Mathematical logic

Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics.

History

Mathematical logic was the name given by Giuseppe Peano to what is also known as symbolic logic. In essentials, it is still the logic of Aristotle, but from the point of view of notation it is written as a branch of abstract algebra.

Related Topics:
Giuseppe Peano - Aristotle - Abstract algebra

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Attempts to treat the operations of formal logic in a symbolic or algebraic way were made by some of the more philosophical mathematicians, such as Leibniz and Lambert; but their labors remained little known and isolated. It was George Boole and then Augustus De Morgan, in the middle of the nineteenth century, who presented a systematic mathematical (of course non-quantitative) way of regarding logic. The traditional, Aristotelian doctrine of logic was reformed and completed; and out of it developed an adequate instrument for investigating the fundamental concepts of mathematics. It would be misleading to say that the foundational controversies that were alive in the period 1900-1925 have all been settled; but philosophy of mathematics was greatly clarified by the 'new' logic.

Related Topics:
Leibniz - Lambert - George Boole - Augustus De Morgan - Quantitative - Fundamental concepts of mathematics - Philosophy of mathematics

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While the traditional development of logic (see list of topics in logic) put heavy emphasis on forms of arguments, the attitude of current mathematical logic might be summed up as the combinatorial study of content. This covers both the syntactic (for example, sending a string from a formal language to a compiler program to write it as sequence of machine instructions), and the semantic (constructing specific models or whole sets of them, in model theory).

Related Topics:
List of topics in logic - Formal language - Compiler - Model theory

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Some landmark publications were the Begriffsschrift by Gottlob Frege, Studies in Logic by Charles Peirce, and Principia Mathematica by Bertrand Russell and Alfred North Whitehead.

Related Topics:
Begriffsschrift - Gottlob Frege - Studies in Logic - Charles Peirce - Principia Mathematica - Bertrand Russell - Alfred North Whitehead

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