Combining Texts

Ideas for 'The Epistemology of Modality', 'The Intentional Fallacy' and 'What Required for Foundation for Maths?'

unexpand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

---

9 ideas

5. Theory of Logic / A. Overview of Logic / 2. History of Logic

17786	The mainstream of modern logic sees it as a branch of mathematics [Mayberry]
	Full Idea: In the mainstream tradition of modern logic, beginning with Boole, Peirce and Schröder, descending through Löwenheim and Skolem to reach maturity with Tarski and his school ...saw logic as a branch of mathematics.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.410-1)
	A reaction: [The lesser tradition, of Frege and Russell, says mathematics is a branch of logic]. Mayberry says the Fregean tradition 'has almost died out'.

5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic

17788	First-order logic only has its main theorems because it is so weak [Mayberry]
	Full Idea: First-order logic is very weak, but therein lies its strength. Its principle tools (Compactness, Completeness, Löwenheim-Skolem Theorems) can be established only because it is too weak to axiomatize either arithmetic or analysis.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.411-2)
	A reaction: He adds the proviso that this is 'unless we are dealing with structures on whose size we have placed an explicit, finite bound' (p.412-1).

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

17791	Only second-order logic can capture mathematical structure up to isomorphism [Mayberry]
	Full Idea: Second-order logic is a powerful tool of definition: by means of it alone we can capture mathematical structure up to isomorphism using simple axiom systems.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1)

5. Theory of Logic / G. Quantification / 2. Domain of Quantification

17787	Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry]
	Full Idea: The 'logica magna' [of the Fregean tradition] has quantifiers ranging over a fixed domain, namely everything there is. In the Boolean tradition the domains differ from interpretation to interpretation.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.410-2)
	A reaction: Modal logic displays both approaches, with different systems for global and local domains.

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

17790	No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry]
	Full Idea: No logic which can axiomatize real analysis can have the Löwenheim-Skolem property.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1)

5. Theory of Logic / K. Features of Logics / 1. Axiomatisation

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)

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.

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?

5. Theory of Logic / K. Features of Logics / 6. Compactness

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).