101 ideas
| 19441 | All philosophies presuppose their historical moment, and arise from it [Feuerbach] |
| Full Idea: Every philosophy originates as a manifestation of its time; its origin presupposes its historical time. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.59) | |
| A reaction: There seems to be widespread agreement among continental philosophers about this idea, whereas analytic philosophers largely ignore, and treat Plato as if he were a current professor in Chicago. |
| 19456 | Philosophy is distinguished from other sciences by its complete lack of presuppositions [Feuerbach] |
| Full Idea: Philosophy does not presuppose anything. It is precisely in this fact of non-presupposition that its beginning lies - a beginning by virtue of which it is set apart from all the other sciences. | |
| From: Ludwig Feuerbach (On 'The Beginning of Philosophy' [1841], p.135) | |
| A reaction: Most modern philosophers seem to laugh at such an idea, because everything is theory-laden, culture-laden, language-laden etc. As an aspiration I love it, and think good philosophers get quite close to the goal (which, I admit, is not fully attainable). |
| 19442 | I don't study Plato for his own sake; the primary aim is always understanding [Feuerbach] |
| Full Idea: Plato in writing is only a means for me; that which is primary and a priori, that which is the ground to which all is ultimately referred, is understanding. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.63) | |
| A reaction: It always seems to that the main aim of philosophy is understanding - which is why its central activity is explanation. |
| 6928 | Only that which can be an object of religion is an object of philosophy [Feuerbach] |
| Full Idea: Only that which can be an object of religion is an object of philosophy. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §35) | |
| A reaction: The temple of Pythagoras at Solon sounds like an embodiment of this idea. The obvious candidate would be truth, to which philosophers must show almost religious respect. Some what motivates the philosophy of a minimalist (Idea 3750)? |
| 6918 | Philosophy should not focus on names, but on the determined nature of things [Feuerbach] |
| Full Idea: Philosophy need not care about the conceptions that common usage or misuse attaches to a name; philosophy, however, has to bind itself to the determined nature of things, whose signs are names. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §23) | |
| A reaction: I like this attempt to nip ordinary language philosophy in the bud. Indeed I like the notion of philosophy binding itself to the 'determined nature of things' (which sound like essences to me), rather than to their names or descriptions. |
| 6904 | Modern philosophy begins with Descartes' abstraction from sensation and matter [Feuerbach] |
| Full Idea: The beginning of Descartes' philosophy, namely, the abstraction from sensation and matter, is the beginning of modern speculative philosophy. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §10) | |
| A reaction: In Britain it might be said that modern philosophy begins with a rebellion against Descartes' move. Feuerbach is charting the movement towards idealism. |
| 6931 | Empiricism is right about ideas, but forgets man himself as one of our objects [Feuerbach] |
| Full Idea: Empiricism rightly derives the origin of our ideas from the senses; only it forgets that the most important and essential object of man is man himself. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §41) | |
| A reaction: This seems to nicely pinpoint the objection of most 'continental' philosophy to British empiricism and analytic philosophy. It seems to point towards Husserl's phenomenology as the next step. It is true that empiricists divided person from world. |
| 6933 | The laws of reality are also the laws of thought [Feuerbach] |
| Full Idea: The laws of reality are also the laws of thought. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §45) | |
| A reaction: I like this a lot, though it runs contrary to a lot of conventionalist thinking in the twentieth century. Russell, though, agrees with Feuerbach (Idea 5405). There is not much point to thought if it doesn't plug into reality at the roots. |
| 19444 | Each proposition has an antithesis, and truth exists as its refutation [Feuerbach] |
| Full Idea: Every intellectual determination has its antithesis, its contradiction. Truth exists not in unity with, but in refutation of its opposite. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.72) | |
| A reaction: This appears to be a rejection of the 'synthesis' in Hegel, in favour of what strikes me as a rather more sensible interpretation of the modern dialectic. Being exists in contrast to nothingness, and truth exists in contrast to its negation? |
| 19445 | A dialectician has to be his own opponent [Feuerbach] |
| Full Idea: A thinker is a dialectician only insofar as he is his own opponent. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.72) | |
| A reaction: Quite an inspirational slogan for beginners in philosophy. How many non-philosophers are willing to be their own opponent. In law courts and the House of Commons we assign the roles to separate persons. Hence rhetoric replaces reason? |
| 19443 | Truth forges an impersonal unity between people [Feuerbach] |
| Full Idea: The urge to communicate is a fundamental urge - the urge for truth. ...That which is true belongs neither to me nor exclusively to you, but is common to all. The thought in which 'I' and 'You' are united is a true thought. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.65) | |
