118 ideas
| 15010 | Your metaphysics is 'cheating' if your ontology won't support the beliefs you accept [Sider] |
| Full Idea: Ontological 'cheaters' are those ne'er-do-well metaphysicians (such as presentists, phenomenalists, or solipsists) who refuse to countenance a sufficiently robust conception of the fundamental to underwrite the truths they accept. | |
| From: Theodore Sider (Writing the Book of the World [2011], 08.4) | |
| A reaction: Presentists are placed in rather insalubrious company here, The notion of 'cheaters' is nice, and I associate it with Australian philosophy, and the reason that was admired by David Lewis. |
| 14977 | Metaphysics is not about what exists or is true or essential; it is about the structure of reality [Sider] |
| Full Idea: Metaphysics, at bottom, is about the fundamental structure of reality. Not about what's necessarily true. Not about what properties are essential. Not about conceptual analysis. Not about what there is. Structure. | |
| From: Theodore Sider (Writing the Book of the World [2011], 01) | |
| A reaction: The opening words of his book. I take them to be absolutely correct, and to articulate the new orthodoxy about metaphysics which has emerged since about 1995. He expands this as being about patterns, categories and joints. |
| 14994 | Extreme doubts about metaphysics also threaten to undermine the science of unobservables [Sider] |
| Full Idea: The most extreme critics of metaphysics base their critique on sweeping views about language (logical positivism), or knowledge (empiricism), ...but this notoriously threatens the science of unobservables as much as it threatens metaphysics. | |
| From: Theodore Sider (Writing the Book of the World [2011], 05.1) | |
| A reaction: These criticisms also threaten speculative physics (even about what is possibly observable). |
| 15003 | It seems unlikely that the way we speak will give insights into the universe [Sider] |
| Full Idea: It has always seemed odd that insight into the fundamental workings of the universe should be gained by reflection on how we think and speak. | |
| From: Theodore Sider (Writing the Book of the World [2011], 07.8) | |
| A reaction: A nice expression of what should by now be obvious to all philosophers - that analysis of language is not going to reveal very much. It is merely clearing the undergrowth so that we can go somewhere. |
| 14986 | Conceptual analysts trust particular intuitions much more than general ones [Sider] |
| Full Idea: Conceptual analysts generally regard intuitive judgements about particular cases as being far more diagnostic than intuitive judgements about general principles. | |
| From: Theodore Sider (Writing the Book of the World [2011], 02.4 n7) | |
| A reaction: Since I take the aim to be the building up an accurate picture about general truths, it would be daft to just leap to our intuitions about those general truths. Equally you can't cut intuition out of the picture (pace Ladyman). |
| 10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
| Full Idea: I take 'coherence' to be a primitive, intuitive notion, not reduced to something formal, and so I do not venture a rigorous definition of it. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8) | |
| A reaction: I agree strongly with this. Best to talk of 'the space of reasons', or some such. Rationality extends far beyond what can be formally defined. Coherence is the last court of appeal in rational thought. |
| 10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
| Full Idea: An 'implicit definition' characterizes a structure or class of structures by giving a direct description of the relations that hold among the places of the structure. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
| A reaction: This might also be thought of as a 'functional definition', since it seems to say what the structure or entity does, rather than give the intrinsic characteristics that make its relations and actions possible. |
| 14981 | Philosophical concepts are rarely defined, and are not understood by means of definitions [Sider] |
| Full Idea: Philosophical concepts of interest are rarely reductively defined; still more rarely does our understanding of such concepts rest on definitions. ...(We generally understand concepts to the extent that we know what role they play in thinking). | |
| From: Theodore Sider (Writing the Book of the World [2011], 02.1) | |
| A reaction: I'm not sure that I agree with this. I suspect that Sider has the notion of definition in mind that is influenced by lexicography. Aristotle's concept of definition I take to be lengthy and expansive, and that is very relevant to philosophy. |
| 15015 | It seems possible for a correct definition to be factually incorrect, as in defining 'contact' [Sider] |
| Full Idea: Arguably, 'there is absolutely no space between two objects in contact' is false, but definitional of 'contact'. ...We need a word for true definitional sentences. I propose: 'analytic'. | |
| From: Theodore Sider (Writing the Book of the World [2011], 09.8) |
| 14992 | We don't care about plain truth, but truth in joint-carving terms [Sider] |
| Full Idea: What we care about is truth in joint-carving terms, not just truth. | |
| From: Theodore Sider (Writing the Book of the World [2011], 04.5) | |
| A reaction: The thought is that it matters what conceptual scheme is used to express the truth (the 'ideology'). Truths can be true but uninformative or unexplanatory. |
| 15012 | Orthodox truthmaker theories make entities fundamental, but that is poor for explanation [Sider] |
| Full Idea: According to the entrenched truthmaker theorist, the fundamental facts consist just of facts citing the existence of entities. It's hard to see how all the complexity we experience could possibly be explained from that sparse basis. | |
| From: Theodore Sider (Writing the Book of the World [2011], 08.5) | |
| A reaction: This may be the 'entrenched' truthmaker view, but it is not clear why there could not be more complicated fundamental truthmakers, with structure as well as entities. And powers. |
| 10206 | Modal operators are usually treated as quantifiers [Shapiro] |
| Full Idea: It is common now, and throughout the history of philosophy, to interpret modal operators as quantifiers. This is an analysis of modality in terms of ontology. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) |
| 15023 | The Barcan schema implies if X might have fathered something, there is something X might have fathered [Sider] |
| Full Idea: If we accept the Barcan and converse Barcan schemas, this leads to surprising ontological consequences. Wittgenstein might have fathered something, so, by the Barcan schema, there is something that Wittgenstein might have fathered. | |
| From: Theodore Sider (Writing the Book of the World [2011], 11.9) | |
| A reaction: [He cites Tim Williamson for this line of thought] I was liking the Barcan picture, by now I am backing away fast. They cannot be serious! |
| 10208 | Axiom of Choice: some function has a value for every set in a given set [Shapiro] |
| Full Idea: One version of the Axiom of Choice says that for every set A of nonempty sets, there is a function whose domain is A and whose value, for every a ∈ A, is a member of a. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 1) |
| 10252 | The Axiom of Choice seems to license an infinite amount of choosing [Shapiro] |
| Full Idea: If the Axiom of Choice says we can choose one member from each of a set of non-empty sets and put the chosen elements together in a set, this licenses the constructor to do an infinite amount of choosing. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3) | |
