100 ideas
| 3811 | Entailment and validity are relations, but inference is a human activity [Searle] |
| Full Idea: We must distinguish between entailment and validity as logical relations on the one hand, and inferring as a voluntary human activity on the other. | |
| From: John Searle (Rationality in Action [2001], Ch.1.II) |
| 3822 | Theory involves accepting conclusions, and so is a special case of practical reason [Searle] |
| Full Idea: Theoretical reason is typically a matter of accepting a conclusion or hypothesis on the basis of argument or evidence, and is thus a special case of practical reason. | |
| From: John Searle (Rationality in Action [2001], Ch.3.VII) |
| 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. |
| 3812 | Rationality is the way we coordinate our intentionality [Searle] |
| Full Idea: The constraints of rationality ought to be thought of adverbially; they are a matter of the way in which we coordinate our intentionality. | |
| From: John Searle (Rationality in Action [2001], Ch.1.II) |
| 3806 | Rationality is built into the intentionality of the mind, and its means of expression [Searle] |
| Full Idea: Constraints of rationality are built into the structure of mind and language, specifically into the structure of intentionality and speech acts. | |
| From: John Searle (Rationality in Action [2001], Int xiv) |
| 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. |
| 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) |
| 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. |
| 3809 | If complex logic requires rules, then so does basic logic [Searle] |
| Full Idea: If you think you need a rule to infer q from 'p and (if p then q)', then you would also need a rule to infer p from p. | |
| From: John Searle (Rationality in Action [2001], Ch.1.II) |
| 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. |
| 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] |
| 3810 | In real reasoning semantics gives validity, not syntax [Searle] |
| Full Idea: In real-life reasoning it is the semantic content that guarantees the validity of the inference, not the syntactical rule. | |
| From: John Searle (Rationality in Action [2001], Ch.1.II) |
| 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. |
| 3841 | Users of 'supervenience' blur its causal and constitutive meanings [Searle] |
| Full Idea: I am no fan of the concept of supervenience. Its uncritical use is a sign of philosophical confusion, because the concept oscillates between causal supervenience and constitutive supervenience. | |
| From: John Searle (Rationality in Action [2001], Ch.9 n5) | |
| A reaction: I don't see why you shouldn't assert the supervenience of one thing on another, while saying that you are not sure whether it is causal or constitutive. The confusion seems to me to be in understandings of the causal version. |
| 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. |
| 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. |
| 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). |
| 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. |
| 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? |
| 3833 | A belief is a commitment to truth [Searle] |
| Full Idea: A belief is a commitment to truth. | |
| From: John Searle (Rationality in Action [2001], Ch.4.III) |
| 3837 | We can't understand something as a lie if beliefs aren't commitment to truth [Searle] |
| Full Idea: If I lie and say "It is raining", my utterance is intelligible to me as a lie precisely because I understand that the utterance commits me to the truth of a proposition I do not believe to be true. | |
| From: John Searle (Rationality in Action [2001], Ch.6.II) |
| 3816 | Our beliefs are about things, not propositions (which are the content of the belief) [Searle] |
| Full Idea: The terminology of "propositional attitudes" is confused, because it suggests that a belief is an attitude towards a propositions, …but the proposition is the content, not the object, of my belief. | |
| From: John Searle (Rationality in Action [2001], Ch.2) |
| 3828 | Thinking must involve a self, not just an "it" [Searle] |
| Full Idea: We should not say "It thinks" in preference to "I think". If thinking is an active, voluntary process, there must be a self who thinks. | |
| From: John Searle (Rationality in Action [2001], Ch.3.IX) |
| 3831 | Reasons can either be facts in the world, or intentional states [Searle] |
| Full Idea: Both reasons and the things they are reasons for can be either facts in the world or intentional states such as beliefs, desires, and intentions. | |
| From: John Searle (Rationality in Action [2001], Ch.4.I) | |
| A reaction: One might point out that beliefs, desires and intentions are facts in the world too. Implicit dualism. One can ask, what turns a fact into a reason? |
| 3830 | In the past people had a reason not to smoke, but didn't realise it [Searle] |
