36 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. |
| 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. |
| 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 |
| 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. |
| 22595 | Liberty is the triumph of the individual, over both despotic government and enslaving majorities [Constant] |
| Full Idea: Lliberty is the triumph of the individual, as much over a government which seeks to rule by despotic methods, as over the masses who seek to render the minority the slave of the majority. | |
| From: Benjamin Constant (Principles of Politics [1806]), quoted by Ian Dunt - How to be a Liberal 4 | |
| A reaction: [No page given] Dunt describes Constant's book as the first really systematic account of liberalism. Very important to have rights against the majority, as well as against government. |
| 22597 | Minority rights are everyone's rights, because we all have turns in the minority [Constant] |
| Full Idea: To defend the rights of minorities is to defend the rights of all. Everyone in turn finds himself in the minority. | |
| From: Benjamin Constant (Principles of Politics [1806]), quoted by Ian Dunt - How to be a Liberal 4 | |
| A reaction: Very conformist people, who are often the most oppressive, are rarely in the minority, and are unlikely to be impressed by this idea. |