61 ideas
| 125 | Is a gifted philosopher unmanly if he avoids the strife of the communal world? [Plato] |
| Full Idea: Callicles: Even a naturally gifted philosopher isn't going to develop into a real man, because he's avoiding the heart of his community and the thick of the agora. | |
| From: Plato (Gorgias [c.387 BCE], 485d) | |
| A reaction: A serious charge against philosophy. An attraction of the subject is its purity, its necessity, its timelessness, and in some ways these are just nicer and easier and more understandable than the hard mess of real life. But understanding has to be good. |
| 1848 | We are coerced into assent to a truth by reason's violence [Aquinas] |
| Full Idea: We are coerced into assent to a truth by reason's violence. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.10) |
| 1858 | The mind is compelled by necessary truths, but not by contingent truths [Aquinas] |
| Full Idea: Mind is compelled by necessary truths that can't be regarded as false, but not by contingent ones that might be false. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 12) |
| 1654 | In "Gorgias" Socrates is confident that his 'elenchus' will decide moral truth [Vlastos on Plato] |
| Full Idea: In the 'Gorgias' Socrates is still supremely confident that the elenchus is the final arbiter of moral truth. | |
| From: comment on Plato (Gorgias [c.387 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.117 |
| 4321 | We should test one another, by asking and answering questions [Plato] |
| Full Idea: Test me, and let yourself be tested as well, by asking and answering questions. | |
| From: Plato (Gorgias [c.387 BCE], 462a) | |
| A reaction: The idea must be to avoid wild speculation, by continually filtering ideas through rival critical intelligences. The best philosophical method ever devised. |
| 1852 | For the mind Good is one truth among many, and Truth is one good among many [Aquinas] |
| Full Idea: Good itself as taken in by mind is one truth among others, and truth itself as goal of mind's activity is one good among others. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) |
| 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
| Full Idea: While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: [The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc. |
| 10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
| Full Idea: In standard ZFC ('Zermelo-Fraenkel with Choice') set theory we deal merely with pure sets, not with additional urelements. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: The 'urelements' would the actual objects that are members of the sets, be they physical or abstract. This idea is crucial to understanding philosophy of mathematics, and especially logicism. Must the sets exist, just as the urelements do? |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| Full Idea: In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types. |
| 10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
| Full Idea: 'Analysis' is the theory of the real numbers. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: 'Analysis' began with the infinitesimal calculus, which later built on the concept of 'limit'. A continuum of numbers seems to be required to make that work. |
| 10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
| Full Idea: The difficulties for a nominalistic mereological approach to arithmetic is that an infinity of physical objects are needed (space-time points? strokes?), and it must define functions, such as 'successor'. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: Many ontologically austere accounts of arithmetic are faced with the problem of infinity. The obvious non-platonist response seems to be a modal or if-then approach. To postulate infinite abstract or physical entities so that we can add 3 and 2 is mad. |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| Full Idea: A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'! |
| 10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
| Full Idea: The merits of basing an account of mathematics on set theory are that it allows for a comprehensive unified treatment of many otherwise separate branches of mathematics, and that all assumption, including existence, are explicit in the axioms. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I am forming the impression that set-theory provides one rather good model (maybe the best available) for mathematics, but that doesn't mean that mathematics is set-theory. The best map of a landscape isn't a landscape. |
| 10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
| Full Idea: Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent. |
| 10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
| Full Idea: Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight? |
| 10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
| Full Idea: The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6) | |
| A reaction: This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start. |
| 10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