| A reaction: Sceptics may doubt that there are such truths, but this is certainly how we experience agreement - that there is some truth shared between us which is no longer the possession of either of us. Nice idea. |
| 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. |
| 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 |
| 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. |
| 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 |
| 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. |
| 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. |
| 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. |
| 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'. |
| 6919 | Absolute thought remains in another world from being [Feuerbach] |
| Full Idea: Absolute thought never extricates itself from itself to become being. Being remains in another world. …If being is to be added to an object of thought, so must something distinct from thought be added to thought itself. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §24/5) | |
| A reaction: This sounds a bit like a child wishing for the moon. Is he saying he doesn't just want to think about reality - he wants his mental states to BE external reality? The distinction between a thought and its content or intentionality would help here. |
| 19457 | Being is what is undetermined, and hence indistinguishable [Feuerbach] |
| Full Idea: Being in the sense in which it is an object of speculative thought is that which is purely and simply unmediated, that is, undetermined; in other words, there is nothing to distinguish and nothing to think of in being. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], 26) | |
| A reaction: This sounds remarkably like the idea of 'prime matter' used in scholastic Aristotelian philosophy. Matter existing without form is somehow ungraspable, but presented from Hegel onwards as the ultimate mystery. |
| 6920 | Being posits essence, and my essence is my being [Feuerbach] |
| Full Idea: Being is the positing of essence. That which is my essence is my being. The fish exists in water; you cannot, however, separate its essence from this being. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §27) | |
| A reaction: This throws a different light on later (e.g. Heidegger) discussions of 'being', which may map onto Aristotelian discussions of essences. |
| 6921 | Particularity belongs to being, whereas generality belongs to thought [Feuerbach] |
| Full Idea: Particularity and individuality belong to being, whereas generality belongs to thought. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §29) | |
| A reaction: This agrees with Russell's view that every sentence (and proposition) must contain a universal (i.e a generality). The very notion of thinking 'about' a horse seems to require a move to the universal concept of a horse. |
| 6926 | The only true being is of the senses, perception, feeling and love [Feuerbach] |
| Full Idea: Being as an object of being - and only this being is being and deserves the name of being - is the being of the senses, perception, feeling, and love. …Only passion is the hallmark of existence. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §33) | |
| A reaction: This remark seems to make Feuerbach a romantic and anti-Enlightenment figure. I don't see why there shouldn't be just as much 'being' in doing maths as in admiring a landscape. The mention of love links him to Empedocles (Ideas 459 + 630). |
| 19446 | To our consciousness it is language which looks unreal [Feuerbach] |
| Full Idea: To sensuous consciousness it is precisely language that is unreal, nothing. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.77) | |
| A reaction: Offered as a corrective to the view that our ontological commitments entirely concern what we are willing to say. |
| 6908 | Consciousness is absolute reality, and everything exists through consciousness [Feuerbach] |
| Full Idea: Consciousness is the absolute reality, the measure of all existence; all that exists, exists only as being for consciousness, as comprehended in consciousness; for consciousness is first and foremost being. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §17) | |
| A reaction: This is Feuerbach declaring himself in favour of idealism even as he was trying to rebel against it, and move towards a more sensuous and human view of the world. I just see idealists as confusing ontology and epistemology. |
| 19447 | The Absolute is the 'and' which unites 'spirit and nature' [Feuerbach] |
| Full Idea: The Absolute is spirit and nature. ...But what then is the Absolute? Nothing other than this 'and', that is, the unity of spirit and nature. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.82) | |
| A reaction: This is Feuerbach's spin on Hegel. He has been outlining idealist philosophy and the philosophy of nature in Schelling. Is this Spinoza's one substance? |
| 6932 | Ideas arise through communication, and reason is reached through community [Feuerbach] |
| Full Idea: Only through communication and conversation between man and man do ideas arise; not alone, but only with others, does one reach notions and reason in general. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §41) | |
| A reaction: This is a strikingly modern view of the solipsism problem, and is close in spirit to Wittgenstein's Private Language Argument (Ideas 4143 +4158). Feuerbach is interested in universals rather than rules. I prefer Feuerbach. |
| 6935 | In man the lowest senses of smell and taste elevate themselves to intellectual acts [Feuerbach] |
| Full Idea: Even the lowest senses, smell and taste, elevate themselves in man to intellectual and scientific acts. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §53) | |
| A reaction: Since Darwin we have, I am glad to say, lost this need to distinguish what is 'low' or 'high', and to try to show that even our 'lowest' functions are on the 'high' side. Personally, though, I still need the low/high distinction in moral thinking. |
| 12790 | Generalisations must be invariant to explain anything [Leuridan] |
| Full Idea: A generalisation is explanatory if and only if it is invariant. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §4) | |
| A reaction: [He cites Jim Woodward 2003] I dislike the idea that generalisations and regularities explain anything at all, but this rule sounds like a bare minimum for being taken seriously in the space of explanations. |
| 12789 | Biological functions are explained by disposition, or by causal role [Leuridan] |
| Full Idea: The main alternative to the dispositional theory of biological functions (which confer a survival-enhancing propensity) is the etiological theory (effects are functions if they play a role in the causal history of that very component). | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §3) | |
| A reaction: [Bigelow/Pargetter 1987 for the first, Mitchell 2003 for the second] The second one sounds a bit circular, but on the whole a I prefer causal explanations to dispositional explanations. |
| 14388 | Mechanisms must produce macro-level regularities, but that needs micro-level regularities [Leuridan] |
| Full Idea: Nothing can count as a mechanism unless it produces some macro-level regular behaviour. To produce macro-level regular behaviour, it has to rely on micro-level regularities. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §5) | |
| A reaction: This is the core of Leuridan's argument that regularities are more basic than mechanisms. It doesn't follow, though, that the more basic a thing is the more explanatory work it can do. I say mechanisms explain more than low-level regularities do. |
| 12787 | Mechanisms can't explain on their own, as their models rest on pragmatic regularities [Leuridan] |
| Full Idea: To model a mechanism one must incorporate pragmatic laws. ...As valuable as the concept of mechanism and mechanistic explanation are, they cannot replace regularities nor undermine their relevance for scientific explanation. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1) | |
| A reaction: [See Idea 12786 for 'pragmatic laws'] I just don't see how the observation of a regularity is any sort of explanation. I just take a regularity to be something interesting which needs to be explained. |
| 14384 | We can show that regularities and pragmatic laws are more basic than mechanisms [Leuridan] |
| Full Idea: Summary: mechanisms depend on regularities, there may be regularities without mechanisms, models of mechanisms must incorporate pragmatic laws, and pragmatic laws do not depend epistemologically on mechanistic models. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1) | |
| A reaction: See Idea 14382 for 'pragmatic' laws. I'm quite keen on mechanisms, so this is an arrow close to the heart, but at this point I say that my ultimate allegiance is to powers, not to mechanisms. |
| 14386 | Mechanisms are ontologically dependent on regularities [Leuridan] |
| Full Idea: Mechanisms are ontologically dependent on the existence of regularities. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §3) | |
| A reaction: This seems to be the Humean rearguard action in favour of the regularity account of laws. Wrong, but a nice paper. This point shows why only powers (despite their vagueness!) are the only candidate for the bottom level of explanation. |
| 14389 | There is nothing wrong with an infinite regress of mechanisms and regularities [Leuridan] |
| Full Idea: I see nothing metaphysically wrong in an infinite ontological regress of mechanisms and regularities. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §5) | |
| A reaction: This is a pretty unusual view, and I can't accept it. My revulsion at this regress is precisely the reason why I believe in powers, as the bottom level of explanation. |
| 19451 | When absorbed in deep reflection, is your reason in control, or is it you? [Feuerbach] |
| Full Idea: When, submerged in deep reflection, you forget both yourself and your surroundings, is it you who controls reason, or is it rather reason that controls and absorbs you? | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], I) | |
| A reaction: A delightful question, even if it looks like a false dichotomy. I'm not sure what to make of 'me', if my reason can be subtracted from it. Aquinas was one the same wavelength here. |
| 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? |
| 6925 | The new philosophy thinks of the concrete in a concrete (not a abstract) manner [Feuerbach] |
| Full Idea: The new philosophy is the philosophy that thinks of the concrete not in an abstract, but in a concrete manner. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §31) | |
| A reaction: This leads to placing a high value on art, and on virtuous action through particulars rather than principles, and on empirical science. The only problem is that what he proposes is impossible. To think 'about' is to abstract from the particulars. |
| 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. |
| 6924 | Plotinus was ashamed to have a body [Feuerbach] |
| Full Idea: Plotinus, according to his biographers, was ashamed to have a body. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §29) | |
