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. |
| 14650 | Maybe proper names involve essentialism [Plantinga] |
| Full Idea: Perhaps the notion of a proper name itself involves essentialism. | |
| From: Alvin Plantinga (De Re and De Dicto [1969], p.43) | |
| A reaction: This is just before Kripke's announcement of 'rigid designation', which seems to have relaunched modern essentialism. The thought is that you can't name something, if you don't have a stable notion of what is (and isn't) being named. |
| 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. |
| 14648 | Could I name all of the real numbers in one fell swoop? Call them all 'Charley'? [Plantinga] |
| Full Idea: Can't I name all the real numbers in the interval (0,1) at once? Couldn't I name them all 'Charley', for example? | |
| From: Alvin Plantinga (De Re and De Dicto [1969], p.40) | |
| A reaction: Plantinga is nervous about such a sweeping move, but can't think of an objection. This addresses a big problem, I think - that you are supposed to accept the real numbers when we cannot possibly name them all. |
| 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. |
| 14647 | Surely self-identity is essential to Socrates? [Plantinga] |
| Full Idea: If anything is essential to Socrates, surely self-identity is. | |
| From: Alvin Plantinga (De Re and De Dicto [1969], p.37) | |
| A reaction: This is the modern move of Plantinga and Adams, to make 'is identical with Socrates' the one property which assures the identity of Socrates (his 'haecceity'). My view is that self-identity is not a property. Plantinga wonders about that on p.44. |
| 14646 | An object has a property essentially if it couldn't conceivably have lacked it [Plantinga] |
| Full Idea: An object has a property essentially just in case it couldn't conceivably have lacked that property. | |
| From: Alvin Plantinga (De Re and De Dicto [1969], p.35) | |
| A reaction: Making it depend on what we can conceive seems a bit dubious, for someone committed to real essences. The key issue is how narrowly or broadly you interpret the word 'property'. The word 'object' needs a bit of thought, too! |
| 14649 | Can we find an appropriate 'de dicto' paraphrase for any 'de re' proposition? [Plantinga] |
| Full Idea: To explain the 'de re' via the 'de dicto' is to provide a rule enabling us to find, for each de re proposition, an equivalent de dicto proposition. | |
| From: Alvin Plantinga (De Re and De Dicto [1969], p.41) | |
| A reaction: Many 'de dicto' paraphrases will change the modality of a 'de re' statement, so the challenge is to find the right equivalent version. Plantinga takes up this challenge. The 'de dicto' statement says the object has the property, and must have it. |
| 14642 | Expressing modality about a statement is 'de dicto'; expressing it of property-possession is 'de re' [Plantinga] |
| Full Idea: Some statements predicate modality of another statement (modality 'de dicto'); but others predicate of an object the necessary or essential possession of a property; these latter express modality 'de re'. | |
| From: Alvin Plantinga (De Re and De Dicto [1969], p.26) | |
| A reaction: The distinction seems to originate in Aquinas, concerning whether God knows the future (or, how he knows the future). 'De dicto' is straightforward, but possibly the result of convention. 'De re' is controversial, and implies deep metaphysics. |
| 14643 | 'De dicto' true and 'de re' false is possible, and so is 'de dicto' false and 'de re' true [Plantinga] |
| Full Idea: Aquinas says if a 'de dicto' statement is true, the 'de re' version may be false. The opposite also applies: 'What I am thinking of [17] is essentially prime' is true, but 'The proposition "what I am thinking of is prime" is necessarily true' is false. | |
| From: Alvin Plantinga (De Re and De Dicto [1969], p.27) | |
| A reaction: In his examples the first is 'de re' (about the number), and the second is 'de dicto' (about that proposition). |
| 14651 | What Socrates could have been, and could have become, are different? [Plantinga] |
| Full Idea: Is there a difference between what Socrates could have been, and what he could have become? | |
| From: Alvin Plantinga (De Re and De Dicto [1969], p.44) | |
| A reaction: That is, I take it, 1) how different might he have been in the past, given how he is now?, and 2) how different might he have been in the past, and now, if he had permanently diverged from how he is now? 1) has tight constraints on it. |
| 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. |
| 8409 | Probabilistic causal concepts are widely used in everyday life and in science [Salmon] |
| Full Idea: Probabilistic causal concepts are used in innumerable contexts of everyday life and science. ...In causes of cancer, road accidents, or food poisoning, for example. | |
| From: Wesley Salmon (Probabilistic Causality [1980], p.137) | |
| A reaction: [Second half compresses his examples] This strikes me as rather a weak point. No one ever thought that a particular road accident was actually caused by the high probability of it at a particular location. Causes are in the mechanisms. |