112 ideas
| 6200 | Wisdom is knowing the highest good, and conforming the will to it [Kant] |
| Full Idea: Wisdom, theoretically regarded, means the knowledge of the highest good and, practically, the conformability of the will to the highest good. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.V) | |
| A reaction: This seems a narrow account of wisdom, focusing entirely on goodness rather than truth. A mind that valued nothing but understood everything would have a considerable degree of wisdom, in the normal use of that word. |
| 6207 | What fills me with awe are the starry heavens above me and the moral law within me [Kant] |
| Full Idea: Two things fill the mind with ever new and increasing wonder and awe, the oftener and the more steadily we reflect on them: the starry heavens above me and the moral law within me. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], Concl) | |
| A reaction: I am beginning to think that the two major issues of all philosophy are ontology and metaethics, and Kant is close to agreeing with me. He certainly wasn't implying that astronomy was a key aspect of philosophy. |
| 6184 | Consistency is the highest obligation of a philosopher [Kant] |
| Full Idea: Consistency is the highest obligation of a philosopher and yet the most rarely found. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.1.§3) | |
| A reaction: I agree with this, and it also strikes me as the single most important principle of Kant's philosophy, which is the key to his whole moral theory. |
| 6203 | Metaphysics is just a priori universal principles of physics [Kant] |
| Full Idea: Metaphysics only contains the pure a priori principles of physics in their universal import. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.VI) | |
| A reaction: 'Universal' seems to imply 'necessary'. If you thought that no a priori universal principles were possible, you would be left with physics. I quite like the definition, except that I think there would still be metaphysics even if there were no physics. |
| 224 | When questions are doubtful we should concentrate not on objects but on ideas of the intellect [Plato] |
| Full Idea: Doubtful questions should not be discussed in terms of visible objects or in relation to them, but only with reference to ideas conceived by the intellect. | |
| From: Plato (Parmenides [c.364 BCE], 135e) |
| 232 | Opposites are as unlike as possible [Plato] |
| Full Idea: Opposites are as unlike as possible. | |
| From: Plato (Parmenides [c.364 BCE], 159a) |
| 8937 | Plato's 'Parmenides' is the greatest artistic achievement of the ancient dialectic [Hegel on Plato] |
| Full Idea: Plato's 'Parmenides' is the greatest artistic achievement of the ancient dialectic. | |
| From: comment on Plato (Parmenides [c.364 BCE]) by Georg W.F.Hegel - Phenomenology of Spirit Pref 71 | |
| A reaction: It is a long way from the analytic tradition of philosophy to be singling out a classic text for its 'artistic' achievement. Eventually we may even look back on, say, Kripke's 'Naming and Necessity' and see it in that light. |
| 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] |
| 13986 | Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle] |
| Full Idea: Plato (in 'Parmenides') shows that the theory that 'Eide' are substances, and Kant that space and time are substances, and Bradley that relations are substances, all lead to aninomies. | |
| From: report of Plato (Parmenides [c.364 BCE]) by Gilbert Ryle - Are there propositions? 'Objections' |
| 14150 | Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato] |
| Full Idea: Plato's 'Parmenides' is perhaps the best collection of antinomies ever made. | |
| From: comment on Plato (Parmenides [c.364 BCE]) by Bertrand Russell - The Principles of Mathematics §337 |
| 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). |
| 16150 | One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato] |
| Full Idea: If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it. | |
| From: Plato (Parmenides [c.364 BCE], 144a) | |
| A reaction: This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed. |
| 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'. |
| 229 | The one was and is and will be and was becoming and is becoming and will become [Plato] |
| Full Idea: The one was and is and will be and was becoming and is becoming and will become. | |
| From: Plato (Parmenides [c.364 BCE], 155d) |
| 21821 | Plato's Parmenides has a three-part theory, of Primal One, a One-Many, and a One-and-Many [Plato, by Plotinus] |
| Full Idea: The Platonic Parmenides is more exact [than Parmenides himself]; the distinction is made between the Primal One, a strictly pure Unity, and a secondary One which is a One-Many, and a third which is a One-and-Many. | |
| From: report of Plato (Parmenides [c.364 BCE]) by Plotinus - The Enneads 5.1.08 | |
| A reaction: Plotinus approves of this three-part theory. Parmenides has the problem that the highest Being contains no movement. By placing the One outside Being you can give it powers which an existent thing cannot have. Cf the concept of God. |
| 221 | Absolute ideas, such as the Good and the Beautiful, cannot be known by us [Plato] |
| Full Idea: The absolute good and the beautiful and all which we conceive to be absolute ideas are unknown to us. | |
| From: Plato (Parmenides [c.364 BCE], 134c) |
| 223 | If you deny that each thing always stays the same, you destroy the possibility of discussion [Plato] |
| Full Idea: If a person denies that the idea of each thing is always the same, he will utterly destroy the power of carrying on discussion. | |
| From: Plato (Parmenides [c.364 BCE], 135c) |
| 227 | You must always mean the same thing when you utter the same name [Plato] |
| Full Idea: You must always mean the same thing when you utter the same name. | |
| From: Plato (Parmenides [c.364 BCE], 147d) |
| 210 | It would be absurd to think there were abstract Forms for vile things like hair, mud and dirt [Plato] |
| Full Idea: Are there abstract ideas for such things as hair, mud and dirt, which are particularly vile and worthless? That would be quite absurd. | |
| From: Plato (Parmenides [c.364 BCE], 130d) |
| 220 | The concept of a master includes the concept of a slave [Plato] |
| Full Idea: Mastership in the abstract is mastership of slavery in the abstract. | |
| From: Plato (Parmenides [c.364 BCE], 133e) |
| 211 | If admirable things have Forms, maybe everything else does as well [Plato] |
| Full Idea: It is troubling that if admirable things have abstract ideas, then perhaps everything else must have ideas as well. | |
| From: Plato (Parmenides [c.364 BCE], 130d) |
| 219 | If absolute ideas existed in us, they would cease to be absolute [Plato] |
| Full Idea: None of the absolute ideas exists in us, because then it would no longer be absolute. | |
| From: Plato (Parmenides [c.364 BCE], 133c) |
| 228 | Greatness and smallness must exist, to be opposed to one another, and come into being in things [Plato] |
| Full Idea: These two ideas, greatness and smallness, exist, do they not? For if they did not exist, they could not be opposites of one another, and could not come into being in things. | |
| From: Plato (Parmenides [c.364 BCE], 149e) |
| 16151 | Plato moves from Forms to a theory of genera and principles in his later work [Plato, by Frede,M] |
| Full Idea: It seems to me that Plato in the later dialogues, beginning with the second half of 'Parmenides', wants to substitute a theory of genera and theory of principles that constitute these genera for the earlier theory of forms. | |
| From: report of Plato (Parmenides [c.364 BCE]) by Michael Frede - Title, Unity, Authenticity of the 'Categories' V | |
| A reaction: My theory is that the later Plato came under the influence of the brilliant young Aristotle, and this idea is a symptom of it. The theory of 'principles' sounds like hylomorphism to me. |
| 215 | If things partake of ideas, this implies either that everything thinks, or that everything actually is thought [Plato] |
| Full Idea: If all things partake of ideas, must either everything be made of thoughts and everything thinks, or everything is thought, and so can't think? | |
| From: Plato (Parmenides [c.364 BCE], 132c) |
| 212 | The whole idea of each Form must be found in each thing which participates in it [Plato] |
| Full Idea: The whole idea of each form (of beauty, justice etc) must be found in each thing which participates in it. | |
| From: Plato (Parmenides [c.364 BCE], 131a) |
| 218 | Participation is not by means of similarity, so we are looking for some other method of participation [Plato] |
| Full Idea: Participation is not by means of likeness, so we must seek some other method of participation. | |
| From: Plato (Parmenides [c.364 BCE], 133a) |
| 213 | Each idea is in all its participants at once, just as daytime is a unity but in many separate places at once [Plato] |
| Full Idea: Just as day is in many places at once, but not separated from itself, so each idea might be in all its participants at once. | |
| From: Plato (Parmenides [c.364 BCE], 131b) |
| 216 | If things are made alike by participating in something, that thing will be the absolute idea [Plato] |
| Full Idea: That by participation in which like things are made like, will be the absolute idea, will it not? | |
| From: Plato (Parmenides [c.364 BCE], 132e) |
| 217 | Nothing can be like an absolute idea, because a third idea intervenes to make them alike (leading to a regress) [Plato] |
| Full Idea: It is impossible for anything to be like an absolute idea, because a third idea will appear to make them alike, and if that is like anything, it will lead to another idea, and so on. | |
| From: Plato (Parmenides [c.364 BCE], 133a) |
| 214 | If absolute greatness and great things are seen as the same, another thing appears which makes them seem great [Plato] |
| Full Idea: If you regard the absolute great and the many great things in the same way, will not another appear beyond, by which all these must appear to be great? | |
| From: Plato (Parmenides [c.364 BCE], 132a) |
| 15851 | Parts must belong to a created thing with a distinct form [Plato] |
| Full Idea: The part would not be the part of many things or all, but of some one character ['ideas'] and of some one thing, which we call a 'whole', since it has come to be one complete [perfected] thing composed [created] of all. | |
| From: Plato (Parmenides [c.364 BCE], 157d) | |
| A reaction: A serious shot by Plato at what identity is. Harte quotes it (125) and shows that 'character' is Gk 'idea', and 'composed' will translate as 'created'. 'Form' links this Platonic passage to Aristotle's hylomorphism. |
| 15846 | In Parmenides, if composition is identity, a whole is nothing more than its parts [Plato, by Harte,V] |
| Full Idea: At the heart of the 'Parmenides' puzzles about composition is the thesis that composition is identity. Considered thus, a whole adds nothing to an ontology that already includes its parts | |
| From: report of Plato (Parmenides [c.364 BCE]) by Verity Harte - Plato on Parts and Wholes 2.5 | |
| A reaction: There has to be more to a unified identity that mere proximity of the parts. When do parts come together, and when do they actually 'compose' something? |
| 15849 | Plato says only a one has parts, and a many does not [Plato, by Harte,V] |
| Full Idea: In 'Parmenides' it is argued that a part cannot be part of a many, but must be part of something one. | |
| From: report of Plato (Parmenides [c.364 BCE], 157c) by Verity Harte - Plato on Parts and Wholes 3.2 | |
| A reaction: This looks like the right way to go with the term 'part'. We presuppose a unity before we even talk of its parts, so we can't get into contradictions and paradoxes about their relationships. |
| 15850 | Anything which has parts must be one thing, and parts are of a one, not of a many [Plato] |
| Full Idea: The whole of which the parts are parts must be one thing composed of many; for each of the parts must be part, not of a many, but of a whole. | |
| From: Plato (Parmenides [c.364 BCE], 157c) | |
| A reaction: This is a key move of metaphysics, and we should hang on to it. The other way madness lies. |
| 13259 | It seems that the One must be composed of parts, which contradicts its being one [Plato] |
| Full Idea: The One must be composed of parts, both being a whole and having parts. So on both grounds the One would thus be many and not one. But it must be not many, but one. So if the One will be one, it will neither be a whole, nor have parts. | |
| From: Plato (Parmenides [c.364 BCE], 137c09), quoted by Kathrin Koslicki - The Structure of Objects 5.2 | |
| A reaction: This is the starting point for Plato's metaphysical discussion of objects. It seems to begin a line of thought which is completed by Aristotle, surmising that only an essential structure can bestow identity on a bunch of parts. |
| 15847 | Two things relate either as same or different, or part of a whole, or the whole of the part [Plato] |
| Full Idea: Everything is surely related to everything as follows: either it is the same or different; or, if it is not the same or different, it would be related as part to whole or as whole to part. | |
| From: Plato (Parmenides [c.364 BCE], 146b) | |
| A reaction: This strikes me as a really helpful first step in trying to analyse the nature of identity. Two things are either two or (actually) one, or related mereologically. |
| 6181 | Necessity cannot be extracted from an empirical proposition [Kant] |
| Full Idea: It is a clear contradiction to try to extract necessity from an empirical proposition. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], Pref) | |
| A reaction: This is precisely the idea which Kripke challenged, claiming that the necessary essences of natural kinds such as gold have to be discovered empirically. All my intuitions are with Kant (and Hume) on this, but it is a complex issue… |
| 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. |
| 6183 | Can pure reason determine the will, or are empirical conditions relevant? [Kant] |
| Full Idea: This is the first question: Is pure reason sufficient of itself to determine the will, or is it only as empirically conditioned that it can do so? | |
| From: Immanuel Kant (Critique of Practical Reason [1788], Intro) | |
| A reaction: This seems to be the core question of intellectualism, which goes back to Socrates. You can only accept the question if you accept the concept of 'pure' reason. Values seem to be needed for action, as well as empirical circumstances. |
| 6191 | The will is the faculty of purposes, which guide desires according to principles [Kant] |
| Full Idea: The will could be defined as the faculty of purposes, since they are always determining grounds of the faculty of desire according to principles. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.II) | |
| A reaction: Do animals have wills? Kant implies that you can only have a will if you have principles. Compare Hobbes' rather less elevated definition of the will (Idea 2362). |
| 6190 | The sole objects of practical reason are the good and the evil [Kant] |
| Full Idea: The sole objects of a practical reason are thus those of the good and the evil. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.II) | |
| A reaction: Of course, you may aim to achieve x because it is good, while I judge x to be evil. |
| 18235 | Only human reason can confer value on our choices [Kant, by Korsgaard] |
| Full Idea: Kant argues that only human reason is in a position to confer value on the objects of human choice. | |
| From: report of Immanuel Kant (Critique of Practical Reason [1788]) by Christine M. Korsgaard - Aristotle and Kant on the Source of Value 8 'Kant' | |
| A reaction: If the source of value is humans, then it is not immediately clear why it is only our reason that does the conferring. What is the status of a choice on which reason fails to confer value? The idea is that reason, unlike desire, has intrinsic value. |
| 6196 | People cannot come to morality through feeling, because morality must not be sensuous [Kant] |
| Full Idea: In the subject there is no antecedent feeling tending to morality; that is impossible, because all feeling is sensuous, and the drives of the moral disposition must be free from every sensuous condition. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.III) | |
| A reaction: I'm not quite clear (even after reading Kant) why moral drives 'must' be free of sensuousness. Aristotle gives a much better account, when he tells us that the sensuous drives must be trained in the right way, and must be in harmony with the reason. |
| 18675 | Kant may rate two things as finally valuable: having a good will, and deserving happiness [Orsi on Kant] |
| Full Idea: In some interpretations it appears that for Kant two things are finally valuable: good will (unconditionally), and deserved happiness (conditionally on the value of good will). | |
| From: comment on Immanuel Kant (Critique of Practical Reason [1788]) by Francesco Orsi - Value Theory 2.2 | |
| A reaction: It doesn't sound difficult to reconcile these two. Just ask 'what is required of someone to deserve happiness?'. |
| 22007 | An autonomous agent has dignity [Würde], which has absolute worth [Kant, by Pinkard] |
| Full Idea: For Kant, there is something about beings that can act autonomously that is itself of 'absolute worth', which Kant calls the 'dignity' [Würde] of each such agent. | |
| From: report of Immanuel Kant (Critique of Practical Reason [1788]) by Terry Pinkard - German Philosophy 1760-1860 02 | |
| A reaction: This answers my puzzle about where Kant's fundamental values come from. Surely wicked actions can be autonomous? Autonomous actions aren't thereby good actions. A 'good' will, course, whatever that is. Rational? My problem with existentialist ethics. |
| 18234 | The good will is unconditionally good, because it is the only possible source of value [Kant, by Korsgaard] |
| Full Idea: Kant argues that the good will is unconditionally good because it is the only thing able to be a source of value. | |
| From: report of Immanuel Kant (Critique of Practical Reason [1788]) by Christine M. Korsgaard - Aristotle and Kant on the Source of Value 8 'Kant' | |
| A reaction: The obvious worry is the circularity of resting a theory of value on identifying a 'good' will as its source. |
| 6192 | Good or evil cannot be a thing, but only a maxim of action, making the person good or evil [Kant] |
| Full Idea: If something is held to be absolutely good or evil in all respects and without qualification, it could not be a thing but only the manner of acting, i.e., it could only be the maxim of the will, and consequently the acting person himself is good or evil. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.II) | |