| Full Idea: According to 'pattern' structuralism, what we study are not the various particular isomorphic models of arithmetic, but something in addition to them: a corresponding pattern. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §7) | |
| A reaction: Put like that, we have to feel a temptation to wield Ockham's Razor. It's bad enough trying to give the structure of all the isomorphic models, without seeking an even more abstract account of underlying patterns. But patterns connect to minds.. |
| 10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
| Full Idea: There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9) | |
| A reaction: I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind? |
| 10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
| Full Idea: Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3) | |
| A reaction: [very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations. |
| 10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
| Full Idea: It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: [compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent. |
| 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
| Full Idea: Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator. |
| 10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
| Full Idea: Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra. |
| 10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
| Full Idea: Relativist Structuralism must first assume the existence of an infinite set, otherwise there would be no model to pick, and arithmetical terms would have no reference. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: See Idea 10169 for Relativist Structuralism. They point out that ZFC has an Axiom of Infinity. |
| 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
| Full Idea: One way for a nominalist to reject appeal to all abstract objects, including sets, is to only appeal to nominalistically acceptable objects, including mereological sums. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I'm suddenly thinking that this looks very interesting and might be the way to go. The issue seems to be whether mereological sums should be seen as constrained by nature, or whether they are unrestricted. See Mereology in Ontology...|Intrinsic Identity. |
| 1860 | Knowledge may be based on senses, but we needn't sense all our knowledge [Aquinas] |
| Full Idea: All our knowledge comes through our senses, but that doesn't mean that everything we know is sensed. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 18) |
| 1855 | If we saw something as totally and utterly good, we would be compelled to will it [Aquinas] |
| Full Idea: Something apprehended to be good and appropriate in any and every circumstance that could be thought of would compel us to will it. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) |
| 1856 | Nothing can be willed except what is good, but good is very varied, and so choices are unpredictable [Aquinas] |
| Full Idea: Nothing can be willed except good, but many and various things are good, and you can't conclude from this that wills are compelled to choose this or that one. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 05) |
| 1862 | However habituated you are, given time to ponder you can go against a habit [Aquinas] |
| Full Idea: However habituated you are, given time to ponder you can go against a habit. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 24) |
| 1849 | Since will is a reasoning power, it can entertain opposites, so it is not compelled to embrace one of them [Aquinas] |
| Full Idea: Reasoning powers can entertain opposite objects. Now will is a reasoning power, so will can entertain opposites and is not compelled to embrace one of them. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.x2) |
| 1861 | The will is not compelled to move, even if pleasant things are set before it [Aquinas] |
| Full Idea: The will is not compelled to move, for it doesn't have to want the pleasant things set before it. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 21) |
| 1853 | Because the will moves by examining alternatives, it doesn't compel itself to will [Aquinas] |
| Full Idea: Because will moves itself by deliberation - a kind of investigation which doesn't prove some one way correct but examines the alternatives - will doesn't compel itself to will. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) |
| 1854 | We must admit that when the will is not willing something, the first movement to will must come from outside the will [Aquinas] |
| Full Idea: We are forced to admit that, in any will that is not always willing, the very first movement to will must come from outside, stimulating the will to start willing. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) | |
| A reaction: cf Nietzsche |
| 116 | Rhetoric is irrational about its means and its ends [Plato] |
| Full Idea: Rhetoric is a knack, because it lacks rational understanding of its object or what it dispenses (and can't explain the reason anything happens). | |
| From: Plato (Gorgias [c.387 BCE], 465a) | |
| A reaction: If there are cunning people who have the wrong sort of intelligence for morality, there must be cunning users of rhetoric who know exactly what they are doing. |
| 114 | Rhetoric can produce conviction, but not educate people about right and wrong [Plato] |
| Full Idea: Rhetoric is an agent of the kind of persuasion which is designed to produce conviction, but not to educate people about right and wrong. | |
| From: Plato (Gorgias [c.387 BCE], 455a) | |
| A reaction: Surely there must be good rhetoric (or at least it is an open question)? |
| 135 | All activity aims at the good [Plato] |
| Full Idea: All activity aims at the good. | |
| From: Plato (Gorgias [c.387 BCE], 499e) | |
| A reaction: He includes non-conscious activity, so this is the 'teleological' view of nature, which seems a bit optimistic to the modern mind. |
| 1846 | The will can only want what it thinks is good [Aquinas] |
| Full Idea: Will's object is what is good, and so it cannot will anything but what is good. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.06) |
| 1847 | The will must aim at happiness, but can choose the means [Aquinas] |
| Full Idea: The will is compelled by its ultimate goal (to achieve happiness), but not by the means to achieve it. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.07) |
| 1857 | We don't have to will even perfect good, because we can choose not to think of it [Aquinas] |
| Full Idea: The will can avoid actually willing something by avoiding thinking of it, since mental activity is subject to will. In this respect we aren't compelled to will even total happiness, which is the only perfect good. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 07) |
| 1850 | Without free will not only is ethical action meaningless, but also planning, commanding, praising and blaming [Aquinas] |
| Full Idea: If we are not free to will in any way, but are compelled, everything that makes up ethics vanishes: pondering action, exhorting, commanding, punishing, praising, condemning. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) | |
| A reaction: If doesn't require some magical 'free will' to avoid compulsions. All that is needed is freedom to enact your own willing, rather than someone else's. |
| 122 | Moral rules are made by the weak members of humanity [Plato] |
| Full Idea: Callicles: It's the weaklings who constitute the majority of the human race who make the rules. | |
| From: Plato (Gorgias [c.387 BCE], 483b) | |
| A reaction: An aristocrat bemoans democracy. Presumably the qualification for being a 'weakling' is shortage of money. How strong are the scions of the aristocrats? |
| 139 | A good person is bound to act well, and this brings happiness [Plato] |
| Full Idea: A good person is bound to do whatever he does well and successfully, and success brings fulfilment and happiness. | |
| From: Plato (Gorgias [c.387 BCE], 507c) | |
| A reaction: Not how we would see it, I guess, but this is the Greek idea that a good person is one who functions well. Anyone who functions well is probably having a good time. |
| 128 | Is it natural to simply indulge our selfish desires? [Plato] |
| Full Idea: Callicles: Nature says the only authentic way of life is to do nothing to hinder or restrain the expansion of one's desires. | |
| From: Plato (Gorgias [c.387 BCE], 491e) | |
| A reaction: Sounds like the natural desires of a young single man. Parents and spouses have natural desires that focus on other people's desires. |
| 4322 | In slaking our thirst the goodness of the action and the pleasure are clearly separate [Plato] |
| Full Idea: When we drink to quench thirst, we lose the distress of the thirst and the pleasure of drinking at the same moment, but one loss is good and the other bad, so the pleasure and the goodness must be separate. | |
| From: Plato (Gorgias [c.387 BCE], 497d) | |
| A reaction: This is open to the objection that the good of slaking one's thirst is a long-term pleasure, where the drinking is short-term, so pleasure is still the good. |
| 136 | Good should be the aim of pleasant activity, not the other way round [Plato] |
| Full Idea: Good should be the goal of pleasant activities, rather than pleasure being the goal of good activities. | |
| From: Plato (Gorgias [c.387 BCE], 500a) | |
| A reaction: Nice. Not far off what Aristotle says on the topic. So what activities should we seek out? Narrow the pleasures down to the good ones, or narrow the good ones down to the pleasurable? |
| 1851 | Good applies to goals, just as truth applies to ideas in the mind [Aquinas] |
| Full Idea: Good applies to all goals, just as truth applies to all forms mind takes in. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) | |
| A reaction: In danger of being tautological, if good is understood as no more than the goal of actions. It seems perfectly possibly to pursue a wicked end, and perhaps feel guilty about it. |
| 134 | Good and bad people seem to experience equal amounts of pleasure and pain [Plato] |
| Full Idea: There is little to tell between good and bad people (e.g. cowards) in terms of how much pleasure and distress they experience. | |
| From: Plato (Gorgias [c.387 BCE], 498c) | |
| A reaction: A very perceptive remark. If the good are people with empathy for others, then they may suffer more distress than the insensitive wicked. |
