33 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. |
| 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. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |
| 20712 | God is 'eternal' either by being non-temporal, or by enduring forever [Davies,B] |
| Full Idea: Saying 'God is eternal' means either that God is non-temporal or timeless, or that God has no beginning and no end. The first ('classical') view is found in Anselm, Augustine, Boethius, Aquinas, Calvin and Descartes. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 8 'Meaning') | |
| A reaction: A God who is outside of time but performs actions is a bit of a puzzle. It seems that Augustine started the idea of a timeless God. |
| 20701 | Can God be good, if he has not maximised goodness? [Davies,B] |
| Full Idea: We may wonder whether God can be good since he has not produced more moral goodness than he has. We may wonder whether God is guilty by neglect. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 3 'Freedom') | |
| A reaction: The orthodox response is that we cannot possibly know what the maximum of moral goodness would look like, so we can't make this judgement. Atheists say that God fails by human standards, which are not particularly high. |
| 20702 | The goodness of God may be a higher form than the goodness of moral agents [Davies,B] |
| Full Idea: If we can know that God exists and if God's goodness is not moral goodness, then moral goodness is not the highest form of goodness we know. There is the goodness of God to be reckoned with. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 3 'Goodness') | |
| A reaction: This idea is to counter the charge that God fails to meet human standards for an ideal moral agent. But it sounds hand-wavy, since we presumably cannot comprehend the sort of goodness that is postulated here. |
| 20703 | How could God have obligations? What law could possibly impose them? [Davies,B] |
| Full Idea: We have good reason for resisting the suggestion that God has any duties or obligations. …What can oblige God in relation to his creatures? Could there be a law saying God has such obligations? Where does such a law come from? | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 3 'Goodness') | |
| A reaction: Plato can answer this question. Greek gods are not so supreme that nothing could put them under an obligation, but 'God' has to be supreme in every respect. |
| 20694 | 'Natural theology' aims to prove God to anyone (not just believers) by reason or argument [Davies,B] |
| Full Idea: 'Natural theology' is the attempt to show that belief in God's existence can be defended with reference to reason or argument which ought to be acceptable to anyone, not simply to those who believe in God's existence. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 1 'Other') | |
| A reaction: I assume by 'reason or argument' he primarily means evidence (plus the ontological argument). He cites Karl Barth as objecting to the assumption of natural theology (preferring revelation). Presumably Kierkegaard offers a rival view too. |
| 20706 | A distinct cause of the universe can't be material (which would be part of the universe) [Davies,B] |
| Full Idea: If the universe was caused to come into being, it presumably could not have been caused to do so by anything material. For a material object would be part of the universe, and we are now asking for a cause distinct from the universe. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 5 'God') | |
| A reaction: We're out of our depth here. We only have two modes of existence to offer, material and spiritual, and 'spiritual' means little more than non-material. |
| 20707 | The universe exhibits design either in its sense of purpose, or in its regularity [Davies,B] |
| Full Idea: The design argument offers two lines: the first states that the universe displays design in the sense of purpose; the second that it displays design in the sense of regularity. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 6 'Versions') | |
| A reaction: I would have thought that you would infer the purpose from the regularity. How could you see purpose in a totally chaotic universe? |
| 20708 | If God is an orderly being, he cannot be the explanation of order [Davies,B] |
| Full Idea: If God is an instance of something orderly, how can he serve to account for the order of orderly things? | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 6 'b Has') | |
| A reaction: You can at least explain the tidiness of a house by the tidiness of its owner, but obviously that won't explain the phenomenon of tidiness. |
| 20710 | Maybe an abnormal state of mind is needed to experience God? [Davies,B] |
| Full Idea: Might it not be possible that experience of God requires an unusual state or psychological abnormality, just as an aerial view of Paris requires that one be in the unusual state of being abnormally elevated? | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 7 'Are the') | |
| A reaction: That would make sense if it were analogous to great mathematical or musical ability, but it sounds more like ouija boards in darkened rooms. Talent has a wonderful output, but people in mystical states don't return with proofs. |
| 20711 | A believer can experience the world as infused with God [Davies,B] |
| Full Idea: Maybe someone who believes in God can be regarded as experiencing everything as something behind which God lies. Believers see the world as a world in which God is present. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 7 'Experiencing') | |
| A reaction: [Attributed to John Hick] This would count as supporting evidence for God, perhaps, if seeing reality as infused with God produces a consistent and plausible picture. But seeing reality as infused with other things might pass the same test. |
| 20709 | The experiences of God are inconsistent, not universal, and untestable [Davies,B] |
| Full Idea: A proclaimed experience of God must be rejected because a) there is no agreed test that it is such an experience, b) some people experience God's absence, and c) there is no uniformity of testimony about the experience. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 7 'Objections') | |
| A reaction: [compressed] I'm not sure that absence of an experience is experience of an absence. Compare it with experiencing the greatness of Beethoven's Ninth. |
| 20697 | One does not need a full understanding of God in order to speak of God [Davies,B] |
| Full Idea: In order to speak meaningfully about God, it is not necessary that one should understand exactly the import of one's statements about him. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 2 'Sayng') | |
| A reaction: Perfectly reasonable. To insist that all discussion of a thing requires exact understanding of the thing is ridiculous. Equally, though, to discuss God while denying all understanding of God is just as ridiculous. |
| 20699 | Paradise would not contain some virtues, such as courage [Davies,B] |
| Full Idea: There are virtues (such as courage) that would not be present in a paradise. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 3 'Evil') | |
| A reaction: Part of a suggestion that morality would be entirely inapplicable in paradise, and so we need dangers etc in the world. |