43 ideas
| 354 | Wisdom makes virtue and true goodness possible [Plato] |
| Full Idea: It is wisdom that makes possible courage and self-control and integrity or, in a word, true goodness. | |
| From: Plato (Phaedo [c.382 BCE], 069b) | |
| A reaction: Aristotle also says that prudence (phronesis) makes virtue possible. |
| 370 | Philosophy is a purification of the soul ready for the afterlife [Plato] |
| Full Idea: Souls which have purified themselves sufficiently by philosophy will live after death without bodies. | |
| From: Plato (Phaedo [c.382 BCE], 114b) | |
| A reaction: Purifying it of what? Error, or desire, or narrow-mindedness, or the physical? |
| 350 | In investigation the body leads us astray, but the soul gets a clear view of the facts [Plato] |
| Full Idea: When philosophers investigate with the help of the body they are led astray, but through reflection the soul gets a clear view of the facts. | |
| From: Plato (Phaedo [c.382 BCE], 065c) |
| 362 | The greatest misfortune for a person is to develop a dislike for argument [Plato] |
| Full Idea: No greater misfortune could happen to anyone than developing a dislike for argument. | |
| From: Plato (Phaedo [c.382 BCE], 089d) |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 13155 | If you add one to one, which one becomes two, or do they both become two? [Plato] |
| Full Idea: I cannot convince myself that when you add one to one either the first or the second one becomes two, or they both become two by the addition of the one to the other, ...or that when you divide one, the cause of becoming two is now the division. | |
| From: Plato (Phaedo [c.382 BCE], 097d) | |
| A reaction: Lovely questions, all leading to the conclusion that two consists of partaking in duality, to which you can come by several different routes. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 21347 | If Simmias is taller than Socrates, that isn't a feature that is just in Simmias [Plato] |
| Full Idea: When you say Simmias is taller than Socrates but shorter than Phaedo, so you mean there is in Simmias both tallness and shortness? - I do. ...But surely he is not taller than Socrates because he is Simmias but because of the tallness he happens to have? | |
| From: Plato (Phaedo [c.382 BCE], 102b-c) | |
| A reaction: He adds that both people must be cited. This appears to be what we now call a rejection relative height as an 'internal' relation, which is it would presumably be if it was a feature of one or of both men. |
| 360 | We must have a prior knowledge of equality, if we see 'equal' things and realise they fall short of it [Plato] |
| Full Idea: We must have some previous knowledge of equality, before the time when we saw equal things, but realised that they fell short of it. | |
| From: Plato (Phaedo [c.382 BCE], 075a) |
| 1 | There is only one source for all beauty [Plato] |
| Full Idea: If anything is beautiful other than beauty itself, it is beautiful for no other reason but because it participates in that beautiful. | |
| From: Plato (Phaedo [c.382 BCE], 100c) | |
| A reaction: The Greek word will be 'kalon' (beautiful, fine, noble). Like Aristotle, I find it baffling that such diversity could have a single source. Beautiful things have diverse aims. |
| 368 | Other things are named after the Forms because they participate in them [Plato] |
| Full Idea: The reason why other things are called after the forms is that they participate in the forms. | |
| From: Plato (Phaedo [c.382 BCE], 102a) |
| 16516 | The ship which Theseus took to Crete is now sent to Delos crowned with flowers [Plato] |
| Full Idea: The day before the trial the prow of the ship that the Athenians send to Delos had been crowned with garlands. - Which ship is that? - It is the ship in which, the Athenians say, Theseus once sailed to Crete, taking the victims. | |
| From: Plato (Phaedo [c.382 BCE], 058a) | |
| A reaction: Not philosophical, but this is the Ship of Theseus whose subsequent identity, Plutarch tells us, became a matter of dispute. |
| 357 | People are obviously recollecting when they react to a geometrical diagram [Plato] |
| Full Idea: The way in which people react to a geometrical diagram or anything like that is unmistakable proof of the theory of recollection. | |
| From: Plato (Phaedo [c.382 BCE], 073a) |
| 359 | If we feel the inadequacy of a resemblance, we must recollect the original [Plato] |
| Full Idea: If someone sees a resemblance, but feels that it falls far short of the original, they must therefore have a recollection of the original. | |
| From: Plato (Phaedo [c.382 BCE], 074e) |
| 9343 | To achieve pure knowledge, we must get rid of the body and contemplate things with the soul [Plato] |
| Full Idea: We are convinced that if we are ever to have pure knowledge of anything, we must get rid of the body and contemplate things by themselves with the soul by itself. | |
| From: Plato (Phaedo [c.382 BCE], 066c) | |
| A reaction: This seems to be the original ideal which motivates the devotion to a priori knowledge - that it will lead to a 'pure' knowledge, which in Plato's case will be eternal and necessary knowledge, like taking lessons from the gods. Wrong. |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 15859 | To investigate the causes of things, study what is best for them [Plato] |
| Full Idea: If one wished to know the cause of each thing, why it comes to be or perishes or exists, one had to find what was the best way for it to be, or to be acted upon, or to act. Then it befitted a man to investigate only ...what is best. | |
| From: Plato (Phaedo [c.382 BCE], 097d) | |
| A reaction: A reversal of the modern idea of 'best explanation'. Socrates is citing Anaxagoras's proposal to understand things by interpreting the workings of a supreme Mind. It is the religious version of best explanation. |
| 13154 | Do we think and experience with blood, air or fire, or could it be our brain? [Plato] |
| Full Idea: Is it with the blood that we think, or with the air or the fire that is in us? Or is it none of these, but the brain that supplies our senses of hearing and sight and smell. | |
| From: Plato (Phaedo [c.382 BCE], 097a) | |
| A reaction: In retrospect it seems surprising that such clever people hadn't worked this one out, given the evidence of anatomy, in animals and people, and given brain injuries. By the time of Galen they appear to have got the answer. |
| 364 | One soul can't be more or less of a soul than another [Plato] |
| Full Idea: Is one soul, even minutely, more or less of a soul than another? Not in the least. | |
| From: Plato (Phaedo [c.382 BCE], 093b) | |
| A reaction: This idea is attractive because unconsciousness and death seem to be abrupt procedures, and so appear to be all-or-nothing, but I would personally view extreme Alzheimer's as an erasing of the soul, though a minimum level of it seems all-or-nothing. |
| 361 | It is a mistake to think that the most violent pleasure or pain is therefore the truest reality [Plato] |
| Full Idea: When anyone's soul feels a keen pleasure or pain it cannot help supposing that whatever causes the most violent emotion is the plainest and truest reality - which it is not. | |
| From: Plato (Phaedo [c.382 BCE], 084c) | |
| A reaction: Do people think that? Most people distinguish subjective from objective. Wounded soldiers are also aware of victory or defeat. |
| 351 | War aims at the acquisition of wealth, because we are enslaved to the body [Plato] |
| Full Idea: All wars are undertaken for the acquisition of wealth, and we want this because of the body, to which we are slave. | |
| From: Plato (Phaedo [c.382 BCE], 066c) |
| 20937 | The state should produce higher civilisations for all, in tune with the economic apparatus [Gramsci] |
| Full Idea: The role of the State is always that of creating new and higher types of civilisation; of adapting the 'civilisation' and the morality of the broades popular masses to the necessities of the continuous development of the economic apparatus of production. | |
| From: Antonio Gramsci (Selections from Prison Notebooks [1971], 2 'Collective') | |
| A reaction: This makes education virtually the prime role of the state. Reminiscent of Sir John Reith's original dream, in the 1930s, for the BBC. Many marxists feel that the economy is in direct conflict with morality and civilisation. |
| 20935 | Eventually political parties lose touch with the class they represent, which is dangerous [Gramsci] |
| Full Idea: At a certain point in their lives, social classes become detached from their traditional parties. In that particular form ...the parties are no longer recognised by their class as its exopression. ...The field is then open for violent solutions. | |
| From: Antonio Gramsci (Selections from Prison Notebooks [1971], 2 'Parties') | |
| A reaction: Left wing parties pursue ideologies that don't connect with the actual current interests of the working class, and righ wing parties are taken over by rich elites who don't value safe traditonal communities. (This thought is resonant in the 2018 UK). |
| 20936 | Caesarism emerges when two forces in society are paralysed in conflict [Gramsci] |
| Full Idea: Caesarism (as the emergence of a 'heroic' personality) expresses a situation in which the forces in conflict balance each other in a catastrophic manner ...which can only terminate in their reciprocal destruction. | |
| From: Antonio Gramsci (Selections from Prison Notebooks [1971], 2 'Caesarism') | |
| A reaction: He goes on to distinguish progressive and reactionary versions of Caesarism. Gramsci's interest is in the circumstances that throw up such people. Marx had identified 'Bonapartism'. |
| 20941 | Totalitarian parties cut their members off from other cultural organisations [Gramsci] |
| Full Idea: A totalitarian party ensures that members find in that particular party all the satisfactions that they formerly found in a multiplicity of organisations. They break the threads that bind them to extraneous cultural organisms. | |
| From: Antonio Gramsci (Selections from Prison Notebooks [1971], 2 'Organisation') | |
| A reaction: British parties traditionally had a 'club house', where you could do most of your socialising. Presumably Nazis left the church, and various interest groups. |
| 20939 | What is the function of a parliament? Does it even constitute a part of the State structure? [Gramsci] |
| Full Idea: The question has to be asked: do parliaments, even in fact constitute a part of the State structure? In other words, what is the real function? | |
| From: Antonio Gramsci (Selections from Prison Notebooks [1971], 2 'Parliament') | |
| A reaction: Nice question. In the UK it is only the cabinet which has active power. Backbench MPs are usually very frustrated, especially if their party has a comfortable majority, and their vote is not precious. They are privileged lobbyists. |
| 20938 | Liberalism's weakness is its powerful rigid bureaucracy [Gramsci] |
| Full Idea: Liberalism's weakness is the bureacracy - the crystallisation of the leading personnel - which exercises power, and at a certain point it becomes a caste. | |
| From: Antonio Gramsci (Selections from Prison Notebooks [1971], 2 'Hegemony') | |
| A reaction: This sounds more like what is called 'the Establishment' in Britain, which is the hidden controllers of power, rather than the administrators (whose role is only despised by right-wingers). |
| 20940 | Perfect political equality requires economic equality [Gramsci] |
| Full Idea: The idea that complete and perfect political equality cannot exist without economic equality ...remains correct. | |
| From: Antonio Gramsci (Selections from Prison Notebooks [1971], 2 'The State') | |
| A reaction: In the west we are living in a period (2018) when the top 0.1% of the wealthy are racing away, creating huge inequality. Their wealth controls the media, and it seems unrestrainable. The belief that we live in a 'democracy' is an illusion. |
| 13156 | Fancy being unable to distinguish a cause from its necessary background conditions! [Plato] |
| Full Idea: Fancy being unable to distinguish between the cause of a thing, and the condition without which it could not be a cause. | |
| From: Plato (Phaedo [c.382 BCE], 099c) | |
| A reaction: Not as simple as he thinks. It seems fairly easy to construct a case where the immediately impacting event remains constant, and the background condition is changed. Even worse when negligence is held to be the cause. |
| 369 | If the Earth is spherical and in the centre, it is kept in place by universal symmetry, not by force [Plato] |
| Full Idea: If the earth is spherical and in the middle of the heavens, it needs neither air nor force to keep it from falling. The uniformity of heaven and equilibrium of earth are sufficient support. | |
| From: Plato (Phaedo [c.382 BCE], 108e) |
| 363 | Whether the soul pre-exists our body depends on whether it contains the ultimate standard of reality [Plato] |
| Full Idea: The theory that our soul exists even before it enters the body surely stands or falls with the soul's possession of the ultimate standard of reality. | |
| From: Plato (Phaedo [c.382 BCE], 092d) |