33 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. |
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. |
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. |
6402 | In 1927, Russell analysed force and matter in terms of events [Russell, by Grayling] |
Full Idea: In his 'Analysis of Matter' (1927), Russell sought to analyse the chief concepts of physics, such as force and matter, in terms of events. | |
From: report of Bertrand Russell (The Analysis of Matter [1927]) by A.C. Grayling - Russell Ch.2 | |
A reaction: My immediate reaction is that this is not very promising, simply because we can always ask why a particular event occurred, and this seems to point to a deeper level in the analysis. See Idea 4779, for example. |
14732 | A perceived physical object is events grouped around a centre [Russell] |
Full Idea: The physical object, as inferred from perception, is a group of events arranged about a centre. | |
From: Bertrand Russell (The Analysis of Matter [1927], 23) | |
A reaction: At least I like the active aspect of this definition. You then have to explain what an event is, without mentioning objects. You'd better no mention properties either, since they will probably depend on the dreaded objects. |
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. |
14733 | An object produces the same percepts with or without a substance, so that is irrelevant to science [Russell] |
Full Idea: There may be a substance at the centre of an object, but is no reason to think so, since the group of events making up the object will produce exactly the same percepts; so the substance, if there is one, is an abstract possibility irrelevant to science. | |
From: Bertrand Russell (The Analysis of Matter [1927], 23) | |
A reaction: All empiricists (as Russell is in this passage) seem to neglect inference to the best explanation. Things can be indirectly testable, and I would say that there are genuine general entities which are too close to abstraction to ever be testable. |
6418 | Russell rejected phenomenalism because it couldn't account for causal relations [Russell, by Grayling] |
Full Idea: Russell reverted to realism when he recognised that the notion of causality is problematic for phenomenalism; things in the world seem to affect one another causally in ways that are difficult to account for properly by mere reports of sense-experiences. | |
From: report of Bertrand Russell (The Analysis of Matter [1927]) by A.C. Grayling - Russell Ch.3 | |
A reaction: This is very interesting, and doesn't seem to have been enough to make A.J. Ayer eschew phenomenalism (Idea 5170). Once your metaphysics becomes realist (like Russell), your account of perception and objects has to change too. |
7658 | Obviously there can't be a functional anaylsis of qualia if they are defined by intrinsic properties [Dennett] |
Full Idea: If you define qualia as intrinsic properties of experiences considered in isolation from all their causes and effects, logically independent of all dispositional properties, then they are logically guaranteed to elude all broad functional analysis. | |
From: Daniel C. Dennett (Sweet Dreams [2005], Ch.8) | |
A reaction: This is a good point - it seems daft to reify qualia and imagine them dangling in mid-air with all their vibrant qualities - but that is a long way from saying there is nothing more to qualia than functional roles. Functions must be exlained too. |
7655 | The work done by the 'homunculus in the theatre' must be spread amongst non-conscious agencies [Dennett] |
Full Idea: All the work done by the imagined homunculus in the Cartesian Theater must be distributed among various lesser agencies in the brain, none of which is conscious. | |
From: Daniel C. Dennett (Sweet Dreams [2005], Ch.3) | |
A reaction: Dennett's account crucially depends on consciousness being much more fragmentary than most philosophers claim it to be. It is actually full of joints, which can come apart. He may be right. |
7657 | Intelligent agents are composed of nested homunculi, of decreasing intelligence, ending in machines [Dennett] |
Full Idea: As long as your homunculi are more stupid and ignorant than the intelligent agent they compose, the nesting of homunculi within homunculi can be finite, bottoming out, eventually, with agents so unimpressive they can be replaced by machines. | |
From: Daniel C. Dennett (Sweet Dreams [2005], Ch.6) | |
A reaction: [Dennett first proposed this in 'Brainstorms' 1978]. This view was developed well by Lycan. I rate it as one of the most illuminating ideas in the modern philosophy of mind. All complex systems (like aeroplanes) have this structure. |
7656 | I don't deny consciousness; it just isn't what people think it is [Dennett] |
Full Idea: I don't maintain, of course, that human consciousness does not exist; I maintain that it is not what people often think it is. | |
From: Daniel C. Dennett (Sweet Dreams [2005], Ch.3) | |
A reaction: I consider Dennett to be as near as you can get to an eliminativist, but he is not stupid. As far as I can see, the modern philosopher's bogey-man, the true total eliminativist, simply doesn't exist. Eliminativists usually deny propositional attitudes. |
7654 | What matters about neuro-science is the discovery of the functional role of the chemistry [Dennett] |
Full Idea: Neuro-science matters because - and only because - we have discovered that the many different neuromodulators and other chemical messengers that diffuse throughout the brain have functional roles that make important differences. | |
From: Daniel C. Dennett (Sweet Dreams [2005], Ch.1) | |
A reaction: I agree with Dennett that this is the true ground for pessimism about spectacular breakthroughs in artificial intelligence, rather than abstract concerns about irreducible features of the mind like 'qualia' and 'rationality'. |
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. |
21706 | At first matter is basic and known by sense-data; later Russell says matter is constructed [Russell, by Linsky,B] |
Full Idea: In the beginning Russell's ontology included matter as basic, to be known, however, only by inference from sense-data. By the end he wanted to 'contruct' matter from sense-data. | |
From: report of Bertrand Russell (The Analysis of Matter [1927]) by Bernard Linsky - Russell's Metaphysical Logic 1 | |
A reaction: [see also p.133] Russell always seems to have been a robust realist about the external world, but the later view seems a lot less realist than the earlier view. |