77 ideas
| 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) |
| 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. |
| 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) |
| 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. |
| 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. |
| 17518 | Counting 'coin in this box' may have coin as the unit, with 'in this box' merely as the scope [Ayers] |
| Full Idea: If we count the concept 'coin in this box', we could regard coin as the 'unit', while taking 'in this box' to limit the scope. Counting coins in two boxes would be not a difference in unit (kind of object), but in scope. | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Counting') | |
| A reaction: This is a very nice alternative to the Fregean view of counting, depending totally on the concept, and rests more on a natural concept of object. I prefer Ayers. Compare 'count coins till I tell you to stop'. |
| 17516 | If counting needs a sortal, what of things which fall under two sortals? [Ayers] |
| Full Idea: If we accepted that counting objects always presupposes some sortal, it is surely clear that the class of objects to be counted could be designated by two sortals rather than one. | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Realist' vii) | |
| A reaction: His nice example is an object which is both 'a single piece of wool' and a 'sweater', which had better not be counted twice. Wiggins struggles to argue that there is always one 'substance sortal' which predominates. |
| 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. |
| 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) |
| 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. |
| 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) |
| 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) |
| 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. |
| 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. |
| 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) |
| 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) |
| 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) |
| 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) |
| 17520 | Events do not have natural boundaries, and we have to set them [Ayers] |
| Full Idea: In order to know which event has been ostensively identified by a speaker, the auditor must know the limits intended by the speaker. ...Events do not have natural boundaries. | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Concl') | |
| A reaction: He distinguishes events thus from natural objects, where the world, to a large extent, offers us the boundaries. Nice point. |
| 17519 | To express borderline cases of objects, you need the concept of an 'object' [Ayers] |
| Full Idea: The only explanation of the power to produce borderline examples like 'Is this hazelnut one object or two?' is the possession of the concept of an object. | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Counting') |
| 12132 | Indiscernibility is a necessary and sufficient condition for identity [Brody] |
| Full Idea: Enduring objects should be taken as fundamental in an ontology, and for all such objects indiscernibility is both a necessary and sufficient condition for identity. | |
| From: Baruch Brody (Identity and Essence [1980], 3) | |
| A reaction: Brody offers a substantial defence, but I don't find it plausible. Apart from Black's well known twin spheres example (Idea 10195), discernibility is relative to the powers of the observer. Two similar people in the mist aren't thereby identical. |
| 17510 | Speakers need the very general category of a thing, if they are to think about it [Ayers] |
| Full Idea: If a speaker indicates something, then in order for others to catch his reference they must know, at some level of generality, what kind of thing is indicated. They must categorise it as event, object, or quality. Thinking about something needs that much. | |
| From: M.R. Ayers (Individuals without Sortals [1974], Intro) | |
| A reaction: Ayers defends the view that such general categories are required, but not the much narrower sortal terms defended by Geach and Wiggins. I'm with Ayers all the way. 'What the hell is that?' |
| 17522 | We use sortals to classify physical objects by the nature and origin of their unity [Ayers] |
| Full Idea: Sortals are the terms by which we intend to classify physical objects according to the nature and origin of their unity. | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Concl') | |
| A reaction: This is as opposed to using sortals for the initial individuation. I take the perception of the unity to come first, so resemblance must be mentioned, though it can be an underlying (essentialist) resemblance. |
| 17515 | Seeing caterpillar and moth as the same needs continuity, not identity of sortal concepts [Ayers] |
| Full Idea: It is unnecessary to call moths 'caterpillars' or caterpillars 'moths' to see that they can be the same individual. It may be that our sortal concepts reflect our beliefs about continuity, but our beliefs about continuity need not reflect our sortals. | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Realist' vi) | |
| A reaction: Something that metamorphosed through 15 different stages could hardly required 15 different sortals before we recognised the fact. Ayers is right. |
| 15834 | Brody bases sortal essentialism on properties required throughout something's existence [Brody, by Mackie,P] |
