display all the ideas for this combination of philosophers
4 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) |
11026 | Classical logic is deliberately extensional, in order to model mathematics [Fitting] |
Full Idea: Mathematics is typically extensional throughout (we write 3+2=2+3 despite the two terms having different meanings). ..Classical first-order logic is extensional by design since it primarily evolved to model the reasoning of mathematics. | |
From: Melvin Fitting (Intensional Logic [2007], §1) |