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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 9626 A structure is an abstraction, focussing on relationships, and ignoring other features
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 [Lavine]
Pref p.-17 13624 The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed
Pref p.-17 13625 Mathematics and logic have no border, and logic must involve mathematics and its ontology
Pref p.-16 13626 Semantic consequence is ineffective in second-order logic
Pref p.-15 13627 There is no 'correct' logic for natural languages
Pref p.-14 13628 We can live well without completeness in logic
Pref p.-13 13629 Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics?
Pref p.-12 13630 Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures
Pref p.-9 13631 Are sets part of logic, or part of mathematics?
1.1 p.3 13632 Finding the logical form of a sentence is difficult, and there are no criteria of correctness
1.1 p.5 13633 'Satisfaction' is a function from models, assignments, and formulas to {true,false}
1.1 p.6 13634 Satisfaction is 'truth in a model', which is 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
1.2.1 p.12 13636 An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation
1.2.1 p.12 13637 If a logic is incomplete, its semantic consequence relation is not effective
1.3 p.16 13638 Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects
1.3 p.19 13640 Russell's paradox shows that there are classes which are not iterative sets
2.1 p.26 13641 Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals
2.3.1 p.36 13642 Logic is the ideal for learning new propositions on the basis of others
2.5 p.43 13643 Aristotelian logic is complete
2.5.1 p.44 13644 Semantics for models uses set-theory
3.3 p.73 13650 Henkin semantics has separate variables ranging over the relations and over the functions
3.3 p.73 13645 In standard semantics for second-order logic, a single domain fixes the ranges for the variables
4.1 p.79 13646 Compactness is derived from soundness and completeness
4.1 p.80 13648 The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity
4.1 p.80 13649 Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics
4.1 p.80 13647 Choice is essential for proving downward Löwenheim-Skolem
4.2 p.85 13651 A set is 'transitive' if contains every member of each of its members
5 n28 p.132 13657 First-order arithmetic can't even represent basic number theory
5.1.2 p.105 13652 The 'continuum' is the cardinality of the powerset of a denumerably infinite set
5.1.3 p.106 13653 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element
5.1.4 p.109 13654 It is central to the iterative conception that membership is well-founded, with no infinite descending chains
5.3.3 p.123 13656 Some sets of natural numbers are definable in set-theory but not in arithmetic
6.5 p.158 13658 Downward Löwenheim-Skolem: each satisfiable countable set always has countable models
6.5 p.158 13659 Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes
6.5 p.158 13661 A language is 'semantically effective' if its logical truths are recursively enumerable
6.5 p.159 13660 Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable
7.1 p.173 13662 First-order logic was an afterthought in the development of modern logic
7.1 p.174 13663 Some reject formal properties if they are not defined, or defined impredicatively
7.1 p.176 13664 Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions
7.1 p.177 13666 Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets
7.2 p.178 13667 Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order
7.2.1 p.180 13668 Bernays (1918) formulated and proved the completeness of propositional logic
7.2.2 p.182 13669 Can one develop set theory first, then derive numbers, or are numbers more basic?
7.3 p.196 13670 Categoricity can't be reached in a first-order language
9.1 p.238 13673 The notion of finitude is actually built into first-order languages
9.1.4 p.243 13674 We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models
9.1.4 p.245 13675 Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails
9.1.4 p.246 13676 Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are
9.3 p.251 13677 Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals
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?
Intro p.4 10200 We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false)
Intro p.5 10202 Natural numbers just need an initial object, successors, and an induction principle
Intro p.5 10201 Virtually all of mathematics can be modeled in set theory
Intro p.11 10203 We apprehend small, finite mathematical structures by abstraction from patterns
Intro p.13 10205 Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic)
Intro p.13 10204 An 'implicit definition' gives a direct description of the relations of an entity
Intro p.16 10206 Modal operators are usually treated as quantifiers
Intro p.17 10207 Anti-realists reject set theory
1 p.24 10208 Axiom of Choice: some function has a value for every set in a given set
1 p.24 10209 A function is just an arbitrary correspondence between collections
1 p.25 10210 If mathematical objects are accepted, then a number of standard principles will follow
2.5 p.53 10213 Real numbers are thought of as either Cauchy sequences or Dedekind cuts
2.5 p.55 10214 Theory ontology is never complete, but is only determined 'up to isomorphism'
3 p.42 10212 Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and'
3.1 p.72 10215 Platonists claim we can state the essence of a number without reference to the others
3.1 p.74 10217 We can apprehend structures by focusing on or ignoring features of patterns
3.1 p.76 10218 Baseball positions and chess pieces depend entirely on context
3.3 p.84 10220 Because one structure exemplifies several systems, a structure is a one-over-many
3.3 p.85 10221 Is there is no more to structures than the systems that exemplify them?
3.3 p.87 10222 Mathematical foundations may not be sets; categories are a popular rival
3.4 p.95 10223 There is no 'structure of all structures', just as there is no set of all sets
3.5 p.100 10224 The even numbers have the natural-number structure, with 6 playing the role of 3
4.1 p.111 10227 The abstract/concrete boundary now seems blurred, and would need a defence
4.1 p.112 10228 Could infinite structures be apprehended by pattern recognition?
4.1 n1 p.109 10226 Mathematicians regard arithmetic as concrete, and group theory as abstract
4.2 p.113 10229 Simple types can be apprehended through their tokens, via abstraction
4.2 p.115 10230 The 4-pattern is the structure common to all collections of four objects
4.4 p.96 8703 Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Friend]
4.5 p.124 10231 Abstract objects might come by abstraction over an equivalence class of base entities
4.7 p.131 10233 Platonism must accept that the Peano Axioms could all be false
4.8 p.132 10234 Any theory with an infinite model has a model of every infinite cardinality
4.8 p.135 10235 A sentence is 'satisfiable' if it has a model
4.8 p.135 10237 Coherence is a primitive, intuitive notion, not reduced to something formal
4.8 p.135 10236 There is no grounding for mathematics that is more secure than mathematics
4.8 p.136 10238 The set-theoretical hierarchy contains as many isomorphism types as possible
4.9 p.138 18243 Understanding the real-number structure is knowing usage of the axiomatic language of analysis
4.9 p.139 10240 Model theory deals with relations, reference and extensions
4.9 p.139 10239 The central notion of model theory is the relation of 'satisfaction'
5.2 p.150 10244 Intuition is an outright hindrance to five-dimensional geometry
5.3.4 p.166 10248 Number statements are generalizations about number sequences, and are bound variables
5.4 p.171 18245 Cuts are made by the smallest upper or largest lower number, some of them not rational
5.5 p.176 10249 The main mathematical structures are algebraic, ordered, and topological
6.3 p.187 10251 The law of excluded middle might be seen as a principle of omniscience
6.3 p.188 10252 The Axiom of Choice seems to license an infinite amount of choosing
6.4 p.192 10253 Either logic determines objects, or objects determine logic, or they are separate
6.5 p.194 10254 Can the ideal constructor also destroy objects?
6.5 p.194 10255 Presumably nothing can block a possible dynamic operation?
6.7 p.205 10256 For intuitionists, proof is inherently informal
6.7 p.207 10257 Intuitionism only sanctions modus ponens if all three components are proved
7.1 p.216 10258 Logical modalities may be acceptable, because they are reducible to satisfaction in models
7.2 p.222 10259 The two standard explanations of consequence are semantic (in models) and deductive
7.2 p.223 10262 Fictionalism eschews the abstract, but it still needs the possible (without model theory)
7.4 p.233 10266 Why does the 'myth' of possible worlds produce correct modal logic?
7.4 p.235 10268 Maybe plural quantifiers should be understood in terms of classes or sets
7.5 p.242 10270 The main versions of structuralism are all definitionally equivalent
8.1 p.245 10272 The notion of 'object' is at least partially structural and mathematical
8.2 p.248 10273 Some structures are exemplified by both abstract and concrete
8.2 p.254 10274 Does someone using small numbers really need to know the infinite structure of arithmetic?
8.3 p.255 10276 Mathematical structures are defined by axioms, or in set theory
8.3 p.255 10275 A blurry border is still a border
8.4 p.256 10277 Structuralism blurs the distinction between mathematical and ordinary objects
8.4 p.259 10280 A stone is a position in some pattern, and can be viewed as an object, or as a location
p.74 p.79 9554 We can focus on relations between objects (like baseballers), ignoring their other features
2000 Thinking About Mathematics
1.1 p.3 8725 Rationalism tries to apply mathematical methodology to all of knowledge
1.2 p.9 8730 'Impredicative' definitions refer to the thing being described
1.2 p.9 8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects
2.2.1 p.26 8731 Conceptualist are just realists or idealist or nominalists, depending on their view of concepts
5.1 p.113 8744 Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own
6.1.1 p.142 8749 Term Formalism says mathematics is just about symbols - but real numbers have no names
6.1.2 p.144 8750 Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
6.2 p.149 8752 Deductivism says mathematics is logical consequences of uninterpreted axioms
7.1 p.174 8753 Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
10.1 p.258 8760 Numbers do not exist independently; the essence of a number is its relations to other numbers
10.1 p.259 8761 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them
10.2 p.265 8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3
10.2 p.267 8763 The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
10.3 n7 p.272 8764 Categories are the best foundation for mathematics
7.2 n4 p.181 18249 Cauchy gave a formal definition of a converging sequence.
2001 Higher-Order Logic
2.1 p.33 10290 Second-order variables also range over properties, sets, relations or functions
2.1 p.34 10588 First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems
2.1 p.34 10292 Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model
2.1 p.34 10590 Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them
2.1 p.34 10294 Second-order logic has the expressive power for mathematics, but an unworkable model theory
2.2.1 p.36 10591 Logicians use 'property' and 'set' interchangeably, with little hanging on it
2.3.2 p.47 10296 The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics
2.4 p.49 10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic
2.4 p.50 10298 Some say that second-order logic is mathematics, not logic
2.4 p.51 10299 If the aim of logic is to codify inferences, second-order logic is useless
2.4 p.51 10300 Logical consequence can be defined in terms of the logical terminology
n 3 p.52 10301 The axiom of choice is controversial, but it could be replaced