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Single Idea 9828

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

Full Idea

The number 0 is not differentiated from 1 by its position in a progression, otherwise there would be no difference between starting with 0 and starting with 1. That is enough to show that numbers are not identifiable just as positions in structures.

Gist of Idea

Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1

Source

Michael Dummett (Frege philosophy of mathematics [1991], Ch. 5)

Book Ref

Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.53

A Reaction

This sounds conclusive, but doesn't feel right. If numbers are a structure, then where you 'start' seems unimportant. Where do you 'start' in St Paul's Cathedral? Starting sounds like a constructivist concept for number theory.

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]