44 ideas
| 7278 | Words of wisdom are precise and clear [Zhuangzi (Chuang Tzu)] |
| Full Idea: Words of wisdom are precise and clear. | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.2) | |
| A reaction: I can only approve of this. The issue of clarity is much discussed amongs philosophers, especially in the analytic v continental debate. Note, therefore, the additional requirement to be 'precise'. Should we be less clear in order to be precise? |
| 7281 | Don't even start, let's just stay put [Zhuangzi (Chuang Tzu)] |
| Full Idea: Don't even start, let's just stay put. | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.2) | |
| A reaction: What a remarkable proposal! He seems frightened to make an omelette, because he will have to break an egg, or he might burn himself. I can't relate to this idea, but it's existence must be noted, like other scepticisms. |
| 7282 | Disagreement means you do not understand at all [Zhuangzi (Chuang Tzu)] |
| Full Idea: The sage encompasses everything, while ordinary people just argue about things. Disagreement means you do not understand at all. | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.2) | |
| A reaction: This is why democracy and western analytical philosophy come as a package. We can't assume that our government is always right, and we can't assume that a 'sage' has managed to encompass everything. Criticism is essential! |
| 7284 | If you beat me in argument, does that mean you are right? [Zhuangzi (Chuang Tzu)] |
| Full Idea: If you get the better of me in a disagreement, rather than me getting the better of you, does this mean that you are automatically right and I am automatically wrong? | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.2) | |
| A reaction: Very nice. I don't, though, think that this invalidates the process of argument. What matters about such an exchange is the resulting reflection by the two parties. Only a fool thinks that he is right because he won, or wrong because he lost. |
| 22289 | Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Dedekind, by Potter] |
| Full Idea: Dedkind gave a rigorous proof of the principle of definition by recursion, permitting recursive definitions of addition and multiplication, and hence proofs of the familiar arithmetical laws. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 13 'Deriv' |
| 10183 | An infinite set maps into its own proper subset [Dedekind, by Reck/Price] |
| Full Idea: A set is 'Dedekind-infinite' iff there exists a one-to-one function that maps a set into a proper subset of itself. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], §64) by E Reck / M Price - Structures and Structuralism in Phil of Maths n 7 | |
| A reaction: Sounds as if it is only infinite if it is contradictory, or doesn't know how big it is! |
| 22288 | We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Dedekind, by Potter] |
| Full Idea: Dedekind had an interesting proof of the Axiom of Infinity. He held that I have an a priori grasp of the idea of my self, and that every idea I can form the idea of that idea. Hence there are infinitely many objects available to me a priori. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], no. 66) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 12 'Numb' | |
| A reaction: Who said that Descartes' Cogito was of no use? Frege endorsed this, as long as the ideas are objective and not subjective. |
| 10706 | Dedekind originally thought more in terms of mereology than of sets [Dedekind, by Potter] |
| Full Idea: Dedekind plainly had fusions, not collections, in mind when he avoided the empty set and used the same symbol for membership and inclusion - two tell-tale signs of a mereological conception. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], 2-3) by Michael Potter - Set Theory and Its Philosophy 02.1 | |
| A reaction: Potter suggests that mathematicians were torn between mereology and sets, and eventually opted whole-heartedly for sets. Maybe this is only because set theory was axiomatised by Zermelo some years before Lezniewski got to mereology. |
| 9823 | Numbers are free creations of the human mind, to understand differences [Dedekind] |
| Full Idea: Numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], Pref) | |
| A reaction: Does this fit real numbers and complex numbers, as well as natural numbers? Frege was concerned by the lack of objectivity in this sort of view. What sort of arithmetic might the Martians have created? Numbers register sameness too. |
| 10090 | Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman] |
| Full Idea: It was primarily Dedekind's accomplishment to define the integers, rationals and reals, taking only the system of natural numbers for granted. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by A.George / D.J.Velleman - Philosophies of Mathematics Intro |
| 7524 | Order, not quantity, is central to defining numbers [Dedekind, by Monk] |
| Full Idea: Dedekind said that the notion of order, rather than that of quantity, is the central notion in the definition of number. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.4 | |
| A reaction: Compare Aristotle's nice question in Idea 646. My intuition is that quantity comes first, because I'm not sure HOW you could count, if you didn't think you were changing the quantity each time. Why does counting go in THAT particular order? Cf. Idea 8661. |
| 17452 | Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck] |
| Full Idea: Dedekind and Cantor said the cardinals may be defined in terms of the ordinals: The cardinal number of a set S is the least ordinal onto whose predecessors the members of S can be mapped one-one. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 5 |
| 14131 | Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell] |
| Full Idea: Dedekind's ordinals are not essentially either ordinals or cardinals, but the members of any progression whatever. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - The Principles of Mathematics §243 | |
| A reaction: This is part of Russell's objection to Dedekind's structuralism. The question is always why these beautiful structures should actually be considered as numbers. I say, unlike Russell, that the connection to counting is crucial. |
| 17611 | We want the essence of continuity, by showing its origin in arithmetic [Dedekind] |
| Full Idea: It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], Intro) | |
| A reaction: [He seeks the origin of the theorem that differential calculus deals with continuous magnitude, and he wants an arithmetical rather than geometrical demonstration; the result is his famous 'cut']. |
| 10572 | A cut between rational numbers creates and defines an irrational number [Dedekind] |
| Full Idea: Whenever we have to do a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], §4) | |
| A reaction: Fine quotes this to show that the Dedekind Cut creates the irrational numbers, rather than hitting them. A consequence is that the irrational numbers depend on the rational numbers, and so can never be identical with any of them. See Idea 10573. |
| 14437 | Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell] |
| Full Idea: Dedekind set up the axiom that the gap in his 'cut' must always be filled …The method of 'postulating' what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - Introduction to Mathematical Philosophy VII | |
| A reaction: This remark of Russell's is famous, and much quoted in other contexts, but I have seen the modern comment that it is grossly unfair to Dedekind. |
| 18094 | Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock] |
| Full Idea: One view, favoured by Dedekind, is that the cut postulates a real number for each cut in the rationals; it does not identify real numbers with cuts. ....A view favoured by later logicists is simply to identify a real number with a cut. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by David Bostock - Philosophy of Mathematics 4.4 | |
| A reaction: Dedekind is the patriarch of structuralism about mathematics, so he has little interest in the existenc of 'objects'. |
| 18244 | I say the irrational is not the cut itself, but a new creation which corresponds to the cut [Dedekind] |
| Full Idea: Of my theory of irrationals you say that the irrational number is nothing else than the cut itself, whereas I prefer to create something new (different from the cut), which corresponds to the cut. We have the right to claim such a creative power. | |
| From: Richard Dedekind (Letter to Weber [1888], 1888 Jan), quoted by Stewart Shapiro - Philosophy of Mathematics 5.4 | |
| A reaction: Clearly a cut will not locate a unique irrational number, so something more needs to be done. Shapiro remarks here that for Dedekind numbers are objects. |
| 9824 | In counting we see the human ability to relate, correspond and represent [Dedekind] |
| Full Idea: If we scrutinize closely what is done in counting an aggregate of things, we see the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, without which no thinking is possible. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], Pref) | |
| A reaction: I don't suppose it occurred to Dedekind that he was reasserting Hume's observation about the fundamental psychology of thought. Is the origin of our numerical ability of philosophical interest? |
| 17612 | Arithmetic is just the consequence of counting, which is the successor operation [Dedekind] |
| Full Idea: I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself is nothing else than the successive creation of the infinite series of positive integers. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], §1) | |
| A reaction: Thus counting roots arithmetic in the world, the successor operation is the essence of counting, and the Dedekind-Peano axioms are built around successors, and give the essence of arithmetic. Unfashionable now, but I love it. Intransitive counting? |
| 9826 | A system S is said to be infinite when it is similar to a proper part of itself [Dedekind] |
| Full Idea: A system S is said to be infinite when it is similar to a proper part of itself. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], V.64) |
| 18087 | If x changes by less and less, it must approach a limit [Dedekind] |
| Full Idea: If in the variation of a magnitude x we can for every positive magnitude δ assign a corresponding position from and after which x changes by less than δ then x approaches a limiting value. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], p.27), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.7 | |
| A reaction: [Kitcher says he 'showed' this, rather than just stating it] |
| 13508 | Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD] |
| Full Idea: Dedekind's natural numbers: an object is in a set (0 is a number), a function sends the set one-one into itself (numbers have unique successors), the object isn't a value of the function (it isn't a successor), plus induction. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by William D. Hart - The Evolution of Logic 5 | |
| A reaction: Hart notes that since this refers to sets of individuals, it is a second-order account of numbers, what we now call 'Second-Order Peano Arithmetic'. |
| 18096 | Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock] |
| Full Idea: Dedekind's idea is that the set of natural numbers has zero as a member, and also has as a member the successor of each of its members, and it is the smallest set satisfying this condition. It is the intersection of all sets satisfying the condition. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by David Bostock - Philosophy of Mathematics 4.4 |
| 18841 | Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind] |
| Full Idea: It is Dedekind's categoricity result that convinces most of us that he has articulated our implicit conception of the natural numbers, since it entitles us to speak of 'the' domain (in the singular, up to isomorphism) of natural numbers. | |
| From: comment on Richard Dedekind (Nature and Meaning of Numbers [1888]) by Ian Rumfitt - The Boundary Stones of Thought 9.1 | |
| A reaction: The main rival is set theory, but that has an endlessly expanding domain. He points out that Dedekind needs second-order logic to achieve categoricity. Rumfitt says one could also add to the 1st-order version that successor is an ancestral relation. |
| 14130 | Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell] |
| Full Idea: Dedekind proves mathematical induction, while Peano regards it as an axiom, ...and Peano's method has the advantage of simplicity, and a clearer separation between the particular and the general propositions of arithmetic. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - The Principles of Mathematics §241 |
| 8924 | Dedekind originated the structuralist conception of mathematics [Dedekind, by MacBride] |
| Full Idea: Dedekind is the philosopher-mathematician with whom the structuralist conception originates. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], §3 n13) by Fraser MacBride - Structuralism Reconsidered | |
| A reaction: Hellman says the idea grew naturally out of modern mathematics, and cites Hilbert's belief that furniture would do as mathematical objects. |
| 9153 | Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K] |
| Full Idea: Dedekindian abstraction says mathematical objects are 'positions' in a model, while Cantorian abstraction says they are the result of abstracting on structurally similar objects. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Kit Fine - Cantorian Abstraction: Recon. and Defence §6 | |
| A reaction: The key debate among structuralists seems to be whether or not they are committed to 'objects'. Fine rejects the 'austere' version, which says that objects have no properties. Either version of structuralism can have abstraction as its basis. |
| 9825 | A thing is completely determined by all that can be thought concerning it [Dedekind] |
| Full Idea: A thing (an object of our thought) is completely determined by all that can be affirmed or thought concerning it. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], I.1) | |
| A reaction: How could you justify this as an observation? Why can't there be unthinkable things (even by God)? Presumably Dedekind is offering a stipulative definition, but we may then be confusing epistemology with ontology. |
| 7289 | Do not try to do things, or to master knowledge; just be empty [Zhuangzi (Chuang Tzu)] |
| Full Idea: Do not try to do things. Do not try to master knowledge. ...Just be empty. | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.7) | |
| A reaction: Stands as a nice challenge to the assumption that knowledge is a good thing. Aristotle's views make a nice contrast (Ideas 548 and 549). Personally I totally agree with Aristotle, and with the western tradition. |
| 23403 | You know you were dreaming when you wake, but there might then be a greater awakening from that [Zhuangzi (Chuang Tzu)] |
| Full Idea: Often after waking do you know that your dream was a dream. Still, there may be an even greater awakening after which you will know that this, too, was just a greater dream. | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], 02), quoted by Bryan van Norden - Intro to Classical Chinese Philosophy 9.2 | |
| A reaction: This is the key to the full horror of dream scepticism (as dramatised in the film 'The Matrix'). We can never know whether there is yet another awakening about to occur. |
| 7285 | Did Chuang Tzu dream he was a butterfly, or a butterfly dream he was Chuang Tzu? [Zhuangzi (Chuang Tzu)] |
| Full Idea: Once I, Chuang Tzu, dreamt that I was a butterfly, flitting around and enjoying myself. Suddenly I woke and was Chuang Tzu again. But had I been Chuang Tzu dreaming I was a butterfly, or a butterfly dreaming I was now Chuang Tzu? | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.2) | |
| A reaction: Plato (Idea 2047) also spotted this problem, later made famous by Descartes (Idea 2250). Given the size of a butterfly's brain, this suggests that Chuang Tzu was a dualist. What can't I take the idea seriously, when reason says I should? |
| 7277 | The perfect man has no self [Zhuangzi (Chuang Tzu)] |
| Full Idea: As the saying goes, 'The perfect man has no self' | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.1) | |
| A reaction: This seems to be quoted with approval. This is interesting because it implies that lesser beings do have a self, and that having a self is a moral issue, and one which can be controlled. One could, I suppose, concentrate on externals. |
| 7286 | To see with true clarity, your self must be irrelevant [Zhuangzi (Chuang Tzu)] |