| A reaction: It goes on to deny that pain is intrinsically evil, but his reason for the claim is not clear. Nevetheless, I think he is right. This remark is an important bridge between Enlightenment concerns with law and Greek concerns with character. |
| 6197 | Morality involves duty and respect for law, not love of the outcome [Kant] |
| Full Idea: All the morality of actions may be placed in their necessity from duty and from respect for the law, and not from love for or leaning toward that which the action is to produce. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.III) | |
| A reaction: Kant tries to reject consequentialism, but you cannot assess your duty or the universal law without an assessment of probable consequences, and we could never choose between laws if we did not already see value in the outcome. |
| 6193 | Our happiness is all that matters, not as a sensation, but as satisfaction with our whole existence [Kant] |
| Full Idea: Our happiness is the only thing of importance, provided this is judged, as reason requires, not according to transitory sensation but according to the influence which this contingency has on our whole existence and our satisfaction with it. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.II) | |
| A reaction: This is closer to the Greek eudaimonia than to the modern conception of happiness, which is largely just a feeling. Kant's view seems more like a private judgement on your whole life, where the Greek idea seems more public and objective. |
| 1452 | Happiness is the condition of a rational being for whom everything goes as they wish [Kant] |
| Full Idea: Happiness is the condition of a rational being in the world with whom everything goes according to his wish and will. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.V) | |
| A reaction: For such a sophisticated and rational philosopher this seems a rather crude notion. Reluctant alcoholics don't fit. Bradley has a much better definition (Idea 5655). |
| 1454 | Morality is not about making ourselves happy, but about being worthy of happiness [Kant] |
| Full Idea: Morality is not properly the doctrine of how we should make ourselves happy, but how we should become worthy of happiness. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.V) | |
| A reaction: Whatever else you think of Kant's moral theory, this remark is a clarion call we can all recognise. Suppose we all somehow ended up in a state of maximal happiness by systematically betraying one another. |
| 6194 | The highest worth for human beings lies in dispositions, not just actions [Kant] |
| Full Idea: The highest worth which human beings can and should procure for themselves lies in dispositions and not in actions only. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.II) | |
| A reaction: This leaves the problem of the well-meaning fool, who has wonderful dispositions but poor judgement. What Kant is describing here is better known as virtue. See Idea 58. |
| 6198 | Virtue is the supreme state of our pursuit of happiness, and so is supreme good [Kant] |
| Full Idea: Virtue (as the worthiness to be happy) is the supreme condition of whatever appears to us to be desirable and thus of all our pursuit of happiness and, consequently, the supreme good. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II) | |
| A reaction: Thus Kant can claim to be a virtue theorist, but giving us a very different account of how virtue arises. He emphasises elsewhere (Idea 6197) that the supreme good must be in the will, not in the outcome. 'Virtue' is here a rather thin concept. |
| 1456 | Moral law is holy, and the best we can do is achieve virtue through respect for the law [Kant] |
| Full Idea: The moral law is holy (unyielding), although all the moral perfection to which man can attain is still only virtue, that is, a rightful disposition arising from respect for the law. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.V) | |
| A reaction: In comparison with Aristotle's view of virtue this is very passive and external. Aristotle doesn't need laws for virtue, he needs inner harmony and a grasp of what has high value. |
| 6185 | No one would lend money unless a universal law made it secure, even after death [Kant] |
| Full Idea: If my maxim is 'augment my property by all safe means', I can't make that a law allowing me to keep a dead man's loan, because no one would make a loan if that were the moral law. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.1.§4) | |
