34 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. |
| 15592 | The usual Tarskian interpretation of variables is to specify their range of values [Fine,K] |
| Full Idea: The usual Tarskian way of indicating how a variable is to be interpreted is to simply specify its range of values. | |
| From: Kit Fine (Semantic Relationism [2007], 1.B) |
| 15593 | Variables can be viewed as special terms - functions taking assignments into individuals [Fine,K] |
| Full Idea: The alternative Tarskian way of indicating how a variable is to be interpreted is that a variable x will be a special case of the semantic value of the term; it will be a function which takes each assignment into the individual which it assigns to x. | |
| From: Kit Fine (Semantic Relationism [2007], 1.B) |
| 15590 | It seemed that Frege gave the syntax for variables, and Tarski the semantics, and that was that [Fine,K] |
| Full Idea: Once Frege had provided a clear syntactic account of variables and once Tarski had supplemented this with a rigorous semantic account, it would appear that there was nothing more of significance to be said. | |
| From: Kit Fine (Semantic Relationism [2007], 1) | |
| A reaction: He later remarks that there are now three semantic accounts: the Tarskian, the instantial, and the algebraic [see xref ideas]. He offers a fourth account in his Semantic Relationism. This grows from his puzzles about variables. |
| 15591 | In separate expressions variables seem identical in role, but in the same expression they aren't [Fine,K] |
| Full Idea: When we consider the semantic role of 'x' and 'y' in two distinct expressions x>0 and y>0, their semantic roles seems the same. But in the same expression, such as x>y, their roles seem to be different. | |
| From: Kit Fine (Semantic Relationism [2007], 1.A) | |
| A reaction: [compressed] This new puzzle about variables leads Fine to say that the semantics of variables, and other expressions, is not intrinsic to them, but depends on their external relations. Variables denote any term - unless another variable got there first. |
| 15595 | The 'algebraic' account of variables reduces quantification to the algebra of its component parts [Fine,K] |
| Full Idea: In the 'algebraic' approach to variables, we move from a quantified sentence to the term specifying a property (the λ-term), and then reducing to the algebraic operations for atomic formulas. | |
| From: Kit Fine (Semantic Relationism [2007], 1.C) | |
| A reaction: [Bealer is a source for this view] Fine describes it as an 'algebra of operations'. I presume this is a thoroughly formalist approach to the matter, which doesn't seem to get to the heart of the semantic question. |
| 15594 | 'Instantial' accounts of variables say we grasp arbitrary instances from their use in quantification [Fine,K] |
| Full Idea: According to the 'instantial' approach to variables, a closed quantified sentence is to be understood on the basis of one of its instances; from an understanding of an instance we understand satisfaction by an arbitrary individual. | |
| From: Kit Fine (Semantic Relationism [2007], 1.D) | |
| A reaction: Fine comments that this is intuitively plausible, but not very precise, because it depends on 'abstraction' of the individual from the expression. |
| 15599 | Cicero/Cicero and Cicero/Tully may differ in relationship, despite being semantically the same [Fine,K] |
| Full Idea: There may be a semantic relationship between 'Cicero' and 'Cicero' that does not hold between 'Cicero' and 'Tully', despite the lack of an intrinsic semantic difference between the names themselves. | |
| From: Kit Fine (Semantic Relationism [2007], 2.E) | |
| A reaction: This is the key idea of Fine's book, and a most original and promising approach to a rather intractable problem in reference. He goes on to distinguish names which are 'strictly' coreferential (the first pair) from those that are 'accidentally' so. |
| 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. |
| 15603 | I can only represent individuals as the same if I do not already represent them as the same [Fine,K] |
| Full Idea: I can only represent two individuals as being the same if I do not already represent them as the same. | |
| From: Kit Fine (Semantic Relationism [2007], 3.A) | |
| A reaction: A very nice simple point. If I say 'Hesperus is Hesperus' I am unable to comment on the object, but 'Hesperus is Phosphorus' has a different expressive power. Start from contexts where it is necessary to say that two things are actually one. |
| 15604 | If Cicero=Tully refers to the man twice, then surely Cicero=Cicero does as well? [Fine,K] |
| Full Idea: 'Cicero=Cicero' and 'Cicero=Tully' are both dyadic predications. It is unnatural to suppose that the use of the same name converts a dyadic predicate into a reflexive predicate, or that there is one reference to Cicero in the first and two in the second. | |
| From: Kit Fine (Semantic Relationism [2007], 3.A) | |
| A reaction: I am deeply suspicious of the supposed 'property' of being self-identical, but that may not deny that it could be a genuine truth (shorthand for 'the C you saw is the same as the C I saw'). Having an identity makes equality with self possible. |
| 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. |
| 15602 | Mental files are devices for keeping track of basic coordination of objects [Fine,K] |
| Full Idea: Mental files should be seen as a device for keeping track of when objects are coordinated (represented as-the-same) and, rather than understand coordination in terms of mental files, we should understand mental files in terms of coordination. | |