| Full Idea: When a man discerns his own self as irrelevant, he sees with true clarity. | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.6) | |
| A reaction: Seeing 'with clarity' is only one of the ways of seeing, and one mustn't unquestioningly assume that it is the best. Wisdom should contemplate vision with and without the self, and then rise higher and compare the two views. Compare Parfit (Idea 5518). |
| 9189 | Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dedekind, by Dummett] |
| Full Idea: By applying the operation of abstraction to a system of objects isomorphic to the natural numbers, Dedekind believed that we obtained the abstract system of natural numbers, each member having only properties consequent upon its position. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Michael Dummett - The Philosophy of Mathematics | |
| A reaction: Dummett is scornful of the abstractionism. He cites Benacerraf as a modern non-abstractionist follower of Dedekind's view. There seems to be a suspicion of circularity in it. How many objects will you abstract from to get seven? |
| 9827 | We derive the natural numbers, by neglecting everything of a system except distinctness and order [Dedekind] |
| Full Idea: If in an infinite system, set in order, we neglect the special character of the elements, simply retaining their distinguishability and their order-relations to one another, then the elements are the natural numbers, created by the human mind. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], VI.73) | |
| A reaction: [compressed] This is the classic abstractionist view of the origin of number, but with the added feature that the order is first imposed, so that ordinals remain after the abstraction. This, of course, sounds a bit circular, as well as subjective. |
| 9979 | Dedekind has a conception of abstraction which is not psychologistic [Dedekind, by Tait] |
| Full Idea: Dedekind's conception is psychologistic only if that is the only way to understand the abstraction that is involved, which it is not. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by William W. Tait - Frege versus Cantor and Dedekind IV | |
| A reaction: This is a very important suggestion, implying that we can retain some notion of abstractionism, while jettisoning the hated subjective character of private psychologism, which seems to undermine truth and logic. |
| 7279 | If words can't be defined, they may just be the chirruping of chicks [Zhuangzi (Chuang Tzu)] |
| Full Idea: Our words are not just hot air. Words work because they are something, but the problem is that, if we cannot define a word's meaning, it doesn't really say anything. Can we make a case for it being anything different from the chirruping of chicks? | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.2) | |
| A reaction: This obviously points us towards Quine's challenge to analyticity, and hence the value of definitions (Ideas 1622 and 1624). Even for Chuang Tzu, it seems naïve to think that you cannot use a word well if you cannot define it. |
| 23404 | Words are for meaning, and once you have that you can forget the words [Zhuangzi (Chuang Tzu)] |
| Full Idea: Words are for meaning: when you've gotten the meaning, you can forget the words. | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], 26), quoted by Bryan van Norden - Intro to Classical Chinese Philosophy 9.VI | |
| A reaction: 'What exactly did this person say?' 'Don't know, but I've given you the accurate gist'. This is such an obvious phenomenon that I amazed by modern philosophers who deny propositions, or deny meaning entirely. |
| 7283 | Great courage is not violent [Zhuangzi (Chuang Tzu)] |
| Full Idea: Great courage is not violent. | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.2) | |
| A reaction: A very nice remark. This, I think, is what the Greeks were struggling to say about courage, but they never quite pinned it down as Chuang Tzu does. |
| 7280 | As all life is one, what need is there for words? [Zhuangzi (Chuang Tzu)] |
| Full Idea: As all life is one, what need is there for words? | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.2) | |
| A reaction: In a sense this is nonsense, but it has an appeal. I presume that God would not need words, any more than he would need arithmetic. Life is obviously a complex one, with parts which can be discussed. |
| 7288 | Go with the flow, and be one with the void of Heaven [Zhuangzi (Chuang Tzu)] |
| Full Idea: Don't struggle, go with the flow, and you will find yourself at one with the vastness of the void of Heaven. | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.6) | |
| A reaction: Ugh. I've got all eternity to do that. The underlying assumption of Taoism seems to be that it is better not to have been born, and if you are thus unfortunate, you should try to pretend that it never happened. Much too negative for my taste. |
| 7287 | Fish forget about each other in the pond and forget each other in the Tao [Zhuangzi (Chuang Tzu)] |
| Full Idea: Fish forget about each other in the pond and forget each other in the Tao. | |
| From: Zhuangzi (Chuang Tzu) (The Book of Chuang Tzu [c.329 BCE], Ch.6) | |
| A reaction: Strikingly different from Christianity. No wonder Europeans used to describe orientals as 'enigmatic'; the faces of Taoists presumably express indifference. Not for me, I'm afraid. I identify with my fellow humans, because of our shared predicaments. |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
| Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
| From: Herodotus (The Histories [c.435 BCE], 2.123.2) |