117 ideas
| 9414 | Metaphysics is the mapping of possibilities [Lowe, by Mumford] |
| Full Idea: Metaphysics can be judged as the mapping of possibilities. | |
| From: report of E.J. Lowe (The Possibility of Metaphysics [1998], 1) by Stephen Mumford - Laws in Nature 2.2 |
| 16414 | Science needs metaphysics to weed out its presuppositions [Lowe, by Hofweber] |
| Full Idea: Lowe argues that the sciences need metaphysics to discharge the assumptions that they simply made at the outset. | |
| From: report of E.J. Lowe (The Possibility of Metaphysics [1998]) by Thomas Hofweber - Ambitious, yet modest, Metaphysics 1.2 | |
| A reaction: Hofweber doesn't buy this, and neither do I. I don't think science 'needs' metaphysics (or barely needs it), but I do think metaphysics needs a fair degree of science. It is high-level abstraction based on the facts. |
| 8282 | Only metaphysics can decide whether identity survives through change [Lowe] |
| Full Idea: Only metaphysics can vindicate a judgement that a caterpillar survives to become a butterfly, but a pig does not survive to become pork, or that water survives as ice, but that paper does not survive as ash. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 8.2) | |
| A reaction: The works of Lowe, and other modern heroes, have shown that these real questions can be pursued intensively into areas where no scientist, or even theologian, would dare to tread. |
| 16127 | Metaphysics tells us what there could be, rather than what there is [Lowe] |
| Full Idea: I do not claim that metaphysics on its own can, in general, tell us what there is. Rather - to a first approximation - I hold that metaphysics by itself only tells us what there could be. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 1.3) | |
| A reaction: If there is going to be a modern defence of metaphysics, in an age dominated by empirical science, this sounds pretty good to me. Presumably it also says what there couldn't be. The challenge is to offer authority for any claims made. |
| 18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
| Full Idea: Poincaré suggested that what is wrong with an impredicative definition is that it allows the set defined to alter its composition as more sets are added to the theory. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) |
| 8262 | How can a theory of meaning show the ontological commitments of two paraphrases of one idea? [Lowe] |
| Full Idea: Nothing purely within the theory of meaning is capable of telling us which of two sentences which are paraphrases of one another more accurately reflects the ontological commitments of those who utter them. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 2.3) | |
| A reaction: This is an attack on the semantic approach to ontology, associated with Quine. Cf. Idea 7923. I have always had an aversion to that approach, and received opinion is beginning to agree. "There are more things in heaven and earth, Horatio..." |
| 8315 | Maybe facts are just true propositions [Lowe] |
| Full Idea: If facts are 'proposition-like' or 'thinkable' (we speak of 'knowing' or 'understanding' facts) might they not simply be true propositions? | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 11.2) | |
| A reaction: They certainly can't be if we are going to use facts as what makes propositions true. The proposal would be empty without out some other account of truth (probably a dubious one). Facts are truth-makers? |
| 8319 | One-to-one correspondence would need countable, individuable items [Lowe] |
| Full Idea: Where there is one-to-one correspondence there must certainly be countable, and therefore individuable items of some kind. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 11.6) | |
| A reaction: Lowe is criticising precise notions of 'a fact'. We can respond by relaxing the notion of 'one-to-one', if critics are going to be fussy about exactly what the items are. "There is a huge wave coming" doesn't need a precise notion of a wave to be true. |
| 18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
| Full Idea: None of the classical ways of defining one logical constant in terms of others is available in intuitionist logic (and this includes the two quantifiers). | |
| From: David Bostock (Philosophy of Mathematics [2009], 7.2) |
| 18114 | There is no single agreed structure for set theory [Bostock] |
| Full Idea: There is so far no agreed set of axioms for set theory which is categorical, i.e. which does pick just one structure. | |
| From: David Bostock (Philosophy of Mathematics [2009], 6.4) | |
| A reaction: This contrasts with Peano Arithmetic, which is categorical in its second-order version. |
| 8309 | A set is a 'number of things', not a 'collection', because nothing actually collects the members [Lowe] |
| Full Idea: A set is 'a number of things', not a 'collection'. Nothing literally 'collects' the members of a set, such as the set of planets of the sun, unless it be a Fregean 'concept' under which they fall. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.6) | |
| A reaction: I'm tempted to say that the sun has collected a set of planets (they're the ones that rotate around it). Why can't we have natural sets, which have been collected by nature? A question of the intension, as well as the extension.... |
| 18107 | A 'proper class' cannot be a member of anything [Bostock] |
| Full Idea: A 'proper class' cannot be a member of anything, neither of a set nor of another proper class. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.4) |
| 8322 | I don't believe in the empty set, because (lacking members) it lacks identity-conditions [Lowe] |
| Full Idea: It is not clear to me that the empty set has well-defined identity-conditions. A set has these only to the extent that its members do - but the empty set has none. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 12.3 n8) | |
| A reaction: The empty set is widely used by those who base their metaphysics of maths on sets. It defines zero, and hence is the starting poing for Peano's Postulates (Idea 5897). It might not have identity in itself, but you know where you have arrived after 2 - 2. |
| 18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
| Full Idea: We could add the axiom that all sets are constructible (V = L), making the universe of sets as small as possible, or add the axiom that there is a supercompact cardinal (SC), making the universe as large as we no know how to. | |
| From: David Bostock (Philosophy of Mathematics [2009], 6.4) | |
| A reaction: Bostock says most mathematicians reject the first option, and are undecided about the second option. |
| 18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
| Full Idea: The usual accounts of ZF are not restricted to subsets that we can describe, and that is what justifies the axiom of choice. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.4 n36) | |
| A reaction: This contrasts interestingly with predicativism, which says we can only discuss things which we can describe or define. Something like verificationism hovers in the background. |
| 18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
| Full Idea: The Axiom of Replacement (or the Axiom of Subsets, 'Aussonderung', Fraenkel 1922) in effect enforces the idea that 'limitation of size' is a crucial factor when deciding whether a proposed set or does not not exist. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.4) |
| 18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
| Full Idea: First-order logic is not decidable. That is, there is no test which can be applied to any arbitrary formula of that logic and which will tell one whether the formula is or is not valid (as proved by Church in 1936). | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) |
| 18109 | The completeness of first-order logic implies its compactness [Bostock] |
| Full Idea: From the fact that the usual rules for first-level logic are complete (as proved by Gödel 1930), it follows that this logic is 'compact'. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) | |
| A reaction: The point is that the completeness requires finite proofs. |
| 8312 | It is better if the existential quantifier refers to 'something', rather than a 'thing' which needs individuation [Lowe] |
| Full Idea: If we take the existential quantifier to mean 'there is at least one thing that' then its value must qualify as one thing, individuable in principle. ...So I propose to read it as 'there is something that', which implies nothing about individuability. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 11) | |
| A reaction: All sorts of doubts about the existential quantifier seem to be creeping in nowadays (e.g. Ideas 6067, 6069, 8250). Personally I am drawn to the sound of 'free logic', Idea 8250, which drops existential claims. This would reduce metaphysical confusion. |
| 18123 | Substitutional quantification is just standard if all objects in the domain have a name [Bostock] |
| Full Idea: Substitutional quantification and quantification understood in the usual 'ontological' way will coincide when every object in the (ontological) domain has a name. | |
| From: David Bostock (Philosophy of Mathematics [2009], 7.3 n23) |
| 18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
| Full Idea: The Deduction Theorem is what licenses a system of 'natural deduction' in the first place. | |
| From: David Bostock (Philosophy of Mathematics [2009], 7.2) |
| 18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
| Full Idea: Berry's Paradox can be put in this form, by considering the alleged name 'The least number not named by this name'. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.1) |
| 18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
| Full Idea: If you add to the ordinals you produce many different ordinals, each measuring the length of the sequence of ordinals less than it. They each have cardinality aleph-0. The cardinality eventually increases, but we can't say where this break comes. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.5) |
| 18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
| Full Idea: If we add ω onto the end of 0,1,2,3,4..., it then has a different length, of ω+1. It has a different ordinal (since it can't be matched with its first part), but the same cardinal (since adding 1 makes no difference). | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.5) | |
| A reaction: [compressed] The ordinals and cardinals coincide up to ω, but this is the point at which they come apart. |
| 18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
| Full Idea: It is the usual procedure these days to identify a cardinal number with the earliest ordinal number that has that number of predecessors. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.5) | |
| A reaction: This sounds circular, since you need to know the cardinal in order to decide which ordinal is the one you want, but, hey, what do I know? |
| 18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
| Full Idea: The cardinal aleph-1 is identified with the first ordinal to have more than aleph-0 members, and so on. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.4) | |
| A reaction: That is, the succeeding infinite ordinals all have the same cardinal number of members (aleph-0), until the new total is triggered (at the number of the reals). This is Continuum Hypothesis territory. |
| 18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
| Full Idea: In addition to cuts, or converging series, Cantor suggests we can simply lay down a set of axioms for the real numbers, and this can be done without any explicit mention of the rational numbers [note: the axioms are those for a complete ordered field]. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.4) | |
| A reaction: It is interesting when axioms are best, and when not. Set theory depends entirely on axioms. Horsten and Halbach are now exploring treating truth as axiomatic. You don't give the 'nature' of the thing - just rules for its operation. |
| 18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
| Full Idea: It is not difficult to show that the number of the real numbers is the same as the number of all the subsets of the natural numbers. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.5) | |
| A reaction: The Continuum Hypothesis is that this is the next infinite number after the number of natural numbers. Why can't there be a number which is 'most' of the subsets of the natural numbers? |
| 18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
| Full Idea: As Eudoxus claimed, two distinct real numbers cannot both make the same cut in the rationals, for any two real numbers must be separated by a rational number. He did not say, though, that for every such cut there is a real number that makes it. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.4) | |
| A reaction: This is in Bostock's discussion of Dedekind's cuts. It seems that every cut is guaranteed to produce a real. Fine challenges the later assumption. |
| 17447 | Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck] |
| Full Idea: In Parsons's demonstrative model of counting, '1' means the first, and counting says 'the first, the second, the third', where one is supposed to 'tag' each object exactly once, and report how many by converting the last ordinal into a cardinal. | |
| From: report of Charles Parsons (Frege's Theory of Numbers [1965]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 3 | |
| A reaction: This sounds good. Counting seems to rely on that fact that numbers can be both ordinals and cardinals. You don't 'convert' at the end, though, because all the way you mean 'this cardinality in this order'. |
| 18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
| Full Idea: Non-standard natural numbers will yield non-standard rational and real numbers. These will include reciprocals which will be closer to 0 than any standard real number. These are like 'infinitesimals', so that notion is not actually a contradiction. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) |
| 18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
| Full Idea: A modern axiomatisation of geometry, such as Hilbert's (1899), does not need to claim the existence of real numbers anywhere in its axioms. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.B.5.ii) | |
| A reaction: This is despite the fact that geometry is reduced to algebra, and the real numbers are the equivalent of continuous lines. Bostock votes for a Greek theory of proportion in this role. |
| 18097 | The Peano Axioms describe a unique structure [Bostock] |
| Full Idea: The Peano Axioms are categorical, meaning that they describe a unique structure. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.4 n20) | |
| A reaction: So if you think there is nothing more to the natural numbers than their structure, then the Peano Axioms give the essence of arithmetic. If you think that 'objects' must exist to generate a structure, there must be more to the numbers. |
| 8297 | Numbers are universals, being sets whose instances are sets of appropriate cardinality [Lowe] |
| Full Idea: My view is that numbers are universals, beings kinds of sets (that is, kinds whose particular instances are individual sets of appropriate cardinality). | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10) | |
| A reaction: [That is, 12 is the set of all sets which have 12 members] This would mean, I take it, that if the number of objects in existence was reduced to 11, 12 would cease to exist, which sounds wrong. Or are we allowed imagined instances? |
| 18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
| Full Idea: Hume's Principle will not do as an implicit definition because it makes a positive claim about the size of the universe (which no mere definition can do), and because it does not by itself explain what the numbers are. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) |
| 18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
| Full Idea: Hume's Principle gives a criterion of identity for numbers, but it is obvious that many other things satisfy that criterion. The simplest example is probably the numerals (in any notation, decimal, binary etc.), giving many different interpretations. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) |
| 8266 | Simple counting is more basic than spotting that one-to-one correlation makes sets equinumerous [Lowe] |
| Full Idea: That one-to-one correlated sets of objects are equinumerous is a more sophisticated achievement than the simple ability to count sets of objects. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 2.9) | |
| A reaction: This is an objection to Frege's way of defining numbers, in terms of equinumerous sets. I take pattern-recognition to be the foundation of number, and so spotting a pattern would have to precede spotting that two patterns were identical. |
| 8302 | Fs and Gs are identical in number if they one-to-one correlate with one another [Lowe] |
| Full Idea: What is now known as Hume's Principle says the number of Fs is identical with the number of Gs if and only if the Fs and the Gs are one-to-one correlated with one another. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.3) | |
| A reaction: This seems popular as a tool in attempts to get the concept of number off the ground. Although correlations don't seem to require numbers ('find yourself a partner'), at some point you have to count the correlations. Sets come first, to identify the Fs. |
| 18149 | There are many criteria for the identity of numbers [Bostock] |
| Full Idea: There is not just one way of giving a criterion of identity for numbers. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) |
| 18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
| Full Idea: The Julius Caesar problem was one reason that led Frege to give an explicit definition of numbers as special sets. He does not appear to notice that the same problem affects his Axiom V for introducing sets (whether Caesar is or is not a set). | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) | |
| A reaction: The Julius Caesar problem is a sceptical acid that eats into everything in philosophy of mathematics. You give all sorts of wonderful accounts of numbers, but at what point do you know that you now have a number, and not something else? |
| 18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
| Full Idea: There is no ground for saying that a number IS a position, if the truth is that there is nothing to determine which number is which position. | |
| From: David Bostock (Philosophy of Mathematics [2009], 6.4) | |
| A reaction: If numbers lose touch with the empirical ability to count physical objects, they drift off into a mad world where they crumble away. |
| 18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
| Full Idea: Structuralism begins from a false premise, namely that numbers have no properties other than their relations to other numbers. | |
| From: David Bostock (Philosophy of Mathematics [2009], 6.5) | |
| A reaction: Well said. Describing anything purely relationally strikes me as doomed, because you have to say why those things relate in those ways. |
| 8298 | Sets are instances of numbers (rather than 'collections'); numbers explain sets, not vice versa [Lowe] |
| Full Idea: I favour an account of sets which sees them as being instances of numbers, thereby avoiding the unhelpful metaphor which speaks of a set as being a 'collection' of things. This reverses the normal view, which explains numbers in terms of sets. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10) | |
| A reaction: Cf. Idea 8297. Either a set is basic, or a number is. We might graft onto Lowe's view an account of numbers in terms of patterns, which would give an empirical basis to the picture, and give us numbers which could be used to explain sets. |
| 8311 | If 2 is a particular, then adding particulars to themselves does nothing, and 2+2=2 [Lowe] |
| Full Idea: If 2 is a particular, 'adding' it to itself can, it would seem, only leave us with 2, not another number. (If 'Socrates + Socrates' denotes anything, it most plausibly just denotes Socrates). | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.7) | |
| A reaction: This suggest Kant's claim that arithmetical sums are synthetic (Idea 5558). It is a nice question why, when you put two 2s together, they come up with something new. Addition is movement. Among patterns, or along abstract sequences. |
| 8310 | Does the existence of numbers matter, in the way space, time and persons do? [Lowe] |
| Full Idea: Does it really matter whether the numbers actually exist - in anything like the way in which it matters that space and time or persons actually exist? | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.6) | |
| A reaction: Nice question! It might matter a lot. I take the question of numbers to be a key test case, popular with philosophers because they are the simplest and commonest candidates for abstract existence. The ontological status of values is the real issue. |
| 18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
| Full Idea: The style of nominalism which aims to reduce statements about numbers to statements about their applications does not work for the natural numbers, because they have many applications, and it is arbitrary to choose just one of them. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.B.5.iii) |
| 18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
| Full Idea: Nominalism has two main versions, one which tries to 'reduce' the objects of mathematics to something simpler (Russell and Wittgenstein), and another which claims that such objects are mere 'fictions' which have no reality (Field). | |
| From: David Bostock (Philosophy of Mathematics [2009], 9) |
| 18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
| Full Idea: We all know that in practice no physical measurement can be 100 per cent accurate, and so it cannot require the existence of a genuinely irrational number, rather than some of the rational numbers close to it. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.3) |
| 18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
| Full Idea: The basic use of the ordinal numbers is their use as ordinal adjectives, in phrases such as 'the first', 'the second' and so on. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.5.iii) | |
| A reaction: That is because ordinals seem to attach to particulars, whereas cardinals seem to attach to groups. Then you say 'three is greater than four', it is not clear which type you are talking about. |
| 18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
| Full Idea: The simple theory of types distinguishes sets into different 'levels', but this is quite different from the distinction into 'orders' which is imposed by the ramified theory. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.1) | |
| A reaction: The ramified theory has both levels and orders (p.235). Russell's terminology is, apparently, inconsistent. |
| 18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
| Full Idea: The response of neo-logicists to the Julius Caesar problem is to strengthen Hume's Principle in the hope of ensuring that only numbers will satisfy it. They say the criterion of identity provided by HP is essential to number, and not to anything else. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) |
| 18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
| Full Idea: The neo-logicists take up Frege's claim that Hume's Principle introduces a new concept (of a number), but unlike Frege they go on to claim that it by itself gives a complete account of that concept. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) | |
| A reaction: So the big difference between Frege and neo-logicists is the Julius Caesar problem. |
| 18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
| Full Idea: If logic is neutral on the number of objects there are, then logicists can't construe numbers as objects, for arithmetic is certainly not neutral on the number of numbers there are. They must be treated in some other way, perhaps as numerical quantifiers. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) |
| 18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
| Full Idea: If Hume's Principle is all we are given, by way of explanation of what the numbers are, the only conclusion to draw would seem to be the structuralists' conclusion, ...studying all systems that satisfy that principle. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) | |
| A reaction: Any approach that implies a set of matching interpretations will always imply structuralism. To avoid it, you need to pin the target down uniquely. |
| 18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
| Full Idea: Many of the crucial definitions in the logicist programme are in fact impredicative. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.2) |
| 18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
| Full Idea: In its higher reaches, which posit sets of huge cardinalities, set theory is just a fairy story. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.5.iii) | |
| A reaction: You can't say the higher reaches are fairy stories but the lower reaches aren't, if the higher is directly derived from the lower. The empty set and the singleton are fairy stories too. Bostock says the axiom of infinity triggers the fairy stories. |
| 18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
| Full Idea: A common view is that although a fairy tale may provide very useful predictions, it cannot provide explanations for why things happen as they do. In order to do that a theory must also be true (or, at least, an approximation to the truth). | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.B.5) | |
| A reaction: Of course, fictionalism offers an explanation of mathematics as a whole, but not of the details (except as the implications of the initial fictional assumptions). |
| 18140 | The best version of conceptualism is predicativism [Bostock] |
| Full Idea: In my personal opinion, predicativism is the best version of conceptualism that we have yet discovered. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.4) | |
| A reaction: Since conceptualism is a major player in the field, this makes predicativism a very important view. I won't vote Predicativist quite yet, but I'm tempted. |
| 18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
| Full Idea: Three simple objections to conceptualism in mathematics are that we do not ascribe mathematical properties to our ideas, that our ideas are presumably finite, and we don't think mathematics lacks truthvalue before we thought of it. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.4) | |
| A reaction: [compressed; Bostock refers back to his Ch 2] Plus Idea 18134. On the whole I sympathise with conceptualism, so I will not allow myself to be impressed by any of these objections. (So, what's actually wrong with them.....?). |
| 18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
| Full Idea: If an abstract object exists only when there is some suitable way of expressing it, then there are at most denumerably many abstract objects. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.2) | |
| A reaction: Fine by me. What an odd view, to think there are uncountably many abstract objects in existence, only a countable portion of which will ever be expressed! [ah! most people agree with me, p.243-4] |
| 18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
| Full Idea: Classical mathematicians say predicative mathematics omits areas of great interest, all concerning non-denumerable real numbers, such as claims about huge cardinals. There cannot be a predicative version of this theory. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) | |
| A reaction: I'm not sure that anyone will really miss huge cardinals if they are prohibited, though cryptography seems to flirt with such things. Are we ever allowed to say that some entity conjured up by mathematicians is actually impossible? |
| 18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
| Full Idea: It has been claimed that only predicative mathematics has a justification through its usefulness to science (an empiricist approach). | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) | |
| A reaction: [compressed. Quine is the obvious candidate] I suppose predicativism gives your theory roots, whereas impredicativism is playing an abstract game. |
| 18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
| Full Idea: If we accept the initial idea that we can think only of what we ourselves can describe, then something like the theory of predicativism quite naturally results | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) | |
| A reaction: I hate the idea that we can only talk of what falls under a sortal, but 'what we can describe' is much more plausible. Whether or not you agree with this approach (I'm pondering it), this makes predicativism important. |
| 18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
| Full Idea: The predicative approach cannot accept either the usual definition of identity or the usual definition of the natural numbers, for both of these definitions are impredicative. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) | |
| A reaction: [Bostock 237-8 gives details] |
| 18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
| Full Idea: It is with the real numbers that the restrictions imposed by predicativity begin to make a real difference. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) |
| 8321 | All possible worlds contain abstracta (e.g. numbers), which means they contain concrete objects [Lowe] |
| Full Idea: One could argue that some abstract objects exist in all possible worlds (e.g. natural numbers) and that abstract objects always depend for their existence upon concrete objects, and conclude that some concrete objects exist in all possible worlds. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 12.1) | |
| A reaction: We're all in the dark on this one, but I quite like this argument. I can't conceive of a reality that lacks natural numbers, and the truths that accompany them, and I personally think that numbers arise from the patterns of physical reality. |
| 8300 | Perhaps possession of causal power is the hallmark of existence (and a reason to deny the void) [Lowe] |
| Full Idea: For some metaphysicians, possession of causal power is the very hallmark of real existence (and is one reason, for instance, why some have denied the existence of the void or absolute space). | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.2) | |
| A reaction: You could try saying that space has the power of making movement possible. The 'hallmark' of something doesn't define what it is. Existence without causal power seems logically possible and imaginable, but unlikely. Epiphenomena have this problem. |
| 8281 | Heraclitus says change is new creation, and Spinoza that it is just phases of the one substance [Lowe] |
| Full Idea: The extreme views on change are the Heraclitan view - that every change brings into existence an entirely new entity, and destroys what existed before, and the Spinozan view - that all changes are phase changes within a single substance. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 8.2) | |
| A reaction: The views in between are that bundles of properties shift their contents, or that many substances undergo changes in their properties. The unification of physics might be aiming to vindicate Spinoza. Temporal parts (Lewis) are close to Heraclitus. |
| 8270 | Events are changes or non-changes in properties and relations of persisting objects [Lowe] |
| Full Idea: My own broadly Aristotelian view is that events are changes (and unchanges) in the properties and relations of persisting objects. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 4.4) | |
| A reaction: This needs an account of what it is that persists, and the philosophers' (but not physicists') concept of 'substance' fills this role. It is rather hard to give identity-conditions for an event if it is an 'unchange'. How would you count such events? |
| 8308 | Events are ontologically indispensable for singular causal explanations [Lowe] |
| Full Idea: We must include events in our ontology because they figure indispensably in singular causal explanations. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.5) | |
| A reaction: Hm. Spirits figure indispensably in supernatural explanations. It would be quite a task to prove that events really are indispensable to causal explanations. Why would nomological or counterfactual causal explanations not have the same need? |
| 8314 | Are facts wholly abstract, or can they contain some concrete constituents? [Lowe] |
| Full Idea: Philosophers who invoke facts are divided over whether facts are wholly abstract entities or are complexes capable of containing concrete objects as constituents. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 11.2) | |
| A reaction: If externalism about concepts was true (see Thought|Content|Broad Content), this would support the second (more concrete) view of facts. The correspondence theory of truth would love to plug belief into the concrete world. Me too. |
| 8316 | Facts cannot be wholly abstract if they enter into causal relations [Lowe] |
| Full Idea: There is a difficulty for any view of facts which sees them as being wholly abstract entities, and yet also being causal relata; for it seems that only concrete entities, existing in time and space, can enter into causal relations. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 11.3) | |
| A reaction: There seems a lot of ambiguity in the air here, between epistemology and ontology (surprise!). I take causation to be a physical activity in the concrete world. Our understanding of it is expressed with abstractions. 'Fact' seems to have two meanings. |
| 8318 | The problem with the structured complex view of facts is what binds the constituents [Lowe] |
|
Full Idea:
The most notorious problem besetting the view that facts are structured complexes of constituents is the question of what it is that binds the supposed constituents into the fact. The ordered triple |
|
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 11.5) | |
| A reaction: Lowe denies that facts are complex entities on this basis. You only have the problem if Mars and its redness are two 'things'. If redness is intrinsically a dependent item, we may escape. I wish they wouldn't use colours as examples. See Idea 5456. |
| 8323 | It is whimsical to try to count facts - how many facts did I learn before breakfast? [Lowe] |
| Full Idea: Although 'fact' is grammatically a count noun, it strikes us as being at best whimsical to talk about enumerating facts - to talk, for instance, about how many facts I learned today before breakfast. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 12.4) | |
| A reaction: I always liked the question 'how many facts are there in this room?' One might make a serious attempt to decide how many facts I learned before breakfast, and reach a reasonable approximation, especially if one didn't open the newspaper. |
| 8313 | Facts are needed for truth-making and causation, but they seem to lack identity criteria [Lowe] |
| Full Idea: Facts seem to be indispensable as truth-makers and perhaps as causal relata, ..but if we must only include in our ontology things for which we can state a criterion of identity (Quine), ..we seem to be faced with a dilemma. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 11) | |
| A reaction: Lowe proposes to relax the identification requirement (see Idea 8312). This seems a good strategy. An awful lot of strange philosophy arises from insisting on strict conditions for our understanding, and then finding everywhere failure to achieve it. |
| 8258 | Two of the main rivals for the foundations of ontology are substances, and facts or states-of-affairs [Lowe] |
| Full Idea: One of the chief rivals to my own substance-based ontology is the view that holds facts or states of affairs to be the building-blocks of the world. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], Pref) | |
| A reaction: I think I side with Lowe, even though I am uneasy about the gap between the philosopher's 'substance' and the basic entities of physics. Facts are hard to individuate, and seem to be composed of more basic elements. |
| 8301 | Some abstractions exist despite lacking causal powers, because explanation needs them [Lowe] |
| Full Idea: Some abstract objects, notably certain universals, need to be invoked for explanatory purposes, even if it cannot be said that they themselves possess causal powers or enter into causal relations. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.2) | |
| A reaction: I am unconvinced that an entity with no causal powers could be any kind of explanation, given that, by definition, it can't do anything. You would have to think that the world of pure reason functioned without the aid of causal powers. |
| 8283 | Ontological categories are not natural kinds: the latter can only be distinguished using the former [Lowe] |
| Full Idea: Ontological categories should not be confused with natural kinds: for natural kinds can only be differentiated in a principled way relative to an accepted framework of ontological categories. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 8.2) | |
| A reaction: I presume that the natural kinds are likely to be contingent facts about the actual world (though they may entail necessary laws), whereas I like to think, unfashionably, that categories aim at deconstructing the mind of God (roughly). |
| 8284 | The top division of categories is either abstract/concrete, or universal/particular, or necessary/contingent [Lowe] |
| Full Idea: Some metaphysicians take the highest division to be between abstract and concrete entities, others take it to be between universals and particulars (my own preference, though it is not crucial), and others between necessary and contingent entities. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 8.3) | |
| A reaction: The first division may be blurred, and I am doubtful about universals, so I favour the third. Intuition tells me that there is nothing more basic than the distinction between what is true in all worlds and what is only true in some. The former is bedrock. |
