39 ideas
| 2557 | Analytical philosophy seems to have little interest in how to tell a good analysis from a bad one [Rorty] |
| Full Idea: There is nowadays little attempt to bring "analytic philosophy" to self-consciousness by explaining how to tell a successful from an unsuccessful analysis. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 4.1) |
| 2556 | Rational certainty may be victory in argument rather than knowledge of facts [Rorty] |
| Full Idea: We can think of "rational certainty" as a matter of victory in argument rather than relation to an object known. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 3.4) |
| 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' |
| 4726 | Rorty seems to view truth as simply being able to hold one's view against all comers [Rorty, by O'Grady] |
| Full Idea: Rorty seems to view truth as simply being able to hold one's view against all comers. | |
| From: report of Richard Rorty (Philosophy and the Mirror of Nature [1980]) by Paul O'Grady - Relativism Ch.4 | |
| A reaction: This may be a caricature of Rorty, but he certainly seems to be in the business of denying truth as much as possible. This strikes me as the essence of pragmatism, and as a kind of philosophical nihilism. |
| 2549 | For James truth is "what it is better for us to believe" rather than a correct picture of reality [Rorty] |
| Full Idea: Truth is, in James' phrase, "what it is better for us to believe", rather than "the accurate representation of reality". | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], Intro) |
| 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 |
| 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. |
| 14296 | Dispositions are physical states of mechanism; when known, these replace the old disposition term [Quine] |
| Full Idea: Each disposition, in my view, is a physical state or mechanism. ...In some cases nowadays we understand the physical details and set them forth explicitly in terms of the arrangement and interaction of small bodies. This replaces the old disposition. | |
| From: Willard Quine (The Roots of Reference [1990], p.11), quoted by Stephen Mumford - Dispositions 01.3 | |
| A reaction: A challenge to the dispositions and powers view of nature, one which rests on the 'categorical' structural properties, rather than the 'hypothetical' dispositions. But can we define a mechanism without mentioning its powers? |
| 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. |
| 2548 | If knowledge is merely justified belief, justification is social [Rorty] |
| Full Idea: If we have a Deweyan conception of knowledge, as what we are justified in believing, we will see "justification" as a social phenomenon. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], Intro) | |
| A reaction: I find this observation highly illuminating (though I probably need to study Dewey to understand it). There just is no absolute about whether someone is justified. How justified do you want to be? |
| 6599 | Knowing has no definable essence, but is a social right, found in the context of conversations [Rorty] |
| Full Idea: If we see knowing not as having an essence, described by scientists or philosophers, but rather as a right, by current standards, to believe, then we see conversation as the ultimate context within which knowledge is to be understood. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], Ch.5), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5 | |
| A reaction: This teeters towards ridiculous relativism (e.g. what if the conversation is among a group of fools? - Ah, there are no fools! Politically incorrect!). However, knowledge can be social, provided we are healthily elitist. Scientists know more than us. |
| 2566 | You can't debate about whether to have higher standards for the application of words [Rorty] |
| Full Idea: The decision about whether to have higher than usual standards for the application of words like "true" or "good" or "red" is, as far as I can see, not a debatable issue. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.6) |
| 2553 | The mind is a property, or it is baffling [Rorty] |
| Full Idea: All that is needed for the mind-body problem to be unintelligible is for us to be nominalist, to refuse firmly to hypostasize individual properties. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 1.3) | |
| A reaction: Edelman says the mind is a process rather than a property. It might vanish if the clockspeed was turned right down? Nominalism here sounds like behaviourism or instrumentalism. Would Dennett plead guilty? |
| 2550 | Pain lacks intentionality; beliefs lack qualia [Rorty] |
| Full Idea: We can't define the mental as intentional because pains aren't about anything, and we can't define it as phenomenal because beliefs don't feel like anything. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 1.2) | |
| A reaction: Nice, but simplistic? There is usually an intentional object for a pain, and the concepts which we use to build beliefs contain the residue of remembered qualia. It seems unlikely that any mind could have one without the other (even a computer). |
| 2554 | Is intentionality a special sort of function? [Rorty] |
| Full Idea: Following Wittgenstein, we shall treat the intentional as merely a subspecies of the functional. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 1.3) | |
| A reaction: Intriguing but obscure. Sounds wrong to me. The intentional refers to the content of thoughts, but function concerns their role. They have roles because they have content, so they can't be the same. |
| 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. |
| 2565 | Nature has no preferred way of being represented [Rorty] |
| Full Idea: Nature has no preferred way of being represented. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.5) | |
| A reaction: Tree rings accidentally represent the passing of the years. If God went back and started again would she or he opt for a 'preferred way'? |
| 2560 | Can meanings remain the same when beliefs change? [Rorty] |
| Full Idea: For cooler heads there must be some middle view between "meanings remain and beliefs change" and "meanings change whenever beliefs do". | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.2) | |
| A reaction: The second one seems blatanty false. How could we otherwise explain a change in belief? But obviously some changes in belief (e.g. about electrons) produce a change in meaning. |
| 2562 | A theory of reference seems needed to pick out objects without ghostly inner states [Rorty] |
| Full Idea: The need to pick out objects without the help of definitions, essences, and meanings of terms produced, philosophers thought, a need for a "theory of reference". | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.3) | |
| A reaction: Frege's was very perceptive in noting that meaning and reference are not the same. Whether we need a 'theory' of reference is unclear. It is worth describing how it occurs. |
| 2559 | Davidson's theory of meaning focuses not on terms, but on relations between sentences [Rorty] |
| Full Idea: A theory of meaning, for Davidson, is not an assemblage of "analyses" of the meanings of individual terms, but rather an understanding of the inferential relations between sentences. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.1) | |
| A reaction: Put that way, the influence of Frege on Davidson is obvious. Purely algebraic expressions can have inferential relations, using variables and formal 'sentences'. |
| 2558 | Since Hegel we have tended to see a human as merely animal if it is outside a society [Rorty] |
| Full Idea: Only since Hegel have philosophers begun toying with the idea that the individual apart from his society is just one more animal. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 4.3) |