27 ideas
| 8251 | The logical space of reasons is a natural phenomenon, and it is the realm of freedom [McDowell] |
| Full Idea: The logical space of reasons is just part of the logical space of nature. ...And, in a Kantian slogan, the space of reasons is the realm of freedom. | |
| From: John McDowell (Mind and World [1994], Intro 7) | |
| A reaction: [second half on p.5] This is a modern have-your-cake-and-eat-it view of which I am becoming very suspicious. The modern Kantians (Davidson, Nagel, McDowell) are struggling to naturalise free will, but it won't work. Just dump it! |
| 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. |
| 11984 | Asserting a possible property is to say it would have had the property if that world had been actual [Plantinga] |
| Full Idea: To say than x has a property in a possible world is simply to say that x would have had the property if that world had been actual. | |
| From: Alvin Plantinga (Transworld Identity or worldbound Individuals? [1973], I) | |
| A reaction: Plantinga tries to defuse all the problems with identity across possible worlds, by hanging on to subjunctive verbs and modal modifiers. The point, though, was to explain these, or at least to try to give their logical form. |
| 11980 | A possible world is a maximal possible state of affairs [Plantinga] |
| Full Idea: A possible world is just a maximal possible state of affairs. | |
| From: Alvin Plantinga (Transworld Identity or worldbound Individuals? [1973], I) | |
| A reaction: The key point here is that Plantinga includes the word 'possible' in his definition. Possibility defines the worlds, and so worlds cannot be used on their own to define possibility. |
| 11982 | If possible Socrates differs from actual Socrates, the Indiscernibility of Identicals says they are different [Plantinga] |
| Full Idea: If the Socrates of the actual world has snubnosedness but Socrates-in-W does not, this is surely inconsistent with the Indiscernibility of Identicals, a principle than which none sounder can be conceived. | |
| From: Alvin Plantinga (Transworld Identity or worldbound Individuals? [1973], I) | |
| A reaction: However, we allow Socrates to differ over time while remaining the same Socrates, so some similar approach should apply here. In both cases we need some notion of what is essential to Socrates. But what unites aged 3 with aged 70? |
| 11983 | It doesn't matter that we can't identify the possible Socrates; we can't identify adults from baby photos [Plantinga] |
| Full Idea: We may say it makes no sense to say that Socrates exists at a world, if there is in principle no way of identifying him. ...But this is confused. To suppose Agnew was a precocious baby, we needn't be able to pick him from a gallery of babies. | |
| From: Alvin Plantinga (Transworld Identity or worldbound Individuals? [1973], I) | |
| A reaction: This seems a good point, and yet we have a space-time line joining adult Agnew with baby Agnew, and no such causal link is available between persons in different possible worlds. What would be the criterion in each case? |
| 11985 | If individuals can only exist in one world, then they can never lack any of their properties [Plantinga] |
| Full Idea: The Theory of Worldbound Individuals contends that no object exists in more than one possible world; this implies the outrageous view that - taking properties in the broadest sense - no object could have lacked any property that it in fact has. | |
| From: Alvin Plantinga (Transworld Identity or worldbound Individuals? [1973], II) | |
| A reaction: Leibniz is the best known exponent of this 'outrageous view', though Plantinga shows that Lewis may be seen in the same light, since only counterparts are found in possible worlds, not the real thing. The Theory does seem wrong. |
| 11986 | The counterparts of Socrates have self-identity, but only the actual Socrates has identity-with-Socrates [Plantinga] |
| Full Idea: While Socrates has no counterparts that lack self-identity, he does have counterparts that lack identity-with-Socrates. He alone has that - the property, that is, of being identical with the object that in fact instantiates Socrateity. | |
| From: Alvin Plantinga (Transworld Identity or worldbound Individuals? [1973], II) | |
| A reaction: I am never persuaded by arguments which rest on such dubious pseudo-properties. Whether or not a counterpart of Socrates has any sort of identity with Socrates cannot be prejudged, as it would beg the question. |
| 11987 | Counterpart Theory absurdly says I would be someone else if things went differently [Plantinga] |
| Full Idea: It makes no sense to say I could have been someone else, yet Counterpart Theory implies not merely that I could have been distinct from myself, but that I would have been distinct from myself had things gone differently in even the most miniscule detail. | |
| From: Alvin Plantinga (Transworld Identity or worldbound Individuals? [1973], II) | |
| A reaction: A counterpart doesn't appear to be 'me being distinct from myself'. We have to combine counterparts over possible worlds with perdurance over time. I am a 'worm' of time-slices. Anything not in that worm is not strictly me. |
| 8128 | Representation must be propositional if it can give reasons and be epistemological [McDowell, by Burge] |
| Full Idea: McDowell has claimed that one cannot make sense of representation that plays a role in epistemology unless one takes the representation to be propositional, and thus capable of yielding reasons. | |
| From: report of John McDowell (Mind and World [1994]) by Tyler Burge - Philosophy of Mind: 1950-2000 p.456 | |
| A reaction: A transcendental argument leads back to a somewhat implausible conclusion. I suspect that McDowell has a slightly inflated (Kantian) notion of the purity of the 'space of reasons'. Do philosophers just imagine their problems? |
| 19092 | There is no pure Given, but it is cultured, rather than entirely relative [McDowell, by Macbeth] |
| Full Idea: McDowell argues that the Myth of the Given shows not that there is no content to a concept that is not a matter of its inferential relations to other concepts but only that awareness of the sort that we enjoy ...is acquired in the course of acculturation. | |
| From: report of John McDowell (Mind and World [1994]) by Danielle Macbeth - Pragmatism and Objective Truth p.185 | |
| A reaction: The first view is of Wilfred Sellars, who derives pragmatic relativism from his rejection of the Myth. This idea is helpful is seeing why McDowell has a good proposal. As I look out of my window, my immediate experience seems 'cultured'. |
| 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. |
| 8253 | Sense impressions already have conceptual content [McDowell] |
| Full Idea: The world's impressions on our senses are already possessed of conceptual content. | |
| From: John McDowell (Mind and World [1994], I.6) | |
| A reaction: This is a key idea of McDowell's, which challenges most traditional empiricist views, and (maybe) offers a solution to the rationalist/empiricist debate. His commitment to the 'space of reasons' strikes me as an optional extra. |
| 8254 | Forming concepts by abstraction from the Given is private definition, which the Private Lang. Arg. attacks [McDowell] |
| Full Idea: The idea that concepts can be formed by abstraction from the Given just is the idea of private ostensive definition. So the Private Language Argument just is the rejection of the Given, in so far as it bears on the possibilities for language. | |
| From: John McDowell (Mind and World [1994], I.7) | |
| A reaction: I'm not clear why the process of abstraction from raw impressions shouldn't be a matter of public, explicit, community negotiation. We seem to be able to share and compare fairly raw impressions without much trouble (discussing sunsets). |