| A reaction: This is a simple illustration of Kant's strategy and it shows clearly how, for all his talk of 'pure reason', his moral law is strongly guided by consequences, and that these can only judged by prior values - for example, that loans are a good thing. |
| 6187 | Universality determines the will, and hence extends self-love into altruism [Kant] |
| Full Idea: The form of universality is itself the determining ground of the will, …and from this limitation alone, and not from the addition of any exernal drive, the concept of obligation arises to extend the maxim of self-love also to the happiness of others. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.1.§8) | |
| A reaction: This is the heroic and optimistic part of Kant's philosophy, the attempt to derive altruism from pure reason. The claim seems to be that maxims don't motivate until they have been universalised. I fear that only altruism could add such motivation. |
| 6201 | Everyone (even God) must treat rational beings as ends in themselves, and not just as means [Kant] |
| Full Idea: In the order of ends, man (and every rational being) is an end in himself, i.e., he is never to be used merely as a means for someone (even for God) without at the same time being himself an end. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.V) | |
| A reaction: The worry here is that Kant has set up an exam that you have to pass before you can be treated as a moral end. Animals and the ecosystem will fail the exam, and even some human beings will be borderline cases. We should respect everything. |
| 6186 | A holy will is incapable of any maxims which conflict with the moral law [Kant] |
| Full Idea: A holy will is one which is incapable of any maxims which conflict with the moral law | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.1.§7) | |
| A reaction: If such a will is 'incapable' of conflicting with moral law, it will not need to think or assess before action. This means that Kant's moral ideal can ultimately exclude the free-thinking intellect. Kant is describing a state of true Aristotelian virtue. |
| 6195 | Reason cannot solve the problem of why a law should motivate the will [Kant] |
| Full Idea: How a law in itself can be the direct motive of the will (which is the essence of morality) is an insoluble problem for the human reason. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.III) | |
| A reaction: If that is the great man's final word, then it is tempting to switch to an empirical moral theory, such as that of Hobbes or Hume or E.O. Wilson, which starts from what motivations are available, and builds morality up from that. |
| 222 | Only a great person can understand the essence of things, and an even greater person can teach it [Plato] |
| Full Idea: Only a man of very great natural gifts will be able to understand that everything has a class and absolute essence, and an even more wonderful man can teach this. | |
| From: Plato (Parmenides [c.364 BCE], 135a) |
| 6188 | A permanent natural order could not universalise a rule permitting suicide [Kant] |
| Full Idea: The maxim of freely disposing of my life could not hold as a universal law of nature, …because no one could choose to end his life, for such an arrangement could not constitute a permanent natural order. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.1.1.I) | |
| A reaction: This sort of claim brings out the advantanges of Aristotelian 'particularism' (expounded by Dancy). Obviously universal suicide isn't promising, but no one wants that. A few suicides in extreme cases will have no effect at all on the natural order. |
| 225 | The unlimited has no shape and is endless [Plato] |
| Full Idea: The unlimited partakes neither of the round nor of the straight, because it has no ends nor edges. | |
| From: Plato (Parmenides [c.364 BCE], 137e) |
| 233 | Some things do not partake of the One [Plato] |
| Full Idea: The others cannot partake of the one in any way; they can neither partake of it nor of the whole. | |
| From: Plato (Parmenides [c.364 BCE], 159d) | |
| A reaction: Compare Idea 231 |
| 2062 | The only movement possible for the One is in space or in alteration [Plato] |
| Full Idea: If the One moves it either moves spatially or it is altered, since these are the only motions. | |
| From: Plato (Parmenides [c.364 BCE], 138b) |
| 231 | Everything partakes of the One in some way [Plato] |
| Full Idea: The others are not altogether deprived of the one, for they partake of it in some way. | |
| From: Plato (Parmenides [c.364 BCE], 157c) | |
| A reaction: Compare Idea 233. |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |
| 6199 | Obligation does not rest on the existence of God, but on the autonomy of reason [Kant] |
| Full Idea: It is not to be understood that the assumption of the existence of God is necessary as a ground for all obligation in general (for this rests, as has been shown, solely on the autonomy of reason itself). | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.V) | |
| A reaction: This shows that Kant agrees with Plato about the Euthyphro Question - that is, they both think that morality is logically and naturally prior to any gods. I agree. Why would we admire or worship or obey gods if we didn't think they were good? |
| 234 | We couldn't discuss the non-existence of the One without knowledge of it [Plato] |
| Full Idea: There must be knowledge of the one, or else not even the meaning of the words 'if the one does not exist' would be known. | |
| From: Plato (Parmenides [c.364 BCE], 160d) |
| 1453 | We have to postulate something outside nature which makes happiness coincide with morality [Kant] |
| Full Idea: The existence must be postulated of a cause of the whole of nature, itself distinct from nature, which contains the ground of the exact coincidence of happiness with morality. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.V) | |
| A reaction: I can see that we need a concept of how happiness could be made proportional to morality, but I can't make sense of the assumption that it is actually possible, and hence something must exist that would achieve it. |
| 1455 | Belief in justice requires belief in a place for justice (heaven), a time (eternity), and a cause (God) [Kant, by PG] |
| Full Idea: To believe in justice in an unjust world, you have to believe in a place of perfect justice (heaven), a time for perfect justice (eternity), and a cause of perfect justice (God). | |
| From: report of Immanuel Kant (Critique of Practical Reason [1788], I.II.II.V) by PG - Db (ideas) | |
| A reaction: Compare Boethius in Idea 5765. I can see that we might need to grasp the ideals of eternal justice in order to understand morality, but belief in their genuine possibility, or even actuality, doesn't seem to follow. |
| 6205 | To know if this world must have been created by God, we would need to know all other possible worlds [Kant] |
| Full Idea: We can't infer the existence of God from knowledge of this world, because we should have to know all possible worlds in order to compare them - in short, we should have to be omniscient - in order to say that it is possible only through a God. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.VI) | |
| A reaction: A nice remark, but not wholly convincing. This argument would block all attempts to work out necessities a priori, such as those of maths and logic. Must we know all possible worlds intimately to know that 2+2 is always 4? |
| 6204 | Using God to explain nature is referring to something inconceivable to explain what is in front of you [Kant] |
| Full Idea: To have recourse to God in explaining the arrangements of nature is not a physical explanation but a confession that one has come to the end of philosophy, since one assumes something of which one has no concept to conceive what is before one's eyes. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.VI) | |
| A reaction: Hume had many objections to the design argument, some of them positively sarcastic, but none as ruthless as this, since Kant (here) seems to find God to be a totally empty concept, and hence a complete non-starter as explanation for anything. |
| 6206 | From our limited knowledge we can infer great virtues in God, but not ultimate ones [Kant] |
| Full Idea: Since we know only a small part of the world, and cannot compare it with all possible worlds, we can infer from the order, design and magnitude to a wise, beneficent and powerful Author, but not that He is all-knowing, all-good, and all-powerful. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.VI) | |
| A reaction: This is very much in the spirit of David Hume, who inferred from the flaws in the world that God did not seem to be entirely competent. Hume is also more imaginative, in seeing that God might be a committee, or a hired workman. |
| 6202 | In all naturalistic concepts of God, if you remove the human qualities there is nothing left [Kant] |
| Full Idea: One can confidently challenge all pretended natural theologians to cite one single definitive attribute of their object, of which one could not irrefutably show that, when everything anthropomorphic is removed, only the word remains. | |
| From: Immanuel Kant (Critique of Practical Reason [1788], I.II.II.VI) | |
| A reaction: This idea derives from Hume's very empiricist view of our understanding of God (Idea 2185), but Kant is (remarkably) more hostile than Hume, because he actually implies that most people's concept of God is totally vacuous. |