| Full Idea: For a long time people had a reason not to smoke cigarettes, without knowing that they had such a reason. | |
| From: John Searle (Rationality in Action [2001], Ch.4) | |
| A reaction: What does 'had' a reason mean here? If I wish you dead, there is a reason why you should be dead, but you don't 'have' the reason, and never will have. There's probably a reason why I should never have been born. |
| 3832 | Causes (usually events) are not the same as reasons (which are never events) [Searle] |
| Full Idea: Causes are typically events, reasons are never events. You can give a reason by stating a cause, but it does not follow that the reason and the cause are the same thing. | |
| From: John Searle (Rationality in Action [2001], Ch.4.I) | |
| A reaction: This is against Davidson. I'm with Searle here; my having a reason to do something is not the cause of my doing it. I don't, unlike Searle, believe in free will, but doing something for a reason is not just the operation of the reason. |
| 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. |
| 3823 | Being held responsible for past actions makes no sense without personal identity [Searle] |
| Full Idea: I am held responsible now for things that I did in the distant past. But that only makes sense if there is some entity that is both the agent of the action in the past and me now. | |
| From: John Searle (Rationality in Action [2001], Ch.3.VII) | |
| A reaction: A possible response, of course, is that you are held responsible for your past deeds, but you shouldn't be. The idea that you are the same as when you committed the crime is a convenient fiction for people who desire revenge. Responsibility fades. |
| 3821 | Giving reasons for action requires reference to a self [Searle] |
| Full Idea: The requirement that I state reasons that I acted on requires a reference to the self. …Only for a self can something be a reason for an action. | |
| From: John Searle (Rationality in Action [2001], Ch.3.VII) | |
| A reaction: Why can't we just say that this reason, given this desire and this belief, led to this action, and never mention the self? Admittedly leaving out 'I' is an odd circumlocution, but I don't find this particular argument very convincing. |
| 3824 | A 'self' must be capable of conscious reasonings about action [Searle] |
| Full Idea: In order to be a self the entity that acts as an agent must also be capable of conscious reasoning about its actions. | |
| From: John Searle (Rationality in Action [2001], Ch.3.VIII) | |
| A reaction: I can't accept this all-or-nothing account. A chimpanzee is some sort of 'agent', and there are bad chimpanzees you wouldn't want in your colony. Why does Searle want to cut us off in some special compound where our actions are totally different? |
| 3834 | An intentional, acting, rational being must have a self [Searle] |
| Full Idea: Selfhood in my sense comes for free once you have a conscious intentional being capable of engaging in free actions on the basis of reasons. | |
| From: John Searle (Rationality in Action [2001], Ch.5.II) | |
| A reaction: The concept of an 'action' is probably the thing that most clearly needs a self, because it implies co-ordination and purpose, and there must be some item which benefits. Personally I think you can drop 'free actions' and still have a self. |
| 3825 | Action requires a self, even though perception doesn't [Searle] |
| Full Idea: It is a formal requirement on rational action that there must be a self who acts, in a way that it is not a formal requirement on perception that there be an agent or a self who perceives. | |
| From: John Searle (Rationality in Action [2001], Ch.3.IX) | |
| A reaction: I don't find this persuasive. I don't see how we can rule out a priori the possibility of a set of desires and reasons within an organism which generate an action, without any intervening 'self' to add something. Ockham's Razor. |
| 3829 | Selfs are conscious, enduring, reasonable, active, free, and responsible [Searle] |
| Full Idea: A self is conscious, persists through time, operates with reasons, carries out free actions, and is responsible. | |
| From: John Searle (Rationality in Action [2001], Ch.3.X) | |
| A reaction: Personally I would substitute 'makes decisions' for 'carries out free actions', but otherwise I agree, though he seems to miss a key aspect, which is that the self is in charge of the mind, and directs its focus and co-ordinates its inputs and outputs. |
| 3826 | A self must at least be capable of consciousness [Searle] |
| Full Idea: The first condition on the self is that it should be capable of consciousness. | |
| From: John Searle (Rationality in Action [2001], Ch.3.IX) | |
| A reaction: This strikes me as a stipulative definition. It raises the question of whether it is possible that a lizard (say) is not actually conscious, but has some sort of propriotreptic awareness, and a 'central controller' for its decision-making. |
