126 ideas
| 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. |
| 6052 | Definitions identify two concepts, so they presuppose identity [McGinn] |
| Full Idea: Any definition must presuppose the notion of identity precisely because a definition affirms the identity of two concepts. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: McGinn is arguing that identity is fundamental to thought, and this seems persuasive. It may be, though, that while identities are inescapable, definitions are impossible. |
| 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. |
| 6064 | Regresses are only vicious in the context of an explanation [McGinn] |
| Full Idea: Regresses are only vicious in the context of some explanatory aim, not in themselves. | |
| From: Colin McGinn (Logical Properties [2000], Ch.2 n11) | |
| A reaction: A nice point. It is not quite clear how 'pure' reason could ever be vicious, or charming, or sycophantic. The problem about a vicious regress is precisely that it fails to explain anything. Now benign regresses are something else… (see Idea 2523) |
| 6088 | Truth is a method of deducing facts from propositions [McGinn] |
| Full Idea: Truth is essentially a method of deducing facts from propositions. | |
| From: Colin McGinn (Logical Properties [2000], Ch.5) | |
| A reaction: Very persuasive. McGinn is offering a disquotational account of truth, but in a robust form. Of course, deduction normally takes the form of moving infallibly from one truth to another, but that model of deduction won't fit this particular proposal. |
| 6084 | 'Snow does not fall' corresponds to snow does fall [McGinn] |
| Full Idea: We can say that the proposition that snow does not fall from the sky corresponds to the fact that snow does fall from the sky - in the sense that there is a mapping from fact to proposition. | |
| From: Colin McGinn (Logical Properties [2000], Ch.5) | |
| A reaction: A very nice difficulty for the correspondence theory. It becomes essential to say how the two things correspond before it can offer any sort of account of the truth-relation. |
| 6085 | The idea of truth is built into the idea of correspondence [McGinn] |
| Full Idea: The correspondence theory has an air of triviality, and hence undeniability, but this is because it implicitly builds the idea of truth into the notion of correspondence. | |
| From: Colin McGinn (Logical Properties [2000], Ch.5) | |
| A reaction: If this is accepted, it is a really fatal objection to the theory. Russell tried to use the idea of 'congruency' between beliefs and reality, but that may be open to the same objection. McGinn is claiming that truth is essentially indefinable. |
| 6083 | The coherence theory of truth implies idealism, because facts are just coherent beliefs [McGinn] |
| Full Idea: If 'snow falls from the sky' is true iff it coheres with other beliefs, this is a form of idealism; snow could surely fall from sky even if there were no beliefs in the world to cohere with each other. | |
| From: Colin McGinn (Logical Properties [2000], Ch.5) | |
| A reaction: The coherence theory of truth strikes me as yet another blunder involving a confusion of ontology and epistemology. Of course, idealism may be true, but I have yet to hear a good reason why I should abandon commonsense realism. |
| 6086 | Truth is the property of propositions that makes it possible to deduce facts [McGinn] |
| Full Idea: Truth is a property of a proposition from which one can deduce the fact stated by the proposition. | |
| From: Colin McGinn (Logical Properties [2000], Ch.5) | |
| A reaction: This is McGinn's explanation of the disquotational account of truth ('p' is true iff p). The redundancy theorist would reply that you can deduce p from 'p' without mentioning truth, but it remains to ask why this deduction is possible. |
| 6087 | Without the disquotation device for truth, you could never form beliefs from others' testimony [McGinn] |
| Full Idea: Imagine being in a community which had no concept of truth; ..you cannot disquote on p and hence form beliefs about the world as a result of testimony, since you lack the device of disquotation that is the essence of truth. | |
| From: Colin McGinn (Logical Properties [2000], Ch.5) | |
| A reaction: Whether his theory is right or not, the observation that testimony is the really crucial area where we must have a notion of truth is very good. How about 'truth is what turns propositions into beliefs'? |
| 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) |
| 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. |
| 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) |
| 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. |
| 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. |
| 6051 | In 'x is F and x is G' we must assume the identity of x in the two statements [McGinn] |
| Full Idea: If we say 'for some x, x is F and x is G' we are making tacit appeal to the idea of identity in using 'x' twice here: it has to be the same object that is both F and G. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: This may well be broadened to any utterances whatsoever. The only remaining question is to speculate about whether it is possible to think without identities. The Hopi presumably gave identity to processes rather objects. How does God think? |
| 6055 | Both non-contradiction and excluded middle need identity in their formulation [McGinn] |
| Full Idea: To formulate the law of non-contradiction ('nothing can be both F and non-F') and the law of excluded middle ('everything is either F or it is not-F'), we need the concept of identity (in 'nothing' and 'everything'). | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: Two good examples in McGinn's argument that identity is basic to all thinking. But the argument also works to say that necessity is basic (since both laws claim it) and properties are basic. Let's just declare everything 'basic', and we can all go home. |
| 6059 | Identity is unitary, indefinable, fundamental and a genuine relation [McGinn] |
| Full Idea: I have endorsed four main theses about identity: it is unitary, it is indefinable, it is fundamental, and it is a genuine relation | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: That it is fundamental to our thinking seems certain (but to all possible thought?). That it is a relation looks worth questioning. One might challenge unitary by comparing the identity of numbers, values, electrons and continents. I can't define 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. |
| 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. |
| 6042 | The quantifier is overrated as an analytical tool [McGinn] |
| Full Idea: The quantifier has been overrated as a tool of logical and linguistic analysis. | |
| From: Colin McGinn (Logical Properties [2000], Pref) | |
| A reaction: I find this proposal quite thrilling. Twentieth century analytical philosophy has been in thrall to logic, giving the upper hand in philosophical discussion to the logicians, who are often not very good at philosophy. |
| 6067 | Existential quantifiers just express the quantity of things, leaving existence to the predicate 'exists' [McGinn] |
| Full Idea: What the existential quantifier does is indicate the quantity of things in question - it says that some are; it is left up to the predicate 'exists' to express existence. | |
| From: Colin McGinn (Logical Properties [2000], Ch.2) | |
| A reaction: This seems right. The whole quantification business seems like a conjuring trick to conceal the embarrassingly indefinable and 'metaphysical' notion of 'existence'. Cf Idea 7697. |
| 6069 | 'Partial quantifier' would be a better name than 'existential quantifier', as no existence would be implied [McGinn] |
| Full Idea: We would do much better to call 'some' the 'partial quantifier' (rather than the 'existential quantifier'), on analogy with the universal quantifier - as neither of them logically implies existence. | |
| From: Colin McGinn (Logical Properties [2000], Ch.2) | |
| A reaction: Like McGinn's other suggestions in this chapter, this strikes me as a potentially huge clarification in linguistic analysis. I wait with interest to see whether the philosophical logicians take it up. I bet they don't. |
| 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] |
| 6068 | We need an Intentional Quantifier ("some of the things we talk about.."), so existence goes into the proposition [McGinn] |
| Full Idea: We could introduce an 'intentional quantifier' (Ix) which means 'some of the things we talk about..'; we could then say 'some of the things we talk about are F and exist' (Ix, x is F and x exists). | |
| From: Colin McGinn (Logical Properties [2000], Ch.2) | |
| A reaction: This immediately strikes me as a promising contribution to the analytical toolkit. McGinn is supporting his view that existence is a predicate, and so belongs inside the proposition, not outside. |
| 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) |
| 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. |
| 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] |
| 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. |
| 6070 | Existence is a primary quality, non-existence a secondary quality [McGinn] |
| Full Idea: Existence is like a primary quality; non-existence is like a secondary quality. | |
| From: Colin McGinn (Logical Properties [2000], Ch.2 n29) | |
| A reaction: Since McGinn thinks existence really is a property, and hence, presumably, a predicate, I don't quite see why he uses the word "like". A nicely pithy and thought-provoking remark. |
| 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. |
| 16588 | I prefer a lack of form to mean non-existence, than to think of some quasi-existence [Augustine] |
| Full Idea: I sooner judged that what lacks all form does not exist, than thought of as something in between form and nothing, neither formed nor nothing, unformed and next to nothing. | |
| From: Augustine (Confessions [c.398], XII.6), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 03.1 | |
| A reaction: Scholastics were struck by the contrast between this remark, and the remark of Averroes (Idea 16587) that prime matter was halfway existence. Their two great authorities disagreed! This sort of thing stimulated the revival of metaphysics. |
| 6062 | Existence can't be analysed as instantiating a property, as instantiation requires existence [McGinn] |
| Full Idea: Paraphrasing existence statements into statements about the instantiation of a property does not establish that existence is not a predicate, since the notion of instantiation must be taken to have existence built into it. | |
| From: Colin McGinn (Logical Properties [2000], Ch.2) | |
| A reaction: Thank you, Colin McGinn! This now strikes me as so obvious that it is astonishing that for the whole of the twentieth century no one seems to have said it. For a century philosophers had swept the ontological dirt under the mat. |
| 6065 | We can't analyse the sentence 'something exists' in terms of instantiated properties [McGinn] |
| Full Idea: The problems of the orthodox view are made vivid by analysis of the sentence 'something exists'; this is meaningful and true, but what property are we saying is instantiated here? | |
| From: Colin McGinn (Logical Properties [2000], Ch.2) | |
| A reaction: A very nice point. McGinn claims that existence is a property, a very generalised one. Personally I don't think anyone is even remotely clear what a property is, so the whole discussion is a bit premature. Must properties have causal powers? |
| 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. |
| 22979 | Three main questions seem to be whether a thing is, what it is, and what sort it is [Augustine] |
| Full Idea: I am told that I can ask three sorts of questions - whether a thing is, what it is, and what sort it is. | |
| From: Augustine (Confessions [c.398], X.10) | |
| A reaction: This seems to be a very Aristotelian approach. I am pleased to see that what it is and what sort it is are not conflated. The first one must be its individual essence, and the second its generic essence. |
| 6082 | If causal power is the test for reality, that will exclude necessities and possibilities [McGinn] |
| Full Idea: Whether my body weight is necessary or contingent makes no difference at all to my causal powers, so modality is epiphenomenal; if you took causal potential as a test of reality you would have to declare modes unreal. | |
| From: Colin McGinn (Logical Properties [2000], Ch.4) | |
| A reaction: We could try analysing modality into causal terms, as Lewis proposes with quantification across worlds, or as Quine proposes by reduction to natural regularities. I am not sure what it would mean to declare that modes are 'real'. |
| 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. |
| 6075 | Facts are object-plus-extension, or property-plus-set-of-properties, or object-plus-property [McGinn] |
| Full Idea: A fact may be an object and an extension (Quine's view), or a property and a set of properties, or an object and a property; the view I favour is the third one, which seems the most natural. | |
| From: Colin McGinn (Logical Properties [2000], Ch.3) | |
| A reaction: Personally I tend to use the word 'fact' in a realist and non-linguistic way. There must be innumerable inexpressible facts, such as the single pattern made by all the particles of the universe. McGinn seems to be talking of 'atomic facts'. See Idea 6111. |
| 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). |
| 6058 | Identity propositions are not always tautological, and have a key epistemic role [McGinn] |
| Full Idea: Identity propositions are not always analytic or a priori (as Frege long ago taught us) so there is nothing trivial about such propositions; the claim of redundancy ignores the epistemic role that the concept of identity plays. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: He is referring to Frege's Morning Star/Evening Star distinction (Idea 4972). Wittgenstein wanted to eliminate our basic metaphysics by relabelling it as analytic or tautological, but his project failed. Long live metaphysics! |
| 6053 | Identity is as basic as any concept could ever be [McGinn] |
| Full Idea: Identity has a universality and basicness that is hard to overstate; concepts don't get more basic than this - or more indispensable. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: I agree with this. It seems to me to follow that the natural numbers are just as basic, because they are entailed by the separateness of the identities of things. And the whole of mathematics is the science of the patterns within these numbers. |
| 6043 | Type-identity is close similarity in qualities [McGinn] |
| Full Idea: Two things are said to be type-identical when they are similar enough to be declared qualitatively identical. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: A simple point which brings out the fact that type-identity is unlikely to be any sort of true identity (unless there is absolutely no different at all between two electrons, say). |
| 6045 | It is best to drop types of identity, and speak of 'identity' or 'resemblance' [McGinn] |
| Full Idea: It would be better to drop talk of 'numerical' and 'qualitative' identity altogether, speaking instead simply of identity and resemblance. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1 n4) | |
| A reaction: This is the kind of beautifully simple proposal I pay analytical philosophers to come up with. I will attempt in future to talk either of 'identity' (which is strict), or 'resemblance' (which comes in degrees). |
| 6044 | Qualitative identity is really numerical identity of properties [McGinn] |
| Full Idea: A statement of so-called qualitative identity is really a statement of numerical identity (that is, identity tout court) about the properties of the objects in question - assuming that there are genuine universals. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: We might agree that two cars are type-identical, even though (under the microscope) we decided that none of their properties were absolutely identical. |
| 6046 | Qualitative identity can be analysed into numerical identity of the type involved [McGinn] |
| Full Idea: We can analyse qualitative identity in terms of numerical identity, by saying that x and y are type-identical if there is a single type T that x and y both are, i.e. they both exemplify the same type. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: This just seems to shift the problem onto the words 'are' and 'exemplify'. This takes us back to the problem of things 'partaking' of Plato's Forms. Better to say that qualitative identity isn't identity - it is resemblance (see Idea 6045). |
| 6054 | Sherlock Holmes does not exist, but he is self-identical [McGinn] |
| Full Idea: Sherlock Holmes does not exist, but he is self-identical (he is certainly not indentical to Dr Watson). | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: Most significant. Identity does not entail existence; identity is necessary for existence (I think) but not sufficient. But the notion of existence might be prior to the notion of identity, and the creation of Holmes be parasitic on real existence. |
| 6066 | Existence is a property of all objects, but less universal than self-identity, which covers even conceivable objects [McGinn] |
| Full Idea: Existence is a property universal to all objects that exist, somewhat like self-identity, but less universal, because self-identity holds of all conceivable objects, not merely those that happen to exist. | |
| From: Colin McGinn (Logical Properties [2000], Ch.2) | |
| A reaction: This is a splendidly defiant response to the Kantian slogan that 'existence is not a predicate', and I find McGinn persuasive. I can still not find anyone to explain to me exactly what a property is, so I will reserve judgement. |
| 6047 | All identity is necessary, though identity statements can be contingently true [McGinn] |
| Full Idea: All identity is necessary, although there can be contingently true identity statements - those that contain non-rigid designators. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1 n5) | |
| A reaction: A nice case of the need to keep epistemology and ontology separate. An example might be 'The Prime Minister wears a wig', where 'Prime Minister' may not be a rigid designator. 'Winston wears a wig' will be necessary, if true (which it wasn't). |
| 6048 | Leibniz's Law is so fundamental that it almost defines the concept of identity [McGinn] |
| Full Idea: Leibniz's Law, which a defender of relative identity might opt to reject, is so fundamental to the notion of identity that rejecting it amounts to changing the subject. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1 n8) | |
| A reaction: The Law here is the 'indiscernibility of identicals'. I agree with McGinn, and anyone who loses their grip on this notion of identity strikes me as losing all grip on reality, and threatening their own sanity (well, call it their 'philosophical sanity'). |
| 6049 | Leibniz's Law says 'x = y iff for all P, Px iff Py' [McGinn] |
| Full Idea: Leibniz's Law says 'x = y iff for all P, Px iff Py'. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: That is, two things are the same if when we say that one thing (x) has a property (P), then we are saying that the other thing (y) also has the property. A usefully concise statement of the Law. |
| 6050 | Leibniz's Law presupposes the notion of property identity [McGinn] |
| Full Idea: Leibniz's Law presupposes the notion of property identity. | |
| From: Colin McGinn (Logical Properties [2000], Ch.1) | |
| A reaction: A very important observation, because it leads to recognition of the way in which basic concepts and categories of thought interconnect. Which is more metaphysically basic, identity or properties? It is not easy to say… |
| 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. |
| 6080 | Modality is not objects or properties, but the type of binding of objects to properties [McGinn] |
| Full Idea: Modality has a special ontological category: it consists neither in objects (possible worlds theory) nor in properties (predicate modifier view), but items I have called 'modes', ..which can be hard/soft/rigid/pliable binding of objects to properties. | |
| From: Colin McGinn (Logical Properties [2000], Ch.4) | |
| A reaction: As so often, McGinn is very persuasive. Essentially he is proposing that modality is adverbial. He associates the middle view with David Wiggins. |
| 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? |
| 6079 | If 'possible' is explained as quantification across worlds, there must be possible worlds [McGinn] |
| Full Idea: If we replace modal words like 'possible' with quantification across worlds, clearly the notion of 'world' must exclude impossible worlds, otherwise 'possibly p' will be true if 'p' holds in an impossible world. | |
| From: Colin McGinn (Logical Properties [2000], Ch.4) | |
| A reaction: The point here, of course, is that the question is being begged of what 'possible' and 'impossible' actually mean. |
| 6081 | Necessity and possibility are big threats to the empiricist view of knowledge [McGinn] |
| Full Idea: It is clear that modality is a prima-facie threat to the usual kind of naturalistic-causal-empiricist theory of knowledge. | |
| From: Colin McGinn (Logical Properties [2000], Ch.4) | |
| A reaction: This is why modern empiricists spend of a lot of energy on trying to analyse counterfactuals and laws of nature. Rationalists are much happier to assert necessities a priori, but then they often don't have much basis for their claims. |
| 22980 | Memory contains innumerable principles of maths, as well as past sense experiences [Augustine] |
| Full Idea: The memory contains the innumerable principles and laws of numbers and dimensions. None of these can have been conveyed to me by the bodily senses. | |
| From: Augustine (Confessions [c.398], X.12) | |
| A reaction: Even if you have a fairly empirical view of the sources of mathematics (a view with which I sympathise), it must by admitted that our endless extrapolations from the sources also reside in memory. So we remember thoughts as well as experiences. |
| 22981 | Mind and memory are the same, as shown in 'bear it in mind' or 'it slipped from mind' [Augustine] |
| Full Idea: The mind and the memory are one and the same. We even call the memory the mind, for when we tell a person to remember something, we tell them to 'bear this in mind', and when we forget something 'it slipped out of my mind'. | |
| From: Augustine (Confessions [c.398], X.14) | |
| A reaction: This idea has become familiar in modern neuroscience, I think, presumably because we do not find distinct types of neurons for consciousness and for memory. |
| 22977 | I can distinguish different smells even when I am not experiencing them [Augustine] |
| Full Idea: I can distinguish the scent of lilies from that of violets, even though there is no scent at all in my nostrils. | |
| From: Augustine (Confessions [c.398], X.08) | |
| A reaction: Augustine has a nice introspective account of how we experience memory, and identifies lots of puzzling features. I know I can identify the smell of vinegar, but I can't bring it to mind, the way I can the appearance of roses. |
| 22983 | We would avoid remembering sorrow or fear if that triggered the emotions afresh [Augustine] |
| Full Idea: If we had to experience sorrow or fear every time that we mentioned these emotions, no one would be willing to speak of them. | |
| From: Augustine (Confessions [c.398], X.14) | |
| A reaction: Remembering the death of a loved one can trigger fresh grief, but remembering their dangerous illness from which they recovered no longer contains the feeling of fear. |
| 22982 | Why does joy in my mind make me happy, but joy in my memory doesn't? [Augustine] |
| Full Idea: How can it be that my mind can be happy because of the joy that is in it, and yet my memory is not sad by reason of the sadness that is in it? | |
| From: Augustine (Confessions [c.398], X.14) | |
| A reaction: This seems to contradict his thought in Idea 22981, that memory and mind are the same. Recall seems to be a part of consciousness which is not fully wired up to the rest of the mind. |
| 6071 | Scepticism about reality is possible because existence isn't part of appearances [McGinn] |
| Full Idea: Scepticism about the external world is possible because you can never build existence into the appearances, so it must always be inferred or assumed. | |
| From: Colin McGinn (Logical Properties [2000], Ch.2) | |
| A reaction: When McGinn's claim that existence is a very universal property begins to produce interesting observations like this, I think we should take it very seriously. |
| 22978 | Memory is so vast that I cannot recognise it as part of my mind [Augustine] |
| Full Idea: The memory is a vast immeasurable sanctuary. It is part of my nature, but I cannot understand all that I am. Hence the mind is too narrow to contain itself entirely. Is the other part outside of itself, and not within it? How then can it be a part? | |
