118 ideas
| 11300 | Agathon: good [PG] |
| Full Idea: Agathon: good, the highest good | |
| From: PG (Db (lexicon) [c.1001 BCE], 01) |
| 11301 | Aisthesis: perception, sensation, consciousness [PG] |
| Full Idea: Aisthesis: perception, sensation, consciousness | |
| From: PG (Db (lexicon) [c.1001 BCE], 02) |
| 11302 | Aitia / aition: cause, explanation [PG] |
| Full Idea: Aitia / aition: cause, explanation | |
| From: PG (Db (lexicon) [c.1001 BCE], 03) | |
| A reaction: The consensus is that 'explanation' is the better translation, and hence that the famous Four Causes (in 'Physics') must really be understood as the Four Modes of Explanation. They then make far more sense. |
| 11303 | Akrasia: lack of control, weakness of will [PG] |
| Full Idea: Akrasia: lack of control, weakness of will | |
| From: PG (Db (lexicon) [c.1001 BCE], 04) | |
| A reaction: The whole Greek debate (and modern debate, I would say) makes much more sense if we stick to 'lack of control' as the translation, and forget about weakness of will - and certainly give up 'incontinence' as a translation. |
| 11304 | Aletheia: truth [PG] |
| Full Idea: Aletheia: truth | |
| From: PG (Db (lexicon) [c.1001 BCE], 05) |
| 11305 | Anamnesis: recollection, remembrance [PG] |
| Full Idea: Anamnesis: recollection, remembrance | |
| From: PG (Db (lexicon) [c.1001 BCE], 06) | |
| A reaction: This is used for Plato's doctrine that we recollect past lives. |
| 11306 | Ananke: necessity [PG] |
| Full Idea: Ananke: necessity | |
| From: PG (Db (lexicon) [c.1001 BCE], 07) |
| 11307 | Antikeimenon: object [PG] |
| Full Idea: Antikeimenon: object | |
| From: PG (Db (lexicon) [c.1001 BCE], 08) |
| 11375 | Apatheia: unemotional [PG] |
| Full Idea: Apatheia: lack of involvement, unemotional | |
| From: PG (Db (lexicon) [c.1001 BCE], 09) |
| 11308 | Apeiron: the unlimited, indefinite [PG] |
| Full Idea: Apeiron: the unlimited, indefinite | |
| From: PG (Db (lexicon) [c.1001 BCE], 10) | |
| A reaction: Key term in the philosophy of Anaximander, the one unknowable underlying element. |
| 11376 | Aphairesis: taking away, abstraction [PG] |
| Full Idea: Aphairesis: taking away, abstraction | |
| From: PG (Db (lexicon) [c.1001 BCE], 11) |
| 11309 | Apodeixis: demonstration [PG] |
| Full Idea: Apodeixis: demonstration, proof | |
| From: PG (Db (lexicon) [c.1001 BCE], 12) |
| 11310 | Aporia: puzzle, question, anomaly [PG] |
| Full Idea: Aporia: puzzle, question, anomaly | |
| From: PG (Db (lexicon) [c.1001 BCE], 13) |
| 11311 | Arche: first principle, the basic [PG] |
| Full Idea: Arché: first principle, the basic | |
| From: PG (Db (lexicon) [c.1001 BCE], 14) | |
| A reaction: Interchangeable with 'aitia' by Aristotle. The first principle and the cause are almost identical. |
| 11312 | Arete: virtue, excellence [PG] |
| Full Idea: Areté: virtue, excellence | |
| From: PG (Db (lexicon) [c.1001 BCE], 15) | |
| A reaction: The word hovers between moral excellence and being good at what you do. Annas defends the older translation as 'virtue', rather than the modern 'excellence'. |
| 11313 | Chronismos: separation [PG] |
| Full Idea: Chronismos: separation | |
| From: PG (Db (lexicon) [c.1001 BCE], 16) |
| 11314 | Diairesis: division [PG] |
| Full Idea: Diairesis: division, distinction | |
| From: PG (Db (lexicon) [c.1001 BCE], 17) |
| 11315 | Dialectic: dialectic, discussion [PG] |
| Full Idea: Dialectic: dialectic, discussion | |
| From: PG (Db (lexicon) [c.1001 BCE], 18) |
| 11316 | Dianoia: intellection [cf. Noesis] [PG] |
| Full Idea: Dianoia: intellection, understanding [cf. Noesis] | |
| From: PG (Db (lexicon) [c.1001 BCE], 21) |
| 11317 | Diaphora: difference [PG] |
| Full Idea: Diaphora: difference | |
| From: PG (Db (lexicon) [c.1001 BCE], 22) |
| 11318 | Dikaiosune: moral goodness, justice [PG] |
| Full Idea: Dikaiosune: moral goodness, justice | |
| From: PG (Db (lexicon) [c.1001 BCE], 23) | |
| A reaction: Usually translated as 'justice' in 'Republic', but it is a general term of moral approbation, not like the modern political and legal notion of 'justice'. 'Justice' actually seems to be bad translation. |
| 11319 | Doxa: opinion, belief [PG] |
