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Single Idea 10220

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism ]

Full Idea

Because the same structure can be exemplified by more than one system, a structure is a one-over-many.

Gist of Idea

Because one structure exemplifies several systems, a structure is a one-over-many

Source

Stewart Shapiro (Philosophy of Mathematics [1997], 3.3)

Book Ref

Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.84

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.

Related Idea

Idea 10217 We can apprehend structures by focusing on or ignoring features of patterns [Shapiro]

The 64 ideas from 'Philosophy of Mathematics'

10279 Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro]
10200 We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro]
10203 We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro]
10204 An 'implicit definition' gives a direct description of the relations of an entity [Shapiro]
10206 Modal operators are usually treated as quantifiers [Shapiro]
10207 Anti-realists reject set theory [Shapiro]
10201 Virtually all of mathematics can be modeled in set theory [Shapiro]
10202 Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
10205 Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro]
10208 Axiom of Choice: some function has a value for every set in a given set [Shapiro]
10209 A function is just an arbitrary correspondence between collections [Shapiro]
10210 If mathematical objects are accepted, then a number of standard principles will follow [Shapiro]
10213 Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
10214 Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
10212 Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro]
10218 Baseball positions and chess pieces depend entirely on context [Shapiro]
10215 Platonists claim we can state the essence of a number without reference to the others [Shapiro]
10217 We can apprehend structures by focusing on or ignoring features of patterns [Shapiro]
10222 Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
10221 Is there is no more to structures than the systems that exemplify them? [Shapiro]
10220 Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
10223 There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
10224 The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro]
10228 Could infinite structures be apprehended by pattern recognition? [Shapiro]
10227 The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro]
10226 Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro]
10229 Simple types can be apprehended through their tokens, via abstraction [Shapiro]
10230 The 4-pattern is the structure common to all collections of four objects [Shapiro]
8703 Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend]
10231 Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro]
10233 Platonism must accept that the Peano Axioms could all be false [Shapiro]
10234 Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
10236 There is no grounding for mathematics that is more secure than mathematics [Shapiro]
10235 A sentence is 'satisfiable' if it has a model [Shapiro]
10238 The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro]
10237 Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro]
10240 Model theory deals with relations, reference and extensions [Shapiro]
10239 The central notion of model theory is the relation of 'satisfaction' [Shapiro]
18243 Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro]
10244 Intuition is an outright hindrance to five-dimensional geometry [Shapiro]
10248 Number statements are generalizations about number sequences, and are bound variables [Shapiro]
18245 Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
10249 The main mathematical structures are algebraic, ordered, and topological [Shapiro]
10251 The law of excluded middle might be seen as a principle of omniscience [Shapiro]
10252 The Axiom of Choice seems to license an infinite amount of choosing [Shapiro]
10253 Either logic determines objects, or objects determine logic, or they are separate [Shapiro]
10254 Can the ideal constructor also destroy objects? [Shapiro]
10255 Presumably nothing can block a possible dynamic operation? [Shapiro]
10257 Intuitionism only sanctions modus ponens if all three components are proved [Shapiro]
10256 For intuitionists, proof is inherently informal [Shapiro]
10258 Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro]
10262 Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro]
10259 The two standard explanations of consequence are semantic (in models) and deductive [Shapiro]
10268 Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro]
10266 Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro]
10270 The main versions of structuralism are all definitionally equivalent [Shapiro]
10272 The notion of 'object' is at least partially structural and mathematical [Shapiro]
10274 Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
10273 Some structures are exemplified by both abstract and concrete [Shapiro]
10276 Mathematical structures are defined by axioms, or in set theory [Shapiro]
10275 A blurry border is still a border [Shapiro]
10277 Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro]
10280 A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro]
9554 We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro]