29 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. |
| 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. |
| 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. |
| 2584 | Lobotomised patients can cease to care about a pain [Block] |
| Full Idea: After frontal lobotomies, patients typically report that they still have pains, though the pains no longer bother them. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 83) | |
| A reaction: I take this to be an endorsement of reductive physicalism, because what matters about pains is that they bother us, not how they feel, so frog pain could do the job, if it felt different from ours, but was disliked by the frog. |
| 2582 | A brain looks no more likely than anything else to cause qualia [Block] |
| Full Idea: NO physical mechanism seems very intuitively plausible as a seat of qualia, least of all a brain. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 78) | |
| A reaction: I'm not sure about "least of all", given the mind-boggling complexity of the brain's connections. Certainly, though, nothing in either folk physics or academic physics suggests that any physical object is likely to be aware of anything. |
| 2574 | Behaviour requires knowledge as well as dispositions [Block] |
| Full Idea: A desire cannot be identified with a disposition to act, since the agent might not know that a particular act leads to the thing desired, and thus might not be disposed to do it. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 69) | |
| A reaction: One might have a disposition to act, but not in a particular way. "Something must be done". To get to the particular act, it seems that indeed a belief must be added to the desire. |
| 2576 | In functionalism, desires are internal states with causal relations [Block] |
| Full Idea: According to functionalism, a system might have the behaviouristic input-output relations, yet not desire something, as this requires internal states with certain causal relations. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 69) | |
| A reaction: Such a system might be Putnam's 'superactor', who only behaves as if he desires something. Of course, the internal states might need more than just 'causal relations'. |
| 2575 | Functionalism is behaviourism, but with mental states as intermediaries [Block] |
| Full Idea: Functionalism is a new incarnation of behaviourism, replacing sensory inputs with sensory inputs plus mental states, and replacing dispositions to act with dispositions plus certain mental states. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 69) | |
| A reaction: I think of functionalism as behaviourism which extends inside the 'black box' between stimulus and response. It proposes internal stimuli and responses. Consequently functionalism inherits some behaviourist problems. |
| 2583 | You might invert colours, but you can't invert beliefs [Block] |
| Full Idea: It is hard to see how to make sense of the analog of color spectrum inversion with respect to non-qualitative states such a beliefs (where they are functionally equivalent but have different beliefs). | |
| From: Ned Block (Troubles with Functionalism [1978], p. 81) | |
| A reaction: I would suggest that beliefs can be 'inverted', because there are all sorts of ways to implement a belief, but colour can't be inverted, because that depends on a particular brain state. It makes good sense to me... |
| 2578 | Could a creature without a brain be in the right functional state for pain? [Block] |
| Full Idea: If pain is a functional state, it cannot be a brain state, because creatures without brains could realise the same Turing machine as creatures with brains. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 70) | |
| A reaction: This strikes me as being a poorly grounded claim. There may be some hypothetical world where brainless creatures implement all our functions, but from here brains look the only plausible option. |
| 2585 | Not just any old functional network will have mental states [Block] |
| Full Idea: If there are any fixed points in the mind-body problem, one of them is that the economy of Bolivia could not have mental states, no matter how it is distorted. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 86) | |
| A reaction: It is hard to disagree with this, but then it can hardly be a serious suggestion that anyone could see how to reconfigure an economy so that it mapped the functional state of the human brain. This is not a crucial problem. |
| 2586 | In functionalism, what are the special inputs and outputs of conscious creatures? [Block] |
| Full Idea: In functionalism, it is very hard to see how there could be a single physical characterization of the inputs and outputs of all and only creatures with mentality. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 87) | |
| A reaction: It would be theoretically possible if the only way to achieve mentality was to have a particular pattern of inputs and outputs. I don't think, though, that 'mentality' is an all-or-nothing concept. |
| 2579 | Physicalism is prejudiced in favour of our neurology, when other systems might have minds [Block] |
| Full Idea: Physicalism is a chauvinist theory: it withholds mental properties from systems that in fact have them. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 71) | |
| A reaction: This criticism interprets physicalism too rigidly. There may be several ways to implement a state. My own view is that other systems might implement our functions, but they won't experience them in a human way. |
| 2577 | Simple machine-functionalism says mind just is a Turing machine [Block] |
| Full Idea: In the simplest Turing-machine version of functionalism (Putnam 1967), mental states are identified with the total Turing-machine state, involving a machine table and its inputs and outputs. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 70) | |
| A reaction: This obviously invites the question of why mental states would be conscious and phenomenal, given that modern computers are devoid of same, despite being classy Turing machines. |
| 2580 | A Turing machine, given a state and input, specifies an output and the next state [Block] |
| Full Idea: In a Turing machine, given any state and input, the machine table specifies an output and the next state. …To have full power the tape must be infinite in at least one direction, and be movable in both directions. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 71) | |
| A reaction: In retrospect, the proposal that this feeble item should be taken as a model for the glorious complexity and richness of human consciousness doesn't look too plausible. |
| 2581 | Intuition may say that a complex sentence is ungrammatical, but linguistics can show that it is not [Block] |
| Full Idea: Linguistics rejects (on theoretical grounds) the intuition that the sentence "the boy the girl the cat bit scratched died" is ungrammatical. | |
| From: Ned Block (Troubles with Functionalism [1978], p. 78) | |
| A reaction: Once we have disentangled it, we practical speakers have no right to say it is ungrammatical. It isn't only theory. The sentence is just stylistically infelicitous. |
| 8659 | The gods alone live forever with Shamash. The days of humans are numbered. [Anon (Gilg)] |
| Full Idea: The gods alone are the ones who live forever with Shamash. / As for humans, their days are numbered. | |
| From: Anon (Gilg) (The Epic of Gilgamesh [c.2300 BCE], 3.2.34), quoted by Michèle Friend - Introducing the Philosophy of Mathematics 1.2 | |
| A reaction: Friend quotes this to show the antiquity of the concept of infinity. It also, of course, shows that Sumerians at that time did not believe in human immortality. |