36 ideas
1749 | If all laws were abolished, philosophers would still live as they do now [Aristippus elder] |
Full Idea: If all laws were abolished, philosophers would still live as they do now. | |
From: Aristippus the elder (fragments/reports [c.395 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.4 | |
A reaction: Presumably philosophers develop inner laws which other people lack. |
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 |
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. |
7720 | Two things can only resemble one another in some respect, and that may reintroduce a universal [Lowe] |
Full Idea: A problem for resemblance nominalism is that in saying that two particulars 'resemble' one another, it is necessary to specify in what respect they do so (e.g. colour, shape, size), and this threatens to reintroduce what appears to be talk of universals. | |
From: E.J. Lowe (Locke on Human Understanding [1995], Ch.7) | |
A reaction: We see resemblance between faces instantly, long before we can specify the 'respects' of the resemblance. This supports the Humean hard-wired view of resemblance, rather than some appeal to Platonic universals. |
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. |
7712 | On substances, Leibniz emphasises unity, Spinoza independence, Locke relations to qualities [Lowe] |
Full Idea: Later philosophers emphasised different strands of Aristotle's concept of substances: Leibniz (in his theory of monads) emphasised their unity; Spinoza emphasised their ontological independence; Locke emphasised their role in relation to qualities. | |
From: E.J. Lowe (Locke on Human Understanding [1995], Ch.4) | |
A reaction: Note that this Aristotelian idea had not been jettisoned in the late seventeenth century, unlike other Aristotelianisms. I think it is only with the success of atomism in chemistry that the idea of substance is forced to recede. |
7710 | Perception is a mode of belief-acquisition, and does not involve sensation [Lowe] |
Full Idea: According to one school of thought, perception is simply a mode of belief-acquisition,and there is no reason to suppose that any element of sensation is literally involved in perception. | |
From: E.J. Lowe (Locke on Human Understanding [1995], Ch.3) | |
A reaction: Blindsight would be an obvious supporting case for this view. I think this point is crucial in understanding what is wrong with Jackson's 'knowledge argument' (involving Mary, see Idea 7377). Sensation gives knowledge, so it can't be knowledge. |
7711 | Science requires a causal theory - perception of an object must be an experience caused by the object [Lowe] |
Full Idea: Only a causal theory of perception will respect the facts of physiology and physics ...meaning a theory which maintains that for a subject to perceive a physical object the subject should enjoy some appropriate perceptual experience caused by the object. | |
From: E.J. Lowe (Locke on Human Understanding [1995], Ch.3) | |
A reaction: If I hallucinate an object, then presumably I am not allowed to say that I 'perceive' it, but that seems to make the causal theory an idle tautology. If we are in virtual reality then there aren't any objects. |
7714 | Personal identity is a problem across time (diachronic) and at an instant (synchronic) [Lowe] |
Full Idea: There is the question of the identity of a person over or across time ('diachronic' personal identity), and there is also the question of what makes for personal identity at a time ('synchronic' personal identity). | |
From: E.J. Lowe (Locke on Human Understanding [1995], Ch.5) | |
A reaction: This seems to me to be the first and most important distinction in the philosophy of personal identity, and they regularly get run together. Locke, for example, has an account of synchronic identity, which is often ignored. It applies to objects too. |
7715 | Mentalese isn't a language, because it isn't conventional, or a means of public communication [Lowe] |
Full Idea: 'Mentalese' would be neither conventional nor a means of public communication so that even to call it a language is seriously misleading. | |
From: E.J. Lowe (Locke on Human Understanding [1995], Ch.7) | |
A reaction: It is, however, supposed to contain symbolic representations which are then used as tokens for computation, so it seems close to a language, if (for example) symbolic logic or mathematics were accepted as languages. But who understands it? |
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. |
7722 | If meaning is mental pictures, explain "the cat (or dog!) is NOT on the mat" [Lowe] |
Full Idea: If meaning is a private mental picture, what does 'the cat is NOT on the mat' mean, and how does it differ from 'the dog is not on the mat?'. | |
From: E.J. Lowe (Locke on Human Understanding [1995], Ch.7) | |
A reaction: Not insurmountable. We picture an empty mat, combined with a cat (or whatever) located somewhere else. A mental 'picture' of something shouldn't be contrued as a single image in a neat black frame. |
3558 | Only the Cyrenaics reject the idea of a final moral end [Aristippus elder, by Annas] |
Full Idea: The Cyrenaics are the most radical ancient moral philosophers, since they are the only school explicitly to reject the importance of achieving an overall final end. | |
From: report of Aristippus the elder (fragments/reports [c.395 BCE]) by Julia Annas - The Morality of Happiness 11.1 | |
A reaction: This looks like dropping out, but it could also be Keats's 'negative capability', of simply participating in existence without needing to do anything about it. |
5835 | The road of freedom is the surest route to happiness [Aristippus elder, by Xenophon] |
Full Idea: The surest road to happiness is not the path through rule nor through servitude, but through liberty. | |
From: report of Aristippus the elder (fragments/reports [c.395 BCE]) by Xenophon - Memorabilia of Socrates 2.1.9 | |
A reaction: The great anarchist slogan. Personally I don't believe it, because I agree a little with Hobbes that authority is required to make cooperation flourish, and that is essential for full happiness. If I were a slave, I would agree with Aristippus. |
3018 | People who object to extravagant pleasures just love money [Aristippus elder, by Diog. Laertius] |
Full Idea: When blamed for buying expensive food he asked "Would you have bought it for just three obols?" When the person said yes, he said,"Then it is not that I am fond of pleasure, but that you are fond of money". | |
From: report of Aristippus the elder (fragments/reports [c.395 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.4 |
1751 | Pleasure is the good, because we always seek it, it satisfies us, and its opposite is the most avoidable thing [Aristippus elder, by Diog. Laertius] |
Full Idea: Pleasure is the good because we desire it from childhood, when we have it we seek nothing further, and the most avoidable thing is its opposite, pain. | |
From: report of Aristippus the elder (fragments/reports [c.395 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.8 |
1755 | Errors result from external influence, and should be corrected, not hated [Aristippus elder, by Diog. Laertius] |
Full Idea: Errors ought to meet with pardon, for a man does not err intentionally, but influenced by some external circumstances. We should not hate someone who has erred, but teach him better. | |
From: report of Aristippus the elder (fragments/reports [c.395 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.9 |