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Single Idea 8702

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

Full Idea

Structuralists give a historical account of why the 'same' number occupies different structures. Numbers are equivalent rather than identical. 8 is the immediate predecessor of 9 in the whole numbers, but in the rationals 9 has no predecessor.

Clarification

For 'rational' numbers see Idea 8668

Gist of Idea

In structuralism the number 8 is not quite the same in different structures, only equivalent

Source

Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4)

Book Ref

Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.93

A Reaction

I don't become a different person if I move from a detached house to a terraced house. This suggests that 8 can't be entirely defined by its relations, and yet it is hard to see what its intrinsic nature could be, apart from the units which compose it.

The 41 ideas from 'Introducing the Philosophy of Mathematics'

8713 In classical/realist logic the connectives are defined by truth-tables [Friend]
8663 Raising omega to successive powers of omega reveal an infinity of infinities [Friend]
8662 The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]
8661 The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]
8664 Cardinal numbers answer 'how many?', with the order being irrelevant [Friend]
8671 The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend]
8669 Between any two rational numbers there is an infinite number of rational numbers [Friend]
8665 A 'proper subset' of A contains only members of A, but not all of them [Friend]
8672 A 'powerset' is all the subsets of a set [Friend]
8666 Infinite sets correspond one-to-one with a subset [Friend]
8667 The 'integers' are the positive and negative natural numbers, plus zero [Friend]
8668 The 'rational' numbers are those representable as fractions [Friend]
8670 A number is 'irrational' if it cannot be represented as a fraction [Friend]
8675 Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend]
8674 The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend]
8677 Set theory makes a minimum ontological claim, that the empty set exists [Friend]
3678 Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend]
8676 Is mathematics based on sets, types, categories, models or topology? [Friend]
8678 Most mathematical theories can be translated into the language of set theory [Friend]
8680 Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend]
8681 The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend]
8682 Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend]
8685 Studying biology presumes the laws of chemistry, and it could never contradict them [Friend]
8688 Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend]
8694 Free logic was developed for fictional or non-existent objects [Friend]
8696 Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend]
8695 Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend]
8701 The number 8 in isolation from the other numbers is of no interest [Friend]
8702 In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend]
8699 Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend]
8700 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend]
8704 Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend]
8706 Constructivism rejects too much mathematics [Friend]
8705 Anti-realists see truth as our servant, and epistemically contrained [Friend]
8708 Double negation elimination is not valid in intuitionist logic [Friend]
8709 The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend]
8707 Intuitionists typically retain bivalence but reject the law of excluded middle [Friend]
8711 Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend]
8712 Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend]
8716 Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend]
8721 An 'impredicative' definition seems circular, because it uses the term being defined [Friend]