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Single Idea 10167
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
]
Full Idea
Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures.
Clarification
[Bourbaki is a notably unusual mathematician - worth investigating!]
Gist of Idea
Structuralism emerged from abstract algebra, axioms, and set theory and its structures
Source
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
A Reaction
In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent.
The
18 ideas
from 'Structures and Structuralism in Phil of Maths'
10166
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ZFC set theory has only 'pure' sets, without 'urelements'
[Reck/Price]
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10165
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'Analysis' is the theory of the real numbers
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10167
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Structuralism emerged from abstract algebra, axioms, and set theory and its structures
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10164
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Peano Arithmetic can have three second-order axioms, plus '1' and 'successor'
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10168
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Formalist Structuralism says the ontology is vacuous, or formal, or inference relations
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10172
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Set-theory gives a unified and an explicit basis for mathematics
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10171
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The existence of an infinite set is assumed by Relativist Structuralism
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10173
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A nominalist might avoid abstract objects by just appealing to mereological sums
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10169
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Relativist Structuralism just stipulates one successful model as its arithmetic
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10174
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Mereological arithmetic needs infinite objects, and function definitions
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10170
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While true-in-a-model seems relative, true-in-all-models seems not to be
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10175
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Three types of variable in second-order logic, for objects, functions, and predicates/sets
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10178
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Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous
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10176
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Universalist Structuralism is based on generalised if-then claims, not one particular model
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10177
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Universalist Structuralism eliminates the base element, as a variable, which is then quantified out
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10179
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There are 'particular' structures, and 'universal' structures (what the former have in common)
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10181
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Pattern Structuralism studies what isomorphic arithmetic models have in common
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10182
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There are Formalist, Relativist, Universalist and Pattern structuralism
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