| Full Idea: Brody bases sortal essentialism on the notion of a property that an individual must possess throughout its existence if it possesses it at any time in its existence. | |
| From: report of Baruch Brody (Identity and Essence [1980]) by Penelope Mackie - How Things Might Have Been 7.1 | |
| A reaction: Brody tends to treat categories as properties, which I dislike. How do you assess 'must' here? A person may possess a mole throughout life without it being essential. |
| 17511 | Recognising continuity is separate from sortals, and must precede their use [Ayers] |
| Full Idea: The recognition of the fact of continuity is logically independent of the possession of sortal concepts, whereas the formation of sortal concepts is at least psychologically dependent upon the recognition of continuity. | |
| From: M.R. Ayers (Individuals without Sortals [1974], Intro) | |
| A reaction: I take this to be entirely correct. I might add that unity must also be recognised. |
| 17517 | Could the same matter have more than one form or principle of unity? [Ayers] |
| Full Idea: The abstract question arises of whether the same matter could be subject to more than one principle of unity simultaneously, or unified by more than one 'form'. | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Realist' vii) | |
| A reaction: He suggests that the unity of the sweater is destroyed by unravelling, and the unity of the thread by cutting. |
| 17513 | If there are two objects, then 'that marble, man-shaped object' is ambiguous [Ayers] |
| Full Idea: The statue is marble and man-shaped, but so is the piece of marble. So not only are the two objects in the same place, but two marble and man-shaped objects in the same place, so 'that marble, man-shaped object' must be ambiguous or indefinite. | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Prob') | |
| A reaction: It strikes me as basic that it can't be a piece of marble if you subtract its shape, and it can't be a statue if you subtract its matter. To treat a statue as an object, separately from its matter, is absurd. |
| 12140 | Modern emphasis is on properties had essentially; traditional emphasis is on sort-defining properties [Brody] |
| Full Idea: The modern emphasis has been on the connection between essential properties and the properties that an object must have essentially. But traditionally there is also the connection between essential properties and the sort of thing that it is. | |
| From: Baruch Brody (Identity and Essence [1980], 5.6) | |
| A reaction: These are the modal essence and the definitional essence. My view is that he has missed out a crucial third (Aristotelian) view, which is that essences are explanatory. This third view can subsume the other two. |
| 11895 | A sortal essence is a property which once possessed always possessed [Brody, by Mackie,P] |
| Full Idea: Brody bases sortal essentialism on the notion of a property that an individual must possess throughout its existence if it possesses it at any time in its existence. ...'Once an F, always an F'. ...Being a parrot is not a temporary occupation. | |
| From: report of Baruch Brody (Identity and Essence [1980]) by Penelope Mackie - How Things Might Have Been 7.1 | |
| A reaction: Hm. Would being less than fifty metres tall qualify as a sortal essence, for a giraffe or a uranium rod? If there is one thing an essential property should be, it is important. How do we assess importance? By explanatory power! Watch this space. |
| 12141 | Maybe essential properties are those which determine a natural kind? [Brody] |
| Full Idea: We can advance the thesis that all essential properties either determine a natural kind or are part of an essential property that does determine a natural kind. | |
| From: Baruch Brody (Identity and Essence [1980]) | |
| A reaction: A useful clear statement of the view. I am opposed to it, because I take it to be of the essence of Socrates that he is philosophical, but humans are not essentially philosophical, and philosophers are unlikely to be a natural kind. |
| 17523 | Sortals basically apply to individuals [Ayers] |
| Full Idea: Sortals, in their primitive use, apply to the individual. | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Concl') | |
| A reaction: If the sortal applies to the individual, any essence must pertain to that individual, and not to the class it has been placed in. |
| 12137 | De re essentialism standardly says all possible objects identical with a have a's essential properties [Brody] |
| Full Idea: To say that an object a has a property P essentially is to say that it has P, and in all of certain worlds (all possible, all in which something identical with it exists, ...) the object identical with it has P. This is the standard de re interpretation. | |
| From: Baruch Brody (Identity and Essence [1980], 5.4) | |
| A reaction: This view always has to be qualified by excluding trivially necessary properties, but that exclusion shows clearly that the notion of essential is more concerned with non-triviality than it is with necessity. |
| 12142 | Essentially, a has P, always had P, must have had P, and has never had a future without P [Brody] |
| Full Idea: 'a has property P essentially' means 'a has P, a always had P, there is no possible past in which P exists without P, and there is no moment of time at which a has had P and at which there is a possible future in which a exists without P' | |
| From: Baruch Brody (Identity and Essence [1980], 6) | |
| A reaction: This is Brody's own final account of essentialism. This is a carefully qualified form of the view that essential properties are, on the whole, the necessary properties, which view I take to be fundamentally mistaken. |
| 12143 | An object having a property essentially is equivalent to its having it necessarily [Brody] |
| Full Idea: An object having a property essentially is equivalent to its having it necessarily. | |
| From: Baruch Brody (Identity and Essence [1980], 6.1) | |
| A reaction: This strikes me as blatantly false. Personally I am toying with the very unorthodox view that essential properties are not at all necessary, and that something can retain its identity while changing its essential character. A philosopher with Alzheimers. |
| 12144 | Essentialism is justified if the essential properties of things explain their other properties [Brody] |
| Full Idea: The reasonableness of the essentialist hypothesis will be proportional to the extent that we can, as a result, use a's possession of P to explain a's other properties, ...and there is an inability to explain otherwise why a has P. | |
| From: Baruch Brody (Identity and Essence [1980], 6.3) | |
| A reaction: Brody as a rather liberal notion of properties. I would hope that we can do rather more than explain a's non-essential properties. If the non-essential properties were entailed by the essential ones, would they not then also be essential? |
| 12139 | Mereological essentialism says that every part that ensures the existence is essential [Brody] |
| Full Idea: Mereological essentialism (whose leading advocate is Chisholm) says that for every x and y, if x is ever part of y, then y is necessarily such that x is part of y at any time that y exists. | |
| From: Baruch Brody (Identity and Essence [1980], 5.6) | |
| A reaction: This sounds implausible, especially given the transitivity of parthood. Not only are the planks that constitute Theseus's Ship now essential to it, but all the parts of the planks, every last chip, are as well. |
| 17521 | You can't have the concept of a 'stage' if you lack the concept of an object [Ayers] |
| Full Idea: It would be impossible for anyone to have the concept of a stage who did not already possess the concept of a physical object. | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Concl') |
| 17514 | Temporal 'parts' cannot be separated or rearranged [Ayers] |
| Full Idea: Temporally extended 'parts' are still mysteriously inseparable and not subject to rearrangement: a thing cannot be cut temporally in half. | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Prob') | |
| A reaction: A nice warning to anyone accepting a glib analogy between spatial parts and temporal parts. |
| 12135 | Interrupted objects have two first moments of existence, which could be two beginnings [Brody] |
| Full Idea: If 'beginning of existence' meant 'first moment of existence after a period of nonexistence', then objects with interrupted existence have two beginnings of existence. | |
| From: Baruch Brody (Identity and Essence [1980], 4.1) | |
| A reaction: One might still maintain that the first beginning was essential to the object, since that is the event that defined it - and that would clarify the reason why we are supposed to think the origins are essential. I say the origin explains it. |
| 17509 | Some say a 'covering concept' completes identity; others place the concept in the reference [Ayers] |
| Full Idea: Some hold that the 'covering concept' completes the incomplete concept of identity, determining the kind of sameness involved. Others strongly deny the identity itself is incomplete, and locate the covering concept within the necessary act of reference. | |
| From: M.R. Ayers (Individuals without Sortals [1974], Intro) | |
| A reaction: [a bit compressed; Geach is the first view, and Quine the second; Wiggins is somewhere between the two] |
| 17512 | If diachronic identities need covering concepts, why not synchronic identities too? [Ayers] |
| Full Idea: Why are covering concepts required for diachronic identities, when they must be supposed unnecessary for synchronic identities? | |
| From: M.R. Ayers (Individuals without Sortals [1974], 'Prob') |
| 12130 | a and b share all properties; so they share being-identical-with-a; so a = b [Brody] |
| Full Idea: Suppose that a and b have all of their properties in common. a certainly has the property of-being-identical-with-a. So, by supposition, does b. Then a = b. | |
| From: Baruch Brody (Identity and Essence [1980], 1.2) | |
| A reaction: Brody defends this argument, and seems to think that it proves the identity of indiscernibles. As far as I can see it totally begs the question, since we can only assume that both have the property of being-identical-with-a if we have assumed a = b. |
| 12138 | Identity across possible worlds is prior to rigid designation [Brody] |
| Full Idea: Identity across possible worlds is prior to rigid designation. | |
| From: Baruch Brody (Identity and Essence [1980], 5.4) | |
| A reaction: An interesting view. We might stipulate that any possible Aristotle is 'our Aristotle', but you would still need criteria for deciding which possible Aristotle's would qualify. Long-frozen Aristotles, stupid Aristotles, alien Aristotle's, deformed... |
| 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). |