| A reaction: When Feuerbach draws our attention to this, we see what an astonishing state it is for a human being to have got into. Modern thought is appalled by it, but it also has something heroic about it, like swimming all the time because you want to be a fish. |
| 6927 | If you love nothing, it doesn't matter whether something exists or not [Feuerbach] |
| Full Idea: To him who loves nothing it is all the same whether something does or does not exist. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §33) | |
| A reaction: This seems to me to be quite a good motto for the aim of education - just get them to love something, no matter what (well, almost!). Loving something, even if it is train-spotting, seems a good route to human happiness. |
| 19450 | Reason, love and will are the highest perfections and essence of man - the purpose of his life [Feuerbach] |
| Full Idea: Reason, love and power of will are perfections of man; they are his highest powers, his absolute essence in so far as he is man, the purpose of his existence. Man exists in order to think, love and will. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], I) | |
| A reaction: Feuerbach was a notable atheist, but adopts a religious style of language which modern atheists would find rather alien. Personally I love talk of ideals and perfections. Ideals have been discredited in modern times, but need a revival. |
| 19458 | Egoism is the only evil, love the only good; genuine love produces all the other virtues [Feuerbach] |
| Full Idea: There is only one evil - egoism; there is only one good - love. ...Love, but truly! All other virtues will automatically come to you. | |
| From: Ludwig Feuerbach (Fragments on My Philosophical Development [1839], 1834-6) | |
| A reaction: This is a rather Christian idea of virtue, coming from the great atheist. Does tough love come from love? |
| 6934 | Man is not a particular being, like animals, but a universal being [Feuerbach] |
| Full Idea: Man is not a particular being, like the animals, but a universal being. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §53) | |
| A reaction: This sounds a bit extravagent. The capacity of man to use universals in thought seems crucial to Feuerbach (though he doesn't directly address the problem). 'We are particulars with access to universals' sounds better. |
| 6936 | The essence of man is in community, but with distinct individuals [Feuerbach] |
| Full Idea: The essence of man is contained only in the community and unity of man and man; it is a unity, however, which rests only on the reality of the distinction between I and thou. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §59) | |
| A reaction: In English provincial suburbs (where I live) it is astonishing how little interest in and need for their neighbours people seem to have. People seem to survive without community. Most of us, though, think full human happiness needs community. |
| 14387 | Rather than dispositions, functions may be the element that brought a thing into existence [Leuridan] |
| Full Idea: The dispositional theory of biological functions is not unquestioned. The main alternative is the etiological theory: a component's effect is a function of that component if it has played an essential role in the causal history of its existence. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §3) | |
| A reaction: [He cites S.D. Mitchell 2003] Presumably this account is meant to fit into a theory of evolution in biology. The obvious problem is where something comes into existence for one reason, and then acquires a new function (such as piano-playing). |
| 14382 | Pragmatic laws allow prediction and explanation, to the extent that reality is stable [Leuridan] |
| Full Idea: A generalization is a 'pragmatic law' if it allows of prediction, explanation and manipulation, even if it fails to satisfy the traditional criteria. To this end, it should describe a stable regularity, but not necessarily a universal and necessary one. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1) | |
| A reaction: I am tempted to say of this that all laws are pragmatic, given that it is rather hard to know whether reality is stable. The universal laws consist of saying that IF reality stays stable in certain ways, certain outcomes will ensue necessarily. |
| 14385 | Strict regularities are rarely discovered in life sciences [Leuridan] |
| Full Idea: Strict regularities are rarely if ever discovered in the life sciences. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §2) | |
| A reaction: This is elementary once it is pointed out, but too much philosophy have science has aimed at the model provided by the equations of fundamental physics. Science is a broad church, to employ an entertaining metaphor. |
| 14383 | A 'law of nature' is just a regularity, not some entity that causes the regularity [Leuridan] |
| Full Idea: By 'law of nature' or 'natural law' I mean a generalization describing a regularity, not some metaphysical entity that produces or is responsible for that regularity. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1 n1) | |
| A reaction: I take the second version to be a relic of a religious world view, and having no place in a naturalistic metaphysic. The regularity view is then the only player in the field, and the question is, can we do more? Can't we explain regularities? |
| 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 |
| 19448 | Consciousness is said to distinguish man from animals - consciousness of his own species [Feuerbach] |
| Full Idea: What constitutes the essential difference between man and animal? The most simple, general, and most widely held answer to this question is consciousness. Consciousness is given only in the case of a being to whom his species ...is an object of thought. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], I) | |
| A reaction: Rather speculative. Since other species cohabit and breed only with their fellow species members, one might have thought they were aware of them. |
| 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'. |
| 19454 | A God needs justice, kindness and wisdom, but those concepts don't depend on the concept of God [Feuerbach] |
| Full Idea: The concept of God depends on the concepts of justice, kindness and wisdom - a God who is not kind, not just, and not wise is no God. But these concepts do not depend on the concept of God. That a quality is possessed by God does not make it divine. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], II) | |
| A reaction: This is part of Feuerbach's argument for atheism, but if you ask for the source of our human concepts of justice, kindness and wisdom, no one, I would have thought, could cite God for the role. |
| 6913 | God's existence cannot be separated from essence and concept, which can only be thought as existing [Feuerbach] |
| Full Idea: God is the being in which existence cannot be separated from essence and concept and which cannot be thought except as existing. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §18) | |
| A reaction: This shows how faith in God endured through the Idealist movement by means of the Ontological Argument, despite the criticisms of Hume and Kant. To me this now appears as an odd abberation in the history of human thought. |
| 19452 | The nature of God is an expression of human nature [Feuerbach] |
| Full Idea: God is the manifestation of man's inner nature, his expressed self. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], II) | |
| A reaction: Even if you are a deeply committed theist, you have to concede some of this point. The perfections attributed to God are usually of human qualities. Leibniz, though, says that God has an infinity of perfection, mostly unknown to us. |
| 6903 | If God is only an object for man, then only the essence of man is revealed in God [Feuerbach] |
| Full Idea: If God is only an object of man, what is revealed to us in his essence? Nothing but the essence of man. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §07) | |
| A reaction: It is important to distinguish here between what we could know about God, and what we think God might actually be like. We may well only be able to read the essence of man into God, but we might speculate that God is more than that. |
| 6923 | God is what man would like to be [Feuerbach] |
| Full Idea: God is what man would like to be. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §29) | |
| A reaction: It is hard to see how even the most devout person could deny the truth of this. Perhaps the essential hallmark of humanity is a desire to be different from the way we are. |
| 6911 | God is for us a mere empty idea, which we fill with our own ego and essence [Feuerbach] |
| Full Idea: God exists, but he is for us a tabula rasa, an empty being, a mere idea; God, as we conceive and think of him, is our ego, our mind, and our essence. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §17) | |
| A reaction: He accepted God's existence because of the Ontological Argument. This is a little stronger than Hume's view (Idea 2185), because Hume seems to be talking about imagining God, but Feuerbach says this is our understanding of God. |
| 19453 | If love, goodness and personality are human, the God who is their source is anthropomorphic [Feuerbach] |
| Full Idea: If love, goodness, and personality are human determinations, the being which constitutes their source and ...their presupposition is also an anthropomorphism; so is the existence of God. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], II) | |
| A reaction: It is certainly a struggle for the imagination to grasp a being which is characterised by idealised versions of human virtues, and yet has an intrinsic nature which is utterly different from humanity. |
| 6902 | Catholicism concerns God in himself, Protestantism what God is for man [Feuerbach] |
| Full Idea: Protestantism is no longer concerned, as Catholicism is, about what God is in himself, but about what he is for man. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §02) | |
| A reaction: It is certainly true that the major religions in their origins seem to be almost exclusively concerned with God alone, and have little interest in human life (or morality). |
| 19449 | Religion is the consciousness of the infinite [Feuerbach] |
| Full Idea: Religion is the consciousness of the infinite. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], I) |
| 6905 | Absolute idealism is the realized divine mind of Leibnizian theism [Feuerbach] |
| Full Idea: Absolute idealism is nothing but the realized divine mind of Leibnizian theism. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §10) | |
| A reaction: In general it seems an accurate commentary that during the eighteenth century philosophers on the continent were designing a religion without God. Kantian duty tries to replace the authority of God with pure reason. |
| 19455 | Today's atheism will tomorrow become a religion [Feuerbach] |
| Full Idea: What is regarded as atheism today will be religion tomorrow. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], II) | |
| A reaction: Modern critics of atheism frequently accuse it of being a new religion. I doubt whether Feuerbach is right, but it is a nice provocative idea. |