| A reaction: This is one reason why the Axiom was originally controversial, and still is for many philosophers. |
| 10207 | Anti-realists reject set theory [Shapiro] |
| Full Idea: Anti-realists reject set theory. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
| A reaction: That is, anti-realists about mathematical objects. I would have thought that one could accept an account of sets as (say) fictions, which provided interesting models of mathematics etc. |
| 15004 | 'Gunk' is an object in which proper parts all endlessly have further proper parts [Sider] |
| Full Idea: An object is 'gunky' if each of its parts has further proper parts; thus gunk involves infinite descent in the part-whole relation. | |
| From: Theodore Sider (Writing the Book of the World [2011], 07.11.2) |
| 14984 | Which should be primitive in mereology - part, or overlap? [Sider] |
| Full Idea: Should our fundamental theory of part and whole take 'part' or 'overlap' as primitive? | |
| From: Theodore Sider (Writing the Book of the World [2011], 02.3) |
| 14980 | There is a real issue over what is the 'correct' logic [Sider] |
| Full Idea: Certain debates over the 'correct' logic are genuine, and not linguistic or conceptual. | |
| From: Theodore Sider (Writing the Book of the World [2011], 01.3) | |
| A reaction: It is rather hard to give arguments in favour of this view, but I am pleased to have the authority of Sider with me. |
| 15000 | 'It is raining' and 'it is not raining' can't be legislated, so we can't legislate 'p or ¬p' [Sider] |
| Full Idea: I cannot legislate-true 'It is raining' and I cannot legislate true 'It is not raining', so if I cannot legislate either true then I cannot legislate-true the disjunction 'it is raining or it is not raining'. | |
| From: Theodore Sider (Writing the Book of the World [2011], 06.5) | |
| A reaction: This strikes me as a very simple and very persuasive argument against the idea that logic is a mere convention. I take disjunction to be an abstract summary of how the world works. Sider seems sympathetic. |
| 15020 | Classical logic is good for mathematics and science, but less good for natural language [Sider] |
| Full Idea: Despite its brilliant success in mathematics and fundamental science, classical logic applies uneasily to natural language. | |
| From: Theodore Sider (Writing the Book of the World [2011], 10.6) | |
| A reaction: He gives examples of the conditional, and debates over the meaning of 'and', 'or' and 'not', and also names and quantifiers. Many modern philosophical problems result from this conflict. |
| 15029 | Modal accounts of logical consequence are simple necessity, or essential use of logical words [Sider] |
| Full Idea: The simplest modal account is that logical consequence is just necessary consequence; another modal account says that logical consequences are modal consequences that involve only logical words essentially. | |
| From: Theodore Sider (Writing the Book of the World [2011], 12.3) | |
| A reaction: [He cites Quine's 'Carnap and Logical Truth' for the second idea] Sider is asserting that Humeans like him dislike modality, and hence need a nonmodal account of logical consequence. |
| 10259 | The two standard explanations of consequence are semantic (in models) and deductive [Shapiro] |
| Full Idea: The two best historical explanations of consequence are the semantic (model-theoretic), and the deductive versions. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.2) | |
| A reaction: Shapiro points out the fictionalists are in trouble here, because the first involves commitment to sets, and the second to the existence of deductions. |
| 10257 | Intuitionism only sanctions modus ponens if all three components are proved [Shapiro] |
| Full Idea: In some intuitionist semantics modus ponens is not sanctioned. At any given time there is likely to be a conditional such that it and its antecedent have been proved, but nobody has bothered to prove the consequent. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.7) | |
| A reaction: [He cites Heyting] This is a bit baffling. In what sense can 'it' (i.e. the conditional implication) have been 'proved' if the consequent doesn't immediately follow? Proving both propositions seems to make the conditional redundant. |
| 10253 | Either logic determines objects, or objects determine logic, or they are separate [Shapiro] |
| Full Idea: Ontology does not depend on language and logic if either one has the objects determining the logic, or the objects are independent of the logic. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.4) | |
| A reaction: I favour the first option. I think we should seek an account of how logic grows from our understanding of the physical world. If this cannot be established, I shall invent a new Mad Logic, and use it for all my future reasoning, with (I trust) impunity. |
| 10251 | The law of excluded middle might be seen as a principle of omniscience [Shapiro] |
| Full Idea: The law of excluded middle might be seen as a principle of omniscience. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3) | |
| A reaction: [E.Bishop 1967 is cited] Put that way, you can see why a lot of people (such as intuitionists in mathematics) might begin to doubt it. |
| 10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro] |
| Full Idea: To some extent, every truth-functional connective differs from its counterpart in ordinary language. Classical conjunction, for example, is timeless, whereas the word 'and' often is not. 'Socrates runs and Socrates stops' cannot be reversed. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 3) | |
| A reaction: Shapiro suggests two interpretations: either the classical connectives are revealing the deeper structure of ordinary language, or else they are a simplification of it. |
| 15019 | Define logical constants by role in proofs, or as fixed in meaning, or as topic-neutral [Sider] |
| Full Idea: Some say that logical constants are those expressions that are defined by their proof-theoretic roles, others that they are the expressions whose semantic values are permutation-invariant, and still others that they are the topic-neutral expressions. | |
| From: Theodore Sider (Writing the Book of the World [2011], 10.3) | |
| A reaction: [He cites MacFarlane 2005 as giving a survey of this] |
| 10209 | A function is just an arbitrary correspondence between collections [Shapiro] |
| Full Idea: The modern extensional notion of function is just an arbitrary correspondence between collections. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 1) | |
| A reaction: Shapiro links this with the idea that a set is just an arbitrary collection. These minimalist concepts seem like a reaction to a general failure to come up with a more useful and common sense definition. |
| 10268 | Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro] |
| Full Idea: Maybe plural quantifiers should themselves be understood in terms of classes (or sets). | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.4) | |
| A reaction: [Shapiro credits Resnik for this criticism] |
| 15001 | 'Tonk' is supposed to follow the elimination and introduction rules, but it can't be so interpreted [Sider] |
| Full Idea: 'Tonk' is stipulated by Prior to stand for a meaning that obeys the elimination and introduction rules; but there simply is no such meaning; 'tonk' cannot be interpreted so as to obey the rules. | |
| From: Theodore Sider (Writing the Book of the World [2011], 06.5) | |
| A reaction: 'Tonk' thus seems to present a problem for so-called 'natural' deduction, if the natural deduction consists of nothing more than obey elimination and introduction rules. |
