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Single Idea 10230
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
]
Full Idea
The 4-pattern is the structure common to all collections of four objects.
Gist of Idea
The 4-pattern is the structure common to all collections of four objects
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 4.2)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.115
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.
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]
|
10214
|
Theory ontology is never complete, but is only determined 'up to isomorphism'
[Shapiro]
|
10213
|
Real numbers are thought of as either Cauchy sequences or Dedekind cuts
[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]
|
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]
|
10222
|
Mathematical foundations may not be sets; categories are a popular rival
[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]
|
10238
|
The set-theoretical hierarchy contains as many isomorphism types as possible
[Shapiro]
|
10236
|
There is no grounding for mathematics that is more secure than mathematics
[Shapiro]
|
10234
|
Any theory with an infinite model has a model of every infinite cardinality
[Shapiro]
|
10235
|
A sentence is 'satisfiable' if it has a model
[Shapiro]
|
10237
|
Coherence is a primitive, intuitive notion, not reduced to something formal
[Shapiro]
|
10239
|
The central notion of model theory is the relation of 'satisfaction'
[Shapiro]
|
10240
|
Model theory deals with relations, reference and extensions
[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]
|