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4 ideas
17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
Full Idea: The central dogma of the axiomatic method is this: isomorphic structures are mathematically indistinguishable in their essential properties. | |
From: John Mayberry (What Required for Foundation for Maths? [1994], p.406-2) | |
A reaction: Hence it is not that we have to settle for the success of a system 'up to isomorphism', since that was the original aim. The structures must differ in their non-essential properties, or they would be the same system. |
17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
Full Idea: The purpose of a 'classificatory' axiomatic theory is to single out an otherwise disparate species of structures by fixing certain features of morphology. ...The aim is to single out common features. | |
From: John Mayberry (What Required for Foundation for Maths? [1994], p.406-2) |
17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
Full Idea: The purpose of what I am calling 'eliminatory' axiomatic theories is precisely to eliminate from mathematics those peculiar ideal and abstract objects that, on the traditional view, constitute its subject matter. | |
From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-1) | |
A reaction: A very interesting idea. I have a natural antipathy to 'abstract objects', because they really mess up what could otherwise be a very tidy ontology. What he describes might be better called 'ignoring' axioms. The objects may 'exist', but who cares? |
17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
Full Idea: No logic which can axiomatise arithmetic can be compact or complete. | |
From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) | |
A reaction: I take this to be because there are new truths in the transfinite level (as well as the problem of incompleteness). |