| 1989 | Structure and Ontology |
| 146 | p.60 | 9626 | A structure is an abstraction, focussing on relationships, and ignoring other features |
| Full Idea: A structure is the abstract form of a system, focussing on the interrelationships among the objects, and ignoring any features of them that do not affect how they relate to other objects in the system. | |||
| From: Stewart Shapiro (Structure and Ontology [1989], 146), quoted by James Robert Brown - Philosophy of Mathematics Ch.4 | |||
| A reaction: I find this account very attractive, even though it appeals to supposedly outmoded psychological abstractionism. It seems pretty close to Aristotle's view of things. Shapiro's account must face up to Frege's worries about these matters. |
| 1991 | Foundations without Foundationalism |
| p.225 | 15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else | |
| Full Idea: Shapiro preferred second-order logic to set theory because second-order logic refers only to the relations and operations in a domain, and not to the other things that set-theory brings with it - other domains, higher-order relations, and so forth. | |||
| From: report of Stewart Shapiro (Foundations without Foundationalism [1991]) by Shaughan Lavine - Understanding the Infinite VII.4 |
| Pref | p.-17 | 13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed |
| Full Idea: The 'triumph' of first-order logic may be related to the remnants of failed foundationalist programmes early this century - logicism and the Hilbert programme. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |||
| A reaction: Being complete must also be one of its attractions, and Quine seems to like it because of its minimal ontological commitment. |
| Pref | p.-17 | 13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology |
| Full Idea: I extend Quinean holism to logic itself; there is no sharp border between mathematics and logic, especially the logic of mathematics. One cannot expect to do logic without incorporating some mathematics and accepting at least some of its ontology. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |||
| A reaction: I have strong sales resistance to this proposal. Mathematics may have hijacked logic and warped it for its own evil purposes, but if logic is just the study of inferences then it must be more general than to apply specifically to mathematics. |
| Pref | p.-16 | 13626 | Semantic consequence is ineffective in second-order logic |
| Full Idea: It follows from Gödel's incompleteness theorem that the semantic consequence relation of second-order logic is not effective. For example, the set of logical truths of any second-order logic is not recursively enumerable. It is not even arithmetic. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |||
| A reaction: I don't fully understand this, but it sounds rather major, and a good reason to avoid second-order logic (despite Shapiro's proselytising). See Peter Smith on 'effectively enumerable'. |
| Pref | p.-15 | 13627 | There is no 'correct' logic for natural languages |
| Full Idea: There is no question of finding the 'correct' or 'true' logic underlying a part of natural language. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |||
| A reaction: One needs the context of Shapiro's defence of second-order logic to see his reasons for this. Call me romantic, but I retain faith that there is one true logic. The Kennedy Assassination problem - can't see the truth because drowning in evidence. |
| Pref | p.-14 | 13628 | We can live well without completeness in logic |
| Full Idea: We can live without completeness in logic, and live well. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |||
| A reaction: This is the kind of heady suggestion that American philosophers love to make. Sounds OK to me, though. Our ability to draw good inferences should be expected to outrun our ability to actually prove them. Completeness is for wimps. |
| Pref | p.-13 | 13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? |
| Full Idea: Three systems of semantics for second-order languages: 'standard semantics' (variables cover all relations and functions), 'Henkin semantics' (relations and functions are a subclass) and 'first-order semantics' (many-sorted domains for variable-types). | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |||
| A reaction: [my summary] |
| Pref | p.-12 | 13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures |
| Full Idea: It is sometimes said that non-compactness is a defect of second-order logic, but it is a consequence of a crucial strength - its ability to give categorical characterisations of infinite structures. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |||
| A reaction: The dispute between fans of first- and second-order may hinge on their attitude to the infinite. I note that Skolem, who was not keen on the infinite, stuck to first-order. Should we launch a new Skolemite Crusade? |
| Pref | p.-9 | 13631 | Are sets part of logic, or part of mathematics? |
| Full Idea: Is there a notion of set in the jurisdiction of logic, or does it belong to mathematics proper? | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |||
| A reaction: It immediately strikes me that they might be neither. I don't see that relations between well-defined groups of things must involve number, and I don't see that mapping the relations must intrinsically involve logical consequence or inference. |
| 1.1 | p.3 | 13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness |
| Full Idea: It is sometimes difficult to find a formula that is a suitable counterpart of a particular sentence of natural language, and there is no acclaimed criterion for what counts as a good, or even acceptable, 'translation'. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) |
| 1.1 | p.5 | 13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} |
| Full Idea: The 'satisfaction' relation may be thought of as a function from models, assignments, and formulas to the truth values {true,false}. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) | |||
| A reaction: This at least makes clear that satisfaction is not the same as truth. Now you have to understand how Tarski can define truth in terms of satisfaction. |
| 1.1 | p.6 | 13634 | Satisfaction is 'truth in a model', which is a model of 'truth' |
| Full Idea: In a sense, satisfaction is the notion of 'truth in a model', and (as Hodes 1984 elegantly puts it) 'truth in a model' is a model of 'truth'. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) | |||
| A reaction: So we can say that Tarski doesn't offer a definition of truth itself, but replaces it with a 'model' of truth. |
| 1.1 | p.8 | 13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence |
| Full Idea: A logic is 'weakly sound' if every theorem is a logical truth, and 'strongly sound', or simply 'sound', if every deduction from Γ is a semantic consequence of Γ. Soundness indicates that the deductive system is faithful to the semantics. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) | |||
| A reaction: Similarly, 'weakly complete' is when every logical truth is a theorem. |
| 1.2.1 | p.12 | 13637 | If a logic is incomplete, its semantic consequence relation is not effective |
| Full Idea: Second-order logic is inherently incomplete, so its semantic consequence relation is not effective. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1) |
| 1.2.1 | p.12 | 13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation |
| Full Idea: An axiomatization is 'categorical' if all its models are isomorphic to one another; ..hence it has 'essentially only one' interpretation [Veblen 1904]. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1) |
| 1.3 | p.16 | 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects |
| Full Idea: Properties are often taken to be intensional; equiangular and equilateral are thought to be different properties of triangles, even though any triangle is equilateral if and only if it is equiangular. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.3) | |||
| A reaction: Many logicians seem to want to treat properties as sets of objects (red being just the set of red things), but this looks like a desperate desire to say everything in first-order logic, where only objects are available to quantify over. |
| 1.3 | p.19 | 13640 | Russell's paradox shows that there are classes which are not iterative sets |
| Full Idea: The argument behind Russell's paradox shows that in set theory there are logical sets (i.e. classes) that are not iterative sets. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.3) | |||
| A reaction: In his preface, Shapiro expresses doubts about the idea of a 'logical set'. Hence the theorists like the iterative hierarchy because it is well-founded and under control, not because it is comprehensive in scope. See all of pp.19-20. |
| 2.1 | p.26 | 13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals |
| Full Idea: 'Definitions' of integers as pairs of naturals, rationals as pairs of integers, reals as Cauchy sequences of rationals, and complex numbers as pairs of reals are reductive foundations of various fields. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.1) | |||
| A reaction: On p.30 (bottom) Shapiro objects that in the process of reduction the numbers acquire properties they didn't have before. |
| 2.3.1 | p.36 | 13642 | Logic is the ideal for learning new propositions on the basis of others |
| Full Idea: A logic can be seen as the ideal of what may be called 'relative justification', the process of coming to know some propositions on the basis of others. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.3.1) | |||
| A reaction: This seems to be the modern idea of logic, as opposed to identification of a set of 'logical truths' from which eternal necessities (such as mathematics) can be derived. 'Know' implies that they are true - which conclusions may not be. |
| 2.5 | p.43 | 13643 | Aristotelian logic is complete |
| Full Idea: Aristotelian logic is complete. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.5) | |||
| A reaction: [He cites Corcoran 1972] |
| 2.5.1 | p.44 | 13644 | Semantics for models uses set-theory |
| Full Idea: Typically, model-theoretic semantics is formulated in set theory. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.5.1) |
| 3.3 | p.73 | 13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables |
| Full Idea: In the standard semantics of second-order logic, by fixing a domain one thereby fixes the range of both the first-order variables and the second-order variables. There is no further 'interpreting' to be done. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 3.3) | |||
| A reaction: This contrasts with 'Henkin' semantics (Idea 13650), or first-order semantics, which involve more than one domain of quantification. |
| 3.3 | p.73 | 13650 | Henkin semantics has separate variables ranging over the relations and over the functions |
|
Full Idea:
In 'Henkin' semantics, in a given model |
|||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 3.3) |
