108 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) |
| 5728 | The concept of truth was originated by the senses [Lucretius] |
| Full Idea: The concept of truth was originated by the senses. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], IV.479) | |
| A reaction: This is a refreshing challenge to the modern view of truth, which seems entirely entangled with language. Truth seems a useful concept when discussing the workings of an animal mind. As you get closer to an object, you see it more 'truly'. |
| 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. |
| 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) |
| 16185 | Causality indicates which properties are real [Cartwright,N] |
| Full Idea: Causality is a clue to what properties are real. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 9.3) | |
| A reaction: An interesting variant on the Shoemaker proposal that properties actually are causal. I'm not sure that there is anything more to causality that the expression in action of properties, which I take to be powers. Structures are not properties. |
| 5702 | The senses are much the best way to distinguish true from false [Lucretius] |
| Full Idea: What can be a surer guide to the distinction of true from false than our own senses? | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.700) | |
| A reaction: This doesn't say they are the only guide, which leaves room for guides such as what is consistent or self-evident or inferred. There is enough here, though, to show that the Epicureans were empiricists in a fairly modern way. |
| 5729 | If the senses are deceptive, reason, which rests on them, is even worse [Lucretius] |
| Full Idea: The structure of your reasoning must be rickety and defective, if the senses on which it rests are themselves deceptive. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], IV.518) | |
| A reaction: This strikes me as one of the most basic tenets of empiricism. It denies the existence of 'pure' reason, and instead asserts that it is built out of complex and abstracted sense experience, which makes it ultimately a second-class citizen. |
| 5697 | The only possible standard for settling doubts is the foundation of the senses [Lucretius] |
| Full Idea: If a belief resting directly on the foundation of the senses is not valid, there will be no standard to which we can refer any doubt on obscure questions for rational confirmation. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.422) | |
| A reaction: A classic statement of empiricist foundationalism. The Epicureans don't appear to have any time for a priori truths at all. I wonder if they settled mathematical disputes by counting objects and drawing diagrams? |
| 5727 | Most supposed delusions of the senses are really misinterpretations by the mind [Lucretius] |
| Full Idea: Paradoxical experiences (such a dreams and illusions) cannot shake our faith in the senses. Most of the illusion is due to the mental assumptions we ourselves superimpose, so that things not perceived by the senses pass for perceptions. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], IV.462) | |
| A reaction: Some misinterpretations of the senses, such as thinking a square tower round, are the result of foolish lack of judgement, but actual delusions within the senses, such as a ringing in the ears, or a pain in a amputated leg, seem like real sense failures. |
| 5714 | Even simple facts are hard to believe at first hearing [Lucretius] |
| Full Idea: No fact is so simple that it is not harder to believe than to doubt at the first presentation. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.1022) | |
| A reaction: Hence induction is just 'drumming it in' until you come to believe it. There are good evolutionary reasons why we should be like this, because we would otherwise believe all sorts of silly half-perceptions in the gloaming. |
| 16182 | Two main types of explanation are by causes, or by citing a theoretical framework [Cartwright,N] |
| Full Idea: In explaining a phenomenon one can cite the causes of that phenomenon; or one can set the phenomenon in a general theoretical framework. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 4.1) | |
| A reaction: The thing is, you need to root an explanation in something taken as basic, and theoretical frameworks need further explanation, whereas causes seem to be basic. |
| 16184 | An explanation is a model that fits a theory and predicts the phenomenological laws [Cartwright,N] |
| Full Idea: To explain a phenomenon is to find a model that fits it into the basic framework of the theory and that thus allows us to derive analogues for the messy and complicated phenomenological laws that are true of it. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 8.3) | |
| A reaction: This summarises the core of her view in this book. She is after models rather than laws, and the models are based on causes. |
| 16167 | Laws get the facts wrong, and explanation rests on improvements and qualifications of laws [Cartwright,N] |
| Full Idea: We explain by ceteris paribus laws, by composition of causes, and by approximations that improve on what the fundamental laws dictate. In all of these cases the fundamental laws patently do not get the facts right. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
