68 ideas
| 14027 | If we are to use words in enquiry, we need their main, unambiguous and uncontested meanings [Epicurus] |
| Full Idea: It is necessary that we look to the primary conception corresponding to each word and that it stand in no need of demonstration, if, that is, we are going to have something to which we can refer the object of search or puzzlement and opinion. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 38) | |
| A reaction: This either points to definition or to consensus, and since definition seems in danger of some sort of Quinean circularity, I favour consensus. Philosophy is, after all, people discussing things, not inscriptions sent to the gods. |
| 14040 | Observation and applied thought are always true [Epicurus] |
| Full Idea: Everything that is observed or grasped by the intellect in an act of application is true. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 62) | |
| A reaction: Not quite clear what he means, but Epicurus is committed to perception as the source of knowledge, with the intellect extending the findings of the senses. He might subscribe to Descartes's 'clear and distinct' perceptions. |
| 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'. |
| 14028 | Nothing comes to be from what doesn't exist [Epicurus] |
| Full Idea: Nothing comes into being from what is not. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 38) | |
| A reaction: King Lear puts it better: Nothing will come of nothing [1.i]. There seems to be an underlying assumption that coming into being out of nothing is much weirder than just existing, but I am not convinced about that. It's all equally weird. |
| 14029 | If disappearing things went to nothingness, nothing could return, and it would all be gone by now [Epicurus] |
| Full Idea: If that which disappears were destroyed into what is not, all things would have been destroyed, since that into which they were dissolved does not exist. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 39) | |
| A reaction: This follows on from Idea 14028. Theologians will immediately spot that this is the underlying principle cited by Aquinas in his Third Way for proving God's existence (Idea 1431). |
| 14030 | The totality is complete, so there is no room for it to change, and nothing extraneous to change it [Epicurus] |
| Full Idea: The totality of things has always been just like it is now and always will be. For there is nothing for it to change into. For there exists nothing in addition to the totality, which could enter into it and produce the change. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 39) | |
| A reaction: This smacks of the sort of dubious arguments that the medieval theologians fell in love with. I never thought I'd say this, but I think Epicurus needs a comprehensive course in set theory before he makes remarks like this. |
| 14048 | Astronomical movements are blessed, but they don't need the help of the gods [Epicurus] |
| Full Idea: Movements, turnings, risings, settings, and related phenomena occur without any god helping out and ordaining or being about to ordain things, and at the same time have complete blessedness and indestructibility. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 76) | |
| A reaction: Epicurus is sometimes accused of atheism for remarks like these, but he is always trying to show piety in his attitudes. We might now call this attitude 'deism' (see alphabetical themes). |
| 14044 | The perceived accidental properties of bodies cannot be conceived of as independent natures [Epicurus] |
| Full Idea: The shapes, colours, sizes and weights which are predicated of body as accidents, ...and are known by sense-perception, must not be thought of as independent natures (for that is inconceivable). | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 68) | |
| A reaction: I take this to be an anti-platonist remark, though he is not denying that the accidental properties may have some universal character. I'm struck by how close the basic metaphysics of Epicurus is to that of Aristotle. |
| 14045 | Accidental properties give a body its nature, but are not themselves bodies or parts of bodies [Epicurus] |
| Full Idea: Accidental qualities are not non-existent, nor are they distinct corporeal entities inhering in the body, nor parts of it. We should think that the whole body throughout derives its permanent nature from these properties, though not as a compound of them. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 69) | |
| A reaction: 'Permanent' nature sounds more like essential than accidental properties. This is uncomfortably negative in its attempt to pin down what accidental properties are. The last bit seems to deny the bundle view of objects. Would he like tropes? |
| 14046 | A 'body' is a conception of an aggregate, with properties defined by application conditions [Epicurus] |
| Full Idea: Properties are known by their peculiar forms of application and comprehension, in close accompaniment with the aggregate [of atoms], which is given the predicate 'body' by reference to the aggregate conception. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 69) | |
| A reaction: There is an interesting hint here of how to think of properties (as both applying and comprehended in some distinctive way), and a suggestion that there is something conventional about bodies, depending on how we conceive them. |
| 14047 | Bodies have impermanent properties, and permanent ones which define its conceived nature [Epicurus] |
| Full Idea: Impermanent properties do not have the nature of an entire thing, which we call a body when we grasp it in aggregate, nor the nature of permanent accompaniments without which it is not possible to conceive of a body. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 70) | |
| A reaction: Epicurus doesn't discuss essences, but this seems to commit to the basic Aristotelian idea, that there there are some properties which actually bestow identity, and then others which are optional for that thing. The 'conception' is always mentioned. |
| 14039 | Above and below us will never appear to be the same, because it is inconceivable [Epicurus] |
| Full Idea: What is over our heads ...or what is below any point which we think of ...will never appear to us as being at the same time and in the same respect both up and down. For it is impossible to conceive of this. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 60) | |
| A reaction: Note that he says it will 'never appear to us' as both - not that it absolutely cannot be both. Both Aristotle and Epicurus are much more focused on how our humanity shapes our metaphysics than the modern pure metaphysicians are. |
| 14050 | We aim to dissolve our fears, by understanding their causes [Epicurus] |
| Full Idea: If we give a correct and complete causal account of the source of our disturbance and fears, we will dissolve them, by accounting for the phenomena to which we are constantly exposed, and which terrify other men most severely. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 82) | |
| A reaction: Notice 'other' men! This eudaimonist aim lies at the heart of Epicurus's physical account of the world. He was primarily interested in living better, rather than in physical science. He seeks 'tranquillity' and 'freedom from disturbance'. |
