92 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) |
| 9570 | In Field's Platonist view, set theory is false because it asserts existence for non-existent things [Field,H, by Chihara] |
| Full Idea: Field commits himself to a Platonic view of mathematics. The theorems of set theory are held to imply or presuppose the existence of things that don't in fact exist. That is why he believes that these theorems are false. | |
| From: report of Hartry Field (Science without Numbers [1980]) by Charles Chihara - A Structural Account of Mathematics 11.1 | |
| A reaction: I am sympathetic to Field, but this sounds wrong. A response that looks appealing is that maths is hypothetical ('if-thenism') - the truth is in the logical consequences, not in the ontological presuppositions. |
| 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. |
| 10260 | Logical consequence is defined by the impossibility of P and ¬q [Field,H, by Shapiro] |
| Full Idea: Field defines logical consequence by taking the notion of 'logical possibility' as primitive. Hence q is a consequence of P if the conjunction of the items in P with the negation of q is not possible. | |
| From: report of Hartry Field (Science without Numbers [1980]) by Stewart Shapiro - Philosophy of Mathematics 7.2 | |
| A reaction: The question would then be whether it is plausible to take logical possibility as primitive. Presumably only intuition could support it. But then intuition will equally support natural and metaphysical possibilities. |
| 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) |
| 8958 | In Field's version of science, space-time points replace real numbers [Field,H, by Szabó] |
| Full Idea: Field's nominalist version of science develops a version of Newtonian gravitational theory, where no quantifiers range over mathematical entities, and space-time points and regions play the role of surrogates for real numbers. | |
| From: report of Hartry Field (Science without Numbers [1980]) by Zoltán Gendler Szabó - Nominalism 5.1 | |
| A reaction: This seems to be a very artificial contrivance, but Field has launched a programme for rewriting science so that numbers can be omitted. All of this is Field's rebellion against the Indispensability Argument for mathematics. I sympathise. |
| 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. |
| 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. |
| 18221 | 'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [Field,H] |
| Full Idea: There are two approaches to axiomatising geometry. The 'metric' approach uses a function which maps a pair of points into the real numbers. The 'synthetic' approach is that of Euclid and Hilbert, which does without real numbers and functions. | |
| From: Hartry Field (Science without Numbers [1980], 5) |
| 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. |
| 8757 | The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H] |
| Full Idea: There is one and only one serious argument for the existence of mathematical entities, and that is the Indispensability Argument of Putnam and Quine. | |
| From: Hartry Field (Science without Numbers [1980], p.5), quoted by Stewart Shapiro - Thinking About Mathematics 9.1 | |
| A reaction: Personally I don't believe (and nor does Field) that this gives a good enough reason to believe in such things. Quine (who likes 'desert landscapes' in ontology) ends up believing that sets are real because of his argument. Not for me. |
| 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) |
| 18212 | Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H] |
| Full Idea: The most popular approach of nominalistically inclined philosophers is to try to reinterpret mathematics, so that its terms and quantifiers only make reference to, say, physical objects, or linguistic expressions, or mental constructions. | |
| From: Hartry Field (Science without Numbers [1980], Prelim) | |
| A reaction: I am keen on naturalism and empiricism, but only referring to physical objects is a non-starter. I think I favour constructions, derived from the experience of patterns, and abstracted, idealised and generalised. Field says application is the problem. |
| 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) |
| 10261 | The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro] |
| Full Idea: Field argues that to account for the applicability of mathematics, we need to assume little more than the possibility of the mathematics, not its truth. | |
| From: report of Hartry Field (Science without Numbers [1980]) by Stewart Shapiro - Philosophy of Mathematics 7.2 | |
| A reaction: Very persuasive. We can apply chess to real military situations, provided that chess isn't self-contradictory (or even naturally impossible?). |
| 18218 | Hilbert explains geometry, by non-numerical facts about space [Field,H] |
| Full Idea: Facts about geometric laws receive satisfying explanations, by the intrinsic facts about physical space, i.e. those laid down without reference to numbers in Hilbert's axioms. | |
| From: Hartry Field (Science without Numbers [1980], 3) | |
| A reaction: Hilbert's axioms mention points, betweenness, segment-congruence and angle-congruence (Field 25-26). Field cites arithmetic and geometry (as well as Newtonian mechanics) as not being dependent on number. |
| 9623 | Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H] |
| Full Idea: Field needs the notion of logical consequence in second-order logic, but (since this is not recursively axiomatizable) this is a semantical notion, which involves the idea of 'true in all models', a set-theoretic idea if there ever was one. | |
| From: comment on Hartry Field (Science without Numbers [1980], Ch.4) by James Robert Brown - Philosophy of Mathematics | |
| A reaction: Brown here summarises a group of critics. Field was arguing for modern nominalism, that actual numbers could (in principle) be written out of the story, as useful fictions. Popper's attempt to dump induction seemed to need induction. |
| 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) |
| 18215 | It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H] |
| Full Idea: No clear explanation of the idea that the conclusion was 'implicitly contained in' the premises was ever given, and I do not believe that any clear explanation is possible. | |
| From: Hartry Field (Science without Numbers [1980], 1) |
| 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) |
| 18210 | Why regard standard mathematics as truths, rather than as interesting fictions? [Field,H] |
| Full Idea: Why regard the axioms of standard mathematics as truths, rather than as fictions that for a variety of reasons mathematicians have become interested in? | |
| From: Hartry Field (Science without Numbers [1980], p.viii) |
| 18216 | Abstractions can form useful counterparts to concrete statements [Field,H] |
| Full Idea: Abstract entities are useful because we can use them to formulate abstract counterparts of concrete statements. | |
| From: Hartry Field (Science without Numbers [1980], 3) | |
| A reaction: He defends the abstract statements as short cuts. If the concrete statements were 'true', then it seems likely that the abstract counterparts will also be true, which is not what fictionalism claims. |
| 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). |
| 18214 | Mathematics is only empirical as regards which theory is useful [Field,H] |
| Full Idea: Mathematics is in a sense empirical, but only in the rather Pickwickian sense that is an empirical question as to which mathematical theory is useful. | |
| From: Hartry Field (Science without Numbers [1980], 1) | |
| A reaction: Field wants mathematics to be fictions, and not to be truths. But can he give an account of 'useful' that does not imply truth? Only in a rather dubiously pragmatist way. A novel is not useful. |
| 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) |
| 458 | Nothing could come out of nothing, and existence could never completely cease [Empedocles] |
| Full Idea: From what in no wise exists, it is impossible for anything to come into being; for Being to perish completely is incapable of fulfilment and unthinkable. | |
| From: Empedocles (fragments/reports [c.453 BCE], B012), quoted by Anon (Lyc) - On Melissus 975b1-4 |
| 5112 | Empedocles says things are at rest, unless love unites them, or hatred splits them [Empedocles, by Aristotle] |
| Full Idea: Empedocles claims that things are alternately changing and at rest - that they are changing whenever love is creating a unity out of plurality, or hatred is creating plurality out of unity, and they are at rest in the times in between. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Physics 250b26 | |
| A reaction: I suppose one must say that this an example of Ruskin's 'pathetic fallacy' - reading human emotions into the cosmos. Being constructive little creatures, we think goodness leads to construction. I'm afraid Empedocles is just wrong. |
| 18211 | You can reduce ontological commitment by expanding the logic [Field,H] |
| Full Idea: One can often reduce one's ontological commitments by expanding one's logic. | |
| From: Hartry Field (Science without Numbers [1980], p.ix) | |
| A reaction: I don't actually understand this idea, but that's never stopped me before. Clearly, this sounds like an extremely interesting thought, and hence I should aspire to understand it. So I do aspire to understand it. First, how do you 'expand' a logic? |
| 8959 | Field presumes properties can be eliminated from science [Field,H, by Szabó] |
| Full Idea: Field regards the eliminability of apparent reference to properties from the language of science as a foregone result. | |
| From: report of Hartry Field (Science without Numbers [1980]) by Zoltán Gendler Szabó - Nominalism 5.1 n50 | |
| A reaction: Field is a nominalist who also denies the existence of mathematics as part of science. He has a taste for ontological 'desert landscapes'. I have no idea what a property really is, so I think he is on to something. |
| 18213 | Abstract objects are only applicable to the world if they are impure, and connect to the physical [Field,H] |
| Full Idea: To be able to apply any postulated abstract entities to the physical world, we need impure abstact entities, e.g. functions that map physical objects into pure abstract objects. | |
| From: Hartry Field (Science without Numbers [1980], 1) | |
| A reaction: I am a fan of 'impure metaphysics', and this pinpoints my reason very nicely. |
| 13209 | There is no coming-to-be of anything, but only mixing and separating [Empedocles, by Aristotle] |
| Full Idea: Empedocles says there is no coming-to-be of anything, but only a mingling and a divorce of what has been mingled. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 314b08 | |
| A reaction: Aristotle comments that this prevents Empedocleans from distinguishing between superficial alteration and fundamental change of identity. Presumably, though, that wouldn't bother them. |
| 457 | Substance is not created or destroyed in mortals, but there is only mixing and exchange [Empedocles] |
| Full Idea: There is no creation of substance in any one of mortal existence, nor any end in execrable death, but only mixing and exchange of what has been mixed. | |
| From: Empedocles (fragments/reports [c.453 BCE], B008), quoted by Plutarch - 74: Reply to Colotes 1111f | |
| A reaction: also Aristotle 314b08 |
| 462 | One vision is produced by both eyes [Empedocles] |
| Full Idea: One vision is produced by both eyes | |
| From: Empedocles (fragments/reports [c.453 BCE], B088), quoted by Strabo - works 8.364.3 |
| 18222 | Beneath every extrinsic explanation there is an intrinsic explanation [Field,H] |
| Full Idea: A plausible methodological principle is that underlying every good extrinsic explanation there is an intrinsic explanation. | |
| From: Hartry Field (Science without Numbers [1980], 5) | |
| A reaction: I'm thinking that Hartry Field is an Aristotelian essentialist, though I bet he would never admit it. |
| 22765 | Wisdom and thought are shared by all things [Empedocles] |
| Full Idea: Wisdom and power of thought, know thou, are shared in by all things. | |
| From: Empedocles (fragments/reports [c.453 BCE]), quoted by Sextus Empiricus - Against the Logicians (two books) II.286 | |
| A reaction: Sextus quotes this, saying that it is 'still more paradoxical', and that it explicitly includes plants. This may mean that Empedocles was not including inanimate matter. |
| 1524 | For Empedocles thinking is almost identical to perception [Empedocles, by Theophrastus] |
| Full Idea: Empedocles assumes that thinking is either identical to or very similar to sense-perception. | |
| From: report of Empedocles (fragments/reports [c.453 BCE], A86) by Theophrastus - On the Senses 9 | |
| A reaction: Not to be sniffed at. We can, of course, control our thinking (though we can't control the controller) and we contemplate abstractions, but that might be seen as a sort of perception. Vision is not as visual as we think. |
| 9917 | 'Abstract' is unclear, but numbers, functions and sets are clearly abstract [Field,H] |
| Full Idea: The term 'abstract entities' may not be entirely clear, but one thing that does seem clear is that such alleged entities as numbers, functions and sets are abstract. | |
| From: Hartry Field (Science without Numbers [1980], p.1), quoted by JP Burgess / G Rosen - A Subject with No Object I.A.1.a | |
| A reaction: Field firmly denies the existence of such things. Sets don't seem a great problem, if the set is a herd of elephants, but the null and singleton sets show up the difficulties. |
| 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). |
| 552 | Empedocles said good and evil were the basic principles [Empedocles, by Aristotle] |
| Full Idea: Empedocles was the first to give evil and good as principles. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Metaphysics 985a | |
| A reaction: Once you start to think that good and evil will only matter if they have causal powers, it is an easy step to the idea of a benevolent god, and a satanic anti-god. Otherwise the 'principles' could be ignored. |
| 589 | 'Nature' is just a word invented by people [Empedocles] |
| Full Idea: Nature is but a word of human framing. | |
| From: Empedocles (fragments/reports [c.453 BCE], B008), quoted by Aristotle - Metaphysics 1015a |
| 21823 | The principle of 'Friendship' in Empedocles is the One, and is bodiless [Empedocles, by Plotinus] |
| Full Idea: In Empedocles we have a dividing principle, 'Strife', set against 'Friendship' - which is the One and is to him bodiless, while the elements represent matter. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Plotinus - The Enneads 5.1.09 | |
| A reaction: The first time I've seen the principle of Love in Empedocles identified with the One of Parmenides. Plotinus is a trustworthy reporter, I think, because he was well read, and had access to lost texts. |
| 2680 | Empedocles said that there are four material elements, and two further creative elements [Empedocles, by Aristotle] |
| Full Idea: Empedocles holds that the corporeal elements are four, but that all the elements, including those which create motion, are six in number. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 314a16 |
| 6002 | Empedocles says bone is water, fire and earth in ratio 2:4:2 [Empedocles, by Inwood] |
| Full Idea: Empedocles used numerical ratios to explain different kinds of matter; for example, bone is two parts water, four parts fire, two parts earth; and blood is an equal blend of all four elements. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Brad Inwood - Empedocles | |
| A reaction: Why isn't the ration 1:2:1? This presumably shows the influence of Pythagoras (who had also been based in Italy, like Empedocles), as well as that of the earlier naturalistic philosophers. It was a very good theory, though wrong. |
| 13207 | Fire, Water, Air and Earth are elements, being simple as well as homoeomerous [Empedocles, by Aristotle] |
| Full Idea: Empedocles says that Fire, Water, Air and Earth are four elements, and are thus 'simple' rather than flesh, bone and bodies which, like these, are 'homoeomeries'. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 314a26 | |
| A reaction: The translation is not quite clear. I take it that flesh and bone may look simple, because they are homoeomerous, but they are not really - but what is his evidence for that? Compare Idea 13208. |
| 459 | All change is unity through love or division through hate [Empedocles] |
| Full Idea: These elements never cease their continuous exchange, sometimes uniting under the influence of Love, so that all become One, at other times again moving apart through the hostile force of Hate. | |
| From: Empedocles (fragments/reports [c.453 BCE], B017), quoted by Simplicius - On Aristotle's 'Physics' 158.1- |
| 13218 | The elements combine in coming-to-be, but how do the elements themselves come-to-be? [Aristotle on Empedocles] |
| Full Idea: Empedocles says it is evident that all the other bodies down to the 'elements' have their coming-to-be and their passing-away: but it is not clear how the 'elements' themselves, severally in their aggregated masses, come-to-be and pass-away. | |
| From: comment on Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 325b20 | |
| A reaction: Presumably the elements are like axioms - and are just given. How do electrons and quarks come-to-be? |
| 13225 | Love and Strife only explain movement if their effects are distinctive [Aristotle on Empedocles] |
| Full Idea: It is not an adequate explanation to say that 'Love and Strife set things moving', unless the very nature of Love is a movement of this kind and the very nature of Strife a movement of that kind. | |
| From: comment on Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 333b23 | |
| A reaction: I take this to be of interest for showing Aristotle's quest for explanations, and his unwillingness to be fobbed off with anything superficial. I take a task of philosophy to be to push explanations further than others wish to go. |
| 460 | If the one Being ever diminishes it would no longer exist, and what could ever increase it? [Empedocles] |
| Full Idea: Besides these elements, nothing else comes into being, nor does anything cease. For if they had been perishing continuously, they would Be no more; and what could increase the Whole? And whence could it have come? | |
| From: Empedocles (fragments/reports [c.453 BCE], B017), quoted by Simplicius - On Aristotle's 'Physics' 158.1- |
| 18223 | In theories of fields, space-time points or regions are causal agents [Field,H] |
| Full Idea: According to theories that take the notion of a field seriously, space-time points or regions are fully-fledge causal agents. | |
| From: Hartry Field (Science without Numbers [1980], n 23) |
| 18220 | Both philosophy and physics now make substantivalism more attractive [Field,H] |
| Full Idea: In general, it seems to me that recent developments in both philosophy and physics have made substantivalism a much more attractive position than it once was. | |
| From: Hartry Field (Science without Numbers [1980], 4) | |
| A reaction: I'm intrigued as to what philosophical developments are involved in this. The arrival of fields is the development in physics. |
| 18219 | Relational space is problematic if you take the idea of a field seriously [Field,H] |
| Full Idea: The problem of the relational view of space is especially acute in the context of physical theories that take the notion of a field seriously, e.g. classical electromagnetic theory. | |
| From: Hartry Field (Science without Numbers [1980], 4) | |
| A reaction: In the Leibniz-Clarke debate I sided with the Newtonian Clarke (defending absolute space), and it looks like modern science agrees with me. Nothing exists purely as relations. |
| 5090 | Maybe bodies are designed by accident, and the creatures that don't work are destroyed [Empedocles, by Aristotle] |
| Full Idea: Is it just an accident that teeth and other parts of the body seem to have some purpose, and creatures survive because they happen to be put together in a useful way? Everything else has been destroyed, as Empedocles says of his 'cow with human head'. | |
| From: report of Empedocles (fragments/reports [c.453 BCE], 61) by Aristotle - Physics 198b29 | |
| A reaction: Good grief! Has no one ever noticed that Empedocles proposed the theory of evolution? It isn't quite natural selection, because we aren't told what does the 'destroying', but it is a little flash of genius that was quietly forgotten. |
| 461 | God is a pure, solitary, and eternal sphere [Empedocles] |
| Full Idea: God is equal in all directions to himself and altogether eternal, a rounded Sphere enjoying a circular solitude. | |
| From: Empedocles (fragments/reports [c.453 BCE], B028), quoted by John Stobaeus - Anthology 1.15.2 |
| 466 | God is pure mind permeating the universe [Empedocles] |
| Full Idea: God is mind, holy and ineffable, and only mind, which darts through the whole cosmos with its swift thought. | |
| From: Empedocles (fragments/reports [c.453 BCE], B134), quoted by Ammonius - On 'De Interpretatione' 4.5.249.6 |
| 1719 | In Empedocles' theory God is ignorant because, unlike humans, he doesn't know one of the elements (strife) [Aristotle on Empedocles] |
| Full Idea: It is a consequence of Empedocles' view that God is the most unintelligent thing, for he alone is ignorant of one of the elements, namely strife, whereas mortal creatures are familiar with them all. | |
| From: comment on Empedocles (fragments/reports [c.453 BCE]) by Aristotle - De Anima 410b08 |
| 1522 | It is wretched not to want to think clearly about the gods [Empedocles] |
| Full Idea: Wretched is he who cares not for clear thinking about the gods. | |
| From: Empedocles (fragments/reports [c.453 BCE], B132), quoted by Clement - Miscellanies 5.140.5.1 |