| 10235 | A sentence is 'satisfiable' if it has a model [Shapiro] |
| Full Idea: Normally, to say that a sentence Φ is 'satisfiable' is to say that there exists a model of Φ. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8) | |
| A reaction: Nothing is said about whether the model is impressive, or founded on good axioms. Tarski builds his account of truth from this initial notion of satisfaction. |
| 10240 | Model theory deals with relations, reference and extensions [Shapiro] |
| Full Idea: Model theory determines only the relations between truth conditions, the reference of singular terms, the extensions of predicates, and the extensions of the logical terminology. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9) |
| 10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
| Full Idea: The central notion of model theory is the relation of 'satisfaction', sometimes called 'truth in a model'. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9) |
| 10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro] |
| Full Idea: No object-language theory determines its ontology by itself. The best possible is that all models are isomorphic, in which case the ontology is determined 'up to isomorphism', but only if the domain is finite, or it is stronger than first-order. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 2.5) | |
| A reaction: This seems highly significant when ontological claims are being made, and is good support for Shapiro's claim that the structures matter, not the objects. There is a parallel in Tarksi's notion of truth-in-all-models. [The Skolem Paradox is the problem] |
| 10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
| Full Idea: Set theorists often point out that the set-theoretical hierarchy contains as many isomorphism types as possible; that is the point of the theory. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8) | |
| A reaction: Hence there are a huge number of models for any theory, which are then reduced to the one we want at the level of isomorphism. |
| 10234 | Any theory with an infinite model has a model of every infinite cardinality [Shapiro] |
| Full Idea: The Löwenheim-Skolem theorems (which apply to first-order formal theories) show that any theory with an infinite model has a model of every infinite cardinality. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8) | |
| A reaction: This aspect of the theorems is the Skolem Paradox. Shapiro argues that in first-order this infinity of models for arithmetic must be accepted, but he defends second-order model theory, where 'standard' models can be selected. |
| 10201 | Virtually all of mathematics can be modeled in set theory [Shapiro] |
| Full Idea: It is well known that virtually every field of mathematics can be reduced to, or modelled in, set theory. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
| A reaction: The word 'virtually' is tantalising. The fact that something can be 'modeled' in set theory doesn't mean it IS set theory. Most weather can be modeled in a computer. |
| 10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro] |
| Full Idea: Real numbers are either Cauchy sequences of rational numbers (interpreted as pairs of integers), or else real numbers can be thought of as Dedekind cuts, certain sets of rational numbers. So π is a Dedekind cut, or an equivalence class of sequences. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 2.5) | |
| A reaction: This question is parallel to the question of whether natural numbers are Zermelo sets or Von Neumann sets. The famous problem is that there seems no way of deciding. Hence, for Shapiro, we are looking at models, not actual objects. |
| 18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro] |
| Full Idea: There is no more to understanding the real-number structure than knowing how to use the language of analysis. .. One learns the axioms of the implicit definition. ...These determine the realtionships between real numbers. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9) | |
| A reaction: This, of course, is the structuralist view of such things, which isn't really interested in the intrinsic nature of anything, but only in its relations. The slogan that 'meaning is use' seems to be in the background. |
| 18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro] |
| Full Idea: A Dedekind Cut is a division of rationals into two set (A1,A2) where every member of A1 is less than every member of A2. If n is the largest A1 or the smallest A2, the cut is produced by n. Some cuts aren't produced by rationals. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.4) |
| 10236 | There is no grounding for mathematics that is more secure than mathematics [Shapiro] |
| Full Idea: We cannot ground mathematics in any domain or theory that is more secure than mathematics itself. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8) | |
| A reaction: This pronouncement comes after a hundred years of hard work, notably by Gödel, so we'd better believe it. It might explain why Putnam rejects the idea that mathematics needs 'foundations'. Personally I'm prepare to found it in countable objects. |
| 10256 | For intuitionists, proof is inherently informal [Shapiro] |
| Full Idea: For intuitionists, proof is inherently informal. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.7) | |
| A reaction: This thought is quite appealing, so I may have to take intuitionism more seriously. It connects with my view of coherence, which I take to be a notion far too complex for precise definition. However, we don't want 'proof' to just mean 'persuasive'. |
| 10202 | Natural numbers just need an initial object, successors, and an induction principle [Shapiro] |
| Full Idea: The natural-number structure is a pattern common to any system of objects that has a distinguished initial object and a successor relation that satisfies the induction principle | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
| A reaction: If you started your number system with 5, and successors were only odd numbers, something would have gone wrong, so a bit more seems to be needed. How do we decided whether the initial object is 0, 1 or 2? |
| 10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
| Full Idea: Originally, the focus of geometry was space - matter and extension - and the subject matter of arithmetic was quantity. Geometry concerned the continuous, whereas arithmetic concerned the discrete. Mathematics left these roots in the nineteenth century. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
| A reaction: Mathematicians can do what they like, but I don't think philosophers of mathematics should lose sight of these two roots. It would be odd if the true nature of mathematics had nothing whatever to do with its origin. |
| 10222 | Mathematical foundations may not be sets; categories are a popular rival [Shapiro] |
| Full Idea: Foundationalists (e.g. Quine and Lewis) have shown that mathematics can be rendered in theories other than the iterative hierarchy of sets. A dedicated contingent hold that the category of categories is the proper foundation (e.g. Lawvere). | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3) | |
| A reaction: I like the sound of that. The categories are presumably concepts that generate sets. Tricky territory, with Frege's disaster as a horrible warning to be careful. |
| 10218 | Baseball positions and chess pieces depend entirely on context [Shapiro] |
| Full Idea: We cannot imagine a shortstop independent of a baseball infield, or a piece that plays the role of black's queen bishop independent of a chess game. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1) | |
| A reaction: This is the basic thought that leads to the structuralist view of things. I must be careful because I like structuralism, but I have attacked the functionalist view in many areas, because it neglects the essences of the functioning entities. |
| 10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro] |
| Full Idea: The even numbers and the natural numbers greater than 4 both exemplify the natural-number structure. In the former, 6 plays the 3 role, and in the latter 8 plays the 3 role. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.5) | |
| A reaction: This begins to sound a bit odd. If you count the even numbers, 6 is the third one. I could count pebbles using only evens, but then presumably '6' would just mean '3'; it wouldn't be the actual number 6 acting in a different role, like Laurence Olivier. |
| 10228 | Could infinite structures be apprehended by pattern recognition? [Shapiro] |
| Full Idea: It is contentious, to say the least, to claim that infinite structures are apprehended by pattern recognition. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1) | |
| A reaction: It only seems contentious for completed infinities. The idea that the pattern continues in same way seems (pace Wittgenstein) fairly self-evident, just like an arithmetical series. |
| 10230 | The 4-pattern is the structure common to all collections of four objects [Shapiro] |
| Full Idea: The 4-pattern is the structure common to all collections of four objects. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.2) | |
| A reaction: This seems open to Frege's objection, that you can have four disparate abstract concepts, or four spatially scattered items of unknown pattern. It certainly isn't a visual pattern, but then if the only detectable pattern is the fourness, it is circular. |
| 10249 | The main mathematical structures are algebraic, ordered, and topological [Shapiro] |
| Full Idea: According to Bourbaki, there are three main types of structure: algebraic structures, such as group, ring, field; order structures, such as partial order, linear order, well-order; topological structures, involving limit, neighbour, continuity, and space. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.5) | |
| A reaction: Bourbaki is mentioned as the main champion of structuralism within mathematics. |
| 10273 | Some structures are exemplified by both abstract and concrete [Shapiro] |
| Full Idea: Some structures are exemplified by both systems of abstracta and systems of concreta. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.2) | |
| A reaction: It at least seems plausible that one might try to build a physical structure that modelled arithmetic (an abacus might be an instance), so the parallel is feasible. Then to say that the abstract arose from modelling the physical seems equally plausible. |
| 10276 | Mathematical structures are defined by axioms, or in set theory [Shapiro] |
| Full Idea: Mathematical structures are characterised axiomatically (as implicit definitions), or they are defined in set theory. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.3) | |
| A reaction: Presumably earlier mathematicians had neither axiomatised their theories, nor expressed them in set theory, but they still had a good working knowledge of the relationships. |
| 10270 | The main versions of structuralism are all definitionally equivalent [Shapiro] |
| Full Idea: Ante rem structuralism, eliminative structuralism formulated over a sufficiently large domain of abstract objects, and modal eliminative structuralism are all definitionally equivalent. Neither is to be ontologically preferred, but the first is clearer. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.5) | |
| A reaction: Since Shapiro's ontology is platonist, I would have thought there were pretty obvious grounds for making a choice between that and eliminativm, even if the grounds are intuitive rather than formal. |
| 10221 | Is there is no more to structures than the systems that exemplify them? [Shapiro] |
| Full Idea: The 'in re' view of structures is that there is no more to structures than the systems that exemplify them. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3) | |
| A reaction: I say there is more than just the systems, because we can abstract from them to a common structure, but that doesn't commit us to the existence of such a common structure. |
| 10248 | Number statements are generalizations about number sequences, and are bound variables [Shapiro] |
| Full Idea: According to 'in re' structuralism, a statement that appears to be about numbers is a disguised generalization about all natural-number sequences; the numbers are bound variables, not singular terms. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.3.4) | |
| A reaction: Any theory of anything which comes out with the thought that 'really it is a variable, not a ...' has my immediate attention and sympathy. |
| 10220 | Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro] |
| Full Idea: Because the same structure can be exemplified by more than one system, a structure is a one-over-many. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3) | |
| A reaction: The phrase 'one-over-many' is a classic Greek hallmark of a universal. Cf. Idea 10217, where Shapiro talks of arriving at structures by abstraction, through focusing and ignoring. This sounds more like a creation than a platonic universal. |
| 10223 | There is no 'structure of all structures', just as there is no set of all sets [Shapiro] |
| Full Idea: There is no 'structure of all structures', just as there is no set of all sets. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.4) | |
| A reaction: If one cannot abstract from all the structures to a higher level, why should Shapiro have abstracted from the systems/models to get the over-arching structures? |
| 8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend] |
| Full Idea: Shapiro's structuralism champions model theory as the branch of mathematics that best describes mathematics. The essence of mathematical activity is seen as an exercise in comparing mathematical structures to each other. | |
| From: report of Stewart Shapiro (Philosophy of Mathematics [1997], 4.4) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Note it 'best describes' it, rather than being foundational. Assessing whether propositional logic is complete is given as an example of model theory. That makes model theory a very high-level activity. Does it capture simple arithmetic? |
| 10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro] |
| Full Idea: According to structuralism, someone who uses small natural numbers in everyday life presupposes an infinite structure. It seems absurd that a child who learns to count his toes applies an infinite structure to reality, and thus presupposes the structure. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.2) | |
| A reaction: Shapiro says we can meet this objection by thinking of smaller structures embedded in larger ones, with the child knowing the smaller ones. |
| 10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro] |
| Full Idea: We must distinguish between 'realism in ontology' - that mathematical objects exist - and 'realism in truth-value', which is suggested by the model-theoretic framework - that each well-formed meaningful sentence is non-vacuously either true or false. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
| A reaction: My inclination is fairly strongly towards realism of the second kind, but not of the first. A view about the notion of a 'truth-maker' might therefore be required. What do the truths refer to? Answer: not objects, but abstractions from objects. |
| 10210 | If mathematical objects are accepted, then a number of standard principles will follow [Shapiro] |
| Full Idea: One who believes in the independent existence of mathematical objects is likely to accept the law of excluded middle, impredicative definitions, the axiom of choice, extensionality, and arbitrary sets and functions. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 1) | |
| A reaction: The underlying thought is that since the objects pre-exist, all of the above simply describe the relations between them, rather than having to actually bring the objects into existence. Personally I would seek a middle ground. |
| 10215 | Platonists claim we can state the essence of a number without reference to the others [Shapiro] |
| Full Idea: The Platonist view may be that one can state the essence of each number, without referring to the other numbers. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1) | |
| A reaction: Frege certainly talks this way (in his 'borehole' analogy). Fine, we are asked to spell out the essence of some number, without making reference either to any 'units' composing it, or to any other number adjacent to it or composing it. Reals? |
| 10233 | Platonism must accept that the Peano Axioms could all be false [Shapiro] |
| Full Idea: A traditional Platonist has to face the possibility that all of the Peano Axioms are false. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.7) | |
| A reaction: This would be because the objects exist independently, and so the Axioms are a mere human attempt at pinning them down. For the Formalist the axioms create the numbers, and so couldn't be false. This makes me, alas, warm to platonism! |
| 10244 | Intuition is an outright hindrance to five-dimensional geometry [Shapiro] |
| Full Idea: Even if spatial intuition provides a little help in the heuristics of four-dimensional geometry, intuition is an outright hindrance for five-dimensional geometry and beyond. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.2) | |
| A reaction: One might respond by saying 'so much the worse for five-dimensional geometry'. One could hardly abolish the subject, though, so the point must be taken. |
| 10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro] |
| Full Idea: For each stone, there is at least one pattern such that the stone is a position in that pattern. The stone can be treated in terms of places-are-objects, or places-are-offices, to be filled with objects drawn from another ontology. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.4) | |
| A reaction: I believe this is the story J.S. Mill had in mind. His view was that the structures move off into abstraction, but it is only at the empirical and physical level that we can possibly learn the structures. |
| 10254 | Can the ideal constructor also destroy objects? [Shapiro] |
| Full Idea: Can we assume that the ideal constructor cannot destroy objects? Presumably the ideal constructor does not have an eraser, and the collection of objects is non-reducing over time. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.5) | |
| A reaction: A very nice question, which platonists should enjoy. |
| 10255 | Presumably nothing can block a possible dynamic operation? [Shapiro] |
| Full Idea: Presumably within a dynamic system, once the constructor has an operation available, then no activity can preclude the performance of the operation? | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.5) | |
| A reaction: There seems to be an interesting assumption in static accounts of mathematics, that all the possible outputs of (say) a function actually exist with a theory. In an actual dynamic account, the constructor may be smitten with lethargy. |
| 10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro] |
| Full Idea: Can we 'discover' whether a deck is really identical with its fifty-two cards, or whether a person is identical with her corresponding time-slices, molecules, or space-time points? This is like Benacerraf's problem about numbers. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997]) | |
| A reaction: Shapiro is defending the structuralist view, that each of these is a model of an agreed reality, so we cannot choose a right model if they all satisfy the necessary criteria. |
| 15017 | Supervenience is a modal connection [Sider] |
| Full Idea: Supervenience is just a kind of modal connection. | |
| From: Theodore Sider (Writing the Book of the World [2011], 09.10) | |
| A reaction: It says what would happen, as well as what does. This is big for Sider because he rejects modality as a feature of actuality. I think the world is crammed full of modal facts, so supervenience should be a handy tool for me. |
| 15008 | Is fundamentality in whole propositions (and holistic), or in concepts (and atomic)? [Sider] |
| Full Idea: The locus of fundamentality for a Finean is the whole proposition, whereas for me it is the proposition-part. Fundamentality is holistic for the Finean, atomistic for me. | |
| From: Theodore Sider (Writing the Book of the World [2011], 08.3) | |
| A reaction: This is because Kit Fine has pushed fundamentality into a relation (grounding), rather than into the particular entities involved (if I understand Sider's reading of him aright). My first intuition is to side with Sider. I'm on Sider's side... |
| 15013 | Tables and chairs have fundamental existence, but not fundamental natures [Sider] |
| Full Idea: The existence of tables and chairs is just as fundamental as the existence of electrons (in contrast, perhaps, with smirks and shadows, which do not exist fundamentally). However, tables and chairs have nonfundamental natures. | |
| From: Theodore Sider (Writing the Book of the World [2011], 08.7) | |
| A reaction: This seems to be a good clarification, and to me the 'nature' of something points towards its essence. However, I suppose he refers here to the place of something in a dependence hierarchy. But then, why does it have that place? What power? |
| 10227 | The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro] |
| Full Idea: The epistemic proposals of ontological realists in mathematics (such as Maddy and Resnik) has resulted in the blurring of the abstract/concrete boundary. ...Perhaps the burden is now on defenders of the boundary. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1) | |
| A reaction: As Shapiro says, 'a vague boundary is still a boundary', so we need not be mesmerised by borderline cases. I would defend the boundary, with the concrete just being physical. A chair is physical, but our concept of a chair may already be abstract. |
| 10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro] |
| Full Idea: Mathematicians use the 'abstract/concrete' label differently, with arithmetic being 'concrete' because it is a single structure (up to isomorphism), while group theory is considered more 'abstract'. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1 n1) | |
| A reaction: I would say that it is the normal distinction, but they have moved the significant boundary up several levels in the hierarchy of abstraction. |
| 15014 | Unlike things, stuff obeys unrestricted composition and mereological essentialism [Sider] |
| Full Idea: Stuff obeys unrestricted composition and mereological essentialism, whereas things do not. | |
| From: Theodore Sider (Writing the Book of the World [2011], 09.6.2) | |
| A reaction: [He cites Markosian 2004] |
| 10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro] |
| Full Idea: Fictionalism takes an epistemology of the concrete to be more promising than concrete-and-abstract, but fictionalism requires an epistemology of the actual and possible, secured without the benefits of model theory. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.2) | |
| A reaction: The idea that possibilities (logical, natural and metaphysical) should be understood as features of the concrete world has always struck me as appealing, so I have (unlike Shapiro) no intuitive problems with this proposal. |
| 10277 | Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro] |
| Full Idea: One result of the structuralist perspective is a healthy blurring of the distinction between mathematical and ordinary objects. ..'According to the structuralist, physical configurations often instantiate mathematical patterns'. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.4) | |
| A reaction: [The quotation is from Penelope Maddy 1988 p.28] This is probably the main reason why I found structuralism interesting, and began to investigate it. |
| 15009 | We must distinguish 'concrete' from 'abstract' and necessary states of affairs. [Sider] |
| Full Idea: The truthmaker theorist's 'concrete' states of affairs must be distinguished from necessarily existing 'abstract' states of affairs. | |
| From: Theodore Sider (Writing the Book of the World [2011], 08.4) | |
| A reaction: [He cites Plantinga's 'Nature of Necessity' for the second one; I presume the first one is Armstrong] |
| 14983 | Accept the ontology of your best theory - and also that it carves nature at the joints [Sider] |
| Full Idea: We can add to the Quinean advice to believe the ontology of your best theory that you should also regard the ideology of your best theory as carving at the joints. | |
| From: Theodore Sider (Writing the Book of the World [2011], 02.3) | |
| A reaction: I've never liked the original Quinean formulation, but this is much better. I just take my ontological commitments to reside in me, not in whatever theory I am currently employing. I may be dubious about my own theory. |
| 14978 | A property is intrinsic if an object alone in the world can instantiate it [Sider] |
| Full Idea: Chisholm and Kim proposed a modal notion of an 'intrinsic' property - that a property is intrinsic if and only if it is possibly instantiated by an object that is alone in the world. | |
| From: Theodore Sider (Writing the Book of the World [2011], 01.2) | |
| A reaction: [He cites Chisholm 1976:127 and Kim 1982:59-60] Sider then gives a counterexample from David Lewis (Idea 14979). |
| 14995 | Predicates can be 'sparse' if there is a universal, or if there is a natural property or relation [Sider] |
| Full Idea: For Armstrong a predicate is sparse when there exists a corresponding universal; for Lewis, a predicate is sparse when there exists a corresponding natural property or relation. | |
| From: Theodore Sider (Writing the Book of the World [2011], 06) | |
| A reaction: I like 'sparse' properties, but have no sympathy with Armstrong, and am cautious about Lewis. I like Shoemaker's account, which makes properties even sparser. 'Abundant' so-called properties are my pet hate. They are 'predicates'! |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| Full Idea: The very notion of 'object' is at least partially structural and mathematical. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.1) | |
| A reaction: [In the context, Shapiro clearly has physical objects in mind] This view seems to fit with Russell's 'relational' view of the physical world, though Russell rejected structuralism in mathematics. I take abstraction to be part of perception. |
| 10275 | A blurry border is still a border [Shapiro] |
| Full Idea: A blurry border is still a border. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.3) | |
| A reaction: This remark deserves to be quoted in almost every area of philosophy, against those who attack a concept by focusing on its vague edges. Philosophers should focus on central cases, not borderline cases (though the latter may be of interest). |
| 15026 | Essence (even if nonmodal) is not fundamental in metaphysics [Sider] |
| Full Idea: We should not regard nonmodal essence as being metaphysically basic: fundamental theories need essence no more than they need modality. | |
| From: Theodore Sider (Writing the Book of the World [2011], 12.1) | |
| A reaction: He is discussing Kit Fine, and notes that Fine offers a nonmodal view of essence, but still doesn't make it fundamental. I am a fan of essences, but making them fundamental in metaphysics seems unlikely. |
| 12295 | 3-D says things are stretched in space but not in time, and entire at a time but not at a location [Fine,K] |
| Full Idea: Three-dimensionalist think a thing is somehow 'stretched out' through its location at a given time though not through the period during which it exists, and it is present in its entirety at a moment when it exists though not at a position of its location. | |
| From: Kit Fine (In Defence of Three-Dimensionalism [2006], p.1) | |
| A reaction: This definition is designed to set up Fine's defence of the 3-D view, by showing that various dubious asymmetries show up if you do not respect the distinctions offered by the 3-D view. |
| 12298 | Genuine motion, rather than variation of position, requires the 'entire presence' of the object [Fine,K] |
| Full Idea: In order to have genuine motion, rather than mere variation in position, it is necessary that the object should be 'entirely present' at each moment of the change. Thus without entire presence, or existence, genuine motion will not be possible. | |
| From: Kit Fine (In Defence of Three-Dimensionalism [2006], p.6) | |
| A reaction: See Idea 4786 for a rival view of motion. Of course, who says we have to have Kit Fine's 'genuine' motion, if some sort of ersatz motion still gets you to work in the morning? |
| 12296 | 4-D says things are stretched in space and in time, and not entire at a time or at a location [Fine,K] |
| Full Idea: Four-dimensionalists have thought that a material thing is as equally 'stretched out' in time as it is in space, and that there is no special way in which it is entirely present at a moment rather than at a position. | |
| From: Kit Fine (In Defence of Three-Dimensionalism [2006], p.1) | |
| A reaction: Compare his definition of 3-D in Idea 12295. The 4-D is contrary to our normal way of thinking. Since I don't think the future exists, I presume that if I am a 4-D object then I have to say that I don't yet exist, and I disapprove of such talk. |
| 18882 | You can ask when the wedding was, but not (usually) when the bride was [Fine,K, by Simons] |
| Full Idea: Fine says it is acceptable to ask when a wedding was and where it was, and it is acceptable to ask or state where the bride was (at a certain time), but not when she was. | |
| From: report of Kit Fine (In Defence of Three-Dimensionalism [2006], p.18) by Peter Simons - Modes of Extension: comment on Fine p.18 | |
| A reaction: This is aimed at three-dimensionalists who seem to think that a bride is a prolonged event, just as a wedding is. Fine is, interestingly, invoking ordinary language. When did the wedding start and end? When was the bride's birth and death? |
| 12297 | Three-dimensionalist can accept temporal parts, as things enduring only for an instant [Fine,K] |
| Full Idea: Even if one is a three-dimensionalist, one might affirm the existence of temporal parts, on the grounds that everything merely endures for an instant. | |
| From: Kit Fine (In Defence of Three-Dimensionalism [2006], p.2) | |
| A reaction: This seems an important point, as belief in temporal parts is normally equated with four-dimensionalism (see Idea 12296). The idea is that a thing might be 'entirely present' at each instant, only to be replaced by a simulacrum. |
| 10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
| Full Idea: For many philosophers the logical notions of possibility and necessity are exceptions to a general scepticism, perhaps because they have been reduced to model theory, via set theory. Thus Φ is logically possible if there is a model that satisfies it. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.1) | |
| A reaction: Initially this looks a bit feeble, like an empiricist only believing what they actually see right now, but the modern analytical philosophy project seems to be the extension of logical accounts further and further into what we intuit about modality. |
| 15030 | Humeans say that we decide what is necessary [Sider] |
| Full Idea: The spirit of Humeanism is that necessity is not a realm to be discovered. We draw the lines around what is necessary. | |
| From: Theodore Sider (Writing the Book of the World [2011], 12.3) | |
| A reaction: I disagree, but it is hard to argue the point. My intuitions are that the obvious necessities of logic and mathematics reflect the way nature has to be. The deepest necessities are patterns (about which God has no choice). |
| 15031 | Modal terms in English are entirely contextual, with no modality outside the language [Sider] |
| Full Idea: English modals are context-dependent through and through; there is no stable 'outer modality'. | |
| From: Theodore Sider (Writing the Book of the World [2011], 12.7) | |
| A reaction: Sider has been doing so well up to here. To me this is swallowing the bait of linguistic approaches to philosophy which he has fought so hard to avoid. |
| 15027 | If truths are necessary 'by convention', that seems to make them contingent [Sider] |
| Full Idea: If □φ says that φ is true by convention, then □φ would apparently turn out to be contingent, since statements about what conventions we adopt are not themselves true by convention. The main axioms of S4 and S5 would be false. | |
| From: Theodore Sider (Writing the Book of the World [2011], 12.1) |
| 15028 | Conventionalism doesn't seem to apply to examples of the necessary a posteriori [Sider] |
| Full Idea: Conventionalism is apparently inapplicable to Kripke's and Putnam's examples of the necessary a posteriori (and, relatedly, to de re modality). | |
| From: Theodore Sider (Writing the Book of the World [2011], 12.1) | |
| A reaction: [Sidelle 1989 discusses this] |
| 15033 | Humeans says mathematics and logic are necessary because that is how our concept of necessity works [Sider] |
| Full Idea: Why are logical (or mathematical, or analytic...) truths necessary? The Humean's answer is that this is just how our concept of necessity works. | |
| From: Theodore Sider (Writing the Book of the World [2011], 12.11) | |
| A reaction: This is why I (unlike Sider) am not a Humean. If we agreed that 'necessary' meant 'whatever is decreed by the Pope', that would so obviously not be necessary that we would have to start searching nature for true necessities. |
| 15025 | The world does not contain necessity and possibility - merely how things are [Sider] |
| Full Idea: At bottom, the world is an amodal place. Necessity and possibility do not carve at the joints; ultimate reality is not 'full of threats and promises' (Goodman). The book of the world says how things are, not how they must or might be. | |
| From: Theodore Sider (Writing the Book of the World [2011], 12) | |
| A reaction: Nice to see this expressed so clearly. I find it much easier to disagree with as a result. At first blush I would say that if you haven't noticed that the world is full of threats and promises, you should wake up and smell the coffee. Actuality is active. |
| 10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
| Full Idea: The fact that the 'myth' of possible worlds happens to produce the correct modal logic is itself a phenomenon in need of explanation. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.4) | |
| A reaction: The claim that it produces 'the' correct modal logic seems to beg a lot of questions, given the profusion of modal systems. This is a problem with any sort of metaphysics which invokes fictionalism - what were those particular fictions responding to? |
| 14988 | A theory which doesn't fit nature is unexplanatory, even if it is true [Sider] |
| Full Idea: 'Theories' based on bizarre, non-joint-carving classifications are unexplanatory even when true. | |
| From: Theodore Sider (Writing the Book of the World [2011], 03.1) | |
| A reaction: This nicely pinpoints why I take explanation to be central to whole metaphysical enterprise. |
| 14982 | If I used Ramsey sentences to eliminate fundamentality from my theory, that would be a real loss [Sider] |
| Full Idea: If the entire theory of this book were replaced by its Ramsey sentence, omitting all mention of fundamentality, something would seem to be lost. | |
| From: Theodore Sider (Writing the Book of the World [2011], 02.2 n2) | |
| A reaction: It is a moot point whether Ramsey sentences actually eliminate anything from the ontology, but trying to wriggle out of ontological commitment looks a rather sad route to follow. |
| 14989 | Problem predicates in induction don't reflect the structure of nature [Sider] |
| Full Idea: 'Is nonblack', 'is a nonraven', and 'grue' fail to carve at the joints. | |
| From: Theodore Sider (Writing the Book of the World [2011], 03.3) | |
| A reaction: A lot more than this needs to said, but this remark encapsulates why I find most of these paradoxes of induction uninteresting. They are all the creations of logicians, rather than of scientists. |
| 14997 | Two applications of 'grue' do not guarantee a similarity between two things [Sider] |
| Full Idea: The applicability of 'grue' to each of a pair of particulars does not guarantee the similarity of those particulars. | |
| From: Theodore Sider (Writing the Book of the World [2011], 06.2) | |
| A reaction: Grue is not a colour but a behaviour. If two things are 'mercurial' or 'erratic', will that ensure a similarity at any given moment? |
| 14990 | Bayes produces weird results if the prior probabilities are bizarre [Sider] |
| Full Idea: In the Bayesian approach, bizarre prior probability distributions will result in bizarre responses to evidence. | |
| From: Theodore Sider (Writing the Book of the World [2011], 03.3) | |
| A reaction: This is exactly what you find when people with weird beliefs encounter ridiculous evidence for things. It doesn't invalidate the formula, but just says rubbish in rubbish out. |
| 15005 | Explanations must cite generalisations [Sider] |
| Full Idea: Explanations must cite generalisations. | |
| From: Theodore Sider (Writing the Book of the World [2011], 07.13) | |
| A reaction: I'm uneasy about this. Presumably some events have a unique explanation - a unique mechanism, perhaps. Language is inescapably general in its nature - which I take to be Aristotle's reason for agreeing the Sider. [Sider adds mechanisms on p.159] |
| 15011 | If the ultimate explanation is a list of entities, no laws, patterns or mechanisms can be cited [Sider] |
| Full Idea: Ultimate explanations always terminate in the citation of entities; but since a mere list of entities is so unstructured, these 'explanations' cannot be systematized with detailed general laws, patterns, or mechanisms. | |
| From: Theodore Sider (Writing the Book of the World [2011], 08.5) | |
| A reaction: We just need to distinguish between ultimate ontology and ultimate explanations. I think explanations peter out at the point where we descend below the mechanisms. Patterns or laws don't explain on their own. Causal mechanisms are the thing. |
| 15018 | Intentionality is too superficial to appear in the catalogue of ultimate physics [Sider] |
| Full Idea: One day the physicists will complete the catalogue of ultimate and irreducible properties of things. When they do, the like of spin, charm and charge will perhaps appear on the list. But aboutness sure won't; intentionality simply doesn't go that deep. | |
| From: Theodore Sider (Writing the Book of the World [2011], 4 Intro) | |
| A reaction: Fodor's project is to give a reductive, and perhaps eliminative, account of intentionality of mind, while leaving open what one might do with the phenomenological aspects. Personally I don't think they will appear on the list either. |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| Full Idea: The epistemological account of mathematical structures depends on the size and complexity of the structure, but small, finite structures are apprehended through abstraction via simple pattern recognition. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
| A reaction: Yes! This I take to be the reason why John Stuart Mill was not a fool in his discussion of the pebbles. Successive abstractions (and fictions) will then get you to more complex structures. |
| 10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
| Full Idea: Some realists argue that simple types can be apprehended through their tokens, via abstraction. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.2) | |
| A reaction: One might rephrase that to say that types are created by abstraction from tokens (and then preserved in language). |
| 10217 | We can apprehend structures by focusing on or ignoring features of patterns [Shapiro] |
| Full Idea: One way to apprehend a particular structure is through a process of pattern recognition, or abstraction. One observes systems in a structure, and focuses attention on the relations among the objects - ignoring features irrelevant to their relations. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1) | |
| A reaction: A lovely statement of the classic Aristotelian abstractionist approach of focusing-and-ignoring. But this is made in 1997, long after Frege and Geach ridiculed it. It just won't go away - not if you want a full and unified account of what is going on. |
| 9554 | We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro] |
| Full Idea: One can observe a system and focus attention on the relations among the objects - ignoring those features of the objects not relevant to the system. For example, we can understand a baseball defense system by going to several games. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], p.74), quoted by Charles Chihara - A Structural Account of Mathematics | |
| A reaction: This is Shapiro perpetrating precisely the wicked abstractionism which Frege and Geach claim is ridiculous. Frege objects that abstract concepts then become private, but baseball defences are discussed in national newspapers. |
| 10231 | Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro] |
| Full Idea: Perhaps we can introduce abstract objects by abstraction over an equivalence relation on a base class of entities, just as Frege suggested that 'direction' be obtained from parallel lines. ..Properties must be equinumerous, but need not be individuated. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.5) | |
| A reaction: [He cites Hale and Wright as the originators of this} It is not entirely clear why this is 'abstraction', rather than just drawing attention to possible groupings of entities. |
| 14999 | Prior to conventions, not all green things were green? [Sider] |
| Full Idea: It is absurd to say that 'before we introduced our conventions, not all green things were green'. | |
| From: Theodore Sider (Writing the Book of the World [2011], 06.5) | |
| A reaction: Well… Different cultures label the colours of the rainbow differently, and many of them omit orange. I suspect the blue/green borderline has shifted. |
| 14998 | Conventions are contingent and analytic truths are necessary, so that isn't their explanation [Sider] |
| Full Idea: To suggest that analytic truths make statements about linguistic conventions is a nonstarter; statements about linguistic conventions are contingent, whereas the statements made by typical analytic sentences are necessary. | |
| From: Theodore Sider (Writing the Book of the World [2011], 06.5) | |
| A reaction: That 'anything yellow is extended' is not just a convention should be fairly obvious, and it is obviously necessary. But we can say that bachelors are necessarily unmarried men - given the current convention. |
| 15016 | Analyticity has lost its traditional role, which relied on truth by convention [Sider] |
| Full Idea: Nothing can fully play the role traditionally associated with analyticity, for much of that traditional role presupposed the doctrine of truth by convention. | |
| From: Theodore Sider (Writing the Book of the World [2011], 09.8) | |
| A reaction: Sider rejects Quine's attack on analyticity, but accepts his critique of truth by convention. |
| 14985 | The notion of law doesn't seem to enhance physical theories [Sider] |
| Full Idea: Adding the notion of law to physical theory doesn't seem to enhance its explanatory power. | |
| From: Theodore Sider (Writing the Book of the World [2011], 02.4) | |
| A reaction: I agree with his scepticism about laws, although Sider offers it as part of his scepticism about modal facts being included in explanations of actuality. Personally I like dispositions, but not laws. See the ideas of Stephen Mumford. |
| 14987 | Many of the key theories of modern physics do not appear to be 'laws' [Sider] |
| Full Idea: That spacetime is 4D Lorentzian manifold, that the universe began with a singularity, and in a state of low entropy, are all central to physics, but it is a stretch to call them 'laws'. ...It has been argued that there are no laws of biology. | |
| From: Theodore Sider (Writing the Book of the World [2011], 03.1) |
| 14991 | Space has real betweenness and congruence structure (though it is not the Euclidean concepts) [Sider] |
| Full Idea: In metaphysics, space is intrinsically structured; the genuine betweenness and congruence relations are privileged in a way that Euclidean-betweenness and Euclidean-congruence are not. | |
| From: Theodore Sider (Writing the Book of the World [2011], 03.4) | |
| A reaction: I note that Einstein requires space to be 'curved', which implies that it is a substance with properties. |
| 15021 | The central question in the philosophy of time is: How alike are time and space? [Sider] |
| Full Idea: The central question in the philosophy of time is: How alike are time and space? | |
| From: Theodore Sider (Writing the Book of the World [2011], 11.1) |
| 15024 | The spotlight theorists accepts eternal time, but with a spotlight of the present moving across it [Sider] |
| Full Idea: The spotlight theorist accepts the block universe, but also something in addition: a joint-carving monadic property of presentness, which is possessed by just one moment of time, and which 'moves', to be possessed by later and later times. | |
| From: Theodore Sider (Writing the Book of the World [2011], 11.9) | |
| A reaction: This seems better than the merely detached eternalist view, which seems to ignore the key phenomenon. I just can't comprehend any theory which makes the future as real as the past. |