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Ideas of Stewart Shapiro, by Text

[American, b.1951, Professor at Ohio State University; visiting Professor at St Andrew's University.]

1989 Structure and Ontology
146 p.60 A structure is an abstraction, focussing on relationships, and ignoring other features
1991 Foundations without Foundationalism
p.225 Second-order logic is better than set theory, since it only adds relations and operations, and nothing else
Pref p.-17 The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed
Pref p.-17 Mathematics and logic have no border, and logic must involve mathematics and its ontology
Pref p.-16 Semantic consequence is ineffective in second-order logic
Pref p.-15 There is no 'correct' logic for natural languages
Pref p.-14 We can live well without completeness in logic
Pref p.-13 Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics?
Pref p.-12 Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures
Pref p.-9 Are sets part of logic, or part of mathematics?
1.1 p.3 Finding the logical form of a sentence is difficult, and there are no criteria of correctness
1.1 p.5 'Satisfaction' is a function from models, assignments, and formulas to {true,false}
1.1 p.6 Satisfaction is 'truth in a model', which is a model of 'truth'
1.1 p.8 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence
1.2.1 p.12 An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation
1.2.1 p.12 If a logic is incomplete, its semantic consequence relation is not effective
1.3 p.16 Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects
1.3 p.19 Russell's paradox shows that there are classes which are not iterative sets
2.1 p.26 Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals
2.3.1 p.36 Logic is the ideal for learning new propositions on the basis of others
2.5 p.43 Aristotelian logic is complete
2.5.1 p.44 Semantics for models uses set-theory
3.3 p.73 Henkin semantics has separate variables ranging over the relations and over the functions
3.3 p.73 In standard semantics for second-order logic, a single domain fixes the ranges for the variables
4.1 p.79 Compactness is derived from soundness and completeness
4.1 p.80 The L÷wenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity
4.1 p.80 Completeness, Compactness and L÷wenheim-Skolem fail in second-order standard semantics
4.1 p.80 Choice is essential for proving downward L÷wenheim-Skolem
4.2 p.85 A set is 'transitive' if contains every member of each of its members
5 n28 p.132 First-order arithmetic can't even represent basic number theory
5.1.2 p.105 The 'continuum' is the cardinality of the powerset of a denumerably infinite set
5.1.3 p.106 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element
5.1.4 p.109 It is central to the iterative conception that membership is well-founded, with no infinite descending chains
5.3.3 p.123 Some sets of natural numbers are definable in set-theory but not in arithmetic
6.5 p.158 Upward L÷wenheim-Skolem: each infinite model has infinite models of all sizes
6.5 p.158 Downward L÷wenheim-Skolem: each satisfiable countable set always has countable models
6.5 p.158 A language is 'semantically effective' if its logical truths are recursively enumerable
6.5 p.159 Maybe compactness, semantic effectiveness, and the L÷wenheim-Skolem properties are desirable
7.1 p.173 First-order logic was an afterthought in the development of modern logic
7.1 p.174 Some reject formal properties if they are not defined, or defined impredicatively
7.1 p.176 Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions
7.1 p.177 Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets
7.2 p.178 Skolem and G÷del championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order
7.2.1 p.180 Bernays (1918) formulated and proved the completeness of propositional logic
7.2.2 p.182 Can one develop set theory first, then derive numbers, or are numbers more basic?
7.3 p.196 Categoricity can't be reached in a first-order language
9.1 p.238 The notion of finitude is actually built into first-order languages
9.1.4 p.243 We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models
9.1.4 p.245 Substitutional semantics only has countably many terms, so Upward L÷wenheim-Skolem trivially fails
9.1.4 p.246 Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are
9.3 p.251 Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals
1997 Philosophy of Mathematics
p.258 Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules?
Intro p.4 We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false)
Intro p.5 Natural numbers just need an initial object, successors, and an induction principle
Intro p.5 Virtually all of mathematics can be modeled in set theory
Intro p.11 We apprehend small, finite mathematical structures by abstraction from patterns
Intro p.13 Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic)
Intro p.13 An 'implicit definition' gives a direct description of the relations of an entity
Intro p.16 Modal operators are usually treated as quantifiers
Intro p.17 Anti-realists reject set theory
1 p.24 Axiom of Choice: some function has a value for every set in a given set
1 p.24 A function is just an arbitrary correspondence between collections
1 p.25 If mathematical objects are accepted, then a number of standard principles will follow
2.5 p.53 Real numbers are thought of as either Cauchy sequences or Dedekind cuts
2.5 p.55 Theory ontology is never complete, but is only determined 'up to isomorphism'
3 p.42 Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and'
3.1 p.72 Platonists claim we can state the essence of a number without reference to the others
3.1 p.74 We can apprehend structures by focusing on or ignoring features of patterns
3.1 p.76 Baseball positions and chess pieces depend entirely on context
3.3 p.84 Because one structure exemplifies several systems, a structure is a one-over-many
3.3 p.85 Is there is no more to structures than the systems that exemplify them?
3.3 p.87 Mathematical foundations may not be sets; categories are a popular rival
3.4 p.95 There is no 'structure of all structures', just as there is no set of all sets
3.5 p.100 The even numbers have the natural-number structure, with 6 playing the role of 3
4.1 p.111 The abstract/concrete boundary now seems blurred, and would need a defence
4.1 p.112 Could infinite structures be apprehended by pattern recognition?
