88 ideas
| 13773 | For the truth you need Prodicus's fifty-drachma course, not his one-drachma course [Socrates] |
| Full Idea: Socrates: If I'd attended Prodicus's fifty-drachma course, I could tell you the truth about names straightway, but as I've only heard the one-drachma course, I don't know the truth about it. | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Cratylus 384b |
| 7421 | A philosopher is one who cares about what other people care about [Socrates, by Foucault] |
| Full Idea: Socrates asks people 'Are you caring for yourself?' He is the man who cares about the care of others; this is the particular position of the philosopher. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Michel Foucault - Ethics of the Concern for Self as Freedom p.287 | |
| A reaction: Priests, politicians and psychiatrists also care quite intensely about the concerns of other people. Someone who was intensely self-absorbed with the critical task of getting their own beliefs right would count for me as a philosopher. |
| 1649 | Socrates opened philosophy to all, but Plato confined moral enquiry to a tiny elite [Vlastos on Socrates] |
| Full Idea: To confine, as Plato does in 'Republic' IV-VII, moral inquiry to a tiny elite, is to obliterate the Socratic vision which opens up the philosophic life to all. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.18 | |
| A reaction: This doesn't mean that Plato is necessarily 'elitist'. It isn't elitist to point out that an activity is very difficult. |
| 5842 | Philosophical discussion involves dividing subject-matter into categories [Socrates, by Xenophon] |
| Full Idea: Self-discipline and avoidance of pleasure makes people most capable of philosophical discussion, which is called 'discussion' (dialegesthai - sort out) because people divide their subject-matter into categories. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 4.5.12 | |
| A reaction: This could be the original slogan for analytical philosophy, as far as I am concerned. I don't think philosophy aims at complete and successful analysis (cf. Idea 2958), but at revealing the structure and interconnection of ideas. This is wisdom. |
| 648 | Socrates began the quest for something universal with his definitions, but he didn't make them separate [Socrates, by Aristotle] |
| Full Idea: Socrates began the quest for something universal in addition to the radical flux of perceptible particulars, with his definitions. But he rightly understood that universals cannot be separated from particulars. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Metaphysics 1086b |
| 164 | It is legitimate to play the devil's advocate [Socrates] |
| Full Idea: It is legitimate to play the devil's advocate. | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Phaedrus 272c |
| 1647 | In Socratic dialogue you must say what you believe, so unasserted premises are not debated [Vlastos on Socrates] |
| Full Idea: Socrates' rule of "say only what you believe"….excluded debate on unasserted premises, thereby distinguishing Socratic from Zenonian and earlier dialectics. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.14 |
| 115 | Socrates was pleased if his mistakes were proved wrong [Socrates] |
| Full Idea: Socrates: I'm happy to have a mistaken idea of mine proved wrong. | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Gorgias 458a |
| 22099 | The method of Socrates shows the student is discovering the truth within himself [Socrates, by Carlisle] |
| Full Idea: Socrates tended to prefer the method of questioning, for this made it clear that the student was discovering the truth within himself. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 7 | |
| A reaction: Sounds like it will only facilitate conceptual analysis, and excludes empirical knowledge. Can you say to Socrates 'I'll just google that'? |
| 5844 | Socrates always proceeded in argument by general agreement at each stage [Socrates, by Xenophon] |
| Full Idea: When Socrates was setting out a detailed argument, he used to proceed by such stages as were generally agreed, because he thought that this was the infallible method of argument. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 4.6.16 | |
| A reaction: This sounds right, and shows how strongly Socrates perceived philosophy to be a group activity, of which I approve. It seems to me that philosophy is clearly a spoken subject before it is a written one. The lonely speculator comes much later. |
| 11389 | Socrates sought essences, which are the basis of formal logic [Socrates, by Aristotle] |
| Full Idea: It is not surprising that Socrates sought essences. His project was to establish formal reasoning, of whose syllogisms essences are the foundations. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Metaphysics 1078b22 | |
| A reaction: This seems to reinforce the definitional view of essences, since definitions seem to be at the centre of most of Socrates's quests. |
| 639 | Socrates developed definitions as the basis of syllogisms, and also inductive arguments [Socrates, by Aristotle] |
| Full Idea: Socrates aimed to establish formal logic, of whose syllogisms essences are the foundations. He developed inductive arguments and also general definitions. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Metaphysics 1078b |
| 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 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 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'. |
| 1652 | Socrates did not consider universals or definitions as having separate existence, but Plato made Forms of them [Socrates, by Aristotle] |
| Full Idea: Socrates did not regard the universals or the objects of definitions as separate existents, while Plato did separate them, and called this sort of entity ideas/forms. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Metaphysics 1078b30 |
| 1650 | For Socrates our soul, though hard to define, is our self [Vlastos on Socrates] |
| Full Idea: For Socrates our soul is our self - whatever that might turn out to be. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.55 | |
| A reaction: The problem with any broad claim like this is that we seem to be able to distinguish between essential and non-essential aspects of the self or of the soul. |
| 23252 | Socrates first proposed that we are run by mind or reason [Socrates, by Frede,M] |
| Full Idea: It would seem that historically the decisive step was taken by Socrates in conceiving of human beings as being run by a mind or reason.. …He postulated an entity whose precision nature and function then was a matter of considerable debate. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Michael Frede - Intro to 'Rationality in Greek Thought' p.19 | |
| A reaction: This is, for me, a rather revelatory idea. I am keen on the fact the animals make judgements which are true and false, and also that we exhibit rationality when walking across uneven ground. So pure rationality is a cultural construct! |
| 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. |
| 20062 | If a desire leads to a satisfactory result by an odd route, the causal theory looks wrong [Chisholm] |
| Full Idea: If someone wants to kill his uncle to inherit a fortune, and having this desire makes him so agitated that he loses control of his car and kills a pedestrian, who turns out to be his uncle, the conditions of the causal theory seem to be satisfied. | |
| From: Roderick Chisholm (Freedom and Action [1966]), quoted by Rowland Stout - Action 6 'Deviant' | |
| A reaction: This line of argument has undermined all sorts of causal theories that were fashionable in the 1960s and 70s. Explanation should lead to understanding, but a deviant causal chain doesn't explain the outcome. The causal theory can be tightened. |
| 20054 | There has to be a brain event which is not caused by another event, but by the agent [Chisholm] |
| Full Idea: There must be some event A, presumably some cerebral event, which is not caused by any other event, but by the agent. | |
| From: Roderick Chisholm (Freedom and Action [1966], p.20), quoted by Rowland Stout - Action 4 'Agent' | |
| A reaction: I'm afraid this thought strikes me as quaintly ridiculous. What kind of metaphysics can allow causation outside the natural nexus, yet occuring within the physical brain? This is a relic of religious dualism. Let it go. |
| 5843 | People do what they think they should do, and only ever do what they think they should do [Socrates, by Xenophon] |
| Full Idea: There is no one who knows what they ought to do, but thinks that they ought not to do it, and no one does anything other than what they think they ought to do. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 4.6.6 | |
| A reaction: This is Socrates' well-known rejection of the possibility of weakness of will (akrasia - lit. 'lack of control'). Aristotle disagreed, and so does almost everyone else. Modern smokers seem to exhibit akrasia. I have some sympathy with Socrates. |
| 5253 | Socrates was shocked by the idea of akrasia, but observation shows that it happens [Aristotle on Socrates] |
| Full Idea: Socrates thought it a shocking idea that when a man actually has knowledge in him something else should overmaster it, ..but this is glaringly inconsistent with the observed facts. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Aristotle - Nicomachean Ethics 1145b24 | |
| A reaction: Aristotle seems very confident, but it is not at all clear (even to the agent) what is going on when apparent weakness of will occurs (e.g. breaking a diet). What exactly does the agent believe at the moment of weakness? |
| 199 | The common belief is that people can know the best without acting on it [Socrates] |
| Full Idea: Most people think there are many who recognise the best but are unwilling to act on it. | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Protagoras 352d |
| 195 | No one willingly commits an evil or base act [Socrates] |
| Full Idea: I am fairly certain that no wise man believes anyone sins willingly or willingly perpetrates any evil or base act. | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Protagoras 345e |
| 1653 | Socrates did not accept the tripartite soul (which permits akrasia) [Vlastos on Socrates] |
| Full Idea: Xenophon indirectly indicates that he does not associate Socrates in any way with the tripartite psychology of the 'Republic', for within that theory akrasia would be all too possible. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.102 |
| 5839 | For Socrates, wisdom and prudence were the same thing [Socrates, by Xenophon] |
| Full Idea: Socrates did not distinguish wisdom from prudence, but judged that the man who recognises and puts into practice what is truly good, and the man who knows and guards against what is disgraceful, are both wise and prudent. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 3.9.3 | |
| A reaction: Compare Aristotle, who separates them, claiming that prudence is essential for moral virtue, but wisdom is pursued at a different level, closer to the gods than to society. |
| 5867 | For Socrates, virtues are forms of knowledge, so knowing justice produces justice [Socrates, by Aristotle] |
| Full Idea: Socrates thought that the virtues were all forms of knowledge, and therefore once a man knew justice, he would be a just man. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Eudemian Ethics 1216b07 | |
| A reaction: The clearest possible statement of Socrates' intellectualism. Aristotle rejected the Socrates view, but I find it sympathetic. Smokers who don't want to die seem to be in denial. To see the victims is to condemn the crime. |
| 5069 | Socrates was the first to base ethics upon reason, and use reason to explain it [Taylor,R on Socrates] |
| Full Idea: Socrates was the first significant thinker to try basing ethics upon reason, and to try uncovering its natural principles solely by the use of reason. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Richard Taylor - Virtue Ethics: an Introduction Ch.7 | |
| A reaction: Interesting. It seems to me that Socrates overemphasised reason, presumably because it was a novelty. Hence his view that akrasia is impossible, and that virtue is simply knowledge. Maybe action is not just rational, but moral action is. |
| 5836 | All human virtues are increased by study and practice [Socrates, by Xenophon] |
| Full Idea: If you consider the virtues that are recognised among human beings, you will find that they are all increased by study and practice. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 2.6.41 | |
| A reaction: 'Study' is the intellectualist part of this remark; the reference to 'practice' fits with Aristotle view that virtue is largely a matter of good habits. The next question would be how theoretical the studies should be. Philosophy, or newspapers? |
| 5840 | The wise perform good actions, and people fail to be good without wisdom [Socrates, by Xenophon] |
| Full Idea: It is the wise who perform truly good actions, and those who are not wise cannot, and, if they try to, fail. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 3.9.6 | |
| A reaction: The essence of Socrates' intellectualism, with which Aristotle firmly disagreed (when he assert that only practical reason was needed for virtuous actions, rather than wisdom or theory). Personally I side more with Socrates than with Aristotle on this. |
| 185 | Socrates despised good looks [Socrates, by Plato] |
| Full Idea: Socrates despises good looks to an almost inconceivable extent. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Plato - The Symposium 216e |
| 5070 | Socrates conservatively assumed that Athenian conventions were natural and true [Taylor,R on Socrates] |
| Full Idea: Socrates' moral philosophy was essentially conservative. He assumed that the principles the Athenians honoured were true and natural, so there was little possibility of conflict between nature and convention in his thinking. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Richard Taylor - Virtue Ethics: an Introduction Ch.8 | |
| A reaction: Taylor contrasts Socrates with Callicles, who claims that conventions oppose nature. This fits with Nietzsche's discontent with Socrates, as the person who endorses conventional good and evil, thus constraining the possibilities of human nature. |
| 5838 | A well-made dung basket is fine, and a badly-made gold shield is base, because of function [Socrates, by Xenophon] |
| Full Idea: A dung-basket is fine, and a golden shield contemptible, if the one is finely and the other badly constructed for carrying out its function. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 3.8.6 | |
| A reaction: This is the basis of a key idea in Aristotle, that virtue (or excellence) arises directly from function. I think it is the most important idea in virtue theory, and seems to have struck most Greeks as being self-evident. |
| 5837 | Things are both good and fine by the same standard [Socrates, by Xenophon] |
| Full Idea: Things are always both good and fine by the same standard. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 3.8.5 | |
| A reaction: This begs many questions, but perhaps it leads to what we call intuitionism, which is an instant ability is perceive a fine action (even in an enemy). This leads to the rather decadent view that the aim of life is the production of beauty. |
| 3017 | The only good is knowledge, and the only evil is ignorance [Socrates, by Diog. Laertius] |
| Full Idea: There is only one good, namely knowledge, and there is only one evil, namely ignorance. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.4.14 | |
| A reaction: Ignorance of how to commit evil sounds quite good. |
| 1646 | Socrates was the first to put 'eudaimonia' at the centre of ethics [Socrates, by Vlastos] |
| Full Idea: Socrates' true place in the development of Greek thought is that he is the first to establish the eudaimonist foundation of ethical theory, which became the foundation of the schools which sprang up around him. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.10 | |
| A reaction: I suspect that he was the first to fully articulate a widely held Greek belief. The only ethical question that they asked was about the nature of a good human life. |
| 1663 | By 'areté' Socrates means just what we mean by moral virtue [Vlastos on Socrates] |
| Full Idea: Socrates uses the word 'areté' to mean precisely what we mean by moral virtue. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.200 |
| 4323 | Socrates is torn between intellectual virtue, which is united and teachable, and natural virtue, which isn't [PG on Socrates] |
| Full Idea: Socrates worries about the unity and teachability of virtue because he is torn between virtue as intellectual (unified and teachable) and virtue as natural (plural and unteachable). | |
| From: comment on Socrates (reports of career [c.420 BCE]) by PG - Db (ideas) | |
| A reaction: Admittedly virtue could be natural but still unified and teachable, but Socrates clearly had a dilemma, and this seems to make sense of it. |
| 8003 | Socrates agrees that virtue is teachable, but then denies that there are teachers [Socrates, by MacIntyre] |
| Full Idea: Socrates' great point of agreement with the sophists is his acceptance of the thesis that areté is teachable. But paradoxically he denies that there are teachers. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Alasdair MacIntyre - A Short History of Ethics Ch.3 | |
| A reaction: This is part of Socrates's presentation of himself as 'not worthy'. Virtue would be teachable, if only anyone knew what it was. He's wrong. Lots of people have a pretty good idea of virtue, and could teach it. The problem is in the pupils. |
| 126 | We should ask what sort of people we want to be [Socrates] |
| Full Idea: Socrates: What sort of person should one be? | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Gorgias 487e |
| 4111 | Socrates believed that basically there is only one virtue, the power of right judgement [Socrates, by Williams,B] |
| Full Idea: Socrates believed that basically there is only one virtue, the power of right judgement. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Bernard Williams - Ethics and the Limits of Philosophy Ch.1 | |
| A reaction: Which links with Aristotle's high place for 'phronesis' (prudence?). The essence of Socrates' intellectualism. Robots and saints make very different judgements, though. |
| 7808 | Socrates made the civic values of justice and friendship paramount [Socrates, by Grayling] |
| Full Idea: In Socrates' thought, the expressly civic values of justice and friendship became paramount. | |
| From: report of Socrates (reports of career [c.420 BCE]) by A.C. Grayling - What is Good? Ch.2 | |
| A reaction: This is the key move in ancient ethics, away from heroism, and towards the standard Aristotelian social virtues. I say this is the essence of what we call morality, and the only one which can be given a decent foundational justification (social health). |
| 23907 | Courage is scientific knowledge [Socrates, by Aristotle] |
| Full Idea: Socrates thought that courage is scientific knowledge. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Eudemian Ethics 1230a06 | |
| A reaction: Aristotle himself says that reason produces courage, but he also says it arises from natural youthful spirits. I favour the view that there is a strong rational component in true courage. |
| 7585 | Socrates emphasises that the knower is an existing individual, with existence his main task [Socrates, by Kierkegaard] |
| Full Idea: The infinite merit of the Socratic position was precisely to accentuate the fact that the knower is an existing individual, and that the task of existing is his essential task. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Søren Kierkegaard - Concluding Unscientific Postscript 'Inwardness' | |
| A reaction: Always claim Socrates as the first spokesman for your movement! It is true that Socrates is always demanding the views of his interlocutors, and not just abstract theories. See Idea 1647. |
| 5841 | Obedience to the law gives the best life, and success in war [Socrates, by Xenophon] |
| Full Idea: A city in which the people are most obedient to the laws has the best life in time of peace and is irresistible in war. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 4.4.15 | |
| A reaction: This is a conservative view, with the obvious problem case of bad laws, but in general it seems to me clearly right. This is why it is so vital that nothing should be done to bring the law into disrepute, such as petty legislation or prosecution. |
| 1661 | Socrates was the first to grasp that a cruelty is not justified by another cruelty [Vlastos on Socrates] |
| Full Idea: Socrates was the first Greek to grasp the truth that if someone has done a nasty thing to me, this does not give the slightest moral justification for doing anything nasty to him. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.190 |
| 5846 | A lover using force is a villain, but a seducer is much worse, because he corrupts character [Socrates, by Xenophon] |
| Full Idea: The fact that a lover uses not force but persuasion makes him more detestable, because a lover who uses force proves himself a villain, but one who uses persuasion ruins the character of the one who consents. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Symposium 8.20 | |
| A reaction: A footnote says that this distinction was enshrined in Athenian law, where seduction was worse than rape. This is a startling and interest contrast to the modern view, which enshrines rights and freedoms, and says seduction is usually no crime at all. |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |
| 1657 | Socrates holds that right reason entails virtue, and this must also apply to the gods [Vlastos on Socrates] |
| Full Idea: It is essential to Socrates' rationalist programme in theology to assume that the entailment of virtue by wisdom binds gods no less than men. He would not tolerate one moral standard for me and another for gods. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.164 |
| 1662 | A new concept of God as unswerving goodness emerges from Socrates' commitment to virtue [Vlastos on Socrates] |
| Full Idea: Undeviating beneficent goodness guides Socrates' thought so deeply that he applies it even to the deity; he projects a new concept of god as a being that can cause only good, never evil. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.197 |