| 3827 | The self is neither an experience nor a thing experienced [Searle] |
| Full Idea: The self is not an experience, nor is it an object that is experienced. | |
| From: John Searle (Rationality in Action [2001], Ch.3.IX) | |
| A reaction: A nice dichotomy, that draws attention to the unique position of the self. Thanks to Descartes for focusing our attention on it. Personally I would say that the self is an object, which cannot be experienced by itself, but can be inferred by others. |
| 3820 | The bundle must also have agency in order to act, and a self to act rationally [Searle] |
| Full Idea: Agency must be added to the bundle to account for how embodied bundles engage in free actions, and selfhood must be added to account for how agents can act rationally. | |
| From: John Searle (Rationality in Action [2001], Ch.3.VII) | |
| A reaction: I don't buy much of this, but I am inclined to say that a will must be added to the bundle to explain why it acts consistently and coherently. It is certainly ridiculous to rest with the picture of a person as a completely unstructured bundle. |
| 3817 | Free will is most obvious when we choose between several reasons for an action [Searle] |
| Full Idea: The most dramatic manifestation of the free will gap is that when one has several reasons for performing an action, one may act on only one of them; one may select which reason one acts on. | |
| From: John Searle (Rationality in Action [2001], Ch.3.II) |
| 3808 | Rational decision making presupposes free will [Searle] |
| Full Idea: In order to engage in rational decision making we have to presuppose free will. | |
| From: John Searle (Rationality in Action [2001], Ch.1.II) |
| 3818 | We freely decide whether to make a reason for action effective [Searle] |
| Full Idea: Where free rational action is concerned, all effective reasons are made effective by the agent. | |
| From: John Searle (Rationality in Action [2001], Ch.3.II) |
| 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. |
| 3814 | Preferences can result from deliberation, not just precede it [Searle] |
| Full Idea: A well-ordered set of preferences is typically the result of successful deliberation, and is not its precondition. | |
| From: John Searle (Rationality in Action [2001], Ch.1.II) |
| 3840 | We don't accept practical reasoning if the conclusion is unpalatable [Searle] |
| Full Idea: If I desire to get rid of my flu symptoms, and believe the only way to do it is death, I am committed to desiring my death. …there is no plausible logic of practical reason. | |
| From: John Searle (Rationality in Action [2001], Ch.8.II) |
| 3815 | The essence of humanity is desire-independent reasons for action [Searle] |
| Full Idea: The single greatest difference between humans and other animals as far as rationality is concerned is our ability to create, recognise and act on desire-independent reasons for action. | |
| From: John Searle (Rationality in Action [2001], Ch.1.II) |
| 3839 | Only an internal reason can actually motivate the agent to act [Searle] |
| Full Idea: Only an internal reason can actually motivate the agent to act. | |
| From: John Searle (Rationality in Action [2001], Ch.6 App) |
| 3835 | If it is true, you ought to believe it [Searle] |
| Full Idea: To say that something is true is already to say that you ought to believe it. | |
| From: John Searle (Rationality in Action [2001], Ch.5.II) | |
| A reaction: I'm sure what Einstein said is true, but I don't understand it. The truth is the thought of how things actually are, but why should I not prefer my private fantasies? I see the point, though. |
| 3836 | If this is a man, you ought to accept similar things as men [Searle] |
| Full Idea: From the fact that an object is truly described as "a man", it follows that you ought to accept relevantly similar objects as men. | |
| From: John Searle (Rationality in Action [2001], Ch.5.IV) | |
| A reaction: 'Similar' rather begs the question. Common speech distinguishes sharply between a man and a 'real man'. You only accept them as men if you see them as men, not as similar to something else. Interesting. |
| 3838 | Promises hold because I give myself a reason, not because it is an institution [Searle] |
| Full Idea: The obligation to keep a promise does not derive from the institution of promising, ..but from the fact that in promising I freely and voluntarily create a reason for myself. | |
| From: John Searle (Rationality in Action [2001], Ch.6.IV) |
| 3813 | 'Ought' implies that there is a reason to do something [Searle] |
| Full Idea: To say that someone 'ought' to do something is to imply that there is a reason for him to do it. | |
| From: John Searle (Rationality in Action [2001], Ch.1.II) |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |