display all the ideas for this combination of texts
11 ideas
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'. |
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). |
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) |
8195 | Undecidable statements result from quantifying over infinites, subjunctive conditionals, and the past tense [Dummett] |
Full Idea: I once wrote that there are three linguistic devices that make it possible for us to frame undecidable statements: quantification over infinity totalities, as expressed by word such as 'never'; the subjunctive conditional form; and the past tense. | |
From: Michael Dummett (Truth and the Past [2001], 4) | |
A reaction: Dummett now repudiates the third one. Statements containing vague concepts also appear to be undecidable. Personally I have no problems with deciding (to a fair extent) about 'never x', and 'if x were true', and 'it was x'. |
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. |
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) |
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? |
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). |
8194 | Surely there is no exact single grain that brings a heap into existence [Dummett] |
Full Idea: There is surely no number n such that "n grains of sand do not make a heap, although n+1 grains of sand do" is true. | |
From: Michael Dummett (Truth and the Past [2001], 4) | |
A reaction: It might be argued that there is such a number, but no human being is capable of determing it. Might God know the value of n? On the whole Dummett's view seems the most plausible. |