41 ideas
| 14052 | Begin philosophy when you are young, and keep going when you are old [Epicurus] |
| Full Idea: Let no one delay the study of philosophy while young nor weary of it when old; for no one is either too young or too old for the health of the soul. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 122) | |
| A reaction: I agree with this on both accounts. I think the correct age to begin the study of philosophy is four, and it is vital to continue its study up to the point where you can no longer remember your own name. 'Health of the soul' sounds right too. |
| 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. |
| 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. |
| 14062 | Sooner follow mythology, than accept the 'fate' of natural philosophers [Epicurus] |
| Full Idea: It would be better to follow the stories told about the gods than to be a slave to the fate of the natural philosophers. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 134) | |
| A reaction: At this point in history there is a blurring between autonomous decisions and what we now call free will, and also between fate and determinism, which we try to keep distinct. |
| 1837 | We should not refer things to irresponsible necessity, but either to fortune or to our own will [Epicurus] |
| Full Idea: The best men have no belief in necessity (set up by some as mistress of all), but refer some things to fortune, some to ourselves, because necessity is irresponsible, and fortune is unstable, while our own will is free. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 133), quoted by Diogenes Laertius - Lives of Eminent Philosophers 10.27 |
| 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. |
| 1836 | Prudence is more valuable than philosophy, because it avoids confusions of the soul [Epicurus] |
| Full Idea: The greatest good in avoiding confusion of the soul is prudence [phronesis], on which account prudence is something more valuable than even philosophy. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 132), quoted by Diogenes Laertius - Lives of Eminent Philosophers 10.27 |
| 14061 | Our own choices are autonomous, and the basis for praise and blame [Epicurus] |
| Full Idea: What occurs by our own agency is autonomous, and it is to this that praise and blame are attached. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 133) | |
| A reaction: I don't think this should be understand as an assertion of free will in the modern sense. The 'swerve' of the atoms just means that decisions can arise out of us - not that they are somehow outside of nature. |
| 14054 | Fearing death is absurd, because we are not present when it occurs [Epicurus] |
| Full Idea: Death, the most frightening of bad things, is nothing to us; since when we exist, death is not yet present, and when death is present, then we do not exist. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 125) | |
| A reaction: This is a fairly accurate observation. To fear not being in this life is a bit like fearing not being in Vancouver next Tuesday. It also involves the paradox of the present moment. E.g. Idea 1904. |
| 14053 | It is absurd to fear the pain of death when you are not even facing it [Epicurus] |
| Full Idea: He is a fool who says that he fears death not because it will be painful when present but because it is painful when it is still to come. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 125) | |
| A reaction: Not very plausible, I'm afraid. It provides a good argument in favour of smoking, if the lung cancer is far in the future. Paralysing fear is daft, but some remote fears should be heeded. |
| 14055 | The wisdom that produces a good life also produces a good death [Epicurus] |
| Full Idea: The same kind of practice produces a good life and a good death. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 126) | |
| A reaction: This is the kind of old fashioned observation which we would do well to hang on to. The ideal of dying well has vanished from our culture. |
| 14057 | All pleasures are good, but it is not always right to choose them [Epicurus] |
| Full Idea: Every pleasure is a good thing, since it has a nature congenial to us, but not every one is to be chosen, just as every pain is a bad thing, but not every one is such as to be always avoided. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 129) | |
| A reaction: This kind of sensible remark would be wholly endorsed by Bentham and Mill. This fits in with the excellent distinction between what is right and what is good. |
| 14058 | Pleasure is the goal, but as lack of pain and calm mind, not as depraved or greedy pleasure [Epicurus] |
| Full Idea: When we say that pleasure is the goal we do not mean the pleasures of the profligate or the pleasures of consumption, but rather the lack of pain in the body and disturbance in the soul. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 131) | |
| A reaction: I don't really understand the aspiration to a 'calm mind'. No one likes stress, but total calmness sounds close to non-existence. The mean! There is no achievement without pain. |
| 1833 | Pleasure is the first good in life [Epicurus] |
| Full Idea: Pleasure is the beginning and end of living happily, and we recognise this as the first good. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 128) | |
| A reaction: We might enquire what we would live for if our capacities for pleasure were surgically removed. Would we still experience intellectual curiosity, or an aspiration to some cold and remote goodness? |
| 14063 | Sooner a good decision going wrong, than a bad one turning out for the good [Epicurus] |
| Full Idea: It is better for a good decision not to turn out right in action than for a bad decision to turn out right because of chance. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 135) | |
| A reaction: This sounds right, and on the whole the law agrees. Notice that what we need is a 'good decision', and not just to 'mean well'. The well-meaning fool is wicked. I am opposed to consequentialism, and agree with this idea. |
| 14059 | The best life is not sensuality, but rational choice and healthy opinion [Epicurus] |
| Full Idea: It is not drinking bouts or enjoying boys and women or consuming fish which produces the pleasant life, but sober calculation which searches out reasons for every choice, and drives out opinions which produce turmoil of the soul. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 132) | |
| A reaction: This more or less sums up what I would call the philosophical life. Spontaneity is good, and some pleasures are killed by excessive thought, but on the whole actions are always better if good reasons are found, and error brings chaos. |
| 1835 | True pleasure is not debauchery, but freedom from physical and mental pain [Epicurus] |
| Full Idea: When we say that pleasure is the chief good, we do not mean debauchery, but freedom of the body from pain, and of the soul from confusion…. which requires sober contemplation. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 131), quoted by Diogenes Laertius - Lives of Eminent Philosophers 10.27 | |
| A reaction: I'm not clear how lack of pain and confusion counts as pleasure. Also the concepts of debauchery held by the puritan and the sybarite are wildly different. |
| 14056 | We only need pleasure when we have the pain of desire [Epicurus] |
| Full Idea: We are in need of pleasure only when we are in pain because of the absence of pleasure, and when we are not in pain, then we no longer need pleasure. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 128) | |
| A reaction: This Buddhist aspiration to eliminate desire has no appeal for me. It just sounds like a recipe for boredom, and an aversion to risk-taking. Start by asking what is best in life; it inevitably involves pleasure of some sort. Anyway, desire isn't painful. |
| 14060 | Prudence is the greatest good, and more valuable than philosophy, because it produces virtue [Epicurus] |
| Full Idea: Prudence is the principle of the rational life and is the greatest good. That is why prudence is more valuable than philosophy, for prudence is the source of all the other virtues. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 132) | |
| A reaction: ['prudence' will be Greek 'phronesis']The interest of this is that it is almost copied straight out of Aristotle's Ethics. Epicurus was an opponent of the Peripatetics, but greatly influenced by them. |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| Full Idea: Philosophers are the forefathers of heretics. | |
| From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
| 6610 | I believe because it is absurd [Tertullian] |
| Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
| From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
| A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |