32 ideas
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. |
10800 | The values of variables can't determine existence, because they are just expressions [Ryle, by Quine] |
Full Idea: Ryle objected somewhere to my dictum that 'to be is to be the value of a variable', arguing that the values of variables are expressions, and hence that my dictum repudiates all things except expressions. | |
From: report of Gilbert Ryle (works [1950]) by Willard Quine - Reply to Professor Marcus p.183 | |
A reaction: I have a lot of sympathy with Ryle's view, and I associate it with the peculiar Millian view that we can somehow replace a name in a sentence with the actual physical object. Objects can't be parts of sentences - and maybe they can't be 'values'. |
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 |
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 |
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. |
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. |
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'. |
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? |
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) |
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. |
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. |
7357 | People who control others with fluent language often end up being hated [Kongzi (Confucius)] |
Full Idea: Of what use is eloquence? He who engages in fluency of words to control men often finds himself hated by them. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], V.5) | |
A reaction: I don't recall Socrates making this very good point to any of the sophists (such as Gorgias). The idea that if you battle or connive your way to dominance over others then you are successful is false. Life is a much longer game than that. |
7358 | All men prefer outward appearance to true excellence [Kongzi (Confucius)] |
Full Idea: I have yet to meet a man as fond of excellence as he is of outward appearances. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], IX.18) | |
A reaction: Interestingly, this cynical view of the love of virtue is put by Plato into the mouths of Glaucon and Adeimantus (in Bk II of 'Republic', e.g. Idea 12), and not into the mouth of Socrates, who goes on to defend the possibility of true virtue. |
7362 | Humans are similar, but social conventions drive us apart (sages and idiots being the exceptions) [Kongzi (Confucius)] |
Full Idea: In our natures we approximate one another; habits put us further and further apart. The only ones who do not change are sages and idiots. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.2) | |
A reaction: I find most of Confucius rather uninteresting, but this is a splendid remark about the influence of social conventions on human nature. Sages can achieve universal morality if they rise above social convention, and seek the true virtues of human nature. |
7360 | Do not do to others what you would not desire yourself [Kongzi (Confucius)] |
Full Idea: Do not do to others what you would not desire yourself. Then you will have no enemies, either in the state or in your home. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XII.2) | |
A reaction: The Golden Rule, but note the second sentence. Logically, it leads to the absurdity of not giving someone an Elvis record for Christmas because you yourself don't like Elvis. Kant (Idea 3733) and Nietzsche (Idea 4560) offer good criticisms. |
7359 | Excess and deficiency are equally at fault [Kongzi (Confucius)] |
Full Idea: Excess and deficiency are equally at fault. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XI.16) | |
A reaction: This is the sort of wisdom we admire in Aristotle (and in any sensible person), but it may also be the deepest motto of conservatism, and it is a long way from romantic philosophy, and the clarion call of Nietzsche to greater excitement in life. |
7363 | The virtues of the best people are humility, maganimity, sincerity, diligence, and graciousness [Kongzi (Confucius)] |
Full Idea: He who in this world can practise five things may indeed be considered Man-at-his-best: humility, maganimity, sincerity, diligence, and graciousness. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.5) | |
A reaction: A very nice list. Who could resist working with a colleague who had such virtues? Who could go wrong if they married a person who had them? I can't think of anything important that is missing. |
7361 | Men of the highest calibre avoid political life completely [Kongzi (Confucius)] |
Full Idea: Men of the highest calibre avoid political life completely. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XIV.37) | |
A reaction: Plato notes that such people tend to avoid political life (and a left sheltering, as if from a wild storm!), but he thinks they should be dragged into the political arena for the common good. Confucius seems to approve of the avoidance. Plato is right. |
23393 | Confucianism assumes that all good developments have happened, and there is only one Way [Norden on Kongzi (Confucius)] |
Full Idea: The two major limitations of Confucianism are that it assumes that all worthwhile cultural, social and ethical innovation has already occurred, and that it does not recognise the plurality of worthwhile ways of life. | |
From: comment on Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE]) by Bryan van Norden - Intro to Classical Chinese Philosophy 3.III | |
A reaction: In modern liberal terms that is about as conservative as it is possible to get. We think of it as the state of mind of an old person who can only long for the way things were when they were young. But 'hold fast to that which is good'! |