44 ideas
| 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. |
| 12723 | The most primitive thing in substances is force, which leads to their actions and dispositions [Leibniz] |
| Full Idea: Since everything that one conceives in substances reduces to their actions and passions and to the dispositions that they have for this effect, I don't see how one can find there anything more primitive than the principle of all of this, which is force. | |
| From: Gottfried Leibniz (Letters to Jacques Lenfant [1693], 1693.11.25), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 4 | |
| A reaction: This is an attempt to connect Aristotelian essentialism with the notion of force in the new physics, and strikes me as an improvement on the original, and as good a basis for metaphysics as any I have heard of. |
| 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. |
| 7222 | It is a crime for someone with a violent disposition to get drunk [Mill] |
| Full Idea: The making himself drunk, in a person whom drunkenness excites to do harm to others, is a crime against others. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This principle (based on knowing your own dispositions) is a very good account of the ethics drunkenness. We have a moral duty to know and remember our own dispositions. Violent people should avoid arguments as well as alcohol. |
| 7214 | Ethics rests on utility, which is the permanent progressive interests of people [Mill] |
| Full Idea: I regard utility as the ultimate appeal on all ethical questions; but it must be utility in the largest sense, grounded on the permanent interests of a man as a progressive being. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: Mill, writing in praise of personal liberty, is desperate to introduce a paternalistic element into his politics, and the 'maximisation of happiness' will justify such paternalism, while his basic liberal principle (Idea 7211) won't. Mill's Dilemma. |
| 7212 | Individuals have sovereignty over their own bodies and minds [Mill] |
| Full Idea: Over himself, over his own body and mind, the individual is sovereign. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: If I should not even think about evil deeds, then neither should you. I would prevent you if I could. I would prevent you from drinking yourself to death, if I could. It is just that intrusions into private lives leads to greater trouble. |
| 7210 | The will of the people is that of the largest or most active part of the people [Mill] |
| Full Idea: The will of the people practically means the will of the most numerous or the most active part of the people. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: Hence the nicely coined modern phrase 'the silent majority', on whose behalf certain politicians, usually conservative, offer to speak. It is unlikely that the silent majority are actually deeply opposed to the views of the very active part. |
| 7227 | It is evil to give a government any more power than is necessary [Mill] |
| Full Idea: Government interference should be restricted because of the great evil of adding unnecessarily to its power. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This would need justification, because it might be replied that individuals should not have unnecessary power either. The main problem is that governments have armies, police and money. |
| 7228 | Individuals often do things better than governments [Mill] |
| Full Idea: Government power should be restricted because things are often done better by individuals. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This contains some truth, but it is obvious that innumerable things can be done better by governments, and also (and more importantly) that innumerable other good things might be done by governments which individuals can't be bothered to do. |
| 7230 | Aim for the maximum dissemination of power consistent with efficiency [Mill] |
| Full Idea: The safest practical ideal is to aim for the greatest dissemination of power consistent with efficiency. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This is a very nice principle, which I would think desirable within an institution as well as on the scale of the state. I am becoming a fan of Mill's politics. I still say that freedom is an overrated virtue, so efficiency must be underrated. |
| 20515 | Maximise happiness by an area of strict privacy, and an area of utilitarian interventions [Mill, by Wolff,J] |
| Full Idea: For Mill the greatest happiness will be achieved by giving people a private sphere of interests where no intervention is permitted, while allowing a public sphere where intervention is possible, but only on utilitarian grounds. | |
| From: report of John Stuart Mill (On Liberty [1857]) by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 4 'Liberty' | |
| A reaction: This is probably standard liberal practice nowadays. Freely consenting adult sexual activity is agreed to be wholly private. At least some lip-service is paid to increasing happiness when government intervenes. |
| 7229 | People who transact their own business will also have the initiative to control their government [Mill] |
| Full Idea: A people accustomed to transacting their own business is certain to be free; it will never let itself be enslaved by any man or body of men because these are able to seize and pull the reins of the central administration. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: He makes reference to Americans. This is an important idea, because it shows that democratic control is not just a matter of elections (which can be abolished or suborned), but is also a characteristic of a certain way of life. |
| 7211 | Prevention of harm to others is the only justification for exercising power over people [Mill] |
| Full Idea: The only purpose for which power can be rightfully exercised over any member of a civilised community, against his will, is to prevent harm to others; his own good, either physical or moral, is not a sufficient warrant. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: This is the key idea in Mill's liberalism, though he goes on to offer some qualifications of this absolute prohibition. I don't disagree with this principle, but there may be a lot more indirect harm than we realise (eg. in allowing liberal sex or drugs). |
| 7231 | The worth of a State, in the long run, is the worth of the individuals composing it [Mill] |
| Full Idea: The worth of a State, in the long run, is the worth of the individuals composing it. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This is a key idea of liberalism, opposed to any idea that we should abandon our own value to that of our state. I agree, but communitarians can subscribe to this too, while disagreeing that maximum freedom is the strategy to follow. |
| 7217 | The main argument for freedom is that interference with it is usually misguided [Mill] |
| Full Idea: The strongest of all the arguments against the interference of the public with purely personal conduct is that, when it does interfere, the odds are that it interferes wrongly, and in the wrong place. | |
| From: John Stuart Mill (On Liberty [1857], Ch.4) | |
| A reaction: This is also a well known objection to capital punishment. Generalised, well established, legal interferences are perhaps more likely to get it right than ad hoc decisions about individuals by individual officials. |
| 7213 | Liberty arises at the point where people can freely and equally discuss things [Mill] |
| Full Idea: Liberty, as a principle, has no application to any state of things anterior to the time when mankind have become capable of being improved by free and equal discussion. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: There is a Victorian (and Enlightenment) optimism here which a glimpse of the freedoms of the early twenty-first century might dampen. I doubt if Mill expected British tabloid newspapers, or porn on cable TV. Education and freedom connect. |
| 20517 | Utilitarianism values liberty, but guides us on which ones we should have or not have [Mill, by Wolff,J] |
| Full Idea: Utilitarianism provides an account of what liberties we should and should not have. Mill argues we should be free to compete in trade, but not to use another's property without consent. Thus he sets limits to liberty, while paying it great respect. | |
| From: report of John Stuart Mill (On Liberty [1857]) by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 4 'Intrinsic' |
| 20516 | Mill defends freedom as increasing happiness, but maybe it is an intrinsic good [Wolff,J on Mill] |
| Full Idea: Mill has presented liberty as instrumentally valuable, as a way of achieving the greatest possible happiness in society. But perhaps he should have argued that liberty is an intrinsic good, good in itself. | |
| From: comment on John Stuart Mill (On Liberty [1857]) by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 4 'Intrinsic' | |
| A reaction: If freedom is intrinsically good, does this leave us (as Wolff warned earlier) unable to defend its value? Freedom isn't an intrinsic good for infants, so why should it be so for adults? Good because it brings happiness, or fulfils our nature? |
| 7215 | True freedom is pursuing our own good, while not impeding others [Mill] |
| Full Idea: The only freedom which deserves the name, is that of pursuing our own good in our own way, so long as we do not attempt to deprive others of theirs, or impede their efforts to obtain it. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: This principle will probably lead up a Prisoner's Dilemma cul-de-sac. The only freedom which deserves the name is the collective agreed freedom of a whole community to live well, when citizens volunteer to restrict their individual freedoms. |
| 7218 | Individuals are not accountable for actions which only concern themselves [Mill] |
| Full Idea: My first maxim is that the individual is not accountable to society for his actions, in so far as these concern the interests of no person but himself. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This is a key idea of liberalism, and one which communitarians have doubts about (because it is almost impossible to perform an action which is of no interest, in the short or long term, to others). I share these doubts. |
| 7221 | Blocking entry to an unsafe bridge does not infringe liberty, since no one wants unsafe bridges [Mill] |
| Full Idea: An official could turn a person back from an unsafe bridge without infringeing their liberty; for liberty consists in doing what one desires, and he does not desire to fall into the river. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: Seems fair enough, but it justifies paternalist interference. The tricky one is where the official and the citizen disagree over what the citizen 'truly' desires. Asking people may involve too much time, but it could also involve too much effort. |
| 7223 | Pimping and running a gambling-house are on the border between toleration and restraint [Mill] |
| Full Idea: A person being free to be a pimp, or to keep a gambling-house, lies on the exact boundary line between two principles, of toleration and of restraint. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: Nothing illuminates a philosopher's principles more than for them to specify cases that lie on their borderlines. Both professions seem, unfortunately, to lead people into worse activities, such as violent bullying, or theft. Tricky.. |
| 7220 | Restraint for its own sake is an evil [Mill] |
| Full Idea: All restraint, qua restraint, is an evil. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: The ultimate justification for this is (presumably) utilitarian, but that would mean that there was nothing wrong with restraint if the person did not mind, or was not aware of the restraint. What is intrinsically wrong with restraint? |
| 7219 | Society can punish actions which it believes to be prejudicial to others [Mill] |
| Full Idea: My second maxim is that for actions that are prejudicial to the interests of others, the individual is accountable, and subject to social or legal punishment, if society believes that this is requisite for its protection. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: (wording compressed). The trouble with this would seem to be the possible disagreement between the individual and the society over whether the actions actually are prejudicial to others. It would justify a conservative society in being repressive. |
| 7226 | Benefits performed by individuals, not by government, help also to educate them [Mill] |
| Full Idea: It is often desirable that beneficial things should be done by individuals, rather than by the government, as a means to their own mental education. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This raises the important danger, which even those on the political left must acknowledge, of the 'nanny state'. It offers a nicely paternalistic, and even patronising reason for giving people freedom, just as a parent might to a child. |
| 7224 | We need individual opinions and conduct, and State education is a means to prevent that [Mill] |
| Full Idea: Individuality of character, and diversity in opinions and modes of conduct, involves diversity of education; a general State education is a mere contrivance for moulding people to be exactly like one another. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This strikes me as being particularly true with the advent in Britain of the National Curriculum in the early 1990s. However, if there is a pressure towards conformity in state education, private education is dominated by class and money. |
| 7225 | It is a crime to create a being who lacks the ordinary chances of a desirable existence [Mill] |
| Full Idea: To bestow a life on someone which may be either a curse or a blessing, unless the being on whom it is to be bestowed will have at least the ordinary chances of a desirable existence, is a crime against that being. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This is the standard utilitarian attitude to engendering people. I think I have to agree. It is no argument against this to say that we value people with poor life prospects, once they have arrived. Altruism towards children may disguise selfish parents. |
| 7216 | The ethics of the Gospel has been supplemented by barbarous Old Testament values [Mill] |
| Full Idea: To extract from the Gospel a body of ethical doctrine, has never been possible withouth eking it out from the Old Testament, that is, from a system elaborate indeed, but in many respects barbarous, and intended only for a barbarous people. | |
| From: John Stuart Mill (On Liberty [1857], Ch.2) | |
| A reaction: 'Barbarous' has a quaint Victorian ring to it, but his point is that the surviving teachings of Jesus are very thin and generalised. Christians would do better to expand their implications, than to borrow from the Old Testament. |