| 4.1 | p.79 | 13646 | Compactness is derived from soundness and completeness |
| Full Idea: Compactness is a corollary of soundness and completeness. If Γ is not satisfiable, then, by completeness, Γ is not consistent. But the deductions contain only finite premises. So a finite subset shows the inconsistency. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) | |||
| A reaction: [this is abbreviated, but a proof of compactness] Since all worthwhile logics are sound, this effectively means that completeness entails compactness. |
| 4.1 | p.80 | 13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity |
| Full Idea: The Löwenheim-Skolem theorems mean that no first-order theory with an infinite model is categorical. If Γ has an infinite model, then it has a model of every infinite cardinality. So first-order languages cannot characterize infinite structures. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) | |||
| A reaction: So much of the debate about different logics hinges on characterizing 'infinite structures' - whatever they are! Shapiro is a leading structuralist in mathematics, so he wants second-order logic to help with his project. |
| 4.1 | p.80 | 13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics |
| Full Idea: The counterparts of Completeness, Compactness and the Löwenheim-Skolem theorems all fail for second-order languages with standard semantics, but hold for Henkin or first-order semantics. Hence such logics are much like first-order logic. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) | |||
| A reaction: Shapiro votes for the standard semantics, because he wants the greater expressive power, especially for the characterization of infinite structures. |
| 4.1 | p.80 | 13647 | Choice is essential for proving downward Löwenheim-Skolem |
| Full Idea: The axiom of choice is essential for proving the downward Löwenheim-Skolem Theorem. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) |
| 4.2 | p.85 | 13651 | A set is 'transitive' if contains every member of each of its members |
| Full Idea: If, for every b∈d, a∈b entails that a∈d, the d is said to be 'transitive'. In other words, d is transitive if it contains every member of each of its members. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.2) | |||
| A reaction: The alternative would be that the members of the set are subsets, but the members of those subsets are not themselves members of the higher-level set. |
| 5 n28 | p.132 | 13657 | First-order arithmetic can't even represent basic number theory |
| Full Idea: Few theorists consider first-order arithmetic to be an adequate representation of even basic number theory. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5 n28) | |||
| A reaction: This will be because of Idea 13656. Even 'basic' number theory will include all sorts of vast infinities, and that seems to be where the trouble is. |
| 5.1.2 | p.105 | 13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set |
| Full Idea: The 'continuum' is the cardinality of the powerset of a denumerably infinite set. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.2) |
| 5.1.3 | p.106 | 13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element |
| Full Idea: A 'well-ordering' of a set X is an irreflexive, transitive, and binary relation on X in which every non-empty subset of X has a least element. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.3) | |||
| A reaction: So there is a beginning, an ongoing sequence, and no retracing of steps. |
| 5.1.4 | p.109 | 13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains |
| Full Idea: In set theory it is central to the iterative conception that the membership relation is well-founded, ...which means there are no infinite descending chains from any relation. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.4) |
| 5.3.3 | p.123 | 13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic |
| Full Idea: There are sets of natural numbers definable in set-theory but not in arithmetic. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.3.3) |
| 6.5 | p.158 | 13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models |
| Full Idea: A language has the Downward Löwenheim-Skolem property if each satisfiable countable set of sentences has a model whose domain is at most countable. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) | |||
| A reaction: This means you can't employ an infinite model to represent a fact about a countable set. |
| 6.5 | p.158 | 13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes |
| Full Idea: A language has the Upward Löwenheim-Skolem property if for each set of sentences whose model has an infinite domain, then it has a model at least as big as each infinite cardinal. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) | |||
| A reaction: This means you can't have a countable model to represent a fact about infinite sets. |
| 6.5 | p.158 | 13661 | A language is 'semantically effective' if its logical truths are recursively enumerable |
| Full Idea: A logical language is 'semantically effective' if the collection of logically true sentences is a recursively enumerable set of strings. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) |
| 6.5 | p.159 | 13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable |
| Full Idea: Tharp (1975) suggested that compactness, semantic effectiveness, and the Löwenheim-Skolem properties are consequences of features one would want a logic to have. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) | |||