4.1 n1 p.109 Mathematicians regard arithmetic as concrete, and group theory as abstract
4.2 p.113 Simple types can be apprehended through their tokens, via abstraction
4.2 p.115 The 4-pattern is the structure common to all collections of four objects
4.4 p.96 Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics
4.5 p.124 Abstract objects might come by abstraction over an equivalence class of base entities
4.6 p.127 Property extensions outstrip objects, so shortage of objects caused the Caesar problem
4.7 p.131 Platonism must accept that the Peano Axioms could all be false
4.8 p.132 Any theory with an infinite model has a model of every infinite cardinality
4.8 p.135 A sentence is 'satisfiable' if it has a model
4.8 p.135 Coherence is a primitive, intuitive notion, not reduced to something formal
4.8 p.135 There is no grounding for mathematics that is more secure than mathematics
4.8 p.136 The set-theoretical hierarchy contains as many isomorphism types as possible
4.9 p.138 Understanding the real-number structure is knowing usage of the axiomatic language of analysis
4.9 p.139 The central notion of model theory is the relation of 'satisfaction'
4.9 p.139 Model theory deals with relations, reference and extensions
5.2 p.150 Intuition is an outright hindrance to five-dimensional geometry
5.3.4 p.166 Number statements are generalizations about number sequences, and are bound variables
5.4 p.171 Cuts are made by the smallest upper or largest lower number, some of them not rational
5.5 p.176 The main mathematical structures are algebraic, ordered, and topological
6.3 p.187 The law of excluded middle might be seen as a principle of omniscience
6.3 p.188 The Axiom of Choice seems to license an infinite amount of choosing
6.4 p.192 Either logic determines objects, or objects determine logic, or they are separate
6.5 p.194 Presumably nothing can block a possible dynamic operation?
6.5 p.194 Can the ideal constructor also destroy objects?
6.7 p.205 For intuitionists, proof is inherently informal
6.7 p.207 Intuitionism only sanctions modus ponens if all three components are proved
7.1 p.216 Logical modalities may be acceptable, because they are reducible to satisfaction in models
7.2 p.222 The two standard explanations of consequence are semantic (in models) and deductive
7.2 p.223 Fictionalism eschews the abstract, but it still needs the possible (without model theory)
7.4 p.233 Why does the 'myth' of possible worlds produce correct modal logic?
7.4 p.235 Maybe plural quantifiers should be understood in terms of classes or sets
7.5 p.242 The main versions of structuralism are all definitionally equivalent
8.1 p.245 The notion of 'object' is at least partially structural and mathematical
8.2 p.248 Some structures are exemplified by both abstract and concrete
8.2 p.254 Does someone using small numbers really need to know the infinite structure of arithmetic?
8.3 p.255 Mathematical structures are defined by axioms, or in set theory
8.3 p.255 A blurry border is still a border
8.4 p.256 Structuralism blurs the distinction between mathematical and ordinary objects
8.4 p.259 A stone is a position in some pattern, and can be viewed as an object, or as a location
p.74 p.79 We can focus on relations between objects (like baseballers), ignoring their other features
2000 Thinking About Mathematics
1.1 p.3 Rationalism tries to apply mathematical methodology to all of knowledge
1.2 p.9 'Impredicative' definitions refer to the thing being described
1.2 p.9 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects
2.2.1 p.26 Conceptualist are just realists or idealist or nominalists, depending on their view of concepts
2.4 p.42 The Continuum Hypothesis says there are no sets between the natural numbers and reals
5.1 p.113 Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own
5.3 p.128 Realists are happy with impredicative definitions, which describe entities in terms of other existing entities
6.1.1 p.142 Term Formalism says mathematics is just about symbols - but real numbers have no names
6.1.2 p.144 Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
6.2 p.149 Deductivism says mathematics is logical consequences of uninterpreted axioms
7.1 p.174 Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
10.1 p.258 Numbers do not exist independently; the essence of a number is its relations to other numbers
10.1 p.259 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them
10.2 p.265 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3
10.2 p.267 The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
10.3 n7 p.272 Categories are the best foundation for mathematics
7.2 n4 p.181 Cauchy gave a formal definition of a converging sequence.
2001 Higher-Order Logic
2.1 p.33 Second-order variables also range over properties, sets, relations or functions
2.1 p.34 Downward L÷wenheim-Skolem: if there's an infinite model, there is a countable model
2.1 p.34 Up L÷wenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them
2.1 p.34 First-order logic is Complete, and Compact, with the L÷wenheim-Skolem Theorems
2.1 p.34 Second-order logic has the expressive power for mathematics, but an unworkable model theory
2.2.1 p.36 Logicians use 'property' and 'set' interchangeably, with little hanging on it
2.3.2 p.47 The L÷wenheim-Skolem Theorems fail for second-order languages with standard semantics
2.4 p.49 The L÷wenheim-Skolem theorem seems to be a defect of first-order logic
2.4 p.50 Some say that second-order logic is mathematics, not logic
2.4 p.51 If the aim of logic is to codify inferences, second-order logic is useless
2.4 p.51 Logical consequence can be defined in terms of the logical terminology
n 3 p.52 The axiom of choice is controversial, but it could be replaced