| Full Idea: Doxa: opinion, belief, judgement | |
| From: PG (Db (lexicon) [c.1001 BCE], 24) |
| 11320 | Dunamis: faculty, potentiality, capacity [PG] |
| Full Idea: Dunamis: faculty, potentiality, capacity | |
| From: PG (Db (lexicon) [c.1001 BCE], 25) |
| 11321 | Eidos: form, idea [PG] |
| Full Idea: Eidos: form, idea | |
| From: PG (Db (lexicon) [c.1001 BCE], 26) | |
| A reaction: In Plato it is the word best translated as 'Form' (Theory of...); in Aritotle's 'Categories' it designates the species, and in 'Metaphysics' it ends up naming the structural form of the species (and hence the essence) [Wedin p.120] |
| 11322 | Elenchos: elenchus, interrogation [PG] |
| Full Idea: Elenchos: elenchus, interrogation | |
| From: PG (Db (lexicon) [c.1001 BCE], 27) |
| 11323 | Empeiron: experience [PG] |
| Full Idea: Empeiron: experience | |
| From: PG (Db (lexicon) [c.1001 BCE], 28) |
| 11324 | Energeia: employment, actuality, power? [PG] |
| Full Idea: Energeia: employment, actuality, power? | |
| From: PG (Db (lexicon) [c.1001 BCE], 31) |
| 11325 | Enkrateia: control [PG] |
| Full Idea: Enkrateia: control | |
| From: PG (Db (lexicon) [c.1001 BCE], 32) | |
| A reaction: See 'akrasia', of which this is the opposite. The enkratic person is controlled. |
| 11326 | Entelecheia: entelechy, having an end [PG] |
| Full Idea: Entelecheia: entelechy, having an end | |
| From: PG (Db (lexicon) [c.1001 BCE], 33) |
| 11327 | Epagoge: induction, explanation [PG] |
| Full Idea: Epagoge: induction, explanation, leading on | |
| From: PG (Db (lexicon) [c.1001 BCE], 34) |
| 11328 | Episteme: knowledge, understanding [PG] |
| Full Idea: Episteme: knowledge, understanding | |
| From: PG (Db (lexicon) [c.1001 BCE], 35) | |
| A reaction: Note that 'episteme' can form a plural in Greek, but we can't say 'knowledges', so we have to say 'branches of knowledge', or 'sciences'. |
| 11329 | Epithumia: appetite [PG] |
| Full Idea: Epithumia: appetite | |
| From: PG (Db (lexicon) [c.1001 BCE], 36) |
| 11330 | Ergon: function [PG] |
| Full Idea: Ergon: function, work | |
| From: PG (Db (lexicon) [c.1001 BCE], 37) |
| 11331 | Eristic: polemic, disputation [PG] |
| Full Idea: Eristic: polemic, disputation | |
| From: PG (Db (lexicon) [c.1001 BCE], 38) | |
| A reaction: This is confrontational argument, rather than the subtle co-operative dialogue of dialectic. British law courts and the House of Commons are founded on eristic, rather than on dialectic. Could there be a dialectical elected assembly? |
| 11332 | Eros: love [PG] |
| Full Idea: Eros: love, desire | |
| From: PG (Db (lexicon) [c.1001 BCE], 41) |
| 11333 | Eudaimonia: flourishing, happiness, fulfilment [PG] |
| Full Idea: Eudaimonia: flourishing, happiness, fulfilment | |
| From: PG (Db (lexicon) [c.1001 BCE], 42) | |
| A reaction: Some people defend 'happiness' as the translation, but that seems to me wildly misleading, since eudaimonia is something like life going well, and certainly isn't a psychological state - and definitely not pleasure. |
| 11334 | Genos: type, genus [PG] |
| Full Idea: Genos: type, genus, kind | |
| From: PG (Db (lexicon) [c.1001 BCE], 43) |
| 11335 | Hexis: state, habit [PG] |
| Full Idea: Hexis: state, habit | |
| From: PG (Db (lexicon) [c.1001 BCE], 44) |
| 11336 | Horismos: definition [PG] |
| Full Idea: Horismos: definition | |
| From: PG (Db (lexicon) [c.1001 BCE], 45) |
| 11337 | Hule: matter [PG] |
| Full Idea: Hule: matter | |
| From: PG (Db (lexicon) [c.1001 BCE], 46) | |
| A reaction: The first half of the 'hylomorphism' of Aristotle. See 'morphe'! |
| 11338 | Hupokeimenon: subject, underlying thing [cf. Tode ti] [PG] |
| Full Idea: Hupokeimenon: subject, underlying thing, substratum [cf. Tode ti] | |
| From: PG (Db (lexicon) [c.1001 BCE], 47) | |
| A reaction: Literally 'that which lies under'. Latin version is 'substratum'. In Aristotle it is the problem, of explaining what lies under. It is not the theory that there is some entity called a 'substratum'. |
| 11339 | Kalos / kalon: beauty, fineness, nobility [PG] |
| Full Idea: Kalos / kalon: beauty, fineness, nobility | |
| From: PG (Db (lexicon) [c.1001 BCE], 48) | |
| A reaction: A revealing Greek word, which is not only our rather pure notion of 'beauty', but also seems to mean something like wow!, and (very suggestive, this) applies as much to actions as to objects. |
| 11340 | Kath' hauto: in virtue of itself, essentially [PG] |
| Full Idea: Kath' hauto: in virtue of itself, essentially | |
| From: PG (Db (lexicon) [c.1001 BCE], 51) |
| 11341 | Kinesis: movement, process [PG] |
| Full Idea: Kinesis: movement, process, change | |
| From: PG (Db (lexicon) [c.1001 BCE], 52) |
| 11342 | Kosmos: order, universe [PG] |
| Full Idea: Kosmos: order, universe | |
| From: PG (Db (lexicon) [c.1001 BCE], 53) |
| 11343 | Logos: reason, account, word [PG] |
| Full Idea: Logos: reason, account, word | |
| From: PG (Db (lexicon) [c.1001 BCE], 54) |
| 11344 | Meson: the mean [PG] |
| Full Idea: Meson: the mean | |
| From: PG (Db (lexicon) [c.1001 BCE], 55) | |
| A reaction: This is not the 'average', and hence not some theoretical mid-point. I would call it the 'appropriate compromise', remembering that an extreme may be appropriate in certain circumstances. |
| 11345 | Metechein: partaking, sharing [PG] |
| Full Idea: Metechein: partaking, sharing | |
| From: PG (Db (lexicon) [c.1001 BCE], 56) | |
| A reaction: The key word in Plato for the difficult question of the relationships between the Forms and the particulars. The latter 'partake' of the former. Hm. Compare modern 'instantiation', which strikes me as being equally problematic. |
| 11377 | Mimesis: imitation, fine art [PG] |
| Full Idea: Mimesis: imitation, fine art | |
| From: PG (Db (lexicon) [c.1001 BCE], 57) |
| 11346 | Morphe: form [PG] |
| Full Idea: Morphe: form | |
| From: PG (Db (lexicon) [c.1001 BCE], 58) |
| 11347 | Noesis: intellection, rational thought [cf. Dianoia] [PG] |
| Full Idea: Noesis: intellection, rational thought [cf. Dianoia] | |
| From: PG (Db (lexicon) [c.1001 BCE], 59) |
| 11348 | Nomos: convention, law, custom [PG] |
| Full Idea: Nomos: convention, law, custom | |
| From: PG (Db (lexicon) [c.1001 BCE], 61) |
| 11349 | Nous: intuition, intellect, understanding [PG] |
| Full Idea: Nous: intuition, intellect | |
| From: PG (Db (lexicon) [c.1001 BCE], 62) | |
| A reaction: There is a condensed discussion of 'nous' in Aristotle's Posterior Analytics B.19 |
| 11350 | Orexis: desire [PG] |
| Full Idea: Orexis: desire | |
| From: PG (Db (lexicon) [c.1001 BCE], 63) |
| 11351 | Ousia: substance, (primary) being, [see 'Prote ousia'] [PG] |
| Full Idea: Ousia: substance, (primary) being [see 'Prote ousia'] | |
| From: PG (Db (lexicon) [c.1001 BCE], 64) | |
| A reaction: It is based on the verb 'to be'. Latin therefore translated it as 'essentia' (esse: to be), and we have ended up translating it as 'essence', but this is wrong! 'Being' is the best translation, and 'substance' is OK. It is the problem, not the answer. |
| 11352 | Pathos: emotion, affection, property [PG] |
| Full Idea: Pathos: emotion, affection, property | |
| From: PG (Db (lexicon) [c.1001 BCE], 65) |
| 11353 | Phantasia: imagination [PG] |
| Full Idea: Phantasia: imagination | |
| From: PG (Db (lexicon) [c.1001 BCE], 66) |
| 11354 | Philia: friendship [PG] |
| Full Idea: Philia: friendship | |
| From: PG (Db (lexicon) [c.1001 BCE], 67) |
| 11355 | Philosophia: philosophy, love of wisdom [PG] |
| Full Idea: Philosophia: philosophy, love of wisdom | |
| From: PG (Db (lexicon) [c.1001 BCE], 68) | |
| A reaction: The point of the word is its claim only to love wisdom, and not actually to be wise. |
| 11356 | Phronesis: prudence, practical reason, common sense [PG] |
| Full Idea: Phronesis: prudence, practical reason, common sense | |
| From: PG (Db (lexicon) [c.1001 BCE], 71) | |
| A reaction: None of the experts use my own translation, which is 'common sense', but that seems to me to perfectly fit all of Aristotle's discussions of the word in 'Ethics'. 'Prudence' seems a daft translation in modern English. |
| 11357 | Physis: nature [PG] |
| Full Idea: Physis: nature | |
| From: PG (Db (lexicon) [c.1001 BCE], 72) |
| 11358 | Praxis: action, activity [PG] |
| Full Idea: Praxis: action, activity | |
| From: PG (Db (lexicon) [c.1001 BCE], 73) |
| 11359 | Prote ousia: primary being [PG] |
| Full Idea: Prote ousia: primary being | |
| From: PG (Db (lexicon) [c.1001 BCE], 74) | |
| A reaction: The main topic of investigation in Aristotle's 'Metaphysics'. 'Ousia' is the central problem of the text, NOT the answer to the problem. |
| 11360 | Psuche: mind, soul, life [PG] |
| Full Idea: Psuche: mind, soul, life | |
| From: PG (Db (lexicon) [c.1001 BCE], 75) | |
| A reaction: The interesting thing about this is that we have tended to translate it as 'soul', but Aristotle says plants have it, and not merely conscious beings. It is something like the 'form' of a living thing, but then 'form' is a misleading translation too. |
| 11361 | Sophia: wisdom [PG] |
| Full Idea: Sophia: wisdom | |
| From: PG (Db (lexicon) [c.1001 BCE], 76) |
| 11362 | Sophrosune: moderation, self-control [PG] |
| Full Idea: Sophrosune: moderation, self-control | |
| From: PG (Db (lexicon) [c.1001 BCE], 77) |
| 11363 | Stoicheia: elements [PG] |
| Full Idea: Stoicheia: elements | |
| From: PG (Db (lexicon) [c.1001 BCE], 78) |
| 11364 | Sullogismos: deduction, syllogism [PG] |
| Full Idea: Sullogismos: deduction, syllogism | |
| From: PG (Db (lexicon) [c.1001 BCE], 81) |
| 11365 | Techne: skill, practical knowledge [PG] |
| Full Idea: Techne: skill, practical knowledge | |
| From: PG (Db (lexicon) [c.1001 BCE], 82) |
| 11366 | Telos: purpose, end [PG] |
| Full Idea: Telos: purpose, end | |
| From: PG (Db (lexicon) [c.1001 BCE], 83) |
| 11367 | Theoria: contemplation [PG] |
| Full Idea: Theoria: contemplation | |
| From: PG (Db (lexicon) [c.1001 BCE], 84) |
| 11368 | Theos: god [PG] |
| Full Idea: Theos: god | |
| From: PG (Db (lexicon) [c.1001 BCE], 85) |
| 11369 | Ti esti: what-something-is, essence [PG] |
| Full Idea: Ti esti: the what-something-is, essence, whatness | |
| From: PG (Db (lexicon) [c.1001 BCE], 86) |
| 11370 | Timoria: vengeance, punishment [PG] |
| Full Idea: Timoria: vengeance, punishment | |
| From: PG (Db (lexicon) [c.1001 BCE], 87) |
| 11371 | To ti en einai: essence, what-it-is-to-be [PG] |
| Full Idea: To ti en einai: essence, what-it-is-to-be | |
| From: PG (Db (lexicon) [c.1001 BCE], 88) | |
| A reaction: This is Aristotle's main term for what we would now call the 'essence'. It is still not a theory of essence, merely an identification of the target. 'Form' is the nearest we get to his actual theory. |
| 11372 | To ti estin: essence [PG] |
| Full Idea: To ti estin: essence | |
| From: PG (Db (lexicon) [c.1001 BCE], 91) |
| 11373 | Tode ti: this-such, subject of predication [cf. hupokeimenon] [PG] |
| Full Idea: Tode ti: this-something, subject of predication, thisness [cf. hupokeimenon] | |
| From: PG (Db (lexicon) [c.1001 BCE], 92) |
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| Full Idea: Second-order set theory is just like first-order set-theory, except that we use the version of Replacement with a universal second-order quantifier over functions from set to sets. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VII.4) |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| Full Idea: A member m of M is an 'upper bound' of a subset N of M if m is not less than any member of N. A member m of M is a 'least upper bound' of N if m is an upper bound of N such that if l is any other upper bound of N, then m is less than l. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: [if you don't follow that, you'll have to keep rereading it till you do] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| Full Idea: Since combinatorial collections are enumerated, some multiplicities may be too large to be gathered into combinatorial collections. But the size of a multiplicity seems quite irrelevant to whether it forms a logical connection. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| Full Idea: Many of those who are skeptical about the existence of infinite combinatorial collections would want to doubt or deny the Axiom of Choice. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| Full Idea: The Power Set is just he codification of the fact that the collection of functions from a mathematical collection to a mathematical collection is itself a mathematical collection that can serve as a domain of mathematical study. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| Full Idea: The Axiom of Replacement (of Skolem and Fraenkel) was remarkable for its universal acceptance, though it seemed to have no consequences except for the properties of the higher reaches of the Cantorian infinite. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| Full Idea: The Axiom of Foundation (Zermelo 1930) says 'Every (descending) chain in which each element is a member of the previous one is of finite length'. ..This forbids circles of membership, or ungrounded sets. ..The iterative conception gives this centre stage. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.4) |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| Full Idea: Combinatorial collections (defined just by the members) obviously obey the Axiom of Choice, while it is at best dubious whether logical connections (defined by a rule) do. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| Full Idea: The controversy was not about Choice per se, but about the correct notion of function - between advocates of taking mathematics to be about arbitrary functions and advocates of taking it to be about functions given by rules. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| Full Idea: The Peano-Russell notion of class is the 'logical' notion, where each collection is associated with some kind of definition or rule that characterises the members of the collection. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.1) |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| Full Idea: The iterative conception of set was not so much as suggested, let alone advocated by anyone, until 1947. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| Full Idea: The iterative conception of sets does not tell us how far to iterate, and so we must start with an Axiom of Infinity. It also presupposes the notion of 'transfinite iteration'. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| Full Idea: The iterative conception does not provide a conception that unifies the axioms of set theory, ...and it has had very little impact on what theorems can be proved. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) | |
| A reaction: He says he would like to reject the iterative conception, but it may turn out that Foundation enables new proofs in mathematics (though it hasn't so far). |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| Full Idea: Limitation of Size has it that if a collection is the same size as a set, then it is a set. The Axiom of Replacement is characteristic of limitation of size. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| Full Idea: A collection M is 'well-ordered' by a relation < if < linearly orders M with a least element, and every subset of M that has an upper bound not in it has an immediate successor. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3) | |
| A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'. |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
| A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku. |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| Full Idea: Mathematics is today thought of as the study of abstract structure, not the study of quantity. That point of view arose directly out of the development of the set-theoretic notion of abstract structure. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.2) | |
| A reaction: It sounds as if Structuralism, which is a controversial view in philosophy, is a fait accompli among mathematicians. |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| Full Idea: One reason to introduce the rational numbers is that it simplifes the theory of division, since every rational number is divisible by every nonzero rational number, while the analogous statement is false for the natural numbers. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.3) | |
| A reaction: That is, with rations every division operation has an answer. |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| Full Idea: The chief importance of the Continuum Hypothesis for Cantor (I believe) was that it would show that the real numbers form a set, and hence that they were encompassed by his theory. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| Full Idea: The Cauchy convergence criterion for a sequence: the sequence S0,S1,... has a limit if |S(n+r) - S(n)| is less than any given quantity for every value of r and sufficiently large values of n. He proved this necessary, but not sufficient. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], 2.5) |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| Full Idea: Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], II.6) | |
| A reaction: Thus there is the problem of whether the contents of the gap are one unique thing, or many. |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| Full Idea: Counting a set produces a well-ordering of it. Conversely, if one has a well-ordering of a set, one can count it by following the well-ordering. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: Cantor didn't mean that you could literally count the set, only in principle. |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| Full Idea: The indiscernibility of indefinitely large sizes will be a critical part of the theory of indefinitely large sizes. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| Full Idea: My proposal is that the concept of the infinite began with an extrapolation from the experience of indefinitely large size. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) | |
| A reaction: I think it might be better to talk of an 'abstraction' than an 'extrapolition', since the latter is just more of the same, which doesn't get you to concept. Lavine spends 100 pages working out his proposal. |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| Full Idea: The intuitionist endorse the actual finite, but only the potential infinite. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| Full Idea: The symbol 'aleph-nought' denotes the cardinal number of the set of natural numbers. The symbol 'aleph-one' denotes the next larger cardinal number. 'Aleph-omega' denotes the omega-th cardinal number. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.3) |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| Full Idea: The ordinals are basic because the transfinite sets are those that can be counted, or (equivalently for Cantor), those that can be numbered by an ordinal or are well-ordered. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: Lavine observes (p.55) that for Cantor 'countable' meant 'countable by God'! |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| Full Idea: The paradox of the largest ordinal (the 'Burali-Forti') is that the class of all ordinal numbers is apparently well-ordered, and so it has an ordinal number as order type, which must be the largest ordinal - but all ordinals can be increased by one. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| Full Idea: The paradox of the largest cardinal ('Cantor's Paradox') says the diagonal argument shows there is no largest cardinal, but the class of all individuals (including the classes) must be the largest cardinal number. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| Full Idea: Every theorem of mathematics has a counterpart with set theory - ...but that theory cannot serve as a basis for the notion of proof. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3) |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| Full Idea: In modern mathematics virtually all work is only up to isomorphism and no one cares what the numbers or points and lines 'really are'. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
| A reaction: At least that leaves the field open for philosophers, because we do care what things really are. So should everybody else, but there is no persuading some people. |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| Full Idea: Intuitionism in philosophy of mathematics rejects set-theoretic foundations. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3 n33) |
| 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'! |