| A reaction: It is rather headline-grabbing to say in this case that laws do not get the facts right. If they were actually 'wrong' and 'lied', there wouldn't be much point in building explanations on them. |
| 16169 | Laws apply to separate domains, but real explanations apply to intersecting domains [Cartwright,N] |
| Full Idea: When different kinds of causes compose, we want to explain what happens in the intersection of different domains. But the laws we use are designed only to tell truly what happens in each domain separately. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
| A reaction: Since presumably the laws are discovered through experiments which try to separate out a single domain, in those circumstances they actually are true, so they don't 'lie'. |
| 16176 | Covering-law explanation lets us explain storms by falling barometers [Cartwright,N] |
| Full Idea: Much criticism of the original covering-law model objects that it lets in too much. It seems we can explain Henry's failure to get pregnant by his taking birth control pills, and we can explain the storm by the falling barometer. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.0) | |
| A reaction: I take these examples to show that true explanations must be largely causal in character. The physicality of causation is what matters, not 'laws'. I'd say the same of attempts to account for causation through counterfactuals. |
| 16177 | I disagree with the covering-law view that there is a law to cover every single case [Cartwright,N] |
| Full Idea: Covering-law theorists tend to think that nature is well-regulated; in the extreme, that there is a law to cover every case. I do not. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.2) | |
| A reaction: The problem of coincidence is somewhere at the back of this thought. Innumerable events have their own explanations, but it is hard to explain their coincidence (see Aristotle's case of bumping into a friend in the market). |
| 16180 | You can't explain one quail's behaviour by just saying that all quails do it [Cartwright,N] |
| Full Idea: 'Why does that quail in the garden bob its head up and down in that funny way whenever it walks?' …'Because they all do'. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 3.5) | |
| A reaction: She cites this as an old complaint against the covering-law model of explanation. It captures beautifully the basic error of the approach. We want to know 'why', rather than just have a description of the pattern. 'They all do' is useful information. |
| 16171 | The covering law view assumes that each phenomenon has a 'right' explanation [Cartwright,N] |
| Full Idea: The covering-law account supposes that there is, in principle, one 'right' explanation for each phenomenon. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
| A reaction: Presumably the law is held to be 'right', but there must be a bit of flexibility in describing the initial conditions, and the explanandum itself. |
| 16183 | In science, best explanations have regularly turned out to be false [Cartwright,N] |
| Full Idea: There are a huge number of cases in the history of science where we now know our best explanations were false. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 5.3) | |
| A reaction: [She cites Laudan 1981 for this] The Ptolemaic system and aether are the standard example cited for this. I believe strongly in the importance of best explanation. Only a fool would just accept the best explanation available. Coherence is needed. |
| 5718 | The mind is in the middle of the breast, because there we experience fear and joy [Lucretius] |
| Full Idea: The guiding principle of the whole body is the mind or intellect, which is firmly lodged in the mid-region of the breast. Here is felt fear and alarm, and the caressing pulse of joy. Here, then is the seat of the intellect and mind. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], III.140) | |
| A reaction: Even by this date thinking people were not clear that the mind is in the brain. They paid insufficient attention to head injuries. The emotions are felt to have a location, but intellect and principles are not. |
| 5717 | The mind is a part of a man, just like a hand or an eye [Lucretius] |
| Full Idea: First, I maintain that the mind, which we often call the intellect, the seat of guidance and control of life, is part of a man, no less than hand or foot or eyes are parts of a whole living creature. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], III.95) | |
| A reaction: Presumably Lucretius asserts this because some people were denying it. Sounds like common sense to me. The only reason I can see for anyone denying what he says is if they are desperate to survive death. |
| 21387 | The separate elements and capacities of a mind cannot be distinguished [Lucretius] |
| Full Idea: No single element [of the soul] can be separated, nor can their capacities be divided spatially; they are like the multiple powers of a single body | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], III.262), quoted by A.A. Long - Hellenistic Philosophy 2.7 | |
| A reaction: It is interesting that this comes from someone with a strongly physicalist view of the mind (though not, if I recall, focusing on the brain). He is still totally impressed by the unified phenomenology of mental experience. He is an empiricist. |
| 5709 | The actions of the mind are not determinate and passive, because atoms can swerve [Lucretius] |
| Full Idea: The fact that the mind itself has no internal necessity to determine its every act and compel it to suffer in helpless passivity - this is due to the slight swerve of the atoms at no determinate time or place. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.294) | |
| A reaction: No one likes this proposal much, but it is very intriguing. The Epicureans had seen a problem, one which doesn't bother me much. If, nowadays, you are a reductive physicalist who believes in free will, you have a philosophical nightmare ahead of you. |
| 5695 | Only bodies can touch one another [Lucretius] |
| Full Idea: Nothing can touch or be touched except body. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.303) | |
| A reaction: This is the key objection to interactionism, and the main reason why the atomists have a thoroughly material view of the mind. It is an induction from a very large number of instances, but the argument is not, of course, conclusive. |
| 5711 | The earth is and always has been an insentient being [Lucretius] |
| Full Idea: The earth is and always has been an insentient being. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.658) | |
| A reaction: The fact that Epicurus needs to deny this shows that some idea close to panpsychism must still have been around in his time. He is discussing gods at the time, so maybe pantheism was the view being attacked, but that is a subset of panpsychism. |
| 5712 | Particles may have sensation, but eggs turning into chicks suggests otherwise [Lucretius] |
| Full Idea: There is the possibility that particles have senses like those of an animate being as a whole, …but from the fact that we perceive eggs turning into live fledglings, we may infer that sense can be generated from the insentient. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.914) | |
| A reaction: He gives other arguments for his view. The egg example is not a strong argument, but is precisely our puzzle of how consciousness can emerge from the process of evolution, and natural selection makes dualism look unlikely. |
| 5719 | The mind moves limbs, wakes the body up, changes facial expressions, which involve touch [Lucretius] |
| Full Idea: Mind and spirit are both composed of matter, as we see them propelling limbs, rousing the body from sleep, changing the expression of the face, and guiding the whole man - activities which clearly involves touch, which involves matter. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], III.164) | |
| A reaction: This is the inverse of Descartes' interaction problem, and strikes me as a straightforward common sense truth. However, if you believe in spiritual gods, this gives you a model for the interaction (however mysterious) of matter and spirit. |
| 5724 | Lions, foxes and deer have distinct characters because their minds share in their bodies [Lucretius] |
| Full Idea: Why are lions ferocious, foxes crafty, and deer timid? It can only be because the mind always shares in the specific growth of the body according to its seed and breed. If it were immortal and reincarnated, living things would have jumbled characters. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], III.743) | |
| A reaction: A nice argument which I have not encountered in modern times. Of course, even Descartes admits that the mind is intermingled with the body, but it seems that the essential character of a mind is dictated by the body. |
| 5713 | You needn't be made of laughing particles to laugh, so why not sensation from senseless seeds? [Lucretius] |
| Full Idea: One can laugh without being composed of laughing particles, ..so why cannot the things that we see gifted with sensation be compounded of seeds that are wholly senseless? | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.988) | |
| A reaction: Lovely argument! You might feel driven to panpsychism in your desperation to explain the 'weirdness' of consciousness, but it would be mad to attribute laughter to basic matter, so comedy has to 'emerge' at some point. |
| 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). |
| 6611 | One man's meat is another man's poison [Lucretius] |
| Full Idea: What is food to one may be literally poison to others. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], IV.638) | |
| A reaction: This seems to be the origin of the well-known saying. This is not relativism of perception, but a relativism of how individuals actually respond to the world. It sums up the position with, say, the operas of Wagner. |
| 5730 | Our bodies weren't created to be used; on the contrary, their creation makes a use possible [Lucretius] |
| Full Idea: Nothing in our bodies was born in order that we might be able to use it, but the thing born creates the use. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], IV.834) | |
| A reaction: This remark (strongly opposed to Aristotle's view of human function and nature) raises the obvious question of why the body is so very useful for staying alive. Most of its uses are not random. Lucretius would abandon this view if he read Darwin. |
| 5726 | The dead are no different from those who were never born [Lucretius] |
| Full Idea: One who no longer is cannot suffer, or differ in any way from one who has never been born. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], III.867) | |
