28 ideas
| 18170 | The Axiom of Reducibility is self-effacing: if true, it isn't needed [Quine] |
| Full Idea: The Axiom of Reducibility is self-effacing: if it is true, the ramification it is meant to cope with was pointless to begin with. | |
| From: Willard Quine (Introduction to Russell's Theory of Types [1967], p.152), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Maddy says the rejection of Reducibility collapsed the ramified theory of types into the simple theory. |
| 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. |
| 23805 | Some explanations offer to explain a mystery by a greater mystery [Schulte] |
| Full Idea: An 'obscurum per obscurius' explanation is explaining something mysterious by something even more mysterious, | |
| From: Peter Schulte (Mental Content [2023], 6) | |
| A reaction: Schulte's example is trying to explain mental content in terms of phenomenal experience. That is, roughly, explaining content by qualia, when the latter is the 'hard problem'. |
| 23806 | Naturalist accounts of representation must match the views of cognitive science [Schulte] |
| Full Idea: Recent naturalisation of content now also has to offer a matching account of representational explanations in cognitive science. | |
| From: Peter Schulte (Mental Content [2023], 08.1) | |
| A reaction: [He cites Cummins, Neander and Shea] This is in addition to the 'status' and 'content' questions of Idea 23796. This seems to be an interesting shift to philosophers working backwards from the theories of empirical science. Few are qualified for this job! |
| 23793 | On the whole, referential content is seen as broad, and sense content as narrow [Schulte] |
| Full Idea: We can say that non-Fregean content [reference] is (virtually) always contrued as broad, while Fregean content [sense] is usually contrued as narrow. | |
| From: Peter Schulte (Mental Content [2023], 3.2) | |
| A reaction: I can't make sense of mental content actually being outside the mind, so I see all content as narrow - but that doesn't mean that externals are irrelevant to it. If I think that is an oak, and it's an elm, the content is oak. |
| 23796 | Naturalists must explain both representation, and what is represented [Schulte] |
| Full Idea: Naturalistic accounts of content ask 1) what makes a state qualify as a representational state?, and 2) what makes a representational state have one specific content rather than another? | |
| From: Peter Schulte (Mental Content [2023], 4) | |
| A reaction: [As often in this collection, the author uses algebraic letters, but I prefer plain English] I would say that the first question looks more amenable to an answer than the second. Do we know the neuronal difference between seeing red and blue? |
| 23792 | Phenomenal and representational character may have links, or even be united [Schulte] |
| Full Idea: Some theorists maintain that all states with representational content or intentionality must have phenomenal character …and we can also ask whether all states with phenomenal character also have representional content. | |
| From: Peter Schulte (Mental Content [2023], 2.4) | |
| A reaction: He mentions that beliefs could involve inner speech. And pains and moods may be phenomenal but lack content. He also asks which determines which. |
| 23795 | Naturalistic accounts of content cannot rely on primitive mental or normative notions [Schulte] |
| Full Idea: A 'naturalistic' explanation of content excludes primitive mental or normative notions, but allows causation, counterfactual dependence, probabilistic dependence or structural similarity. | |
| From: Peter Schulte (Mental Content [2023], 4) | |
| A reaction: Apart from causation, what is permissible to naturalists (like me) all sounds rather superficial (and thus not very explanatory). I'm sure we can do better than this. How about using non-primitive mental notions? |
| 23804 | Maybe we can explain mental content in terms of phenomenal properties [Schulte] |
| Full Idea: The phenomenal intentionality approach says that the content properties of mental states can be explained in terms of the phenomenal properties of mental states. | |
| From: Peter Schulte (Mental Content [2023], 6) | |
| A reaction: [Searle and Loar are cited] Tends to be 'non-naturalistic'. We might decide that content derives from the phenomenal, but still without saying anything interesting about content. Mathematical content? Universally generalised content? |
| 23802 | Conceptual role semantics says content is determined by cognitive role [Schulte] |
| Full Idea: Conceptual role semantics says the content of a representation is determined by the cognitive role it plays with a system. | |
| From: Peter Schulte (Mental Content [2023], 4.5) | |
| A reaction: Obvious problem: if 'swordfish' is the password, its role is quite different from its content. I've never thought that the role of something tells you anything about what it is. Hearts pump blood, but how do they fulfil that role? |
| 23797 | Cause won't explain content, because one cause can produce several contents [Schulte] |
| Full Idea: A simple causal theory of content has the 'content indeterminacy' problem - that the presence of a cow causes 'a cow is present', but also 'an animal is present' and 'a biological organism is present'. | |
| From: Peter Schulte (Mental Content [2023], 4.1) | |
| A reaction: That only rules out the 'simple' version. We just need to add that the cause (cow experience) is shaped by current knowledge and interests. Someone buying cows and someone terrified of them thereby produce different concepts. |
| 23799 | Teleosemantics explains content in terms of successful and unsuccessful functioning [Schulte] |
| Full Idea: The core idea of teleosemantics is that we need to explain how content can be accurate or inaccurate, true or false, realised or unrealised …which must appeal to the distinction between proper functioning and malfunctioning. | |
| From: Peter Schulte (Mental Content [2023], 4.4) | |
| A reaction: My immediate reaction to this is that you don't learn about content by assessing its success. Surely (as with eyesight) you first need to understand what it does, and only then judge its success. …Though success and failure are implicit in function. |
| 23800 | Teleosemantic explanations say content is the causal result of naturally selected functions [Schulte] |
| Full Idea: Teleosemantic theories usually give a causal account of mental functions …where some trait has a particular function if it was selected for that function by a process of natural selection. | |
| From: Peter Schulte (Mental Content [2023], 4.4) | |
| A reaction: This is an idea I like - that something has a specific function if without that function it wouldn't have come into existence (eyes, for example). But presumably the function of a mind is to collect content - which does nothing to explain content! |
| 23798 | Information theories say content is information, such as smoke making fire probable [Schulte] |
| Full Idea: Information theories of content [usually assume that] a column of smoke over there carries the information that fire is over there because it raises the probability of fire being over there. | |
| From: Peter Schulte (Mental Content [2023], 4.2) | |
| A reaction: Theorists usually add further conditions to this basic one. Fred Dretske is the source of this approach. Not promising, in my opinion. Surely the content is just smoke, and fire is one of dozens of possible inferences from it? |