| 13122 | Lowe divides things into universals and particulars, then kinds and properties, and abstract/concrete [Lowe, by Westerhoff] |
| Full Idea: Lowe's Ontological Categories: ENTITIES - {Universals - [Kinds - (Non-natural)(Natural)] [Properties, Relations]} {Particulars - [Abstracta - (Sets)(others)] [Concreta - (Objects)(Non-Objects)]} etc | |
| From: report of E.J. Lowe (The Possibility of Metaphysics [1998], p.181) by Jan Westerhoff - Ontological Categories §01 | |
| A reaction: [my linear representation of a tree diagram; bracket-styles show levels] Lowe's levels below these divide according to whether things are 'substances' or not. I've heard Kit Fine tease Lowe for being too simplistic about ontology. |
| 8273 | Is 'the Thames is broad in London' relational, or adverbial, or segmental? [Lowe] |
| Full Idea: "The Thames is broad in London" might be taken as 'The Thames is broad-in-London', or as 'The Thames is-in-London broad', or as 'The Thames-in-London is broad'. I would urge the superiority of the second one, as an analysis of the normal meaning. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 5.8) | |
| A reaction: He uses the example to attack the perdurance view of objects (i.e. the third analysis). I think I agree with Lowe, but I'm not sure, and I just love the example. Read the second as 'The Thames is (in London) broad'? 'Is' of existence, or predication? |
| 8285 | I prefer 'modes' to 'tropes', because it emphasises their dependence [Lowe] |
| Full Idea: Some philosophers call particularised properties of objects 'tropes', but I prefer the older term 'mode' (or 'individual accident'), because this term rightly has the implication that such entities are existentially dependent ones, depending on objects. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 8.3) | |
| A reaction: A nice illustration of the fact that philosophical terminology is not as metaphysically innocent as it sometimes pretends to be. I agree with Lowe. |
| 8286 | Tropes cannot have clear identity-conditions, so they are not objects [Lowe] |
| Full Idea: I do not believe that tropes or modes can have well-defined or fully determinate identity-conditions, and hence do not believe that they should be thought of as 'objects'. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 8.3) | |
| A reaction: Lowe's account would still allow them to be 'entities'. Any proposal that they have an existence of their own, apart from the objects on which they depend, sounds very misguided. We won't make progress if we don't identify the real properties. |
| 8294 | How can tropes depend on objects for their identity, if objects are just bundles of tropes? [Lowe] |
| Full Idea: It seems that tropes are identity-dependent upon their possessors, but it is difficult to square this claim with the thesis that the possessors of tropes are themselves just bundles of tropes. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 9.8) | |
| A reaction: This circularity in all attempts to individuate tropes is Lowe's main reason for rejecting them. It does seem that the sphericity of a ball must be either identified against other (universal) sphericities, or by the sphere that has the property. |
| 8295 | Why cannot a trope float off and join another bundle? [Lowe] |
| Full Idea: Why cannot a certain trope 'float free' of the trope-bundle to which it belongs and migrate to another bundle? | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 9.8) | |
| A reaction: Tropes are said to be dependent on their possessors, but at the same time to exist as particulars. Lowe's suggestion is that you can't have it both ways. A particular sphericity with no sphere does not even make sense. |
| 8296 | Does a ball snug in plaster have one trope, or two which coincide? [Lowe] |
| Full Idea: If a round ball fits snugly into a round piece of plaster, do they contain the same roundness trope, or do they contain numerically distinct but exactly similar and coinciding roundness tropes? | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 9.8) | |
| A reaction: A microscope would distinguish them, and they are made of different types of matter. Is a hole in a piece of paper a circular cut and a circular area of space? Neither example looks good for tropes. |
| 8288 | Sortal terms for universals involve a substance, whereas adjectival terms do not [Lowe] |
| Full Idea: I want to distinguish 'substantial' universals from 'non-substantial' universals. The former are denoted by sortal terms, such as 'statue' and 'tiger', whereas the latter are denoted by adjectival terms, such as 'red' and 'spherical'. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 9.3) | |
| A reaction: It is an interesting question whether or not (assuming you are committed to universals) a universal necessarily implies an associated substance. If a property is a power, it must be a power of something. Nominalists will deny his distinction. |
| 8293 | Real universals are needed to explain laws of nature [Lowe] |
| Full Idea: I base my case for realism about universals on the need to explain the status of natural laws. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 9.6) | |
| A reaction: I need black magic to explain why my watch has disappeared. The key question, then, would be what we understand by the 'laws of nature'. I am inclined to think that scientific essentialism (qv) can build laws out of natural kinds. Idea 6614. |
| 8307 | Particulars are instantiations, and universals are instantiables [Lowe] |
| Full Idea: A particular is something (not necessarily an object) which instantiates but is not itself instantiated. Universals, on the other hand, necessarily have instances (or, at least, are instantiable). | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.4) | |
| A reaction: This is Lowe's proposal for distinction. It at least establishes the direction of dependency, but I find the notion of 'instantiation' to be as obscure and problematic as the Platonic notion of 'partaking' (see in Ontology|Universals|Platonic Forms). |
| 16130 | To be an object at all requires identity-conditions [Lowe] |
| Full Idea: The only metaphysically defensible notion of an object is precisely that of an entity which possesses determinate identity-conditions. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 1.3) | |
| A reaction: I think he includes abstract objects in this. I suspect this view of muddling epistemology and ontology. Or overemphasising our conventions, rather than reality. |
| 8267 | Perhaps concrete objects are entities which are in space-time and subject to causality [Lowe] |
| Full Idea: An obvious suggestion is that concrete objects are denizens of space-time, and hence subject to causality, though Hale objects that languages are plausibly abstract and yet undergo change and so presumably exist in time. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 2.10) | |
| A reaction: The identity-conditions for a language are pretty loose. Choosing a counterexample from the mental life of human beings begs a billion questions. I can't think of a problem case beyond the world of human culture. |
| 8265 | Our commitment to the existence of objects should depend on their explanatory value [Lowe] |
| Full Idea: Whether objects of a given kind should be thought actually to exist should, in general, be taken to turn on considerations of whether an inclusion of such objects in one's ontology has explanatory value. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 2.3) | |
| A reaction: Blatantly fictional objects, such as fairies, might have wonderful explanatory value (they place dewdrops on flowers). Our ontological commitments cannot be decided one at a time, because consistency of the whole picture is the key value. |
| 8275 | Objects are entities with full identity-conditions, but there are entities other than objects [Lowe] |
| Full Idea: I distinguish objects as those entities - whether abstract or concrete, universal or particular - which possess fully determinate identity-conditions, but there are, or may be, entities other than objects. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 7) | |
| A reaction: A wave on the sea is a candidate for being an entity but not an object. The distinction is probably not quite common usage, but it strikes as one which philosophers should universally adopt. Lots of entities, and some of them are objects. |
| 8263 | An object is an entity which has identity-conditions [Lowe] |
| Full Idea: To be an object is simply to be an entity possessing determinate identity-conditions. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 2.3) | |
| A reaction: This is a nice clear-cut claim, which sounds good, except that there may be a blurring of ontology and epistemology. Presumably the conditions are for the concept, not for an actual act of identification. Maybe we are too stupid to conceive them. |
| 8268 | Some things (such as electrons) can be countable, while lacking proper identity [Lowe] |
| Full Idea: There can be determinate countability even where there is not determinate identity; it is not in dispute that there are two electrons in the shell of a neutral helium atom, even though the identity of electrons is not determinate. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 3.3) | |
| A reaction: If the electrons could merge like water drops, we would be unable to say when they became one object. You can roughly count waves on the sea, but when you seek an exact total, the identity problem intrudes and prevents precise counting. |
| 8303 | Criteria of identity cannot individuate objects, because they are shared among different types [Lowe] |
| Full Idea: Criteria of identity never unambiguously determine the kind of objects to which they apply, since many different types of objects can be governed by the same criteria. Cats and dogs share the criterion of identity for animals in general. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.3) | |
| A reaction: So how do you individuate the type of an object? You could identify 'the thing I dug up yesterday' without being able to individuate it. You can individuate 'the cleverest person in Britain' without being able to identify them. |
| 8292 | Diversity of two tigers is their difference in space-time; difference of matter is a consequence [Lowe] |
| Full Idea: What really makes for the diversity of two tigers is their difference in space-time location, from which their difference in component matter at any time merely follows as a consequence. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 9.5) | |
| A reaction: I daresay this is how we manage to identify the diversity of a pair of tigers (epistemology), but is that what their diversity consists in (ontology)? That they employ different matter seems relevant. If you feed one, the other stays hungry (causation). |
| 8291 | Individuation principles identify what kind it is; identity criteria distinguish items of the same kind [Lowe] |
| Full Idea: A principle of individuation tells us what is to count as one instance of a given kind, such as one ship. A criterion of identity is what makes for the identity or diversity of items of a given kind, to distinguish this ship from that ship. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 9.5) | |
| A reaction: So individuation picks out type/qualitative identity, and identifying picks out token/numerical identity. This agrees with Idea 7926, but is a shift from the usage Lowe mentions in Idea 8290. Common usage makes the technical terms unclear. |
| 16128 | A 'substance' is an object which doesn't depend for existence on other objects [Lowe] |
| Full Idea: A 'substance' might be defined to be an object which does not depend for its existence upon any other object (where dependency is defined in terms of necessity. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 1.3) | |
| A reaction: I'm inclined to leave out 'substance', which has too much historical baggage, and talk of minimal things having 'identity', and proper things having 'essence'. |
| 8279 | The identity of composite objects isn't fixed by original composition, because how do you identify the origin? [Lowe] |
| Full Idea: It is not at all clear that the identity of a composite object can be fixed by the identity of its original composition, since there are good grounds for claiming that the reverse is in fact the case. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 7.5) | |
| A reaction: That is, how could you identify the origin if you didn't know what it was that had originated? Nice point. See also Idea 8274. Vicars must make sure they baptise the right baby. |
| 8271 | An object 'endures' if it is always wholly present, and 'perdures' if different parts exist at different times [Lowe] |
| Full Idea: The 'endurance' view is that an object persists by being 'wholly present' at more than one time, and the 'perdurance' view is that an object has different temporal parts which exist at different times. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 5) | |
| A reaction: It is tempting to say that only a philosopher would come up with a view as bizarre as the second one. Trying to imagine God's view of time has led to a lot of confusion. Endurance seems to need substance, so bundle views of objects encourage perdurance. |
| 8272 | How can you identify temporal parts of tomatoes without referring to tomatoes? [Lowe] |
| Full Idea: The temporal parts approach to identity appears to be viciously circular, for how are the 'temporal parts' of tomatoes to be individuated and identified save by reference to the very tomatoes of which they are parts? | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 5.3) | |
| A reaction: (This attacks the 'perdurance' view - Idea 8271) Something wrong here. Isn't Lowe begging the question, by assuming that a tomato at an instant IS the tomato? To know what a tomato is, you must spend time with it. |
| 8305 | A clear idea of the kind of an object must precede a criterion of identity for it [Lowe] |
| Full Idea: As Locke clearly understood, one must first have a clear conception of what kind of object one is dealing with in order to extract a criterion of identity for objects of that kind from that conception. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.3) | |
| A reaction: Archaeologist face objects which they can number, remember and take pride in, without having a clue what kind of thing they are dealing with. The two processes may not be entirely distinct. |
| 8290 | One view is that two objects of the same type are only distinguished by differing in matter [Lowe] |
| Full Idea: One venerable tradition, exemplified in Aquinas, has it that matter is the 'principle of individuation', that is, that all that can be guaranteed to distinguish two concrete thing of the same kind is the different matter of which they are composed. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 9.5) | |
| A reaction: This seems to be 'identity-conditions' rather than 'individuation', according to Idea 7926. The problem would be how to identify that particular matter, apart from its composing that particular object. Replacing planks on a ship seems unimportant. |
| 15079 | 'Conceptual' necessity is narrow logical necessity, true because of concepts and logical laws [Lowe] |
| Full Idea: I can accept 'conceptual' necessity, as long as it is only identified with 'narrow' logical necessity. For I take it that the 'conceptually' necessary is that which is true solely in virtue of concepts together with the laws of logic. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 1.4) | |
| A reaction: In the narrow version of logical necessity (Idea 8260) some definitions are required in addition to the mere laws of logic. This implies that the concepts are dependent of definitions, which is a bit restrictive. Aren't we allowed undefined concepts? |
| 16063 | Metaphysical necessity is logical necessity 'broadly construed' [Lowe, by Lynch/Glasgow] |
| Full Idea: Lowe (1998) defines metaphysical necessity in terms of logical necessity 'broadly contrued'. | |
| From: report of E.J. Lowe (The Possibility of Metaphysics [1998]) by Lynch,MP/Glasgow,JM - The Impossibility of Superdupervenience n 3 | |
| A reaction: [I seem to have missed this simple thought in Lowe 1998 - must revisit]. Both metaphysical and logical necessity can be taken as 'true in all possible worlds', but that doesn't make them the same truths. |
| 8260 | Logical necessity can be 'strict' (laws), or 'narrow' (laws and definitions), or 'broad' (all logical worlds) [Lowe] |
| Full Idea: 'Strict' logical necessity is true by the laws of logic alone; 'narrow' logical necessity is true by the laws of logic plus definitions of non-logical terms; 'broad' logical necessity is true in every possible world where the laws of logic hold. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 1.4) | |
| A reaction: Lowe then says the third is close to 'metaphysical' necessity. I am unable to distinguish the third from the first. You can't claim that a logical implication holds in this world, but not in another possible world which has the same rules of implication. |
| 16131 | The metaphysically possible is what acceptable principles and categories will permit [Lowe] |
| Full Idea: What is 'metaphysically' possible hinges …on the question of whether acceptable metaphysical principles and categories permit the existence of some state of affairs. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 1.3) | |
| A reaction: Lowe breezes along with confident assertions like this. I once heard Kit Fine tease him for over-confidence. All you do is work out 'acceptable' principles and categories, and you've cracked it! |
| 8320 | Does every abstract possible world exist in every possible world? [Lowe] |
| Full Idea: Possible worlds, conceived of as abstracta, surely exist 'in every possible world'. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 12) | |
| A reaction: A possible very infinite regress, if a particular possible world is distinguished from another only by being perceived from Actual Word 1 or Actual World 2.. How many possible worlds are there? The standard answer is 'lots', rather than infinity. |
| 8280 | While space may just be appearance, time and change can't be, because the appearances change [Lowe] |
| Full Idea: Although the appearance of distance and so of space may conceivably be no more than an appearance (as Berkeley held), the appearance of change and so of time cannot be no more than appearance - for the appearance of change involves change (in minds). | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 7.9) | |
| A reaction: This would seem to place some sort of limit on idealism. Since it doesn't offer a barrier to solipsism, though, it is not much consolation. We mustn't forget that Parmenides and Zeno of Elea proved that change is just an illusion. |
| 8276 | Properties or qualities are essentially adjectival, not objectual [Lowe] |
| Full Idea: I consider properties or qualities to be essentially adjectival rather than objectual in nature (and the same applies to relations, though they are adjectival to more than one object). | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 7.1) | |
| A reaction: Personally I am inclined to say that properties are either real causal powers (functions of objects?), such as being sharp, or else they are subjective ways of distinguishing things (e.g. colours). Or fictions. 'Adjectival' is too vague. |
| 8289 | The idea that Cartesian souls are made of some ghostly 'immaterial' stuff is quite unwarranted [Lowe] |
| Full Idea: The vulgar notion, propagated by some modern physicalist philosophers, that Cartesian souls are supposed to be made of some sort of ghostly, 'immaterial' stuff - a near contradiction in terms - is quite unwarranted. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 9.5) | |
| A reaction: A nice illustration of the service which can be offered by this database. See Idea 3423 for an illustration of the sort of thing which Lowe is attacking. See Idea 5011 for a quotation from Descartes on the subject. I leave the decision with my visitor... |
| 8299 | Abstractions are non-spatial, or dependent, or derived from concepts [Lowe] |
| Full Idea: There are three conceptions of abstractness: 1) non-spatial entities, the opposite of 'concrete' (e.g. numbers and universals); 2) an entity logically incapable of a separate existence (e.g. an apple's colour); 3) Fregean abstractions from concepts. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.1) | |
| A reaction: [Lowe p.218 explains the third one] Lowe rejects the third one, and it is a moot point whether the second one could actually be classed as an entity (do they have identity-conditions?), so the big issue is the first one. |
| 8306 | You can think of a direction without a line, but a direction existing with no lines is inconceivable [Lowe] |
| Full Idea: Although one can separate 'in thought' a direction from any line of which it is the direction, one cannot conceive of a direction existing in the absence of any line possessing that direction. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.3) | |
| A reaction: Intriguing. If I ask you to imagine a line going in a certain direction, don't you need the direction before you can think of the line? 'That line is going in the wrong direction'. Maybe abstract ideas only exist 'in thought'. Lowe is a realist here. |
| 18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
| Full Idea: In Modus Ponens where the first premise is 'P' and the second 'P→Q', in the first premise P is asserted but in the second it is not. Yet it must mean the same in both premises, or it would be guilty of the fallacy of equivocation. | |
| From: David Bostock (Philosophy of Mathematics [2009], 7.2) | |
| A reaction: This is Geach's thought (leading to an objection to expressivism in ethics, that P means the same even if it is not expressed). |
| 8317 | To cite facts as the elements in causation is to confuse states of affairs with states of objects [Lowe] |
| Full Idea: Philosophers who have advocated facts as being causal relata have confused them with states, such as a stone's being heavy; they are guilty of confusing states of affairs with states of objects. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 11.3) | |
| A reaction: A state of an object can be individuated rather more precisely than a fact or state of affairs. There are, of course, vast numbers of states of objects, but only a few states of affairs, involved in (say) the fall of the Berlin Wall. |
| 8269 | Points are limits of parts of space, so parts of space cannot be aggregates of them [Lowe] |
| Full Idea: Points are limits of parts of space, in which case parts of space cannot be aggregates of them. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 3.9) | |
| A reaction: To try to build space out of points (how many per cc?) is fairly obviously asking for trouble, but Lowe articulates nicely why it is a non-starter. |