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Single Idea 17804
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
]
Full Idea
No axiomatic theory, formal or informal, of first or of higher order can logically play a foundational role in mathematics. ...It is obvious that you cannot use the axiomatic method to explain what the axiomatic method is.
Gist of Idea
Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms
Source
John Mayberry (What Required for Foundation for Maths? [1994], p.415-2)
Book Ref
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.415
The
31 ideas
from John Mayberry
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17774
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Definitions make our intuitions mathematically useful
[Mayberry]
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17775
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If proof and definition are central, then mathematics needs and possesses foundations
[Mayberry]
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17776
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The ultimate principles and concepts of mathematics are presumed, or grasped directly
[Mayberry]
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17777
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Foundations need concepts, definition rules, premises, and proof rules
[Mayberry]
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17773
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Proof shows that it is true, but also why it must be true
[Mayberry]
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17779
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'Classificatory' axioms aim at revealing similarity in morphology of structures
[Mayberry]
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17778
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Axiomatiation relies on isomorphic structures being essentially the same
[Mayberry]
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17780
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'Eliminatory' axioms get rid of traditional ideal and abstract objects
[Mayberry]
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17782
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Greek quantities were concrete, and ratio and proportion were their science
[Mayberry]
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17781
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Real numbers were invented, as objects, to simplify and generalise 'quantity'
[Mayberry]
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17785
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Real numbers as abstracted objects are now treated as complete ordered fields
[Mayberry]
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17784
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Real numbers can be eliminated, by axiom systems for complete ordered fields
[Mayberry]
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17786
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The mainstream of modern logic sees it as a branch of mathematics
[Mayberry]
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17787
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Big logic has one fixed domain, but standard logic has a domain for each interpretation
[Mayberry]
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17788
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First-order logic only has its main theorems because it is so weak
[Mayberry]
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17791
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Only second-order logic can capture mathematical structure up to isomorphism
[Mayberry]
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17790
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No Löwenheim-Skolem logic can axiomatise real analysis
[Mayberry]
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17789
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No logic which can axiomatise arithmetic can be compact or complete
[Mayberry]
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17792
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1st-order PA is only interesting because of results which use 2nd-order PA
[Mayberry]
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17793
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It is only 2nd-order isomorphism which suggested first-order PA completeness
[Mayberry]
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17794
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Set theory is not just first-order ZF, because that is inadequate for mathematics
[Mayberry]
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17795
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Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation
[Mayberry]
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17796
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There is a semi-categorical axiomatisation of set-theory
[Mayberry]
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17801
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The set hierarchy doesn't rely on the dubious notion of 'generating' them
[Mayberry]
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17800
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The misnamed Axiom of Infinity says the natural numbers are finite in size
[Mayberry]
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17799
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Cantor's infinite is an absolute, of all the sets or all the ordinal numbers
[Mayberry]
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17797
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Cantor extended the finite (rather than 'taming the infinite')
[Mayberry]
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17802
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We don't translate mathematics into set theory, because it comes embodied in that way
[Mayberry]
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17804
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Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms
[Mayberry]
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17803
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Limitation of size is part of the very conception of a set
[Mayberry]
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17805
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Set theory is not just another axiomatised part of mathematics
[Mayberry]
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