97 ideas
| 16000 | Fixed ideas should be tackled aggressively [Kierkegaard] |
| Full Idea: Fixed ideas are like a cramp in your foot: the best remedy is to stomp on them. | |
| From: Søren Kierkegaard (The Journals of Kierkegaard [1850], JP-III, 635) | |
| A reaction: Sound philosophical advice at any time. [SY] Does this apply in seminars, as well as in private meditation? [PG] |
| 7578 | I conceived it my task to create difficulties everywhere [Kierkegaard] |
| Full Idea: I conceived it my task to create difficulties everywhere. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846], 'Author') | |
| A reaction: Nice. It is like Socrates's image of himself as the 'gadfly' of Athens. The interesting question is always to see what the rest of society makes of having someone in their midst who sees it as their social role to 'create difficulties'. |
| 22087 | Philosophy fails to articulate the continual becoming of existence [Kierkegaard, by Carlisle] |
| Full Idea: Kierkegaard criticise philosophy for its inability to grasp and to articulate the movement, the continual becoming, that characterises existence. | |
| From: report of Søren Kierkegaard (Either/Or: a fragment of life [1843]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 2 | |
| A reaction: Heraclitus had a go, and Hegel's historicism focuses on dynamic thought, but this idea concerns the immediacy of individual life. |
| 22047 | Wherever there is painless contradiction there is also comedy [Kierkegaard] |
| Full Idea: Wherever there is contradiction, the comical is also present. ...The tragic is the suffering contradiction, the comical is the painless contradiction. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846], p.459), quoted by Terry Pinkard - German Philosophy 1760-1860 13 | |
| A reaction: He is not saying that this is the only source of comedy. I once heard an adult say that there is one thing that is always funny, and that is a fart. |
| 16012 | Philosophy can't be unbiased if it ignores language, as that is no more independent than individuals are [Kierkegaard] |
| Full Idea: If the claim of philosophers to be unbiased were all it pretends to be, it would have to take account of language and its significance...Language is partly given and partly develops freely. As individuals cannot be truly independent, so too with language. | |
| From: Søren Kierkegaard (The Journals of Kierkegaard [1850], 1840.07.18) | |
| A reaction: A surprisingly prophetic entry from Kierkegaard anticipating the linguistic turn. [SY] |
| 22092 | Kierkegaard's truth draws on authenticity, fidelity and honesty [Kierkegaard, by Carlisle] |
| Full Idea: Kierkegaard offers a different interpretation of truth, which draws on the notions of authenticity, fidelity and honesty. | |
| From: report of Søren Kierkegaard (Concluding Unscientific Postscript [1846]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 4 | |
| A reaction: This notion of truth, meaning 'the real thing' (as in 'she was a true scholar'), seems to begin with Hegel. I suggest we use the word 'genuine' for that, and save 'truth' for its traditional role. It is disastrous to blur the simple concept of truth. |
| 15999 | Pure truth is for infinite beings only; I prefer endless striving for truth [Kierkegaard] |
| Full Idea: If God held all truth enclosed in his right hand, and in his left hand the ever-striving drive for truth, even if erring forever, and he were to say Choose! I would humbly fall at his left hand and say Father, give! Pure truth is for infinite beings only. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846], p.106) | |
| A reaction: A sobering realistic thought of our own limitations; Kierkegaard allows that there is no limit to how far we can strive for truth. Just that truth is comprehended by infinite beings (if any), not by mere mortals. [SY] |
| 22094 | Subjective truth can only be sustained by repetition [Kierkegaard, by Carlisle] |
| Full Idea: If subjective truth is to be more than momentary, it has to be repeated continually. | |
| From: report of Søren Kierkegaard (Repetition [1843]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 4 | |
| A reaction: This might apply to more traditional concepts of truth, if they are to be part of life, rather than remaining in books. |
| 16005 | I recognise knowledge, but it is the truth by which I can live and die that really matters [Kierkegaard] |
| Full Idea: The thing is to find a truth which is true for me - the idea for which I can live and die. I still recognise an imperative of knowledge, but it must be taken up into my life, which I now recognise as the most important thing. | |
| From: Søren Kierkegaard (Letter to Peter Wilhelm Lund [1835], J-1A) | |
| A reaction: A quintessentially existential idea. Note that he still considers objective knowledge to be quite important, but how we act and relate to those ideas is what really matters for us human beings. [SY] |
| 5651 | Traditional views of truth are tautologies, and truth is empty without a subject [Kierkegaard, by Scruton] |
| Full Idea: Kierkegaard developed the idea of 'truth as subjectivity'; the traditional conceptions of truth - correspondence or coherence - he regarded as equally empty, not because false, but because tautologous; truth ceases to be empty when related to a subject. | |
| From: report of Søren Kierkegaard (Either/Or: a fragment of life [1843]) by Roger Scruton - Short History of Modern Philosophy Ch.13 | |
| A reaction: It strikes me that the correspondence theory of truth also involves a subject. If you become too obsessed with the subject, you lose the concept of truth. You need a concept of the non-subject too. Truth concerns the contents of thought. |
| 20313 | The highest truth we can get is uncertainty held fast by an inward passion [Kierkegaard] |
| Full Idea: An objective uncertainty held fast in an appropriation-process of the most passionate inwardness is the truth, the highest truth available for an existing individual. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846]) | |
| A reaction: [Bk 711] Offered as a definition of truth, knowing how strange and paradoxical it sounds. If we view all life as subjectivity, then there can of course be nothing more to truth than passionate conviction. Personally I think thought can be objective. |
| 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 |
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| Full Idea: Cantor's development of set theory began with his discovery of the progression 0, 1, ....∞, ∞+1, ∞+2, ..∞x2, ∞x3, ...∞^2, ..∞^3, ...∞^∞, ...∞^∞^∞..... | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite VIII.2 |
| 9616 | A set is a collection into a whole of distinct objects of our intuition or thought [Cantor] |
| Full Idea: A set is any collection into a whole M of definite, distinct objects m ... of our intuition or thought. | |
| From: George Cantor (The Theory of Transfinite Numbers [1897], p.85), quoted by James Robert Brown - Philosophy of Mathematics Ch.2 | |
| A reaction: This is the original conception of a set, which hit trouble with Russell's Paradox. Cantor's original definition immediately invites thoughts about the status of vague objects. |
| 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 |
| 17831 | Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake] |
| Full Idea: Cantor gives informal versions of the axioms of ZF as ways of getting from one set to another. | |
| From: report of George Cantor (Later Letters to Dedekind [1899]) by John Lake - Approaches to Set Theory 1.6 | |
| A reaction: Lake suggests that it should therefore be called CZF. |
| 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. |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence). | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 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 |
| 15896 | Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine] |
| Full Idea: Cantor grafted the Power Set axiom onto his theory when he needed it to incorporate the real numbers, ...but his theory was supposed to be theory of collections that can be counted, but he didn't know how to count the new collections. | |
| From: report of George Cantor (The Theory of Transfinite Numbers [1897]) by Shaughan Lavine - Understanding the Infinite I | |
| A reaction: I take this to refer to the countability of the sets, rather than the members of the sets. Lavine notes that counting was Cantor's key principle, but he now had to abandon it. Zermelo came to the rescue. |
| 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? |
| 9992 | The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor] |
| Full Idea: The author entirely overlooks the fact that the 'extension of a concept' in general may be quantitatively completely indeterminate. Only in certain cases is the 'extension of a concept' quantitatively determinate. | |
| From: George Cantor (Review of Frege's 'Grundlagen' [1885], 1932:440), quoted by William W. Tait - Frege versus Cantor and Dedekind | |
| A reaction: Cantor presumably has in mind various infinite sets. Tait is drawing our attention to the fact that this objection long precedes Russell's paradox, which made the objection more formal (a language Frege could understand!). |
| 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'. |
| 16007 | I assume existence, rather than reasoning towards it [Kierkegaard] |
| Full Idea: I always reason from existence, not towards existence. | |
| From: Søren Kierkegaard (Philosophical Fragments [1844], p.40) | |
| A reaction: Kierkegaard's important premise to help show that theistic proofs for God's existence don't actually prove existence, but develop the content of a conception. [SY] |
| 16013 | Nothing necessary can come into existence, since it already 'is' [Kierkegaard] |
| Full Idea: Can the necessary come into existence? That is a change, and everything that comes into existence demonstrates that it is not necessary. The necessary already 'is'. | |
| From: Søren Kierkegaard (Philosophical Fragments [1844], p.74) | |
| A reaction: [SY] |
| 20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
| Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
| From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
| A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |
| 20742 | The real subject is ethical, not cognitive [Kierkegaard] |
| Full Idea: The real subject is not the cognitive subject …the real subject is the ethically existing subject. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846], p.281), quoted by Kevin Aho - Existentialism: an introduction 2 'Subjective' | |
| A reaction: Perhaps we should say the essence of the self is its drive to live, not its drive to know. Just getting through the day is top priority, and ethics don’t figure much for the solitary person. But each activity, such as cooking, has its virtues. |
| 16002 | The self is a combination of pairs of attributes: freedom/necessity, infinite/finite, temporal/eternal [Kierkegaard] |
| Full Idea: A human being is essentially spirit, but what is spirit? Spirit is to be a self. But what is the Self? In short, it is a synthesis of the infinite and the finite, of the temporal and the eternal, of freedom and necessity. | |
| From: Søren Kierkegaard (Sickness unto Death [1849], p.59) | |
| A reaction: The dense language of his first paragraph was to poke fun at fashionable Hegelian writing. The book gets very lucid afterwards! [SY] |
| 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. |
| 9145 | We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor] |
| Full Idea: We call 'cardinal number' the general concept which, by means of our active faculty of thought, arises when we make abstraction from an aggregate of its various elements, and of their order. From this double abstraction the number is an image in our mind. | |
| From: George Cantor (Beitrage [1915], §1), quoted by Kit Fine - Cantorian Abstraction: Recon. and Defence Intro | |
| A reaction: [compressed] This is the great Cantor, creator of set theory, endorsing the traditional abstractionism which Frege and his followers so despise. Fine offers a defence of it. The Frege view is platonist, because it refuses to connect numbers to the world. |
| 22098 | Socrates neglects the gap between knowing what is good and doing good [Kierkegaard, by Carlisle] |
| Full Idea: There is a fundamental weakness in Socrates, that he does not take into account the gap between knowing what is good and actually putting this into action. | |
| From: report of Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 5 | |
| A reaction: This rejects Socrates's intellectualism about weakness of will. It is perhaps a better criticism that Aristotle's view that desires sometimes overcome the will. It is also the problem of motivation in Kantian deontology. Or utilitarianism. |
| 22086 | The most important aspect of a human being is not reason, but passion [Kierkegaard, by Carlisle] |
| Full Idea: Kierkegaard insisted that the most important aspect of a human being is not reason, but passion. | |
| From: report of Søren Kierkegaard (works [1845]) by Clare Carlisle - Kierkegaard: a guide for the perplexed Intro | |
| A reaction: Hume comes to mind for a similar view, but in character Hume was far more rational than Kierkegaard. |
| 16003 | If people marry just because they are lonely, that is self-love, not love [Kierkegaard] |
| Full Idea: People despair about being lonely and therefore get married. But is this love? I should say it is self-love. | |
| From: Søren Kierkegaard (The Journals of Kierkegaard [1850], JP-III, 40-41) | |
| A reaction: If you decide to marry someone because you don't want to be an old maid/bachelor in your elder years, try to actually love the person you're marrying. Not just for money or sex. [SY] |
| 15998 | Perfect love is not in spite of imperfections; the imperfections must be loved as well [Kierkegaard] |
| Full Idea: To love another in spite of his weaknesses and errors and imperfections is not perfect love. No, to love is to find him lovable in spite of, and together with, his weaknesses and errors and imperfections. | |
| From: Søren Kierkegaard (Works of Love [1847], p.158) | |
| A reaction: A true romantic at heart, Kierkegaard ideally posits perfect love as unconditional love, and not just of good attributes, predicates and conditions. However, the real question for both me and Kierkegaard is, is perfect love desirable or even possible?[SY] |
| 7579 | While big metaphysics is complete without ethics, personal philosophy emphasises ethics [Kierkegaard] |
| Full Idea: While the Hegelian philosophy goes on and is finished without having an Ethics, the more simple philosophy which is propounded by an existing individual for existing individuals, will more especially emphasis the ethical. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846], 'Lessing') | |
| A reaction: This is reminiscent of the Socratic revolution, which shifted philosophy from the study of nature to the study of personal virtue. However, if we look for ethical teachings in existentialism, there often seems to be a black hole in the middle. |
| 7581 | Speculative philosophy loses the individual in a vast vision of humanity [Kierkegaard] |
| Full Idea: Being an individual man is a thing that has been abolished, and every speculative philosopher confuses himself with humanity at large, whereby he becomes infinitely great - and at the same time nothing at all. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846], 'Lessing') | |
| A reaction: Compare Idea 4840. This is a beautiful statement of the motivation for existentialism. The sort of philosophers who love mathematics (Plato, Descartes, Leibniz, Russell) love losing themselves in abstractions. This is the rebellion. |
| 22090 | For me time stands still, and I with it [Kierkegaard, by Carlisle] |
| Full Idea: Time flows, life is a stream, people say, and so on. I do not notice it. Time stands still, and I with it. | |
| From: report of Søren Kierkegaard (Either/Or: a fragment of life [1843], I:26) by Clare Carlisle - Kierkegaard: a guide for the perplexed 3 | |
| A reaction: This is from the spokesman for the aesthetic option in life, which is largely pleasure-seeking. No real choices ever occur. |
| 22096 | Anxiety is not a passing mood, but a response to human freedom [Kierkegaard, by Carlisle] |
| Full Idea: For Kierkegaard anxiety is not simply a mood or an emotion that certain people experience at certain times, but a basic response to freedom that is part of the human condition. | |
| From: report of Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 5 | |
| A reaction: Outside of Christianity, this may be Kierkegaard's most influential idea - since existential individualism is floating around in the romantic movement. But the Byronic hero experiences a sort of anxiety. If you can't face anxiety, become a monk or nun. |
| 22097 | The ultimate in life is learning to be anxious in the right way [Kierkegaard] |
| Full Idea: Every human being must learn to be anxious in order that he might not perish either by never having been in anxiety or by succumbing in anxiety. Whoever has learned to be anxious in the right way has learnt the ultimate. | |
| From: Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844], p.154), quoted by Clare Carlisle - Kierkegaard: a guide for the perplexed 5 | |
| A reaction: I think this is the most existentialist quotation I have found in Kierkegaard. It sounds circular. You must be in anxiety because otherwise you won't be able to cope with anxiety? I suppose anxiety is facing up to his concept of truth. |
| 21909 | Ultimate knowledge is being anxious in the right way [Kierkegaard] |
| Full Idea: Whoever learns to be anxious in the right way has learned the ultimate. | |
| From: Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844], p.187), quoted by Alastair Hannay - Kierkegaard 06 | |
| A reaction: This shows us that Kierkegaard had a rather bizarre mental life which the rest of us have little chance of penetrating. I'll have a go at cataloguing my types of anxiety, but I'm not hopeful. |
| 20758 | Anxiety is staring into the yawning abyss of freedom [Kierkegaard] |
| Full Idea: One may liken anxiety to dizziness. He whose eyes chance to look down into a yawning abyss becomes dizzy. Anxiety is the dizziness of freedom which is when freedom gazes down into its own possibility, grasping at finiteness to sustain itself. | |
| From: Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844], p.55), quoted by Kevin Aho - Existentialism: an introduction 6 'Moods' | |
| A reaction: Most of us rapidly retreat from the thought of the infinity of things we might choose. Choosing bizarrely merely to assert one's freedom is simple stupidity. |
| 9305 | The plebeians bore others; only the nobility bore themselves [Kierkegaard] |
| Full Idea: Those who bore others are the plebeians, the crowd, the endless train of humanity in general; those who bore themselves are the chosen ones, the nobility. | |
| From: Søren Kierkegaard (Either/Or: a fragment of life [1843], Pt.1), quoted by Lars Svendsen - A Philosophy of Boredom Ch.2 | |
| A reaction: [p.288 in Princeton Edn] Stunningly elitist, but ask where boredom is most overtly found. "Boring" was once a very fashionable word among the English upper classes. Education and wealth seem to intensify boredom. |
| 21910 | Our destiny is the highest pitch of world-weariness [Kierkegaard] |
| Full Idea: Our destiny in this life is to be brought to the highest pitch of world-weariness. | |
| From: Søren Kierkegaard (The Journals of Kierkegaard [1850], 1855.09.25), quoted by Alastair Hannay - Kierkegaard 10 | |
| A reaction: The beginning of his last entry. Hardly a great general truth, but interesting. Should we aspire to exhaust life? |
| 5650 | Reason is just abstractions, so our essence needs a subjective 'leap of faith' [Kierkegaard, by Scruton] |
| Full Idea: For Kierkegaard, reason, which produces only abstractions, negates our individual essence; this essence is subjectivity, and subjectivity exists only in the 'leap of faith', whereby the individual casts in his lot with eternity. | |
| From: report of Søren Kierkegaard (Either/Or: a fragment of life [1843]) by Roger Scruton - Short History of Modern Philosophy Ch.13 | |
| A reaction: Interesting, but this strikes me as a confusion of reason and logic. A logical life would indeed be a sort of death, and need faith as an escape, but a broad view of the rational life includes emotion, imagination and laughter. Blind faith is disaster. |
| 22095 | There are aesthetic, ethical and religious subjectivity [Kierkegaard, by Carlisle] |
| Full Idea: Kierkegaard distinguishes three main types of subjectivity: aesthetic, ethical and religious. But are these types of people, or different phases of one person's life? | |
| From: report of Søren Kierkegaard (Either/Or: a fragment of life [1843]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 4 | |
| A reaction: His picture of the religious mode holds no appeal for me. I also can't accept that the aesthetic and the moral are somewho distinct. People may discover they have slipped into one of these modes, but no one chooses them, do they? |
| 20314 | People want to lose themselves in movements and history, instead of being individuals [Kierkegaard] |
| Full Idea: Everything must attach itself so as to be part of some movement; men are determined to lose themselves in the totality of things, in world-history, fascinated and deceived by a magic witchery; no one wants to be an individual human being. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846]) | |
| A reaction: [Bk 711] I presume 'world-history' refers to the exhilerating ideas of Hegel. Right now [2017] I would say we have far too much of people only wanting to be individuals, with insufficient attention to our social nature. |
| 7582 | Becoming what one is is a huge difficulty, because we strongly aspire to be something else [Kierkegaard] |
| Full Idea: Striving to become what one already is is a very difficult task, the most difficult of all, because every human being has a strong natural bent and passion to become something more and different. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846], 'Subjective') | |
| A reaction: Presumably most people continually drift between vanity and low self-esteem, and between unattainable daydreams and powerless immediate reality. That creates the stage on which Kierkegaard's interesting battle would have to be fought. |
| 16001 | Life may be understood backwards, but it has to be lived forwards [Kierkegaard] |
| Full Idea: Philosophy is perfectly right in saying that life must be understood backwards. But then it forgets the other side - that it must be lived forwards. | |
| From: Søren Kierkegaard (The Journals of Kierkegaard [1850], JP-III, 635) | |
| A reaction: Some of the best philosophers dwell too much on philosophy, history and the past, while forgetting to actually live and enjoy their lives. [SY] |
| 20747 | What matters is not right choice, but energy, earnestness and pathos in the choosing [Kierkegaard] |
| Full Idea: In making a choice, it is not so much a question of choosing the right way as of the energy, the earnestness, and the pathos with which one chooses. | |
| From: Søren Kierkegaard (Either/Or: a fragment of life [1843], p.106), quoted by Kevin Aho - Existentialism: an introduction 2 'Phenomenology' | |
| A reaction: I'm struggling to identify with the experience he is describing. I can't imagine a more quintessentially existentialist remark than this. Reference to 'energy' in choosing strikes me as very romantic. Is 'the way not taken' crucial (in 'pathos')? |
| 22093 | Life is a repetition when what has been now becomes [Kierkegaard] |
| Full Idea: When one says that life is a repetition one affirms that existence which has been now becomes. | |
| From: Søren Kierkegaard (Repetition [1843], p.49), quoted by Clare Carlisle - Kierkegaard: a guide for the perplexed 4 | |
| A reaction: Not sure I understand this, but it seems very close to Nietzsche's Eternal Recurrence. |
| 16009 | When we seek our own 'freedom' we are just trying to avoid responsibility [Kierkegaard] |
| Full Idea: In all our own 'freedom' we actually seek one thing: to be able to live without responsibility. | |
| From: Søren Kierkegaard (Attack Upon Christendom [1855], p.290) | |
| A reaction: That's the plan when I win the lottery. [SY] |
| 22091 | Kierkegaard prioritises the inward individual, rather than community [Kierkegaard, by Carlisle] |
| Full Idea: Whereas Hegel argues that individuals find fulfilment through participation in their community, Kierkegaard prioritises the inwardness of each person, which is shared only with God. | |
| From: report of Søren Kierkegaard (Either/Or: a fragment of life [1843]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 3 | |
| A reaction: Sounds like the protestant religion opposing the catholic religion (although Hegel was a protestant). Individual v community is the great debate of the last two centuries in Europe. |
| 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 |
| 7586 | God does not think or exist; God creates, and is eternal [Kierkegaard] |
| Full Idea: God does not think, He creates; God does not exist, he is eternal. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846], 'Thinker') | |
| A reaction: The sort of nicely challenging remarks we pay philosophers to come up with. I don't understand the second claim, but the first one certainly avoids all paradoxes that arise if God experiences all the intrinsic problems of thinking. |
| 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'. |
| 16006 | Either Abraham rises higher than universal ethics, or he is a mere murderer [Kierkegaard] |
| Full Idea: Either Abraham was a murderer, or we confront a paradox higher than all mediation. His story therefore contains the teleological suspension of the ethical, and he becomes higher than the universal. If not, he is not a tragic hero or the father of faith. | |
| From: Søren Kierkegaard (Fear and Trembling [1843], p.49) | |
| A reaction: A nice dilemma for Christian thinkers who want to reconcile reason and morality with religion. [SY] |
| 7577 | Abraham was willing to suspend ethics, for a higher idea [Kierkegaard] |
| Full Idea: The story of Abraham (and Isaac) contains a teleological suspension of the ethical. ...In his action he overstepped the ethical altogether, and had a higher idea outside it, in relation to which he suspended it. | |
| From: Søren Kierkegaard (Fear and Trembling [1843], Prob I) | |
| A reaction: My immediate response is to find this proposal very sinister. I can't remotely understand what Abraham's (or God's) 'higher' idea could be that could justify this crime. Maybe ethics is suspended if you are on the beach and a tidal wave arrives? |
| 20312 | God cannot be demonstrated objectively, because God is a subject, only existing inwardly [Kierkegaard] |
| Full Idea: Choosing the objective way enters upon the entire approximation-process by which it is proposed to bring God to light objectively. But this is in all eternity impossible, because God is a subject, and therefore exist only for subjectivity in inwardness. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846]) | |
| A reaction: [pg in 711] This seems to have something like Wittgenstein's problem with a private language - that with no external peer-review it is unclear what the commitment is. |
| 7580 | Pantheism destroys the distinction between good and evil [Kierkegaard] |
| Full Idea: So called pantheistic systems have often been characterised and challenged by the assertion that they abrogate the distinction between good and evil. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846], 'Lessing') | |
| A reaction: He will have Spinoza in mind. Interesting. Obviously this criticism would come from someone who thought that the traditional deity was the only source of goodness. Good/evil isn't all-or-nothing. A monistic system could contain them. |
| 16008 | The best way to be a Christian is without 'Christianity' [Kierkegaard] |
| Full Idea: One best becomes a Christian - without 'Christianity'. | |
| From: Søren Kierkegaard (The Journals of Kierkegaard [1850], JP-1:214) | |
| A reaction: A very healthy attitude for followers of Jesus, given today's television evangelists, religious fundamentalist and zealots. [SY] |
| 20735 | We need to see that Christianity cannot be understood [Kierkegaard] |
| Full Idea: The problem is not to understand Christianity, but to understand that it cannot be understood. | |
| From: Søren Kierkegaard (The Journals of Kierkegaard [1850], p.146), quoted by Kevin Aho - Existentialism: an introduction 1 'Roots' | |
| A reaction: This seems to cut us intellectually adrift. We could say the same of supporting Real Madrid. There has to be some magnetism which holds our attention, and there must be something to say about that. |
| 22088 | Faith is like a dancer's leap, going up to God, but also back to earth [Kierkegaard, by Carlisle] |
| Full Idea: Kierkegaard doesn't use the phrase 'leap of faith'. His metaphor of a dancer's leap expresses the way faith goes 'up' towards God, but also comes back down to earth, and is a way of living in the world. | |
| From: report of Søren Kierkegaard (Either/Or: a fragment of life [1843]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 2 | |
| A reaction: This entirely contradicts what I was taught about this idea many years ago. Memes turn into Chinese whispers. |
| 7584 | Without risk there is no faith [Kierkegaard] |
| Full Idea: Without risk there is no faith. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846], 'Inwardness') | |
| A reaction: Remarks like this make you realise that Kierkegaard is just as much of a romantic as most of the other nineteenth century philosophers. Plunge into the dark unknown of the human psyche, in order to intensify and heighten human life. |
| 7583 | Faith is the highest passion in the sphere of human subjectivity [Kierkegaard] |
| Full Idea: Faith is the highest passion in the sphere of human subjectivity. | |
| From: Søren Kierkegaard (Concluding Unscientific Postscript [1846], 'Subjective') | |
| A reaction: The word 'highest' should always ring alarm bells. The worst sort of religious fanatics seem to be in the grip of this 'high' passion. The early twenty-first century is an echo of eighteenth century England, with its dislike of religious 'enthusiasm'. |