32 ideas
| 9218 | Maybe what distinguishes philosophy from science is its pursuit of necessary truths [Sider] |
| Full Idea: According to one tradition, necessary truth demarcates philosophical from empirical inquiry. Science identifies contingent aspects of the world, whereas philosophical inquiry reveals the essential nature of its objects. | |
| From: Theodore Sider (Reductive Theories of Modality [2003], 1) | |
| A reaction: I don't think there is a clear demarcation, and I would think that lots of generalizations about contingent truths are in philosophical territory, but I quite like this idea - even if it does make scientists laugh at philosophers. |
| 12585 | Most people can't even define a chair [Peacocke] |
| Full Idea: Ordinary speakers are notoriously unsuccessful if asked to offer an explicit definition of the concept 'chair'. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 6.1) |
| 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. |
| 12581 | Perceptual concepts causally influence the content of our experiences [Peacocke] |
| Full Idea: Once a thinker has acquired a perceptually individuated concept, his possession of that concept can causally influence what contents his experiences possess. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 3.3) | |
| A reaction: Like having 35 different words for 'snow', I suppose. I'm never convinced by such claims. Having the concepts may well influence what you look at or listen to, but I don't see the deliverances of the senses being changed by the concepts. |
| 12579 | Perception has proto-propositions, between immediate experience and concepts [Peacocke] |
| Full Idea: Perceptual experience has a second layer of nonconceptual representational content, distinct from immediate 'scenarios' and from conceptual contents. These additional contents I call 'protopropositions', containing an individual and a property/relation. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 3.3) | |
| A reaction: When philosophers start writing this sort of thing, I want to turn to neuroscience and psychology. I suppose the philosopher's justification for this sort of speculation is epistemological, but I see no good coming of it. |
| 12586 | Consciousness of a belief isn't a belief that one has it [Peacocke] |
| Full Idea: I dispute the view that consciousness of a belief consists in some kind of belief that one has the belief. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 6.2) | |
| A reaction: Thus if one is trying to grasp the notion of higher-order thought, it doesn't have to be just more of same but one level up. Any sensible view of the brain would suggest that one sort of activity would lead into an entirely different sort. |
| 18568 | Philosophy should merely give necessary and sufficient conditions for concept possession [Peacocke, by Machery] |
| Full Idea: Peacocke's 'Simple Account' says philosophers should determine the necessary and sufficient conditions for possessing a concept, and psychologists should explain how the human mind meets these conditions. | |
| From: report of Christopher Peacocke (A Study of Concepts [1992]) by Edouard Machery - Doing Without Concepts 2 | |
| A reaction: One can't restrict philosophy so easily. Psychologists could do that job themselves, and dump philosophy. Philosophy is interested in the role of concepts in meaning, experience and judgement. If psychologists can contribute to philosophy, fine. |
| 18571 | Peacocke's account of possession of a concept depends on one view of counterfactuals [Peacocke, by Machery] |
| Full Idea: Peacocke's method for discovering the possession conditions of concepts is committed to a specific account of counterfactual judgements - the Simulation Model (judgements we'd make if the antecedent were actual). | |
| From: report of Christopher Peacocke (A Study of Concepts [1992]) by Edouard Machery - Doing Without Concepts 2.3.4 | |
| A reaction: Machery concludes that the Simulation Model is incorrect. This appears to be Edgington's theory of conditionals, though Machery doesn't mention her. |
| 18572 | Peacocke's account separates psychology from philosophy, and is very sketchy [Machery on Peacocke] |
| Full Idea: Peacocke's Simple Account fails to connect the psychology and philosophy of concepts, it subordinates psychology to specific field of philosophy, it is committed to analytic/synthetic, and (most important) its method is very sketchy. | |
| From: comment on Christopher Peacocke (A Study of Concepts [1992]) by Edouard Machery - Doing Without Concepts 2.3.5 | |
| A reaction: Machery says Peacocke proposes a research programme, and he is not surprised that no one has every followed. Machery is a well-known champion of 'experimental philosophy', makes philosophy respond to the psychology. |
| 12577 | Possessing a concept is being able to make judgements which use it [Peacocke] |
| Full Idea: Possession of any concept requires the capacity to make judgements whose content contain it. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 2.1) | |
| A reaction: Idea 12575 suggested that concept possession was an ability just to think about the concept. Why add that one must actually be able to make a judgement? Presumably to get truth in there somewhere. I may only speculate and fantasise, rather than judge. |
| 12578 | A concept is just what it is to possess that concept [Peacocke] |
| Full Idea: There can be no more to a concept than is determined by a correct account of what it is to possess that concept. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 3.2) | |
| A reaction: He calls this the Principle of Dependence. An odd idea, if you compare 'there is no more to a book than its possession conditions'. If the principle is right, I struggle with the proposal that a philosopher might demonstrate such a principle. |
| 12587 | Employing a concept isn't decided by introspection, but by making judgements using it [Peacocke] |
| Full Idea: On the account I have been developing, what makes it the case that someone is employing one concept rather than another is not constituted by his impression of whether he is, but by complex facts about explanations of his judgements. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 7.2) | |
| A reaction: I presume this brings truth into the picture, and hence establishes a link between the concept and the external world, rather than merely with other concepts. There seems to be a shadowy behaviourism lurking in the background. |
| 12584 | An analysis of concepts must link them to something unconceptualized [Peacocke] |
| Full Idea: At some point a good account of conceptual mastery must tie the mastery to abilities and relations that do not require conceptualization by the thinker. | |
| From: Christopher Peacocke (A Study of Concepts [1992], 5.3) | |
| A reaction: This obviously implies a physicalist commitment. Peacocke seeks, as so many do these days in philosophy of maths, to combine this commitment with some sort of Fregean "platonism without tears" (p.101). I don't buy it. |
| 9335 | Concepts are constituted by their role in a group of propositions to which we are committed [Peacocke, by Greco] |
| Full Idea: Peacocke argues that it may be a condition of possessing a certain concept that one be fundamentally committed to certain propositions which contain it. A concept is constituted by playing a specific role in the cognitive economy of its possessor. | |
| From: report of Christopher Peacocke (A Study of Concepts [1992]) by John Greco - Justification is not Internal §9 | |
| A reaction: Peacocke is talking about thought and propositions rather than language. Good for him. I always have problems with this sort of view: how can something play a role if it doesn't already have intrinsic properties to make the role possible? |
| 9336 | A concept's reference is what makes true the beliefs of its possession conditions [Peacocke, by Horwich] |
| Full Idea: Peacocke has a distinctive view of reference: The reference of a concept is that which will make true the primitively compelling beliefs that provide its possession conditions. | |
| From: report of Christopher Peacocke (A Study of Concepts [1992]) by Paul Horwich - Stipulation, Meaning and Apriority §9 | |
| A reaction: The first thought is that there might occasionally be more than one referent which would do the job. It seems to be a very internal view of reference, where I take reference to be much more contextual and social. |