| A reaction: There is a special kind of pain in being poor if you were once rich, which is not suffered by those who experience only poverty. Lucretius is right, but we are concerned with how we feel now, not with how we won't feel once dead. |
| 5705 | Nature only wants two things: freedom from pain, and pleasure [Lucretius] |
| Full Idea: Nature only clamours for two things, a body free from pain, a mind released from worry and fear for the enjoyment of pleasurable sensation. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.21) | |
| A reaction: I can't help agreeing with those (like Aristotle) who consider this a very demeaning view of human life. See Idea 99. Bentham agrees with Lucretius (Idea 3777). I think they are largely right, but not entirely. Other motives are possible than sensations. |
| 5716 | Nature runs the universe by herself without the aid of gods [Lucretius] |
| Full Idea: Nature is free and uncontrolled by proud masters and runs the universe by herself without the aid of gods. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.1094) | |
| A reaction: A nice remark. This apparent personification of nature implies the application of laws to an essentially passive reality. See Idea 5442 and Nature|Laws of Nature|Essentialism for a different view. |
| 5704 | There can be no centre in infinity [Lucretius] |
| Full Idea: There can be no centre in infinity. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.1069) | |
| A reaction: This is highly significant, because if we can establish that the universe is infinite (as Epicurus believes), it follows that the human race cannot be at the centre of it, as the Ptolemaic/medieval view proposed. |
| 5703 | The universe must be limitless, since there could be nothing outside to limit it [Lucretius] |
| Full Idea: The universe is not bounded in any direction. If it were, it would necessarily have a limit somewhere, but a thing cannot have a limit unless there is something outside to limit it. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.959) | |
| A reaction: This is a subtler argument than the mere enquiry about why you would have to stop at the end of the universe. It still seems a nice argument, though Einstein's curvature of space seems to have thwarted it. |
| 5693 | Everything is created and fed by nature from atoms, and they return to atoms in death [Lucretius] |
| Full Idea: The ultimate realities of heaven and the gods are the atoms, from which nature creates all things and increases and feeds them, and into which, when they perish, nature again resolves them. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.46) | |
| A reaction: Sounds right to me. Nothing in modern particle theory and string theory has refuted this claim. But what makes the atoms move, and what makes them combine in an orderly way? Is the orderliness of atoms made of atoms? Essences? |
| 5701 | If an object is infinitely subdivisible, it will be the same as the whole universe [Lucretius] |
| Full Idea: If there are no atoms, the smallest bodies will have infinite parts, since they can be endlessly halved. ..But then there will be no difference between the smallest thing and the whole universe, as they will equally have infinite parts. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.620) | |
| A reaction: Another argument which remains effective even now. There must surely (intuitively) be more divisions possible in a large object than in a small one? Unless of course there were many different sizes of infinity…. See Cantor. |
| 5708 | In downward motion, atoms occasionally swerve slightly for no reason [Lucretius] |
| Full Idea: When atoms are travelling straight down through empty space by their own weight, at quite indeterminate times and places they swerve ever so little from their course. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.217) | |
| A reaction: Never a popular theory because it seems to breach the Principle of Sufficient Reason (Ideas 306 + 3646). This seems to be the beginning of a strong need for the concept of free will, and an underlying explanation. Most thinkers put mind outside nature. |
| 16175 | A cause won't increase the effect frequency if other causes keep interfering [Cartwright,N] |
| Full Idea: A cause ought to increase the frequency of the effect, but this fact may not show up in the probabilities if other causes are at work. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 1.1) | |
| A reaction: [She cites Patrick Suppes for this one] Presumably in experimental situations you can weed out the interference, but that threatens to eliminate mere 'probability' entirely. |
| 6781 | There are fundamental explanatory laws (false!), and phenomenological laws (regularities) [Cartwright,N, by Bird] |
| Full Idea: Nancy Cartwright distinguishes between 'fundamental explanatory laws', which we should not believe, and 'phenomenological laws', which are regularities established on the basis of observation. | |
| From: report of Nancy Cartwright (How the Laws of Physics Lie [1983]) by Alexander Bird - Philosophy of Science Ch.4 | |
| A reaction: The distinction is helpful, so that we can be clearer about what everyone is claiming. We can probably all agree on the phenomenological laws, which are epistemological. Personally I claim truth for the best fundamental explanatory laws. |
| 16166 | Laws of appearances are 'phenomenological'; laws of reality are 'theoretical' [Cartwright,N] |
| Full Idea: Philosophers distinguish phenomenological from theoretical laws. Phenomenological laws are about appearances; theoretical ones are about the reality behind the appearances. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
| A reaction: I'm suspecting that Humeans only really believe in the phenomenological kind. I'm only interested in the theoretical kind, and I take inference to the best explanation to be the bridge between the two. Cartwright rejects the theoretical laws. |
| 16179 | Good organisation may not be true, and the truth may not organise very much [Cartwright,N] |
| Full Idea: There is no reason to think that the principles that best organise will be true, nor that the principles that are true will organise much. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.5) | |
| A reaction: This is aimed at the Mill-Ramsey-Lewis account of laws, as axiomatisations of the observed patterns in nature. |
| 17004 | Nothing can break the binding laws of eternity [Lucretius] |
| Full Idea: Nothing has power to break the binding laws of eternity. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], 5.56) | |
| A reaction: This seems to be virtually the only remark from the ancient world suggesting that there are 'laws' of nature, so I'm guessing it is a transient metaphor, not a theory about nature. 'Even the gods must bow to necessity'. |
| 16170 | To get from facts to equations, we need a prepared descriptions suited to mathematics [Cartwright,N] |
| Full Idea: To get from a detailed factual knowledge of a situation to an equation, we must prepare the description of the situation to meet the mathematical needs of the theory. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
| A reaction: She is clearly on to something here, as Galileo is blatantly wrong in his claim that the book of nature is written in mathematics. Mathematics is the best we can manage in getting a grip on the chaos. |
| 16181 | Simple laws have quite different outcomes when they act in combinations [Cartwright,N] |
| Full Idea: For explanation simple laws must have the same form when they act together as when they act singly. ..But then what the law states cannot literally be true, for the consequences that occur if it acts alone are not what occurs when they act in combination. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 3.6) | |
| A reaction: This is Cartwright's basic thesis. Her point is that the laws 'lie', because they claim to predict a particular outcome which never ever actually occurs. She says we could know all the laws, and still not be able to explain anything. |
| 16178 | There are few laws for when one theory meets another [Cartwright,N] |
| Full Idea: Where theories intersect, laws are usually hard to come by. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.3) | |
| A reaction: There are attempts at so-called 'bridge laws', to get from complex theories to simple ones, but her point is well made about theories on the same 'level'. |
| 5706 | Atoms move themselves [Lucretius] |
| Full Idea: Atoms move themselves. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.133) | |
| A reaction: Something has to move itself, I suppose, but then that could be psuché, giving us free will (see Idea 1424). Why does Epicurus need the 'swerve' if atoms are self-movers? See Idea 5708. |
| 5696 | If there were no space there could be no movement, or even creation [Lucretius] |
| Full Idea: We see movement everywhere, but if there were no empty space, things would be denied the power of movement - or rather, they could not possibly have come into existence, embedded as they would have been in motionless matter. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.342) | |
| A reaction: This still seems a good argument, if reality is made of particles. People can move in a crowd until it becomes too dense. |
| 5700 | It is quicker to break things up than to assemble them [Lucretius] |
| Full Idea: Anything can be more speedily disintegrated than put together again. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.558) | |
| A reaction: Clearly the concept of entropy was around long before anyone tried to give a systematic or mathematical account of it. |
| 5698 | We can only sense time by means of movement, or its absence [Lucretius] |
| Full Idea: It must not be claimed that anyone can sense time by itself apart from the movement of things or their restful immobility. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.465) | |
| A reaction: This seems a remarkably Einsteinian remark, though he is only talking of the epistemology of the matter, not the ontology. We are not far from the concept of space-time here. |
| 5715 | This earth is very unlikely to be the only one created [Lucretius] |
| Full Idea: It is in the highest degree unlikely that this earth and sky is the only one to have been created. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.1057) | |
| A reaction: I can only admire the science fiction imagination of this, which roughly agrees with the assessment of modern cosmologists. We think imagination was cramped in the ancient world, and now wanders free - but that is not so. |
| 5694 | Nothing can be created by divine power out of nothing [Lucretius] |
| Full Idea: In studying the workings of nature, our starting-point will be this principle: nothing can ever be created by divine power out of nothing. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.152) | |
| A reaction: This claim seems to cry out for a bit of empiricist caution. What observation has convinced Lucretius that creation out of nothing is impossible? The early Christians switched to the view that divine creation is 'ex nihilo' - out of nothing. |
| 5699 | If matter wasn't everlasting, everything would have disappeared by now [Lucretius] |
| Full Idea: If the matter in things had not been everlasting, everything by now would have gone back to nothing, and the things we see would be the product of rebirth out of nothing. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], I.544) | |
| A reaction: See Idea 1431, which is Aquinas's Third Way of proving God. Aquinas thinks there must be a necessary being outside of the system, but Lucretius thinks there must be some necessary existence within the system (as Hume had suggested). |
| 5707 | The universe can't have been created by gods, because it is too imperfect [Lucretius] |
| Full Idea: The universe was certainly not created for us by divine power: it is so full of imperfections. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.180) | |
| A reaction: This is certainly a problem if God is 'supremely perfect', as Descartes proposed, because then the universe would also have to be supremely perfect. See Idea 2114 for a possible answer from Leibniz. Hume agrees with Epicurus about design. |
| 5710 | Gods are tranquil and aloof, and have no need of or interest in us [Lucretius] |
| Full Idea: The nature of deity is to enjoy immortal existence in utter tranquillity, aloof and detached from our affairs. It is free from all pain and peril, strong in its own resources, exempt from any need of us, indifferent to our merits and immune from anger. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], II.652) | |
| A reaction: This seems to be the seed of late seventeenth century deism - the idea of a Creator who is now absent, and ignores our prayers. At that time 'Epicurean' became a synonym for atheist, but Epicureans never quite reached that point. |
| 5731 | Why does Jupiter never hurl lightning from a blue sky? [Lucretius] |
| Full Idea: Why does Jupiter never hurl his thunderbolt upon the earth and let loose his thunder out of a sky that is wholly blue? | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], VI.400) | |
| A reaction: Nice question! It really doesn't take very much to see through superstition, and the fact that most people believed such things shows how staggeringly uncritical they were in their thinking, until philosophers appeared and taught them how to reason. |
| 5722 | For a separated spirit to remain sentient it would need sense organs attached to it [Lucretius] |
| Full Idea: If spirit is immortal and can remain sentient when divorced from our body, we must credit it with possession of five senses; but eyes or nostrils or hand or tongue or ears cannot be attached to a disembodied spirit. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], III.624) | |
| A reaction: This is a powerful argument against immortality. If you are going to see, you must interact with photons; to hear you must respond to compression waves; to smell you must react to certain molecules. Immortality without those would be a bit dull. |
| 5725 | An immortal mind couldn't work harmoniously with a mortal body [Lucretius] |
| Full Idea: It is crazy to couple a mortal object with an eternal and suppose that they can work in harmony and mutually interact. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], III.799) | |
| A reaction: An interesting thought, though not a terrible persuasive argument. A god would indeed be a bit restless if it were chained to a human being, but it would presumably knuckle down to the task if firmly instructed to do it by Zeus. |
| 5720 | Spirit is mortal [Lucretius] |
| Full Idea: Spirit is mortal. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], III.542) | |
| A reaction: This is asserted at an historical moment when immortality is beginning to grip everyone's imagination. |
| 5721 | The mind is very small smooth particles, which evaporate at death [Lucretius] |
| Full Idea: Since the substance of the mind is extraordinarily mobile, it must consist of particles exceptionally small and smooth and round, ..so that, when the spirit has escaped from the body, the outside of the limbs appears intact and there is no loss of weight. | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], III.201) | |
| A reaction: Lucretius is wonderfully attentive to interesting evidence. He goes on to compare it to the evaporation of perfume. The fine-grained connections of the brain are not far off what he is proposing. |
| 5723 | If spirit is immortal and enters us at birth, why don't we remember a previous existence? [Lucretius] |
| Full Idea: If the spirit is by nature immortal and is slipped into the body at birth, why do we retain no memory of an earlier existence, no impress of antecedent events? | |
| From: Lucretius (On the Nature of the Universe [c.60 BCE], III.670) | |
| A reaction: Plato took the view that we do recall previous existence, as seen in our innate ideas. This problem forced the Christian church into the uncomfortable claim that God creates the soul at conception, but that it then goes on to immortality. |