| A reaction: I like this proposal, though Shapiro is strongly against. We keep extending our logic so that we can prove new things, but why should we assume that we can prove everything? That's just what Gödel suggests that we should give up on. |
| 7.1 | p.173 | 13662 | First-order logic was an afterthought in the development of modern logic |
| Full Idea: Almost all the systems developed in the first part of the twentieth century are higher-order; first-order logic was an afterthought. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) |
| 7.1 | p.174 | 13663 | Some reject formal properties if they are not defined, or defined impredicatively |
| Full Idea: Some authors (Poincaré and Russell, for example) were disposed to reject properties that are not definable, or are definable only impredicatively. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) | |||
| A reaction: I take Quine to be the culmination of this line of thought, with his general rejection of 'attributes' in logic and in metaphysics. |
| 7.1 | p.176 | 13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions |
| Full Idea: It is claimed that aiming at a universal language for all contexts, and the thesis that logic does not involve a process of abstraction, separates the logicists from algebraists and mathematicians, and also from modern model theory. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) | |||
| A reaction: I am intuitively drawn to the idea that logic is essentially the result of a series of abstractions, so this gives me a further reason not to be a logicist. Shapiro cites Goldfarb 1979 and van Heijenoort 1967. Logicists reduce abstraction to logic. |
| 7.1 | p.177 | 13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets |
| Full Idea: Iterative sets do not exhibit a Boolean structure, because the complement of an iterative set is not itself an iterative set. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) |
| 7.2 | p.178 | 13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order |
| Full Idea: Skolem and Gödel were the main proponents of first-order languages. The higher-order language 'opposition' was championed by Zermelo, Hilbert, and Bernays. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2) |
| 7.2.1 | p.180 | 13668 | Bernays (1918) formulated and proved the completeness of propositional logic |
| Full Idea: Bernays (1918) formulated and proved the completeness of propositional logic, the first precise solution as part of the Hilbert programme. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.1) |
| 7.2.2 | p.182 | 13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? |
| Full Idea: In 1910 Weyl observed that set theory seemed to presuppose natural numbers, and he regarded numbers as more fundamental than sets, as did Fraenkel. Dedekind had developed set theory independently, and used it to formulate numbers. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.2) |
| 7.3 | p.196 | 13670 | Categoricity can't be reached in a first-order language |
| Full Idea: Categoricity cannot be attained in a first-order language. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.3) |
| 9.1 | p.238 | 13673 | The notion of finitude is actually built into first-order languages |
| Full Idea: The notion of finitude is explicitly 'built in' to the systems of first-order languages in one way or another. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1) | |||
| A reaction: Personally I am inclined to think that they are none the worse for that. No one had even thought of all these lovely infinities before 1870, and now we are supposed to change our logic (our actual logic!) to accommodate them. Cf quantum logic. |
| 9.1.4 | p.243 | 13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models |
| Full Idea: The main role of substitutional semantics is to reduce ontology. As an alternative to model-theoretic semantics for formal languages, the idea is to replace the 'satisfaction' relation of formulas (by objects) with the 'truth' of sentences (using terms). | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4) | |||
| A reaction: I find this very appealing, and Ruth Barcan Marcus is the person to look at. My intuition is that logic should have no ontology at all, as it is just about how inference works, not about how things are. Shapiro offers a compromise. |
| 9.1.4 | p.245 | 13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails |
| Full Idea: The Upward Löwenheim-Skolem theorem fails (trivially) with substitutional semantics. If there are only countably many terms of the language, then there are no uncountable substitution models. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4) | |||
| A reaction: Better and better. See Idea 13674. Why postulate more objects than you can possibly name? I'm even suspicious of all real numbers, because you can't properly define them in finite terms. Shapiro objects that the uncountable can't be characterized. |
| 9.1.4 | p.246 | 13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are |
| Full Idea: The main problem of characterizing the natural numbers is to state, somehow, that 0,1,2,.... are all the numbers that there are. We have seen that this can be accomplished with a higher-order language, but not in a first-order language. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4) |
| 9.3 | p.251 | 13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals |
| Full Idea: By convention, the natural numbers are the finite ordinals, the integers are certain equivalence classes of pairs of finite ordinals, etc. | |||