| From: Kit Fine (Semantic Relationism [2007], 3.A) | |
| A reaction: Personally I think that the metaphor of a 'label' is much closer to the situation than that of a 'file'. Thus my concept of Cicero is labelled 'Tully', 'Roman', 'orator', 'philosophical example'... My problem is to distinguish the concept from its labels. |
| 15588 | You cannot determine the full content from a thought's intrinsic character, as relations are involved [Fine,K] |
| Full Idea: There is no determining the full content of what someone thinks or believes from the individual things that he thinks or believes; we must also look at the threads that tie the contents of these thoughts or beliefs together. | |
| From: Kit Fine (Semantic Relationism [2007], Intro) | |
| A reaction: I'm not sure what 'full' content could possibly mean. Does that include all our background beliefs which we hardly ever articulate. Content comes in degrees, or needs an arbitrary boundary? |
| 15596 | The standard aim of semantics is to assign a semantic value to each expression [Fine,K] |
| Full Idea: The aim of semantics, as standardly conceived, is to assign a semantic value to each (meaningful) expression of the language under consideration. | |
| From: Kit Fine (Semantic Relationism [2007], 1.G) | |
| A reaction: Fine is raising the difficulty that these values can get entangled with one another. He proposes 'semantic connections' as a better aim. |
| 15587 | That two utterances say the same thing may not be intrinsic to them, but involve their relationships [Fine,K] |
| Full Idea: In my 'Semantic Relationism' the fact that two utterances say the same thing is not entirely a matter of their intrinsic semantic features; it may also turn on semantic relationships among the utterances of their parts not reducible to those features. | |
| From: Kit Fine (Semantic Relationism [2007], Intro) | |
| A reaction: You'll need to read the book slowly several times to get the hang of this, but at least it allows that two different utterances might say the same thing (express the same proposition, I would say). |
| 15589 | The two main theories are Holism (which is inferential), and Representational (which is atomistic) [Fine,K] |
| Full Idea: For holists a proper theory will be broadly inferential, while for their opponents it will be representational in character, describing relations between expressions and reality. Representational semantics is atomist, holist semantics inferential. | |
| From: Kit Fine (Semantic Relationism [2007], Intro) | |
| A reaction: Fine presents these as the two main schools in semantics. His own theory then proposes a more holistic version of the Representational view. He seeks the advantages of Frege's position, but without 'sense'. |
| 15598 | We should pursue semantic facts as stated by truths in theories (and not put the theories first!) [Fine,K] |
| Full Idea: A 'semantics' is a body of semantic facts, and a 'semantic theory' is a body of semantic truths. The natural order is a theory being understood as truths, which state facts. Davidson, alas, reversed this order, with facts understood through theories. | |
| From: Kit Fine (Semantic Relationism [2007], 2.C) | |
| A reaction: [compressed; he cites Davidson 1967, and calls it 'one of the most unfortunate tendencies in modern philosophy of language, ..as if chemistry were understood in terms of formulae rather than chemical facts']. |
| 15600 | Referentialist semantics has objects for names, properties for predicates, and propositions for connectives [Fine,K] |
| Full Idea: The standard referentialist semantics for a language with names is that the semantic value of the name is the object, the content of a predicate is a property, and the content of a logical connective is an operation on propositions. | |
| From: Kit Fine (Semantic Relationism [2007], 2.F) | |
| A reaction: My particular bęte noire is the idea that every predicate names a property. It is the tyranny of having to have a comprehensive semantic theory that drives this implausible picture. And I don't see how an object can be a semantic value… |
| 15601 | Fregeans approach the world through sense, Referentialists through reference [Fine,K] |
| Full Idea: Fregeans emphasise an orientation towards the speaker: possession of sense makes language meaningful, and language relates to the world through sense. For the Referentialist its representational relationships make it meaningful, and relate it to the world | |
| From: Kit Fine (Semantic Relationism [2007], 2.G) | |
| A reaction: The Referentialist approach is for Kripkean fans of direct reference, rather than the Fregean reference through descriptions. I am inclined to favour the old-fashioned, deeply discredited, much mocked Fregean approach. |
| 15605 | I take indexicals such as 'this' and 'that' to be linked to some associated demonstration [Fine,K] |
| Full Idea: Demonstrative uses of an indexical such as 'this' or 'that' should be taken to be anaphoric on an associated demonstration. It is a semantic requirement on the use of the indexical that it be coreferential with the demonstration. | |
| From: Kit Fine (Semantic Relationism [2007], Post 'Indexicals') | |
| A reaction: Similarly 'now' must connect to looking at a clock, and 'I' to pointing at some person. The demonstration could be of a verbal event, as much as a physical one. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |