43 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? |
| 10304 | Very few things in set theory remain valid in intuitionist mathematics [Bernays] |
| Full Idea: Very few things in set theory remain valid in intuitionist mathematics. | |
| From: Paul Bernays (On Platonism in Mathematics [1934]) |
| 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. |
| 10303 | Restricted Platonism is just an ideal projection of a domain of thought [Bernays] |
| Full Idea: A restricted Platonism does not claim to be more than, so to speak, an ideal projection of a domain of thought. | |
| From: Paul Bernays (On Platonism in Mathematics [1934], p.261) | |
| A reaction: I have always found Platonism to be congenial when it talks of 'ideals', and ridiculous when it talks of a special form of 'existence'. Ideals only 'exist' because we idealise things. I may declare myself, after all, to be a Restricted Platonist. |
| 10306 | Mathematical abstraction just goes in a different direction from logic [Bernays] |
| Full Idea: Mathematical abstraction does not have a lesser degree than logical abstraction, but rather another direction. | |
| From: Paul Bernays (On Platonism in Mathematics [1934], p.268) | |
| A reaction: His point is that the logicists seem to think that if you increasingly abstract from mathematics, you end up with pure logic. |
| 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. |
| 6346 | The main epistemological theories are foundationalist, coherence, probabilistic and reliabilist [Pollock/Cruz] |
| Full Idea: The most familiar epistemological theories are foundation theories, coherence theories, probabilistic theories, and reliabilist theories. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], Pref) | |
| A reaction: A helpful list. Reliabilism is now the dominant externalist theory. Probability theories will centre on Bayes' Theorem (Idea 2798). The authors want an internalist theory that includes perceptions as well as beliefs. I currently favour coherence. |
| 6351 | Most people now agree that our reasoning proceeds defeasibly, rather than deductively [Pollock/Cruz] |
| Full Idea: One of the most important modern advances in epistemology was the recognition of defeasible reasons; it is now generally acknowledged that most of our reasoning proceeds defeasibly rather than deductively. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §1.2) | |
| A reaction: I agree totally. This is why fallibilism is clearly a correct position in epistemology (e.g. Ideas 2736 and 2755). Deduction is not the only grounds given for certainty - there are rationalist foundations (Descartes) and empiricist foundations (Moore). |
| 6374 | To believe maximum truths, believe everything; to have infallible beliefs, believe nothing [Pollock/Cruz] |
| Full Idea: If we want an agent to believe as many truths as possible, this could be achieved by simply believing everything; if we want an agent to have only true beliefs, this could be achieved by believing nothing. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §6.6) | |
| A reaction: I like this. It highlights the pragmatic need for a middle road, in which a core set of beliefs are going to be approved of as 'knowledge', so that we can get on with life. This has to be a social matter, and needs flexibility of Fallibilism. |
| 6355 | Direct realism says justification is partly a function of pure perceptual states, not of beliefs [Pollock/Cruz] |
| Full Idea: We defend a version of direct realism, saying that justification must be partly a function of perceptual states themselves, and not just a function of our beliefs about perceptual states. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §1.5.3) | |
| A reaction: Judgement suggests that perceptual states give good justification about primary qualities (like mass or shape), but not of secondary qualities (like smell or colour). Perceptions can be downright misleading. |
| 6359 | Phenomenalism offered conclusive perceptual knowledge, but conclusive reasons no longer seem essential [Pollock/Cruz] |
| Full Idea: Phenomenalism offered the prospect of explaining perceptual knowledge within a framework that recognised only conclusive reasons; once it is acknowledged that at least induction uses nonconclusive reasons, it is no longer needed. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §2.3.3.2) | |
| A reaction: I'm not sure that that is the only motivation for phenomenalism, which seemed to be attempting to get as close to 'reality' as intellectual honesty would allow. I certainly favour the modern relaxed attitude to knowledge, which needn't be 'conclusive'. |
| 6366 | Perception causes beliefs in us, without inference or justification [Pollock/Cruz] |
| Full Idea: Perception is a causal process that inputs beliefs into our doxastic system without their being inferred from or justified on the basis of other beliefs we already have. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §3.2.3) | |
| A reaction: This topic is much discussed (e.g. by MacDowell). I don't see how something is going to qualify as a 'belief' if it doesn't involve concepts and propositions. The point that we are caused to have many of our beliefs (rather than judging) seems right. |
| 6362 | Sense evidence is not beliefs, because they are about objective properties, not about appearances [Pollock/Cruz] |
| Full Idea: We think it is a mistake to suppose that the evidence of our senses comes to us in the form of beliefs; in perception, the beliefs we form are almost invariably about the objective properties of physical objects - not about how they appear to us. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §2.5.5) | |
| A reaction: The tricky word here is 'evidence'. At what point in the process of perception does something begin to count as evidence? It must at least involve concepts (and maybe even propositions) if it is going to be thought about in that way. |
| 6371 | Bayesian epistemology is Bayes' Theorem plus the 'simple rule' (believe P if it is probable) [Pollock/Cruz] |
| Full Idea: Bayesian epistemology is based upon the 'simple rule' (believe P if it is sufficiently probable) and Bayes' Theorem. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §4.3.1) | |
| A reaction: For Bayes' Theorem, see Idea 2798. There is the question of whether the proposition is subjectively or objectively probable (I believe in ghosts, so any shadow is probably a ghost). There is also the problem of objective evidence for the calculation. |
| 6373 | Internalism says if anything external varies, the justifiability of the belief does not vary [Pollock/Cruz] |
| Full Idea: Internalist theories make justifiability of a belief a function of the internal states of the believer, in the sense that if we vary anything but his internal states the justifiability of the belief does not vary. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §5.4.3) | |
| A reaction: This seems to be a nice clear definition of internalism (and, by implication, externalism). It favours externalism. I know my car is in the car park; someone takes it for a joyride, then replaces it; my good justification seems thereby weakened. |
| 6353 | People rarely have any basic beliefs, and never enough for good foundations [Pollock/Cruz] |
| Full Idea: We argue that all foundations theories are false, for the simple reason that people rarely have any epistemological basic beliefs, and never have enough to provide a foundation for the rest of our knowledge. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §1.5.3) | |
| A reaction: Once surprising things start to happen in a film, we rapidly jettison our normal basic beliefs, to be ready for surprises. However, it seems to me that quite a lot of beliefs are hard-wired into us, or inescapably arise from the use of our senses. |
| 6361 | Foundationalism requires self-justification, not incorrigibility [Pollock/Cruz] |
| Full Idea: What foundationalism requires is self-justification, which is weaker than incorrigibility. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §2.5.3) | |
| A reaction: The writers oppose foundationalism, but this remark obviously helps the theory. Bonjour votes for a fallible rationalist foundationalism, and an fallible empiricist version seems plausible (because we must check for hallucinations etc.). |
| 6357 | Reason cannot be an ultimate foundation, because rational justification requires prior beliefs [Pollock/Cruz] |
| Full Idea: Reasoning, it seems, can only justify us in holding a belief if we are already justified in holding the beliefs from which we reason, so reasoning cannot provide an ultimate source of justification. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §2.1) | |
| A reaction: This sounds slick and conclusive, but it isn't. If we accept that some truths might be 'self-evident' to reason, they could stand independently. And a large body of rational beliefs might be mutually self-supporting, as in the coherence theory of truth. |
| 6363 | Foundationalism is wrong, because either all beliefs are prima facie justified, or none are [Pollock/Cruz] |
| Full Idea: Either no belief is prima facie justified or all beliefs are prima facie justified; …we regard this as a decisive refutation of foundationalism. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §2.5.5) | |
| A reaction: The full text must he examined, but it is not at all clear to me how my belief that quantum theory is correct could be even remotely as prima facie justified as my belief that this is my hand. I don't think basic beliefs need be sharply divided off. |
| 6365 | Negative coherence theories do not require reasons, so have no regress problem [Pollock/Cruz] |
| Full Idea: The regress argument has no apparent strength against negative coherence theories, because they do not require reasons for beliefs. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §3.2.3) | |
| A reaction: A nice point. Such theories endorse Neurath's picture (Idea 6348). On the whole philosophers like positive support for their beliefs, so the rather passive picture of accepting everything unless it is undermined is not appealing. A fall-back position. |
| 6354 | Coherence theories fail, because they can't accommodate perception as the basis of knowledge [Pollock/Cruz] |
| Full Idea: All coherence theories fail, because they are unable to accommodate perception as the basic source of our knowledge of the world. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §1.5.3) | |
| A reaction: An interesting claim, which the authors attempt to justify. They say it is direct realism, because the perceptions justify, without any intervening beliefs. My immediate thought is that they might justify knowledge of primary qualities, but not secondary. |
| 6367 | Coherence theories isolate justification from the world [Pollock/Cruz] |
| Full Idea: The Isolation Argument objects that coherence theories cut justification off from the world. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §3.2.4) | |
| A reaction: I don't see this as a strong objection. Justification can be in the way beliefs cohere together, but the beliefs themselves consist of holding propositions to be true, and truth asserts a connection to the world (I say). |
| 6370 | Externalism comes as 'probabilism' (probability of truth) and 'reliabilism' (probability of good cognitive process) [Pollock/Cruz] |
| Full Idea: There are two major kinds of externalist theory in the literature - probabilism (which expresses justification in terms of probability of the belief being true), and reliabilism (which refers to the probability of the cognitive processes being right). | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §4.1) | |
| A reaction: A useful clarification. Reliabilism has an obvious problem, that a process can be reliable, but only luckily correct on this occasion (a clock which has, unusually, stopped). A ghost is more probably there if I believe in ghosts. |
| 6358 | One belief may cause another, without being the basis for the second belief [Pollock/Cruz] |
| Full Idea: If I fall flat on my back running to a class, my belief that I was late for class may cause me to have the belief that there are birds in the trees, but I do not believe the latter on the basis of the former. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §2.3.1) | |
| A reaction: A nice example, which fairly conclusively demolishes any causal theory of justification. My example is believing correctly that the phone ring is from mother, because she said she would call. Maybe causation is needed somewhere in the right theory. |
| 6364 | We can't start our beliefs from scratch, because we wouldn't know where to start [Pollock/Cruz] |
| Full Idea: We cannot forsake all of our beliefs and start over again, because then we could not know how to start. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §3.1) | |
| A reaction: A point with which it is hard to disagree, but even Descartes agreed with it (Idea 3604). Presumably all your beliefs can take it in turn to be doubted, while others are held true, or you can whittle the beliefs which can't be abandoned down to a minimum. |
| 6352 | Enumerative induction gives a universal judgement, while statistical induction gives a proportion [Pollock/Cruz] |
| Full Idea: Enumerative induction examines a sample of objects, observes they all have a property, and infers that they all have that property; statistical induction observes a proportion of the objects having the property, and infers that proportion in general. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §1.4.6) | |
| A reaction: There is also induction by elimination, where it is either p or q, and observation keeps saying it isn't p. A small sample is very unreliable, but a huge sample (e.g. cigarettes and cancer) is almost certain, so where is the small/huge boundary? |
| 6372 | Since every tautology has a probability of 1, should we believe all tautologies? [Pollock/Cruz] |
| Full Idea: It follows from the probability calculus that every tautology has probability 1; it then follows in Bayesian epistemology that we are justified in believing every tautology. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §4.3.1.5) | |
| A reaction: If I say 'a bachelor is a small ant' you wouldn't believe it, but if I said 'I define a bachelor as a small ant' you would have to believe it. 'Bachelors are unmarried' men is a description of English usage, so is not really a simple tautology. |
| 6360 | Scientific confirmation is best viewed as inference to the best explanation [Pollock/Cruz] |
| Full Idea: The confirmation of scientific theories is probably best viewed in terms of inference to the best explanation. | |
| From: J Pollock / J Cruz (Contemporary theories of Knowledge (2nd) [1999], §2.3.3.3) | |
| A reaction: A simple claim, but one with which I strongly agree. 'Inference', of course, implies that there is some fairly strict logical thinking going on, which may not be so. I suspect that dogs can move to the best explanation. It is, though, a rational process. |