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5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics

[logic that is used in the practice of mathematics]

8 ideas
Logic (the theory of relations) should be applied to mathematics [Novalis]
     Full Idea: Ought not logic, the theory of relations, be applied to mathematics?
     From: Novalis (Logological Fragments II [1798], 38)
     A reaction: Bolzano was 19 when his was written. I presume Novalis would have been excited by set theory (even though he was a hyper-romantic).
Does some mathematical reasoning (such as mathematical induction) not belong to logic? [Frege]
     Full Idea: Are there perhaps modes of inference peculiar to mathematics which …do not belong to logic? Here one may point to inference by mathematical induction from n to n+1.
     From: Gottlob Frege (Logic in Mathematics [1914], p.203)
     A reaction: He replies that it looks as if induction can be reduced to general laws, and those can be reduced to logic.
The closest subject to logic is mathematics, which does little apart from drawing inferences [Frege]
     Full Idea: Mathematics has closer ties with logic than does almost any other discipline; for almost the entire activity of the mathematician consists in drawing inferences.
     From: Gottlob Frege (Logic in Mathematics [1914], p.203)
     A reaction: The interesting question is who is in charge - the mathematician or the logician?
In modern times, logic has become mathematical, and mathematics has become logical [Russell]
     Full Idea: Logic has become more mathematical, and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one.
     From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII)
     A reaction: This appears to be true even if you reject logicism about mathematics. Logicism is sometimes rejected because it always ends up with a sneaky ontological commitment, but maybe mathematics shares exactly the same commitment.
Mathematical Logic is a non-numerical branch of mathematics, and the supreme science [Gödel]
     Full Idea: 'Mathematical Logic' is a precise and complete formulation of formal logic, and is both a section of mathematics covering classes, relations, symbols etc, and also a science prior to all others, with ideas and principles underlying all sciences.
     From: Kurt Gödel (Russell's Mathematical Logic [1944], p.447)
     A reaction: He cites Leibniz as the ancestor. In this database it is referred to as 'theory of logic', as 'mathematical' seems to be simply misleading. The principles of the subject are standardly applied to mathematical themes.
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)
We should exclude second-order logic, precisely because it captures arithmetic [Read]
     Full Idea: Those who believe mathematics goes beyond logic use that fact to argue that classical logic is right to exclude second-order logic.
     From: Stephen Read (Thinking About Logic [1995], Ch.2)
The model theory of classical predicate logic is mathematics [Beall/Restall]
     Full Idea: The model theory of classical predicate logic is mathematics if anything is.
     From: JC Beall / G Restall (Logical Pluralism [2006], 4.2.1)
     A reaction: This is an interesting contrast to the claim of logicism, that mathematics reduces to logic. This idea explains why students of logic are surprised to find themselves involved in mathematics.