| From: Augustine (Confessions [c.398], X.08) | |
| A reaction: He seems to understand the mind as entirely consisting of consciousness. Nevertheless, this seems to be the first inklings of the modern externalist view of the mind. |
| 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. |
| 22984 | Without memory I could not even speak of myself [Augustine] |
| Full Idea: I do not understand the power of memory that is in myself, although without it I could not even speak of myself. | |
| From: Augustine (Confessions [c.398], X.16) | |
| A reaction: Even if the self is not identical with memory, this idea seems to establish that memory is an essential aspect of the self. This point is neglected by those who see the self as an entity (the 'soul pearl') which persists through all experience. |
| 5982 | If the future does not exist, how can prophets see it? [Augustine] |
| Full Idea: How do prophets see the future, if there is not a future to be seen? | |
| From: Augustine (Confessions [c.398], XI.17) | |
| A reaction: The answer, I suspect, is that prophets can't see the future. The prospect that the future already exists would seem to saboutage human freedom and responsibility, and point to Calvinist predestination, and even fatalism. |
| 22976 | Memories are preserved separately, according to category [Augustine] |
| Full Idea: In memory everything is preserved separately, according to its category. | |
| From: Augustine (Confessions [c.398], X.08) | |
| A reaction: This strikes me as the first seeds of the idea that the mind functions by means of mental files. Our memories of cats are 'close to' or 'linked to' our memories of dogs. |
| 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. |
| 6077 | Semantics should not be based on set-membership, but on instantiation of properties in objects [McGinn] |
| Full Idea: Semantics should not employ the relationship of set-membership between objects and extensions, but rather the relation of instantiation between objects and properties. | |
| From: Colin McGinn (Logical Properties [2000], Ch.3) | |
| A reaction: At least this means that philosophers won't be required to read fat books on set theory, but they will have to think very carefully about 'instantiation'. A good start is the ideas on 'Partaking' of Platonic Forms in this database (in 'Universals'). |
| 6074 | Clearly predicates have extensions (applicable objects), but are the extensions part of their meaning? [McGinn] |
| Full Idea: We are taught that predicates have extensions - the class of objects of which the predicate is true - which seems hard to deny; but a stronger claim is also made - that extensions are semantically relevant features of predicates. | |
| From: Colin McGinn (Logical Properties [2000], Ch.3) | |
| A reaction: He cites Quine as a spokesman for this view. McGinn is going on to challenge it, by defending universals. It seems to fit in with other externalist theories of concepts and meanings, none of which seems very appealing to me. |
| 22985 | Everyone wants happiness [Augustine] |
| Full Idea: Surely happiness is what everyone wants, so much so that there can be none who do not want it? | |
| From: Augustine (Confessions [c.398], X.20) | |
| A reaction: His concept of happiness is, of course, religious. Occasionally you meet habitual grumblers about life who give the impression that they are only happy when they are discontented. So happiness is achieving desires, not feeling good? |
| 5984 | Maybe time is an extension of the mind [Augustine] |
| Full Idea: I begin to wonder whether time is an extension of the mind itself. | |
| From: Augustine (Confessions [c.398], XI.26) | |
| A reaction: The observation that the mind creates a 'specious present' (spreading experience out over a short fraction of second) reinforces this. Personally I like David Marshall's proposal that consciousness is entirely memory, which would deny this idea. |
| 22888 | To be aware of time it can only exist in the mind, as memory or anticipation [Augustine, by Bardon] |
| Full Idea: Augustine answers that for us to be aware of time it must exist only in the mind, …and the difference between past and future is just the difference between memory and anticipation. | |
| From: report of Augustine (Confessions [c.398]) by Adrian Bardon - Brief History of the Philosophy of Time 1 'Augustine's' | |
| A reaction: This is an extreme idealist view. Are we to say that the past consists only of what can be remembered, and the future only of what is anticipated? Absurd anti-realism, in my view. Where do his concepts come from, asks Le Poidevin. |
| 5980 | How can ten days ahead be a short time, if it doesn't exist? [Augustine] |
| Full Idea: A short time ago or a short time ahead we might put at ten days, but how can anything which does not exist be either long or short? | |
| From: Augustine (Confessions [c.398], XI.15) | |
| A reaction: A nice question, which gets at the paradoxical nature of time very nicely. How can it be long, but non-existent? We could break the paradox by concluding '..and therefore time does exist', even though we can't see how. |
| 5979 | If the past is no longer, and the future is not yet, how can they exist? [Augustine] |
| Full Idea: Of the three divisions of time, how can two, the past and the future, be, when the past no longer is, and the future is not yet? | |
| From: Augustine (Confessions [c.398], XI.14) | |
| A reaction: This is the oldest bewilderment about time, which naturally leads us to the thought that time cannot actually 'exist'. The remark implies that at least 'now' is safe, but that also succumbs to paradox pretty quickly. |
| 5981 | The whole of the current year is not present, so how can it exist? [Augustine] |
| Full Idea: We cannot say that the whole of the current year is present, and if the whole of it is not present, the year is not present. | |
| From: Augustine (Confessions [c.398], XI.15) | |
| A reaction: Another nice way of presenting the paradox of time. We are in a particular year, so it has to be real. |
| 5978 | I know what time is, until someone asks me to explain it [Augustine] |
| Full Idea: I know well enough what time is, provided that nobody asks me; but if I am asked what it is and try to explain, I am baffled. | |
| From: Augustine (Confessions [c.398], XI.14) | |
| A reaction: A justly famous remark, even though it adds nothing to our knowledge of time. This sort of thought pushes us towards accepting many things as axiomatic, such as time, space, identity, persons, mind. |
| 5983 | I disagree with the idea that time is nothing but cosmic movement [Augustine] |
| Full Idea: I once heard a learned man say that time is nothing but the movement of the sun and the moon and the stars, but I do not agree. | |
| From: Augustine (Confessions [c.398], XI.22) | |
| A reaction: It is tempting to say that you either take time or movement as axiomatic, and describe one in terms of the other, but you are stuck unable to give the initial statement of the axiom without mentioning the second property you were saving for later. |
| 5977 | Heaven and earth must be created, because they are subject to change [Augustine] |
| Full Idea: The fact that heaven and earth are there proclaims that they were created, for they are subject to change and variation; ..the meaning of change and variation is that something is there which was not there before. | |
| From: Augustine (Confessions [c.398], XI.04) | |
| A reaction: It seems possible that the underlying matter is eternal (as in various conservation laws, such as that of energy), and that all change is in the form rather than the substance. |
| 22887 | If God existed before creation, why would a perfect being desire to change things? [Augustine, by Bardon] |
| Full Idea: If nothing existed by God before creation, then what could have happened to, or within, God that led God to decide to create the universe at that particular moment? Why would an eternal or perfect being want or need to change? | |
| From: report of Augustine (Confessions [c.398]) by Adrian Bardon - Brief History of the Philosophy of Time 1 'Augustine's' | |
| A reaction: I suppose you could reply that change is superior to stasis, but then why did God delay the creation? |
| 5976 | If God is outside time in eternity, can He hear prayers? [Augustine] |
| Full Idea: O Lord, since you are outside time in eternity, are you unaware of the things that I tell you? | |
| From: Augustine (Confessions [c.398], XI.01) | |
| A reaction: This strikes me as the single most difficult and most elusive question about the nature of a supreme divine being. If the being is trapped in time, as we are, it is greatly diminished, and if it is outside, it is hard to see how it could be a participant. |
| 6072 | If Satan is the most imperfect conceivable being, he must have non-existence [McGinn] |
| Full Idea: Satan cannot exist because he is the most imperfect conceivable being, and existence is one of the perfections. | |
| From: Colin McGinn (Logical Properties [2000], Ch.2) | |
| A reaction: The logic of this seems right to me. Presumably the theologians would hastily deny this as a definition of Satan; he must have some positive qualities (like power) in order to enact his supreme moral imperfections. NIce, though. |
| 6073 | I think the fault of the Ontological Argument is taking the original idea to be well-defined [McGinn] |
| Full Idea: My own suspicion about the Ontological Argument is that the fault lies in taking notions like 'the most perfect, impressive and powerful being conceivable' to be well-defined. | |
| From: Colin McGinn (Logical Properties [2000], Ch.2) | |
| A reaction: I'm tempted to put it more strongly: the single greatest challenge for the theist with intellectual integrity is to give a clear and coherent definition of God. There must be no internal contradictions, and it must be within the bounds of possibility. |