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Single Idea 9554

[filed under theme 18. Thought / E. Abstraction / 3. Abstracta by Ignoring ]

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.

Gist of Idea

We can focus on relations between objects (like baseballers), ignoring their other features

Source

Stewart Shapiro (Philosophy of Mathematics [1997], p.74), quoted by Charles Chihara - A Structural Account of Mathematics

Book Ref

Chihara,Charles: 'A Structural Account of Mathematics' [OUP 2004], p.79

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.

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]