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Single Idea 10274

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique ]

Full Idea

According to structuralism, someone who uses small natural numbers in everyday life presupposes an infinite structure. It seems absurd that a child who learns to count his toes applies an infinite structure to reality, and thus presupposes the structure.

Gist of Idea

Does someone using small numbers really need to know the infinite structure of arithmetic?

Source

Stewart Shapiro (Philosophy of Mathematics [1997], 8.2)

Book Ref

Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.254

A Reaction

Shapiro says we can meet this objection by thinking of smaller structures embedded in larger ones, with the child knowing the smaller ones.

The 23 ideas with the same theme [objections to structuralism about mathematics]:

8639 If numbers are supposed to be patterns, each number can have many patterns [Frege]
9977 Ordinals can't be defined just by progression; they have intrinsic qualities [Russell]
9627 Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR]
9829 The identity of a number may be fixed by something outside structure - by counting [Dummett]
9828 Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1 [Dummett]
9192 The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett]
18116 Numbers can't be positions, if nothing decides what position a given number has [Bostock]
18117 Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
10815 We don't need 'abstract structures' to have structural truths about successor functions [Lewis]
10629 If structures are relative, this undermines truth-value and objectivity [Hale/Wright]
10628 The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright]
18500 How could structures be mathematical truthmakers? Maths is just true, without truthmakers [Heil]
10274 Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
10186 If set theory is used to define 'structure', we can't define set theory structurally [Burgess]
10187 Abstract algebra concerns relations between models, not common features of all the models [Burgess]
10188 How can mathematical relations be either internal, or external, or intrinsic? [Burgess]
9628 Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR]
10171 The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price]
8926 For mathematical objects to be positions, positions themselves must exist first [MacBride]
14083 Structuralism is right about algebra, but wrong about sets [Linnebo]
14090 In mathematical structuralism the small depends on the large, which is the opposite of physical structures [Linnebo]
14505 Some questions concern mathematical entities, rather than whole structures [Koslicki]
17932 If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan]