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? |
| 15845 | It seems absurd that seeing a person's limbs, the one is many, and yet the many are one [Plato] |
| Full Idea: Someone first distinguishes a person's limbs and parts and asks your agreement that all the parts are identical with that unity, then ridicules you that you have to admit one is many, and indefinitely many, and again that the many are only only one thing. | |
| From: Plato (Philebus [c.353 BCE], 14e) | |
| A reaction: This is a passing aporia, but actually seems to approach the central mystery of the metaphysics of identity. A thing can't be a 'unity' if there are not things to unify? So what sorts of 'unification' are there? |
| 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. |
| 9867 | It is absurd to define a circle, but not be able to recognise a real one [Plato] |
| Full Idea: It will be ridiculous if our student knows the definition of the circle and of the divine sphere itself, but cannot recognize the human sphere and these our circles, used in housebuilding. | |
| From: Plato (Philebus [c.353 BCE], 62a) | |
| A reaction: This is the equivalent of being able to recite numbers, but not to count objects. It also resembles Molyneux's question (to Locke), of whether recognition by one sense entails recognition by others. Nice (and a bit anti-platonist!). |
| 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. |
| 9865 | Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato] |
| Full Idea: The arithmetic of the many computes sums of unequal units, such as two armies, or two herds, ..but philosopher's arithmetic computes when it is guaranteed that none of those infinitely many units differed in the least from any of the others. | |
| From: Plato (Philebus [c.353 BCE], 56d) | |
| A reaction: But of course 'the many' are ironing out the differences too, when they say there are 'three armies'. Shocking snob, Plato. Even philosophers are interested in the difference between three armies and three platoons. |
| 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. |
| 14503 | If a mixture does not contain measure and proportion, it is corrupted and destroyed [Plato] |
| Full Idea: Any kind of mixture that does not ...possess measure or the nature of proportion will necessarily corrupt its ingredients and most of all itself. For there would be no blending in such cases but really an unconnected medley, and ruin what contains it. | |
| From: Plato (Philebus [c.353 BCE], 64d) | |
| A reaction: My guess is that Plato is thinking of the decay of living things when they die, losing the proportions of psuché, and then applying this to the unity of inanimate objects as well. One might compare Leibniz's monads. |
| 15857 | Any mixture which lacks measure and proportion doesn't even count as a mixture at all [Plato] |
| Full Idea: Any blend [mixture] which does not have measure or the nature of proportion in any way whatsoever, of necessity destroys both its ingredients and, primarily, itself. It is truly no blend at all, but a kind of unblended disaster. | |
| From: Plato (Philebus [c.353 BCE], 64e) | |
| A reaction: Obviously there can be chaotic mixtures, but I guess Plato is picking out mixtures about which we can say something |
| 4447 | If the good is one, is it unchanged when it is in particulars, and is it then separated from itself? [Plato] |
| Full Idea: If man is one, and the good is one, how are they supposed to exist? Do they stay the same even though they are found in many things at the same time, and are they then entirely separated from themselves, which seems most impossible of all? | |
| From: Plato (Philebus [c.353 BCE], 15a) | |
| A reaction: Presumably Plato anguishes over this because he thinks Forms are self-predicating (the Good is good). Big mistake. The Good fathers good particulars which resemble itself, but are diluted? |
| 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. |
| 15856 | A thing can become one or many, depending on how we talk about it [Plato] |
| Full Idea: It is through discourse that the same thing flits around, becoming one and many in all sorts of ways. | |
| From: Plato (Philebus [c.353 BCE], 15d) | |
| A reaction: This is not scepticism about wholes on Plato's part, but a reminder of an obvious fact, that in thought we can break the world up and put it back together again. It is a touchstone of the debate, though. |
| 374 | If one object is divided into its parts, someone can then say that one are many and many is one [Plato] |
| Full Idea: Someone can theoretically divide an object into constituent parts, concede that they are one object, and then claim that therefore the one is many and the many are one. | |
| From: Plato (Philebus [c.353 BCE], 14e) |
| 389 | How can you be certain about aspects of the world if they aren't constant? [Plato] |
| Full Idea: Could we attribute certainty to studying aspects of the world, such as how it arose, or acts, or is acted upon, when none of them ever was or will be constant? Of course not. | |
| From: Plato (Philebus [c.353 BCE], 59b) |
| 390 | If goodness involves moderation and proportion, then it seems to be found in beauty [Plato] |
| Full Idea: Moderation and proportion seem, in effect, to be beauty and excellence. So now this property we're looking for, goodness, has taken refuge in beauty. | |
| From: Plato (Philebus [c.353 BCE], 64e) |
| 392 | Neither intellect nor pleasure are the good, because they are not perfect and self-sufficient [Plato] |
| Full Idea: Both intellect and pleasure are completely absolved of being the good itself, since they both lack independence, that is, sufficiency and perfection. | |
| From: Plato (Philebus [c.353 BCE], 67a) | |
| A reaction: This seems to be Plato disagreeing with Socrates, who sees reason and intellect as central to morality. Presumable he means that the good should be a primitive. Why is pleasure not sufficient? |
| 391 | The good involves beauty, proportion and truth [Plato] |
| Full Idea: If we are unable to net the good in a single concept, three must capture it: namely, beauty, proportion and truth. | |
| From: Plato (Philebus [c.353 BCE], 65a) | |
| A reaction: Very interesting. More illuminating than the discussion of the Good in 'Republic'. Is a handsome and honest murderer good? Is beauty part of the nature of the good, or a hallmark of it? |
| 393 | Good first, then beauty, then reason, then knowledge, then pleasure [Plato, by PG] |
| Full Idea: Good is supreme, followed by beauty, then reason, then knowledge, then pure pleasure, then mixed pleasure. | |
| From: report of Plato (Philebus [c.353 BCE], 67a) by PG - Db (ideas) | |
| A reaction: He tells us that pure pleasures are simple pleasures. Epicurus presumably read this. No mention of truth, unless that is part of reason. Why does he value beauty so highly? |
| 385 | Some of the pleasures and pains we feel are false [Plato] |
| Full Idea: Living beings experience pleasures and pains which seem, and indeed are, false. | |
| From: Plato (Philebus [c.353 BCE], 42c) | |
| A reaction: The idea that there are 'authentic' pleasures and pains needs some investigation. Misguided anger is a false pain? Vanity is a false pleasure? |
| 387 | A small pure pleasure is much finer than a large one contaminated with pain [Plato] |
| Full Idea: A tiny little pleasure is, if uncontaminated by pain, always more pleasant, truer and finer than a large amount. | |
| From: Plato (Philebus [c.353 BCE], 53b) | |
| A reaction: More Platonic puritanism. Is a complete absence of pleasure the highest pleasure of all? I don't think I understand 'truer'. Why would a pleasure be false because it is intense? |
| 373 | Pleasure is certainly very pleasant, but it doesn't follow that all pleasures are good [Plato] |
| Full Idea: The pleasantness of pleasure is not in dispute, but where we say the majority of pleasures are bad, though some are good, you are attributing goodness to all of them. | |
| From: Plato (Philebus [c.353 BCE], 13b) | |
| A reaction: Bentham's plausible view is that the feeling of pleasure is always good, and the badness is in some other aspect of the event. Compare sadistic fantasy with sadistic action. |
| 379 | The good must be sufficient and perfect, and neither intellect nor pleasure are that [Plato] |
| Full Idea: Neither pleasure nor intellect comprises the good. If it did it would have to be sufficient and perfect. | |
| From: Plato (Philebus [c.353 BCE], 22b) | |
| A reaction: Seems sensible. I can't make sense of any vision of the good which consists of suppressing some aspect of human nature. (Hm. Our capacity for violence?) |
| 371 | Reason, memory, truth and wisdom are far better than pleasure, for those who can attain them [Plato] |
| Full Idea: My contention is that reason, intellect, memory - along with correct belief and true calculation - are far better than pleasure for all creatures capable of attaining them. | |
| From: Plato (Philebus [c.353 BCE], 11b) | |
| A reaction: Why? Is it better to understand deeply, or to act well? Can we just say there is objective good and subjective good, and they have little in common? Depressed heroes. |
| 376 | Would you prefer a life of pleasure without reason, or one of reason without pleasure? [Plato] |
| Full Idea: Try thinking about the life of pleasure without reason, and the life of reason without pleasure. | |
| From: Plato (Philebus [c.353 BCE], 20e) | |
| A reaction: I suspect that we see the two as more deeply entangled that Plato did. It would be hard to motivate reasoning if we didn't enjoy it. Pleasure without reason sound dire. |
| 382 | It is unlikely that the gods feel either pleasure or pain [Plato] |
| Full Idea: It is unlikely that the gods feel pleasure or the opposite. | |
| From: Plato (Philebus [c.353 BCE], 33b) | |
| A reaction: Compare Idea 383. |
| 381 | We feel pleasure when we approach our natural state of harmony [Plato] |
| Full Idea: When harmony is being restored, and the natural state of harmony is approached, then pleasure arises. | |
| From: Plato (Philebus [c.353 BCE], 31d) | |
| A reaction: The supreme value of harmony was important to Plato, but most of us are less convinced, I suspect. The way to achieve harmony is to avoid anything stressful. |
| 386 | Intense pleasure and pain are not felt in a good body, but in a worthless one [Plato] |
| Full Idea: Intensity of pleasure and pain is to be found not in a good state of body and soul, but in a worthless one. | |
| From: Plato (Philebus [c.353 BCE], 45e) | |
| A reaction: This just seems to be Plato's puritanism. How can you criticise someone for experience genuine intense pain? Experiencing intense pleasure is no crime, but pursuit of it might be. |
| 388 | Hedonists must say that someone in pain is bad, even if they are virtuous [Plato] |
| Full Idea: A hedonist must say that someone who happens to be feeling pain rather than pleasure is, as long as the pain lasts, a bad man, even if he is the most virtuous man in the world. | |
| From: Plato (Philebus [c.353 BCE], 55b) |
| 377 | If you lived a life of maximum pleasure, would you still be lacking anything? [Plato] |
| Full Idea: Would you, Protarchus, gladly live your whole life experiencing only the greatest pleasure? Would you think you were still lacking anything? | |
| From: Plato (Philebus [c.353 BCE], 21a) | |
| A reaction: the pleasure machine problem |
| 378 | A life of pure pleasure with no intellect is the life of a jellyfish [Plato] |
| Full Idea: A life of pure pleasure with no intellect is not the life of a human being, but the life of a jellyfish. | |
| From: Plato (Philebus [c.353 BCE], 21c) |
| 467 | A virtue is a combination of intelligence, strength and luck [Ion] |
| Full Idea: The virtue of each thing is a Triad: intelligence, strength, luck. | |
| From: Ion (fragments/reports [c.435 BCE], B1), quoted by (who?) - where? |