| 14037 | Atoms only have shape, weight and size, and the properties which accompany shape [Epicurus] |
| Full Idea: One must believe that the atoms bring with them none of the qualities of things which appear except shape, weight, and size and the properties which necessarily accompany shape. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 54) | |
| A reaction: This appears to be fairly precisely a claim that atoms only have primary qualities, though that terminology only came in in the seventeenth century. I take the view to be more or less correct. |
| 6010 | Illusions are not false perceptions, as we accurately perceive the pattern of atoms [Epicurus, by Modrak] |
| Full Idea: Epicurus says illusions are not false perceptions, because the senses accurately report the pattern of atoms; for instance, the edges are worn off the pattern produced by a square tower, so its perception as a round tower is true. | |
| From: report of Epicurus (Letter to Herodotus [c.293 BCE], 47-53) by Deborah K.W. Modrak - Classical theories of Mind | |
| A reaction: As so often, Epicurus got it right, because Democritus got it right, thus demonstrating that good philosophy must be preceded by good physics. However, good physics must be preceded and followed by good philosophy. |
| 14041 | The soul is fine parts distributed through the body, resembling hot breath [Epicurus] |
| Full Idea: The soul is a body made up of fine parts distributed throught the entire aggregate, most closely resembling breath with a certain admixture of heat, in one way resembling breath and in another resembling heat | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 63) | |
| A reaction: Remember that 'psuché' refers as much to the life within a creature as it does to the consciousness. The stoics seem to have held a similar view. |
| 14042 | The soul cannot be incorporeal, because then it could neither act nor be acted upon [Epicurus] |
| Full Idea: Those who say that the soul is incorporeal are speaking to no point; for if it were of that character, it could neither act nor be acted upon at all. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 67) | |
| A reaction: This just is the causal argument, which is espoused by Papineau and other modern physicalists. Personally I am inclined to agree with Papineau, that it is so simple and conclusive that it is hardly worth discussing further. Dualism needs a miracle. |
| 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. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |
| 14032 | Totality has no edge; an edge implies a contrast beyond the edge, and there can't be one [Epicurus] |
| Full Idea: The totality is unlimited. For what is limited has an extreme; but an extreme is seen in contrast to something else, so that since it has no extreme it has no limit. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 41) | |
| A reaction: I presume that the 'limit' is the edge, and the 'extreme' is what is beyond the edge. Why could not the extreme be nothingness, which then contrast dramatically with what exists? |
| 14033 | Bodies are unlimited as well as void, since the two necessarily go together [Epicurus] |
| Full Idea: The number of bodies and the magnitude of the void are unlimited. If void were unlimited, and bodies limited, bodies move in scattered fashion with no support of checking collisions; in limited void, unlimited bodies would not have a place to be in. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 42) | |
| A reaction: Seems good. The point is that without collisions, bodies would not stop relative to one another, and combine to form the objects we perceive. Of course if the started off (anathema!) stuck together, they may not have dispersed yet. |
| 14034 | There exists an infinity of each shape of atom, but the number of shapes is beyond our knowledge [Epicurus] |
| Full Idea: For each type of shape there is an unlimited number of similar atoms, but with respect to the differences they are not simply unlimited but ungraspable. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 42) | |
| A reaction: Epicurus's view of the nature of atoms rests on his empiricism, so while he can reason from experience to how they must be, he admits (impressively) his ignorance of the full facts. He has arguments for the unlimited number. |
| 14035 | Atoms just have shape, size and weight; colour results from their arrangement [Epicurus] |
| Full Idea: There are not even any qualities in atoms, except shape and size and weight; their colour changes according to the arrangement of the atoms. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 44 schol) | |
| A reaction: [This is quoted by a 'scholiast' - an early writer quoting from Epicurus's '12 Basic Principles'] He appears to have got this one wrong, as it is evidently the type of atom, as well as the arrangement, which contributes to the colour. |
| 14038 | There cannot be unlimited division, because it would reduce things to non-existence [Epicurus] |
| Full Idea: One must eliminate unlimited division into smaller pieces (to avoid making everything weak and being forced in our comprehensive grasps of compound things to exhaust the things which exist by reducing them to non-existence). | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 56) | |
| A reaction: A basic argument for atoms, but it seems to rest on Zenonian paradoxes about infinite subdivision. An infinite subdivision of a unit doesn't seem to turn it into zero. |
| 14049 | We aim to know the natures which are observed in natural phenomena [Epicurus] |
| Full Idea: Blessedness lies in knowing the natures which are observed in meteorological phenomena. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 78) | |
| A reaction: This pursuit of 'natures' seems to be at the heart of scientific essentialism. Epicurus demonstrates his proposal, by offering speculations about the natures of all sorts of phenomena (esp. in 'Letter to Pythocles'). |
| 14043 | The void cannot interact, but just gives the possibility of motion [Epicurus] |
| Full Idea: The void can neither act nor be acted upon but merely provides the possibility of motion through itself for bodies. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 67) | |
| A reaction: Epicurus follows this with the anti-dualist Idea 14042, but he is at least offering the notion of something which exists without powers of causal interaction. Does space undermine the causal criterion for existence? |
| 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 |
| 14031 | Space must exist, since movement is obvious, and there must be somewhere to move in [Epicurus] |
| Full Idea: If there did not exist that which we call void and space and intangible nature, bodies would not have any place to be in or move through, as they obviously do move. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 40) | |
| A reaction: The observation that 'they obviously do move' must be aimed at followers of Parmenides. The idea of the void seems to contain a Newtonian commitment to absolute space. |
| 14036 | There are endless cosmoi, some like and some unlike this one [Epicurus] |
| Full Idea: There is an unlimited number of cosmoi, and some are similar to this one and some are dissimilar. | |
| From: Epicurus (Letter to Herodotus [c.293 BCE], 45) |
| 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'. |