46 ideas
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 3005 | 'Jocasta' needs to be distinguished from 'Oedipus's mother' because they are connected by different properties [Fodor] |
| Full Idea: If the concept 'Jocasta' needs to be distinguished from the concept 'Oedipus's mother', that's all right because the two concepts are connected with different properties. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 84) |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 7014 | A particle and a coin heads-or-tails pick out to perfectly well-defined predicates and properties [Fodor] |
| Full Idea: 'Is a particle and my coin is heads' and 'is a particle and my coin is tails' are perfectly well defined predicates and they pick out perfectly well defined (relational) properties of physical particles. | |
| From: Jerry A. Fodor (Psychosemantics [1987], Ch.2) | |
| A reaction: (Somewhat paraphrased). This is a very nice offering for the case that all predicates are properties, and hence that 'properties' is an entirely conventional category. It strikes me as self-evident that Fodor is not picking out 'natural' properties. |
| 5998 | From the necessity of the past we can infer the impossibility of what never happens [Diod.Cronus, by White,MJ] |
| Full Idea: Diodorus' Master Argument inferred that since what is past (i.e. true in the past) is necessary, and the impossible cannot follow from the possible, that therefore if something neither is nor ever will be the case, then it is impossible. | |
| From: report of Diodorus Cronus (fragments/reports [c.300 BCE]) by Michael J. White - Diodorus Cronus | |
| A reaction: The argument is, apparently, no longer fully clear, but it seems to imply determinism, or at least a rejection of the idea that free will and determinism are compatible. (Epictetus 2.19) |
| 20832 | The Master Argument seems to prove that only what will happen is possible [Diod.Cronus, by Epictetus] |
| Full Idea: The Master Argument: these conflict 1) what is past and true is necessary, 2) the impossible does not follow from the possible, 3) something possible neither is nor will be true. Hence only that which is or will be true is possible. | |
| From: report of Diodorus Cronus (fragments/reports [c.300 BCE]) by Epictetus - The Discourses 2.19.1 | |
| A reaction: [Epictetus goes on to discuss views about which of the three should be given up] It is possible there will be a sea fight tomorrow; tomorrow comes, and no sea fight; so there was necessarily no sea fight; so the impossible followed from the possible. |
| 14304 | Conditionals are true when the antecedent is true, and the consequent has to be true [Diod.Cronus] |
| Full Idea: The connected (proposition) is true when it begins with true and neither could nor can end with false. | |
| From: Diodorus Cronus (fragments/reports [c.300 BCE]), quoted by Stephen Mumford - Dispositions 03.4 | |
| A reaction: [Mumford got the quote from Bochenski] This differs from the truth-functional account because it says nothing about when the antecedent is false, which fits in also with the 'supposition' view, where A is presumed. This idea adds necessity. |
| 3008 | Evolution suggests that innate knowledge of human psychology would be beneficial [Fodor] |
| Full Idea: If I had to design homo sapiens, I would have made commonsense knowledge of homo sapiens psychology innate; that way nobody would have to spend time learning it. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.132) |
| 2990 | Contrary to commonsense, most of what is in the mind seems to be unlearned [Fodor] |
| Full Idea: Contrary to commonsense, it looks as though much of what is in the mind is unlearned. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 15) |
| 3009 | Sticklebacks have an innate idea that red things are rivals [Fodor] |
| Full Idea: God gave the male stickleback the idea that whatever is red is a rival. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.133) |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 2994 | In CRTT thought may be represented, content must be [Fodor] |
| Full Idea: In the Representation Theory of Mind, programs (the 'laws of thought') may be explicitly represented, but data structures (the 'contents of thought') have to be. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 25) | |
| A reaction: Presumably this is because content is where mental events actually meet up with the reality being considered. It may be an abstract procedure, but if it doesn't plug into reality then it isn't thought, but merely activity, like that of the liver. |
| 15494 | We can't use propositions to explain intentional attitudes, because they would need explaining [Fodor] |
| Full Idea: It's not clear what the point would be of an explanation of the intentionality of attitudes which presupposes objects that are intentional intrinsically. Why not just say that the attitudes are? | |
| From: Jerry A. Fodor (Psychosemantics [1987], Ch.3) |
| 7326 | Intentionality doesn't go deep enough to appear on the physicists' ultimate list of things [Fodor] |
| Full Idea: Sooner or later the physicists will complete the catalogue of ultimate and irreducible things, with the likes of spin, charm and charge. But aboutness won't be on the list; intentionality simply doesn't go that deep. | |
| From: Jerry A. Fodor (Psychosemantics [1987], 4 Intro) | |
| A reaction: I totally agree with this, which I take to be a warning to John Searle against including something called 'intrinsic intentionality' into his ontology. Intentionality 'emerges' out of certain complex brain activity. |
| 3001 | Behaviourism has no theory of mental causation [Fodor] |
| Full Idea: Behaviourists had trouble providing a robust construal of mental causation (and hence had no logical space for a psychology of mental processes). | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 67) | |
| A reaction: If they could reduce all mental events to stimulus-response, that seems to fall within the normal procedures of physical causation. There is no problem of mental causation if your ontology is entirely physical. |
| 2993 | Any piece of software can always be hard-wired [Fodor] |
| Full Idea: For any machine that computes a function by executing an explicit algorithm, there exists a hard-wired machine that computes the same function by not executing an explicit algorithm. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 23) | |
| A reaction: It is certainly vital for functionalists to understand that software can be hardwired. Presumably we should understand a hardwired alogirthm as 'implicit'? |
| 3011 | Causal powers must be a crucial feature of mental states [Fodor] |
| Full Idea: Everybody is a functionalist, in that we all hold that mental states are individuated, at least in part, by reference to their causal powers. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.138) | |
| A reaction: I might individuate the Prime Minister by the carnation in his buttonhole. However, even a dualist must concede that we individuate mental faculties by their role within the mind. |
| 5498 | Mind is a set of hierarchical 'homunculi', which are made up in turn from subcomponents [Fodor, by Lycan] |
| Full Idea: Fodor sees behaviour as manifestations of psychological capacities, which result from the subject being a set of interconnected 'homunculi', which in turn have subcomponents, all of it arranged in a hierarchy. | |
| From: report of Jerry A. Fodor (Psychosemantics [1987]) by William Lycan - Introduction - Ontology p.9 | |
| A reaction: This may well miss out the most interesting parts of a mind (such as awareness, and personal identity), but it sounds basically right, especially when an evolutionary history is added to the system. Parts of my mind intrude into my trains of thought. |
| 2995 | Supervenience gives good support for mental causation [Fodor] |
| Full Idea: Mind/brain supervenience is the best idea anyone has had so far about how mental causation is possible. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 30) | |
| A reaction: I would have thought that mind brain identity was a much better idea (see Idea 3440). Supervenience seems to prove that 'mental causation' occurs, but doesn't explain it. |
| 2991 | Hume's associationism offers no explanation at all of rational thought [Fodor] |
| Full Idea: With Associationism there proved to be no way to get a rational mental life to emerge from the sorts of causal relations among thoughts that the 'laws of association' recognised. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 18) | |
| A reaction: This might not be true if you add the concept of evolution, which has refined the associations to generate truth (which is vital for survival). |
| 3002 | If mind is just physical, how can it follow the rules required for intelligent thought? [Fodor] |
| Full Idea: Central state identity theorists had trouble providing for the nomological possibility of rational machines (and hence no space for a non-biological, e.g. computational, theory of intelligence). | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 67) | |
| A reaction: I surmise that a more externalist account of the physical mind might do the trick, by explaining intelligence in terms of an evolved relationship between brain and environment. |
| 2992 | We may be able to explain rationality mechanically [Fodor] |
| Full Idea: We are on the verge of solving a great mystery about the mind: how is rationality mechanically possible? | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 20) | |
| A reaction: Optimistic, given that AI has struggled to implement natural languages, mainly because common sense knowledge seems too complex to encode. Can a machine determine logical forms of sentences? |
| 2988 | Folk psychology is the only explanation of behaviour we have [Fodor] |
| Full Idea: Commonsense belief/desire psychology explains vastly more of the facts about behaviour than any of the alternative theories available. It could hardly fail to; there are no alternative theories available. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.x) | |
| A reaction: The alternative view wouldn't expect a clear-cut theory, because it deals with the endless complexity of brain events. The charge is that Fodor and co oversimplify their account, in their desperation for a 'theory'. |
| 3010 | Belief and desire are structured states, which need mentalese [Fodor] |
| Full Idea: A defence of the language of thought has to be an argument that believing and desiring are typically structured states. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.136) | |
| A reaction: A structure is one thing, and a language is another. Both believings and desirings can be extremely vague, to the point where the owner is unsure what is believed or desired. They can, of course, be extremely precise. |
| 2999 | Obsession with narrow content leads to various sorts of hopeless anti-realism [Fodor] |
| Full Idea: People who ask what the narrow content of the thought that water is wet is (for example) get what they deserve: phenomenalism, verificationism, 'procedural' semantics, or scepticism, according to temperament. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 51) | |
| A reaction: The question is whether content IS narrow. We could opt for broad content because then we wouldn't have to worry about scepticism, but I doubt whether we would then sleep well at night. |
| 3012 | Do identical thoughts have identical causal roles? [Fodor] |
| Full Idea: If thoughts have their causal roles in virtue of their contents, then two thoughts with identical contents ought to be identical in their causal roles. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.140) | |
| A reaction: A pencil would presumably have the same causal role if it wrote a love poem or hate mail. But a pencil is also good for scratching your back. 'Causal role' can be a rather vacuous idea. |
| 2998 | Grice thinks meaning is inherited from the propositional attitudes which sentences express [Fodor] |
| Full Idea: According to Gricean theories of meaning, the meaning of a sentence is inherited from the propositional attitudes that the sentence is conventionally used to express. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 50) | |
| A reaction: Since the propositional attitudes contain propositions, this seems like a very plausible idea. If an indexical like 'I' is involved, the meaning of the sentence is not the same as its 'conventional' use. |
| 3006 | Whatever in the mind delivers falsehood is parasitic on what delivers truth [Fodor] |
| Full Idea: The mechanisms that deliver falsehoods are somehow parasitic on the ones that deliver truths. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.107) | |
| A reaction: In the case of a sentence and its negation it is not clear which one is 'parasitic', because that can usually be reversed by paraphrasing. Historically, I very much hope that truth-speaking came first. |
| 3007 | Many different verification procedures can reach 'star', but it only has one semantic value [Fodor] |
| Full Idea: Verification procedures connect terms with their denotations in too many ways. Different routes to 'star' do not determine different semantic values for 'star'. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p.125) | |
| A reaction: This fairly conclusively shows that meaning is not 'the method of verification' - but that wasn't a difficult target to hit. |
| 3004 | The meaning of a sentence derives from its use in expressing an attitude [Fodor] |
| Full Idea: The meaning of a sentence derives from its use in expressing an attitude. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 79) | |
| A reaction: Among other things. It can also arrive from a desire to remember something. A sentence can also acquire meaning compositionally (by assembling) with no use or aim. |
| 3000 | Meaning holism is a crazy doctrine [Fodor] |
| Full Idea: Meaning holism really is a crazy doctrine. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 60) | |
| A reaction: Yes. What is not crazy is a contextualist account of utterances, and a recognition of the contextual and relational ingredient in the meanings of most of our sentences. |
| 3003 | Very different mental states can share their contents, so content doesn't seem to be constructed from functional role [Fodor] |
| Full Idea: It's an embarrassment for attempts to construct content from functional role that quite different sorts of mental states can nevertheless share their contents. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 70) | |
| A reaction: That is, presumably, one content having two different roles. Two contents with the same role is 'multiple realisability'. Pain can tell me I'm damaged, or reveal that my damaged nerves are healing. Problem? |
| 2996 | Mental states may have the same content but different extensions [Fodor] |
| Full Idea: The identity of the content of mental states does not ensure the identity of their extensions. | |
| From: Jerry A. Fodor (Psychosemantics [1987], p. 45) | |
| A reaction: Obviously if I am thinking each day about 'my sheep', that won't change if I am unaware that one of them died this morning. …Because I didn’t have the precise number of sheep in mind. |
| 6024 | Thought is unambiguous, and you should stick to what the speaker thinks they are saying [Diod.Cronus, by Gellius] |
| Full Idea: No one says or thinks anything ambiguous, and nothing should be held to be being said beyond what the speaker thinks he is saying. | |
| From: report of Diodorus Cronus (fragments/reports [c.300 BCE]) by Aulus Gellius - Noctes Atticae 11.12.2 | |
| A reaction: A key argument in favour of propositions, implied in this remark, is that propositions are never ambiguous, though the sentences expressing them may be |