33 ideas
| 9764 | Inspiration and social improvement need wisdom, but not professional philosophy [Quine] |
| Full Idea: Professional philosophers have no peculiar fitness for inspirational and edifying writing, or helping to get society on an even keel (though we should do what we can). Wisdom may fulfil these crying needs: 'sophia' yes, but 'philosophia' not necessarily. | |
| From: Willard Quine (Has Philosophy Lost Contact with People? [1979], p.193) | |
| A reaction: This rather startlingly says that philosophy is unlikely to lead to wisdom, which is rather odd when it is defined as love of that very thing. Does love of horticulture lead to good gardening. I can't agree. Philosophy is the best hope of 'sophia'. |
| 9763 | For a good theory of the world, we must focus on our flabby foundational vocabulary [Quine] |
| Full Idea: Our traditional introspective notions - of meaning, idea, concept, essence, all undisciplined and undefined - afford a hopelessly flabby and unmanageable foundation for a theory of the world. Control is gained by focusing on words. | |
| From: Willard Quine (Has Philosophy Lost Contact with People? [1979], p.192) | |
| A reaction: A very nice statement of the aim of modern language-centred philosophy, though the task offered appears to be that of an under-labourer, when the real target, even according to Quine, is supposed to be a 'theory of the world'. |
| 12463 | Unlike correspondence, truthmaking can be one truth to many truthmakers, or vice versa [Jacobs] |
| Full Idea: I assume a form of truthmaking theory, ..which is a many-many relation, unlike, say correspondence, so that one entity can make multiple truths true and one truth can have multiple truthmakers. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §1) | |
| A reaction: This sounds like common sense, once you think about it. One tree makes many things true, and one statement about trees is made true by many trees. |
| 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. |
| 14375 | If structures result from intrinsic natures of properties, the 'relations' between them can drop out [Jacobs] |
| Full Idea: If a relation holds between two properties as a result of their intrinsic natures, then it appears the relation between the properties is not needed to do the structuring of reality; the properties themselves suffice to fix the structure. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.1) | |
| A reaction: [the first bit quotes Jubien 2007] He cites a group of scientific essentialists as spokesmen for this view. Sounds right to me. No on seems able to pin down what a relation is - which may be because there is no such entity. |
| 14378 | Science aims at identifying the structure and nature of the powers that exist [Jacobs] |
| Full Idea: Scientific practice seems aimed precisely at identifying the structure and nature of the powers that exist. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.3) | |
| A reaction: Good. Friends of powers should look at this nice paper by Jacobs. There is a good degree of support for this view from pronouncements of modern scientists. If scientists don't support it, they should. Otherwise they are trapped in the superficial. |
| 12467 | Powers come from concrete particulars, not from the laws of nature [Jacobs] |
| Full Idea: The source of powers is not the laws of nature; it is the powerful nature of the ordinary properties of concrete particulars. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.2) | |
| A reaction: This pithily summarises my own view. People who think the powers of the world derive from the laws either have an implicit religious framework, or they are giving no thought at all to the ontological status of the laws. |
| 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. |
| 14377 | Possibilities are manifestations of some power, and impossibilies rest on no powers [Jacobs] |
| Full Idea: To be possible is just to be one of the many manifestations of some power, and to be impossible is to be a manifestation of no power. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.2.1) | |
| A reaction: [This remark occurs in a discussion of theistic Aristotelianism] I like this. If we say that something is possible, the correct question is to ask what power could bring it about. |
| 14376 | States of affairs are only possible if some substance could initiate a causal chain to get there [Jacobs] |
| Full Idea: A non-actual state of affairs in possible if there actually was a substance capable of initiating a causal chain, perhaps non-deterministic, that could lead to the state of affairs that we claim is possible. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.2) | |
| A reaction: [He is quoting A.R. Pruss 2002] That seems exactly right. Of course the initial substance(s) might create a further substance, such as a transuranic element, which then produces the state of affairs. I favour this strongly actualist view. |
| 14379 | Counterfactuals invite us to consider the powers picked out by the antecedent [Jacobs] |
| Full Idea: A counterfactual is an invitation to consider what the properties picked out by the antecedent are powers for (where Lewis 1973 took it to be an invitation to consider what goes on in a selected possible world). | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.4.3) | |
| A reaction: A beautifully simple proposal from Jacobs, with which I agree. This seems to be an expansion of the Ramsey test for conditionals, where you consider the antecedent being true, and see what follows. What, we ask Ramsey, would make it follow? |
| 14372 | Possible worlds are just not suitable truthmakers for modality [Jacobs] |
| Full Idea: Possible worlds are just not the sorts of things that could ground modality; they are not suitable truthmakers. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §3) | |
| A reaction: Are possible world theorists actually claiming that the worlds 'ground' modality? Maybe Lewis is, since all those concrete worlds had better do some hard work, but for the ersatzist they just provide a kind of formal semantics, leaving ontology to others. |
| 12466 | All modality is in the properties and relations of the actual world [Jacobs] |
| Full Idea: Properties and the relations between them introduce modal connections in the actual world. ..This is a strong form of actualism, since all of modality is part of the fundamental fabric of the actual world. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4) | |
| A reaction: This is the view of modality which I find most congenial, with the notion of 'powers' giving us the conceptual framework on which to build an account. |
| 14371 | We can base counterfactuals on powers, not possible worlds, and hence define necessity [Jacobs] |
| Full Idea: Together with a definition of possibility and necessity in terms of counterfactuals, the powers semantics of counterfactuals generates a semantics for modality that appeals to causal powers and not possible worlds. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §1) | |
| A reaction: Wonderful. Just what the doctor ordered. The only caveat is that if we say that reality is built up from fundamental powers, then might those powers change their character without losing their identity (e.g. gravity getting weaker)? |
| 12465 | Concrete worlds, unlike fictions, at least offer evidence of how the actual world could be [Jacobs] |
| Full Idea: Lewis's concrete worlds give a better account of modality (than fictional worlds). When I learn that a man like me drives a truck, I gain evidence for the fact that I can drive a truck. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §3) | |
| A reaction: Cf. Idea 12464. Jacobs still rightly rejects this as an account of possibility, since the possibility that I might drive a truck must be rooted in me, not in some other person who drives a truck, even if that person is very like me. |
| 12464 | If some book described a possibe life for you, that isn't what makes such a life possible [Jacobs] |
| Full Idea: Suppose somewhere deep in the rain forest is a book that includes a story about you as a truck-driver. I doubt that you would be inclined the think that that story, that book, is the reason you could have been a truck driver. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §3) | |
| A reaction: This begins to look like a totally overwhelming and obvious reason why possible worlds (especially as stories) don't give a good metaphysical account of possibility. They provide a semantic structure for modal reasoning, but that is entirely different. |
| 12469 | Possible worlds semantics gives little insight into modality [Jacobs] |
| Full Idea: If we want our semantics for modality to give us insight into the truthmakers for modality, then possible worlds semantics is inadequate. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.4) | |
| A reaction: [See the other ideas of Jacobs (and Jubien) for this] It is an interesting question whether a semantics for a logic is meant to give us insight into how things really are, or whether it just builds nice models. Satisfaction, or truth? |