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Single Idea 8926

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

Full Idea

The identification of mathematical objects with positions in structures rests upon the prior credibility of the thesis that positions are objects in their own right.

Gist of Idea

For mathematical objects to be positions, positions themselves must exist first

Source

Fraser MacBride (Structuralism Reconsidered [2007], §3)

Book Ref

'Oxf Handbk of Philosophy of Maths and Logic', ed/tr. Shapiro,Stewart [OUP 2007], p.585

A Reaction

Sounds devastating, but something has to get the whole thing off the ground. This is why Resnik's word 'patterns' is so appealing. Patterns stare you in the face, and they don't change if all the objects making it up are replaced by others.

The 25 ideas from Fraser MacBride

21352 'Multigrade' relations are those lacking a fixed number of relata [MacBride]
21354 It may be that internal relations like proportion exist, because we directly perceive it [MacBride]
21353 Internal relations are fixed by existences, or characters, or supervenience on characters [MacBride]
8923 Numbers are identified by their main properties and relations, involving the successor function [MacBride]
8926 For mathematical objects to be positions, positions themselves must exist first [MacBride]
18466 If truthmaking is classical entailment, then anything whatsoever makes a necessary truth [MacBride]
18471 Different types of 'grounding' seem to have no more than a family resemblance relation [MacBride]
18472 Which has priority - 'grounding' or 'truth-making'? [MacBride]
18473 'Maximalism' says every truth has an actual truthmaker [MacBride]
18474 Does 'this sentence has no truth-maker' have a truth-maker? Reductio suggests it can't have [MacBride]
18475 Russell allows some complex facts, but Wittgenstein only allows atomic facts [MacBride]
18476 'A is F' may not be positive ('is dead'), and 'A is not-F' may not be negative ('is not blind') [MacBride]
18479 There are different types of truthmakers for different types of negative truth [MacBride]
18477 There aren't enough positive states out there to support all the negative truths [MacBride]
18478 Wittgenstein's plan to show there is only logical necessity failed, because of colours [MacBride]
18480 Maybe it only exists if it is a truthmaker (rather than the value of a variable)? [MacBride]
18482 Optimalists say that negative and universal are true 'by default' from the positive truths [MacBride]
18481 Maximalism follows Russell, and optimalism (no negative or universal truthmakers) follows Wittgenstein [MacBride]
18483 The main idea of truth-making is that what a proposition is about is what matters [MacBride]
18484 Phenomenalists, behaviourists and presentists can't supply credible truth-makers [MacBride]
18485 Even idealists could accept truthmakers, as mind-dependent [MacBride]
18486 We might define truth as arising from the truth-maker relation [MacBride]
18490 Maybe 'makes true' is not an active verb, but just a formal connective like 'because'? [MacBride]
18489 Connectives link sentences without linking their meanings [MacBride]
18493 Truthmaker talk of 'something' making sentences true, which presupposes objectual quantification [MacBride]