| 132 | If happiness is the satisfaction of desires, then a life of scratching itches should be happiness [Plato] |
| Full Idea: Socrates: I want to ask whether a lifetime spent scratching, itching and scratching, no end of scratching, is also a life of happiness. | |
| From: Plato (Gorgias [c.387 BCE], 494c) | |
| A reaction: There are plenty of people who think 'fun' is the main aim of life, and who fit what Socrates is referring to. We don't admire such a life, but not many people can be admired. |
| 4319 | In a fool's mind desire is like a leaky jar, insatiable in its desires, and order and contentment are better [Plato] |
| Full Idea: In a fool's mind desire is a leaky jar, …which is an analogy for the mind's insatiability, showing we should prefer an orderly life, in which one is content with whatever is to hand, to a self-indulgent life of insatiable desire. | |
| From: Plato (Gorgias [c.387 BCE], 493b) | |
| A reaction: This points to an interesting paradox, that pleasure requires the misery of desire. And yet absence of desire is like death. An Aristotelian mean, of living according to nature, seems the escape route. |
| 130 | Is the happiest state one of sensual, self-indulgent freedom? [Plato] |
| Full Idea: Callicles: If a person has the means to live a life of sensual, self-indulgent freedom, there's no better or happier state of existence. | |
| From: Plato (Gorgias [c.387 BCE], 492c) |
| 120 | Should we avoid evil because it will bring us bad consequences? [Plato] |
| Full Idea: Socrates: We should avoid doing wrong because of all the bad consequences it will bring us. | |
| From: Plato (Gorgias [c.387 BCE], 480a) |
| 118 | I would rather be a victim of crime than a criminal [Plato] |
| Full Idea: Socrates: If I had to choose between doing wrong and having wrong done to me, I'd prefer the latter to the former. | |
| From: Plato (Gorgias [c.387 BCE], 469c) | |
| A reaction: cf Democritus 68B45 |
| 140 | Self-indulgent desire makes friendship impossible, because it makes a person incapable of co-operation [Plato] |
| Full Idea: Self-indulgent desire makes a person incapable of co-operation, which is a prerequisite of friendship. | |
| From: Plato (Gorgias [c.387 BCE], 507e) |
| 131 | If absence of desire is happiness, then nothing is happier than a stone or a corpse [Plato] |
| Full Idea: Callicles: If people who need nothing are happy, there would be nothing happier than a stone or a corpse. | |
| From: Plato (Gorgias [c.387 BCE], 492e) | |
| A reaction: We aren't really supposed to approve of Callicles, but to me this is a splendidly crushing western response to many of the ideals found in eastern philosophy. |
| 119 | A criminal is worse off if he avoids punishment [Plato] |
| Full Idea: Socrates: A criminal is worse off if he doesn't pay the penalty, and continues to do wrong without getting punished. | |
| From: Plato (Gorgias [c.387 BCE], 472e) |
| 129 | Do most people praise self-discipline and justice because they are too timid to gain their own pleasure? [Plato] |
| Full Idea: Callicles: Why do most people praise self-discipline and justice? Because their own timidity makes them incapable of satisfying their pleasures. | |
| From: Plato (Gorgias [c.387 BCE], 492a) |
| 4320 | The popular view is that health is first, good looks second, and honest wealth third [Plato] |
| Full Idea: I'm sure you know the list of human advantages in the party song: 'The very best is health, Second good looks, and third honest wealth'. | |
| From: Plato (Gorgias [c.387 BCE], 451e) | |
| A reaction: This invites the obvious question of why anyone wants these three things, with the implied answer of 'pleasure'. But we might want them even if we couldn't use them, implying pluralism. |
| 137 | As with other things, a good state is organised and orderly [Plato] |
| Full Idea: As in every case (an artefact, a body, a mind, a creature), a good state is an organised and orderly state. | |
| From: Plato (Gorgias [c.387 BCE], 506e) |
| 141 | A good citizen won't be passive, but will redirect the needs of the state [Plato] |
| Full Idea: The only responsibility of a good member of a community is altering the community's needs rather than going along with them. | |
| From: Plato (Gorgias [c.387 BCE], 517b) |
| 123 | Do most people like equality because they are second-rate? [Plato] |
| Full Idea: Callicles: It's because most people are second-rate that they are happy for things to be distributed equally. | |
| From: Plato (Gorgias [c.387 BCE], 483c) |
| 124 | Does nature imply that it is right for better people to have greater benefits? [Plato] |
| Full Idea: Callicles: We only have to look at nature to find evidence that it is right for better to have a greater share than worse. | |
| From: Plato (Gorgias [c.387 BCE], 483d) |
| 1859 | Even a sufficient cause doesn't compel its effect, because interference could interrupt the process [Aquinas] |
| Full Idea: Even a sufficient cause doesn't always compel its effect, since it can sometimes be interfered with so that its effect doesn't happen | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 15) |