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.3) |
| 1997 | Philosophy of Mathematics |
| p.258 | 10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? | |
| 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. |
| Intro | p.4 | 10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) |
| 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. |
| Intro | p.5 | 10202 | Natural numbers just need an initial object, successors, and an induction principle |
| 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? |
| Intro | p.5 | 10201 | Virtually all of mathematics can be modeled in set theory |
| 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. |
| Intro | p.11 | 10203 | We apprehend small, finite mathematical structures by abstraction from patterns |
| 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. |
| Intro | p.13 | 10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) |
| 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. |
| Intro | p.13 | 10204 | An 'implicit definition' gives a direct description of the relations of an entity |
| 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. |
| Intro | p.16 | 10206 | Modal operators are usually treated as quantifiers |
| 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) |
| Intro | p.17 | 10207 | Anti-realists reject set theory |
| 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. |
| 1 | p.24 | 10208 | Axiom of Choice: some function has a value for every set in a given set |
| 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) |
| 1 | p.24 | 10209 | A function is just an arbitrary correspondence between collections |
| 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. |
| 1 | p.25 | 10210 | If mathematical objects are accepted, then a number of standard principles will follow |
| 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. |
| 2.5 | p.53 | 10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts |
| 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. |
| 2.5 | p.55 | 10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' |
| 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] |
| 3 | p.42 | 10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' |
| 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. |
| 3.1 | p.72 | 10215 | Platonists claim we can state the essence of a number without reference to the others |
| 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? |
| 3.1 | p.74 | 10217 | We can apprehend structures by focusing on or ignoring features of patterns |
| 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. |
| 3.1 | p.76 | 10218 | Baseball positions and chess pieces depend entirely on context |
| 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. |
| 3.3 | p.84 | 10220 | Because one structure exemplifies several systems, a structure is a one-over-many |
| 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. |
| 3.3 | p.85 | 10221 | Is there is no more to structures than the systems that exemplify them? |
| 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. |
| 3.3 | p.87 | 10222 | Mathematical foundations may not be sets; categories are a popular rival |
| 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. |
| 3.4 | p.95 | 10223 | There is no 'structure of all structures', just as there is no set of all sets |
| 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? |
| 3.5 | p.100 | 10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 |
| 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. |
| 4.1 | p.111 | 10227 | The abstract/concrete boundary now seems blurred, and would need a defence |
| 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. |
| 4.1 | p.112 | 10228 | Could infinite structures be apprehended by pattern recognition? |
| 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. |
| 4.1 n1 | p.109 | 10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract |
| 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. |
| 4.2 | p.113 | 10229 | Simple types can be apprehended through their tokens, via abstraction |
| 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). |
| 4.2 | p.115 | 10230 | The 4-pattern is the structure common to all collections of four objects |
| 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. |
| 4.4 | p.96 | 8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics |
| 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? |
| 4.5 | p.124 | 10231 | Abstract objects might come by abstraction over an equivalence class of base entities |
| 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. |
| 4.7 | p.131 | 10233 | Platonism must accept that the Peano Axioms could all be false |
| 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! |
| 4.8 | p.132 | 10234 | Any theory with an infinite model has a model of every infinite cardinality |
| 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. |
| 4.8 | p.135 | 10235 | A sentence is 'satisfiable' if it has a model |
| 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. |
| 4.8 | p.135 | 10237 | Coherence is a primitive, intuitive notion, not reduced to something formal |
| 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. |
| 4.8 | p.135 | 10236 | There is no grounding for mathematics that is more secure than mathematics |
| 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. |
| 4.8 | p.136 | 10238 | The set-theoretical hierarchy contains as many isomorphism types as possible |
| 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. |
| 4.9 | p.138 | 18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis |
| 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. |
| 4.9 | p.139 | 10240 | Model theory deals with relations, reference and extensions |
| 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) |
| 4.9 | p.139 | 10239 | The central notion of model theory is the relation of 'satisfaction' |
| 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) |
| 5.2 | p.150 | 10244 | Intuition is an outright hindrance to five-dimensional geometry |
| 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. |
| 5.3.4 | p.166 | 10248 | Number statements are generalizations about number sequences, and are bound variables |
| 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. |
| 5.4 | p.171 | 18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational |
| 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) |
| 5.5 | p.176 | 10249 | The main mathematical structures are algebraic, ordered, and topological |
| 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. |
| 6.3 | p.187 | 10251 | The law of excluded middle might be seen as a principle of omniscience |
| 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. |
| 6.3 | p.188 | 10252 | The Axiom of Choice seems to license an infinite amount of choosing |
| 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. |
| 6.4 | p.192 | 10253 | Either logic determines objects, or objects determine logic, or they are separate |
| 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. |
| 6.5 | p.194 | 10254 | Can the ideal constructor also destroy objects? |
| 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. |
| 6.5 | p.194 | 10255 | Presumably nothing can block a possible dynamic operation? |
| 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. |
| 6.7 | p.205 | 10256 | For intuitionists, proof is inherently informal |
| 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'. |
| 6.7 | p.207 | 10257 | Intuitionism only sanctions modus ponens if all three components are proved |
| 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. |
| 7.1 | p.216 | 10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models |
| 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. |
| 7.2 | p.222 | 10259 | The two standard explanations of consequence are semantic (in models) and deductive |
| 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. |
| 7.2 | p.223 | 10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) |
| 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. |
| 7.4 | p.233 | 10266 | Why does the 'myth' of possible worlds produce correct modal logic? |
| 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? |
| 7.4 | p.235 | 10268 | Maybe plural quantifiers should be understood in terms of classes or sets |
| 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] |
| 7.5 | p.242 | 10270 | The main versions of structuralism are all definitionally equivalent |
| 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. |
| 8.1 | p.245 | 10272 | The notion of 'object' is at least partially structural and mathematical |
| 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. |
| 8.2 | p.248 | 10273 | Some structures are exemplified by both abstract and concrete |
| 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. |
| 8.2 | p.254 | 10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? |
| 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. |
| 8.3 | p.255 | 10275 | A blurry border is still a border |
| 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). |
| 8.3 | p.255 | 10276 | Mathematical structures are defined by axioms, or in set theory |
| 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. |
| 8.4 | p.256 | 10277 | Structuralism blurs the distinction between mathematical and ordinary objects |
| 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. |
| 8.4 | p.259 | 10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location |
| 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. |
| p.74 | p.79 | 9554 | We can focus on relations between objects (like baseballers), ignoring their other features |
| 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. |
| 2000 | Thinking About Mathematics |
| 1.1 | p.3 | 8725 | Rationalism tries to apply mathematical methodology to all of knowledge |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |||
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 1.2 | p.9 | 8730 | 'Impredicative' definitions refer to the thing being described |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |||
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 1.2 | p.9 | 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |||
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 2.2.1 | p.26 | 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |||
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 5.1 | p.113 | 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |||
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 6.1.1 | p.142 | 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |||
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 6.1.2 | p.144 | 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |||
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 6.2 | p.149 | 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |||
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 7.1 | p.174 | 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |||
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 10.1 | p.258 | 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |||
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 10.1 | p.259 | 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |||
