not think; it is a construction of thinking, like other useful abstractions [Nietzsche] |
| Full Idea: I do not grant to the metaphysicians that the 'I' is what thinks: on the contrary I take the I itself as a construction thinking, of the same rank as 'material',' thing', 'substance', 'purpose', 'number': therefore only as a regulative fiction. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 35[35]) | |
| A reaction: Ah. I have always defended the Self, the thing that is in charge when the mind is directed to something. I suddenly see that this is compatible with the Self not being the thinker! It is just the willer, and the controller of the searchlight. Self = will? |
| 23207 | Appearance is the sole reality of things, to which all predicates refer [Nietzsche] |
| Full Idea: Appearance as I understand it is the actual and single reality of things - that which first merits all existing predicates. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 40[53]) | |
| A reaction: This is the view espoused by John Stuart Mill (a fact which would shock Nietzsche!). Elsewhere he laughs at the concept of the thing-in-itself as a fiction. |
| 23197 | Memory is essential, and is only possible by means of abbreviation signs [Nietzsche] |
| Full Idea: Experience is only possible with the help of memory; memory is only possible by virtue of an abbreviation of an intellectual event as a sign. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[249]) | |
| A reaction: My memory of a town is not formed as a sign, but as a bunch of miscellaneous fragments about it. I think mental files gives a better account of this than do 'signs'. |
| 23206 | Schematic minds think thoughts are truer if they slot into a scheme [Nietzsche] |
| Full Idea: There are schematic minds, those who hold a thought-complex to be truer if it can be sketched into previously drafted schemata or categorical tables. There are countless self-deceptions in this area: nearly all the great 'systems' belong here. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 40[09]) | |
| A reaction: Why 'nearly all'? Aristotle might be a candidate for such a person. Leibniz, perhaps. Nietzsche identified with Becoming and Heraclitus, as opposed to Being and Parmenides. |
| 23209 | Each of our personal drives has its own perspective [Nietzsche] |
| Full Idea: From the standpoint of each of our fundamental drives there is a different perspectival assessment of all events and experiences. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 1[058]) | |
| A reaction: Revealing. Perspectives are not just each individual person's viewpoint, but something more fine-grained than that. Our understanding and response are ambiguous, because we ourselves are intrinsically ambiguous. Super-relativism! |
| 23184 | The mind is a simplifying apparatus [Nietzsche] |
| Full Idea: The intellect and the senses are above all a simplifying apparatus. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[046]) | |
| A reaction: Very plausible, and not an idea I have met elsewhere. There's a PhD here for someone. It fits with my view as universals in language (which is most of language), which capture diverse things by ironing out their differences. |
| 23190 | Consciousness is our awareness of our own mental life [Nietzsche] |
| Full Idea: We have a double brain: our capacity to will, to feel and to think of our willing, feeling, thinking ourselves is what we summarise with the word 'consciousness'. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[087]) | |
| A reaction: Pretty much the modern HOT (higher order thought) theory of consciousness. Higher order thought distinguishes us from the other animals, but I think they too are probably conscious, so I don't agree. Why is level 2 conscious of level 1? |
| 23191 | Minds have an excluding drive to scare things off, and a selecting one to filter facts [Nietzsche] |
| Full Idea: In our conscious intellect there must be an excluding drive that scares things away, a selecting one, which only permits certain facts to present themselves. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[131]) | |
| A reaction: I like this because he is endorsing the idea that philosophy needs faculties, which may not match the views of psychologists and neuroscientists. Quite nice to think of faculties as drives. |
| 23213 | The greatest drive of life is to discharge strength, rather than preservation [Nietzsche] |
| Full Idea: Something that lives wants above all to discharge its strength: 'preservation' is only one of the consequences of this. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 2[063]) | |
| A reaction: This seems to fit a dynamic man like Nietzsche, rather than someone who opts for a quiet and comfortable life. |
| 23210 | That all events are necessary does not mean they are compelled [Nietzsche] |
| Full Idea: The absolute necessity of all events contains nothing of a compulsion. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 1[114]) | |
| A reaction: I like to look for necessity-makers behind necessities. So if the event is not necessary because of its cause, where does it come from? Is it that the whole sequence is a unified necessity? |
| 23189 | Concepts are rough groups of simultaneous sensations [Nietzsche] |
| Full Idea: Concepts are more or less definite groups of sensations that arrive together. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[086]) | |
| A reaction: I like this because I favour accounts of concepts which root them in experience, and largely growing unthinking out of communcal experience. Nietzsche is very empirical here. Hume would probably agree. |
| 23192 | Concepts dont match one thing, but many things a little bit [Nietzsche] |
| Full Idea: A concept is an invention that doesn't correspond entirely to anything; but to many things a little bit. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[131]) | |
| A reaction: This seems to cover some concepts quite well, but others not at all. What else does 'square' correspond to? |
| 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? |
| 23187 | Whatever their origin, concepts survive by being useful [Nietzsche] |
| Full Idea: The most useful concepts have survived: however falsely they may have originated. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[063]) | |
| A reaction: The germ of both pragmatism, and of meaning-as-use, here. The alternative views must be that the concepts are accurate or true, or that they are simply a matter of whim, maintained by authority. |
| 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. |
| 23205 | Thought starts as ambiguity, in need of interpretation and narrowing [Nietzsche] |
| Full Idea: A thought in the shape in which it comes is an ambiguous sign that needs interpretation, more precisely, needs an arbitrary narrowing-down and limitation, until it finally becomes unambiguous. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 38[01]) | |
| A reaction: This is exactly my view of propositions, as mental events. Introspect your thinking process. Track the progress from the first glimmer of a thought to its formulation in a finished sentence. Language, unlike propositions, can be ambiguous. |
| 23198 | Aesthetics can be more basic than morality, in our pleasure in certain patterns of experience [Nietzsche] |
| Full Idea: Some of the aesthetic valuations are more fundamental than the moral ones e.g. the pleasure in what is ordered, surveyable, limited, in repetition. The logical, arithmetical and geometrical good feelings form the ground floor of aesthetic valuations. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 35[02]) | |
| A reaction: Nietzsche's originality is so striking because his novel suggestions are always plausible. Lots of modern philosophers (especially, I fear, in the continental tradition) throw out startling ideas which then fail on closer inspection. |
| 23208 | Caesar and Napoleon point to the future, when they pursue their task regardless of human sacrifice [Nietzsche] |
| Full Idea: In nature's such as Caesar and Napoleon we intuit something of a 'disinterested' laboring on one's marble, regardless of any sacrifice of human beings. The future of the highest human beings lies on here: to bear responsibility and not collapse under it. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 1[056]) | |
| A reaction: Hideous. Nietzsche at his absolute worst. You would think there was some wonderful higher good to which they were leading the human race, when they just strike me as people who liked fighting, and adored winning. |
| 23193 | Napoleon was very focused, and rightly ignored compassion [Nietzsche] |
| Full Idea: With Napoleon only the essential instincts of humanity came into consideration during his calculations, and he had a right not to take notice of the exceptional ones e.g. of compassion. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[131]) | |
| A reaction: Napoleon was notoriously indifferent to casualties, and I find it depressing that Nietzsche supports him. Napoleon brought misery to Europe for nearly twenties, mainly because he loved winning battles. Nothing über about that. |
| 23214 | For the strongest people, nihilism gives you wings! [Nietzsche] |
| Full Idea: In the hands of the strongest every kind of pessimism and nihilism becomes only one more hammer and tool with which one mounts a new pair of wings on oneself. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 2[101]) | |
| A reaction: Not sure how this works. Why is great strength needed? Strength implies forceful overcoming. The wings come from rejecting nihilism, but why does that need strength? Aren't there people with wings who never even thought of nihilism? |
| 23203 | The great question is approaching, of how to govern the earth as a whole [Nietzsche] |
| Full Idea: It is approaching, irrefutably, hesitatingly, terrible as fate, the great task and fate: how should the earth as a whole be governed? | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 37[08]) | |
| A reaction: Two issues have accelerated the question, though we have yet to properly face it. One is the incredible increase in military destructiveness, and other is the damage to the planet caused by the relentless pursuit of wealth. |
| 23200 | The controlling morality of aristocracy is the desire to resemble their ancestors [Nietzsche] |
| Full Idea: The foundation of all aristocracies is to resemble the ancestors as much as possible, which serves as the controlling morality: mourning at the thought of change and variation. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 35[22]) | |
| A reaction: This makes sense of the permanent residence of the family, full of portraits and family trees. Aristocrats preserve records of their predecessors, in a way that most of us don't, going back before grandparents. |
| 23194 | People feel united as a nation by one language, but then want a common ancestry and history [Nietzsche] |
| Full Idea: People who speak one language and read the same newspapers today call themselves 'nations', and also want much too eagerly to be of common ancestry and history. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[203]) | |
| A reaction: This sort of nationalism is still with us, as white supremacy, and as history as mythology. But we can't just shake off a sense of which gene pools we come from, and which lines of history are our personal inheritance. |
| 23204 | To be someone you need property, and wanting more is healthy [Nietzsche] |
| Full Idea: Property owners are to a man of one belief: 'you have to own something to be something'. But this is the oldest and healthiest of all instincts: I would add 'you have to want more than you have in order to become more'. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 37[11]) | |
| A reaction: An odd idea from someone who spent his later years living in one room in a guest house. The context of this is a rejection of socialism. The love of and need for property and possessions must be taken into account in any politics. |
| 23195 | Laws of nature are actually formulas of power relations [Nietzsche] |
| Full Idea: The alleged 'laws of nature' are formulas for power relationships | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[247]) | |
| A reaction: Love it. This is precisely the powers ontology of modern philosophy of science. His Will to Power is not often recognised as closely related to this view. |
| 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 |
| 23185 | In chemistry every substance pushes, and thus creates new substances [Nietzsche] |
| Full Idea: In chemistry is revealed that every substance pushes its force as far as it can, then a third something emerges. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1885-86 [1886], 34[51]) | |
| A reaction: This is the ontology of powers as the basis of science, of which I am a fan. It is Nietzsche's Will to Power in action, which is often mistakenly taken to only refer to human affairs. |
| 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'. |