39 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. |
| 6472 | Continuity is a sufficient criterion for the identity of a rock, but not for part of a smooth fluid [Russell] |
| Full Idea: Continuity is not a sufficient criterion of material identity; it is sufficient in many cases, such as rocks and tables, where the appearances change slowly, but in others, such as the parts of an approximately homogeneous fluid, it fails us utterly. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §XI) | |
| A reaction: It might be debatable to what extent the 'parts' of a homogeneous fluid have identity. How many 'parts' are there in a glass of water? This seems, now, a problem for internalists; externalists can define the identity by the unseen molecules. |
| 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. |
| 6473 | Physical things are series of appearances whose matter obeys physical laws [Russell] |
| Full Idea: We may lay down the following definition: Physical things are those series of appearances whose matter obeys the laws of physics. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §XI) | |
| A reaction: We will then have to define the laws of physic without making any reference to 'physical things'. There is an obvious suspicion of circularity somewhere here. I find it very odd to define objects just in terms of their appearances. |
| 6465 | We need not deny substance, but there seems no reason to assert it [Russell] |
| Full Idea: It is not necessary to deny a substance or substratum underlying appearances; it is merely expedient (by the application of Occam's Razor) to abstain from asserting this unnecessary entity. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §V) | |
| A reaction: Russell then goes on to struggle heroically in attempts to give accounts of 'matter' and 'objects' entirely in terms of 'sense-data'. If he failed, as many think he did, should we go back to belief in Aristotelian substance? |
| 6471 | The assumption by physicists of permanent substance is not metaphysically legitimate [Russell] |
| Full Idea: The assumption of permanent substance, which technically underlies the procedure of physics, cannot of course be regarded as metaphysically legitimate. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §XI) | |
| A reaction: It is a moot point whether physicists still thought this way after the full arrival of quantum theory in 1926. Russell raises all sorts of nice questions about the relationship between physics and philosophy here. I'm on Russell's side. |
| 6466 | Where possible, logical constructions are to be substituted for inferred entities [Russell] |
| Full Idea: The supreme maxim in scientific philosophising is this: Wherever possible, logical constructions are to be substituted for inferred entities. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VI) | |
| A reaction: This seems to represent Russell's first move (in 1914) into what looks like phenomenalism. One might ask what is the difference between 'logical constructions' and 'inferred entities'. The latter appear to have unity, so I prefer them. |
| 6467 | No sensibile is ever a datum to two people at once [Russell] |
| Full Idea: No sensibile is ever a datum to two people at once. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VII) | |
| A reaction: So a loud bang has to broken down into an almost infinite number of sound sensibilia - each one presumably the size of the apperture of a small ear. This is beginning to sound a bit silly. |
| 6483 | Russell held that we are aware of states of our own brain [Russell, by Robinson,H] |
| Full Idea: Russell held that we are aware of states of our own brain. | |
| From: report of Bertrand Russell (The Relation of Sense-Data to Physics [1914]) by Howard Robinson - Perception 1.1 | |
| A reaction: I can't say that I had ever intepreted Russell in this way, but it is a wonderfully thought-provoking idea. All the time that I thought I was looking at a table, I was just looking at my own brain, and drawing an unspoken inference that a table caused it. |
| 8244 | Sense-data are qualities devoid of subjectivity, which are the basis of science [Russell, by Deleuze/Guattari] |
| Full Idea: Rather than oppose sensory knowledge and scientific knowledge, we should identify the sensibilia that are peculiar to science. This is what Russell did when he evoked sense-data, qualities devoid of all subjectivity. | |
| From: report of Bertrand Russell (The Relation of Sense-Data to Physics [1914]) by G Deleuze / F Guattari - What is Philosophy? 2.5 | |
| A reaction: An interesting observation. Russell is striking for his lack of interest in theories of arts and ethics, and his whole work focuses on understanding the scientific view. What is involved in sensibilia is a key modern issue (e.g. McDowell). |
| 6462 | Sense-data are not mental, but are part of the subject-matter of physics [Russell] |
| Full Idea: I regard sense-data as not mental, and as being, in fact, part of the actual subject-matter of physics. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §III) | |
| A reaction: Russell had clearly given himself an ontological problem with the introduction of sense-data, and this is his drastic solution. In 1912 his account seems ambiguous between sense-data being mental and being physical. |
| 6463 | Sense-data are objects, and do not contain the subject as part, the way beliefs do [Russell] |
| Full Idea: Logically a sense-datum is an object, a particular of which the subject is aware; it does not contain the subject as a part, as for example beliefs and volitions do. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §IV) | |
| A reaction: This very firmly rejects any notion that a sense-datum is mental. It is a left as a strange sort of object which gets as close as it is possible to get to the 'borders' of the mind, without actually becoming part of it. |
| 6464 | Sense-data are usually objects within the body, but are not part of the subject [Russell] |
| Full Idea: The sense-datum is an external object of which in sensation the subject is aware; it is true that the sense-datum is in many cases in the subject's body, but the subject's body is as distinct from the subject as tables and chairs are. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §IV) | |
| A reaction: This is probably Russell's clearest statement of the nature of sense-data, which are objects within the subjects body, but are not part of the mind. So once again we come up against the question of their ontology. Are they made of neurons? |
| 6459 | We do not know whether sense-data exist as objects when they are not data [Russell] |
| Full Idea: We do not know, except by means of more or less precarious inferences, whether the objects which are at one time sense-data continue to exist at times when they are not data. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §II) | |
| A reaction: Note that he actually refers to sense-data as 'objects'. It shows how thoroughly reified they are in his theory if they have the possibility of independent existence. This invites the question 'what are they made of?' |
| 6460 | 'Sensibilia' are identical to sense-data, without actually being data for any mind [Russell] |
| Full Idea: I shall give the name 'sensibilia' to those objects which have the same metaphysical and physical status as sense-data without necessarily being data to any mind. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §III) | |
| A reaction: This is his response to the problem of whether sense-data can exist independently of experience, which was unclear in 1912. Presumably sensibilia are objects which are possible sources of experience, but that seems to cover most objects. |
| 6461 | Ungiven sense-data can no more exist than unmarried husbands [Russell] |
| Full Idea: We cannot ask, 'Can sense-data exist without being given?' for that is like asking, 'Can husbands exist without being married?' | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §III) | |
| A reaction: This follows hard on Idea 6460, which introduces the idea of 'sensibilia' for things which are like sense-data, but are not 'given'. This is a new distinction in 1914, which he had not made in 1912. |
| 6458 | Individuating sense-data is difficult, because they divide when closely attended to [Russell] |
| Full Idea: There is some difficulty in deciding what is to be considered one sense-datum: often attention causes divisions to appear where, so far as can be discovered, there were no divisions before. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §II) | |
| A reaction: This was, I suspect, why Russell had dropped the idea of sense-data by 1921. He does, however, say that they are the last unit in analysis, rather than being the most basic unit of perception. In other words, they are purely theoretical. |
| 6469 | Sense-data may be subjective, if closing our eyes can change them [Russell] |
| Full Idea: One reason often alleged for the subjectivity of sense-data is that the appearance of a thing itself may change when we find it hard to suppose that the thing itself has changed - as when we shut our eyes, or screw them up to make things look double. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VIII) | |
| A reaction: Russell firmly denies that they are subjective. These examples are also said to support to proposed existence of sense-data in the first place, since they show the gap between appearance and reality. |
| 7518 | If folk psychology gives a network of causal laws, that fits neatly with functionalism [Churchland,PM] |
| Full Idea: The portrait of folk psychology as a network of causal laws dovetailed neatly with the emerging philosophy of mind called functionalism. | |
| From: Paul M. Churchland (Folk Psychology [1996], II) | |
| A reaction: And from the lower levels functionalism is supported by the notion that the brain is modular. Note the word 'laws'; this implies an underlying precision in folk psychology, which is then easily attacked. Maybe the network is too complex for simple laws. |
| 7519 | Many mental phenomena are totally unexplained by folk psychology [Churchland,PM] |
| Full Idea: Folk psychology fails utterly to explain a considerable variety of central psychological phenomena: mental illness, sleep, creativity, memory, intelligence differences, and many forms of learning, to cite just a few. | |
| From: Paul M. Churchland (Folk Psychology [1996], III) | |
| A reaction: If folk psychology is a theory, it will have been developed to predict behaviour, rather than as a full-blown psychological map. The odd thing is that some people seem to be very bad at folk psychology. |
| 7520 | Folk psychology never makes any progress, and is marginalised by modern science [Churchland,PM] |
| Full Idea: Folk psychology has not progressed significantly in the last 2500 years; if anything, it has been steadily in retreat during this period; it does not integrate with modern science, and its emerging wallflower status bodes ill for its future. | |
| From: Paul M. Churchland (Folk Psychology [1996], III) | |
| A reaction: [compressed] However, while shares in alchemy and astrology have totally collapsed, folk psychology shows not the slightest sign of going away, and it is unclear how it ever could. See Idea 3177. |
| 6470 | Matter is the limit of appearances as distance from the object diminishes [Russell] |
| Full Idea: We offer the following tentative definition: The matter of a given thing is the limit of its appearances as their distance from the thing diminishes. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §IX) | |
| A reaction: This strikes me as empiricism gone mad. Russell is famous for being a 'realist', but you would hardly know it at this point. Personally I put emphasis on 'best explanation', which fairly simply delivers most of our commonsense understandings of reality. |
| 6468 | There is 'private space', and there is also the 'space of perspectives' [Russell] |
| Full Idea: In addition to the private spaces, ..there is the 'space of perspectives', since each private world may be regarded as the appearance which the universe presents from a certain point of view. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VII) | |
| A reaction: This replaces his concept of 'public space', which he introduced in 1912. Russell gradually dropped this, but I like the idea that we somehow directly perceive space in two ways simultaneously (which led him to say that space is six-dimensional). |