24 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. |
| 16209 | How can point-duration slices of people have beliefs or desires? [Thomson] |
| Full Idea: Can one really think that point-duration temporal slices of bodies believe things or want things? | |
| From: Judith (Jarvis) Thomson (People and Their Bodies [1997], p.211), quoted by Katherine Hawley - How Things Persist 2.9 n21 | |
| A reaction: There is a problem with a slice doing anything long-term. The bottom line is that things are said to 'endure', but that is precisely what time-slices are unable to do. Hawley rejects this idea. |
| 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. |
| 2590 | Dispositions need mental terms to define them [Putnam] |
| Full Idea: The chief difficulty with the behaviour-disposition account is the virtual impossibility of specifying a disposition except as a 'disposition of x to behave as though x were in pain'. | |
| From: Hilary Putnam (The Nature of Mental States [1968], p.57) | |
| A reaction: This has become the best-known objection to behaviourism - that you can't specify a piece of behaviour clearly unless you mention the mental state which it is expressing. The defence is to go on endlessly mentioning further behaviour. |
| 2591 | Total paralysis would mean that there were mental states but no behaviour at all [Putnam] |
| Full Idea: Two animals with all motor nerves cut will have the same actual and potential behaviour (i.e. none), but if only one has uncut pain fibres, it will feel pain where the other won't. | |
| From: Hilary Putnam (The Nature of Mental States [1968], p.57) | |
| A reaction: This is a splendidly literal and practical argument against behaviourism - if you prevent all the behaviour, you don't thereby prevent the experience. Clearly we have to say something about what is inside the 'black box' of the mind. |
| 2588 | Is pain a functional state of a complete organism? [Putnam] |
| Full Idea: I propose the hypothesis that pain, or the state of being in pain, is a functional state of a whole organism. | |
| From: Hilary Putnam (The Nature of Mental States [1968], p.54) | |
| A reaction: This sounds wrong right from the start. Pain hurts. The fact that it leads to avoidance behaviour etc. seems much more like a by-product of pain than its essence. |
| 2589 | Functionalism is compatible with dualism, as pure mind could perform the functions [Putnam] |
| Full Idea: The functional-state hypothesis is not incompatible with dualism, as a system consisting of a body and a soul could meet the required conditions. | |
| From: Hilary Putnam (The Nature of Mental States [1968], p.55) | |
| A reaction: He doesn't really believe this, of course. This claim led to all the weak objections to functionalism involving silly implementations of minds. A brain is the only plausible way to implement our mental functions. |
| 2592 | Functional states correlate with AND explain pain behaviour [Putnam] |
| Full Idea: The presence of a certain functional state is not merely 'correlated with' but actually explains the pain behaviour on the part of the organism. | |
| From: Hilary Putnam (The Nature of Mental States [1968], p.58) | |
| A reaction: Does it offer any further explanation beyond saying that it is the brain state that causes the behaviour? The pain is just a link between damage and avoidance. I wish that is all that pain was. |
| 2587 | Temperature is mean molecular kinetic energy, but they are two different concepts [Putnam] |
| Full Idea: The concept of temperature is not the same as the concept of mean molecular kinetic energy. But temperature is mean molecular kinetic energy. | |
| From: Hilary Putnam (The Nature of Mental States [1968], p.52) | |
| A reaction: This is the standard analogy for mind-brain identity, and it seems fair enough to me. The mind is the activity of the brain. It is rather unhelpful to think of weather in terms of chemistry, but it is actions of chemicals. |
| 6376 | Neuroscience does not support multiple realisability, and tends to support identity [Polger on Putnam] |
| Full Idea: Putnam was too quick to assert neuroscientific support for multiple realizability; current evidence does not reveal it, and there is some reason to think the enterprises of neuroscience are premised on the hypothesis of brain-state identity. | |
| From: comment on Hilary Putnam (The Nature of Mental States [1968]) by Thomas W. Polger - Natural Minds Ch.1.4 | |
| A reaction: I have always been suspicious of the glib claim that mental states were multiply realisable. I see no reason to think that octupi see colours as we do, or experience fear as we do, even though their behaviour has to be similar, for survival. |
| 2330 | If humans and molluscs both feel pain, it can't be a single biological state [Putnam, by Kim] |
| Full Idea: Mental states have vastly diverse physical/biological realizations in different species and structures (e.g. pain in humans and in molluscs), so no mental state can be identified with any single physical/biological state. | |
| From: report of Hilary Putnam (The Nature of Mental States [1968]) by Jaegwon Kim - Mind in a Physical World n p.120 | |
| A reaction: But maybe mollusc and human nervous systems ARE the same in the respects that matter. We don't know enough about pain to deny that possibility. |