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 10.2 | p.265 | 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |||
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 10.2 | p.267 | 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |||
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 10.3 n7 | p.272 | 8764 | Categories are the best foundation for mathematics |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |||
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 7.2 n4 | p.181 | 18249 | Cauchy gave a formal definition of a converging sequence. |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |||
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |||
| A reaction: The sequence is 'Cauchy' if N exists. |
| 2001 | Higher-Order Logic |
| 2.1 | p.33 | 10290 | Second-order variables also range over properties, sets, relations or functions |
| Full Idea: Second-order variables can range over properties, sets, or relations on the items in the domain-of-discourse, or over functions from the domain itself. | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 2.1 | p.34 | 10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them |
| Full Idea: Upward Löwenheim-Skolem: if a set of first-order formulas is satisfied by a domain of at least the natural numbers, then it is satisfied by a model of at least some infinite cardinal. | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 2.1 | p.34 | 10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model |
| Full Idea: Downward Löwenheim-Skolem: a finite or denumerable set of first-order formulas that is satisfied by a model whose domain is infinite is satisfied in a model whose domain is the natural numbers | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 2.1 | p.34 | 10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems |
| Full Idea: Early study of first-order logic revealed a number of important features. Gödel showed that there is a complete, sound and effective deductive system. It follows that it is Compact, and there are also the downward and upward Löwenheim-Skolem Theorems. | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 2.1 | p.34 | 10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory |
| Full Idea: Full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory. | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) | |||
| A reaction: [he credits Cowles for this remark] Having an unworkable model theory sounds pretty serious to me, as I'm not inclined to be interested in languages which don't produce models of some sort. Surely models are the whole point? |
| 2.2.1 | p.36 | 10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it |
| Full Idea: In studying second-order logic one can think of relations and functions as extensional or intensional, or one can leave it open. Little turns on this here, and so words like 'property', 'class', and 'set' are used interchangeably. | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.2.1) | |||
| A reaction: Important. Students of the metaphysics of properties, who arrive with limited experience of logic, are bewildered by this attitude. Note that the metaphysics is left wide open, so never let logicians hijack the metaphysical problem of properties. |
| 2.3.2 | p.47 | 10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics |
| Full Idea: Both of the Löwenheim-Skolem Theorems fail for second-order languages with a standard semantics | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.3.2) |
| 2.4 | p.49 | 10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic |
| Full Idea: The Löwenheim-Skolem theorem is usually taken as a sort of defect (often thought to be inevitable) of the first-order logic. | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |||
| A reaction: [He is quoting Wang 1974 p.154] |
| 2.4 | p.50 | 10298 | Some say that second-order logic is mathematics, not logic |
| Full Idea: Some authors argue that second-order logic (with standard semantics) is not logic at all, but is a rather obscure form of mathematics. | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) |
| 2.4 | p.51 | 10299 | If the aim of logic is to codify inferences, second-order logic is useless |
| Full Idea: If the goal of logical study is to present a canon of inference, a calculus which codifies correct inference patterns, then second-order logic is a non-starter. | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |||
| A reaction: This seems to be because it is not 'complete'. However, moves like plural quantification seem aimed at capturing ordinary language inferences, so the difficulty is only that there isn't a precise 'calculus'. |
| 2.4 | p.51 | 10300 | Logical consequence can be defined in terms of the logical terminology |
| Full Idea: Informally, logical consequence is sometimes defined in terms of the meanings of a certain collection of terms, the so-called 'logical terminology'. | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |||
| A reaction: This seems to be a compositional account, where we build a full account from an account of the atomic bits, perhaps presented as truth-tables. |
| n 3 | p.52 | 10301 | The axiom of choice is controversial, but it could be replaced |
| Full Idea: The axiom of choice has a troubled history, but is now standard in mathematics. It could be replaced with a principle of comprehension for functions), or one could omit the variables ranging over functions. | |||
| From: Stewart Shapiro (Higher-Order Logic [2001], n 3) |