Mathematical logic is best understood as a branch of logic or mathematics. Mathematical logic is often divided into the subfields of model theory, proof theory, set theory and recursion theory. Research in mathematical logic has contributed to, and been motivated by, the study of foundations of mathematics, but mathematical logic also contains areas of pure mathematics not directly related to foundational questions.
One unifying theme in mathematical logic is the study of the expressive power of formal logics and formal proof systems. This power is measured both in terms of what these formal systems are able to prove and in terms of what they are able to define. Thus it can be said that "mathematical logic has become the general study of the logical structure of axiomatic theories." (Parsons, p. 206)
Earlier names for mathematical logic were symbolic logic (as opposed to philosophical logic) and metamathematics. The former term is still used (as in the Association for Symbolic Logic), but the latter term is now used for certain aspects of proof theory.
Mathematical logic was the name given by Giuseppe Peano to what is also known as symbolic logic. In its classical version, the basic aspects resemble the logic of Aristotle, but written using symbolic notation rather than natural language. 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 way of regarding logic. The traditional, Aristotelian doctrine of logic was reformed and completed; and out of it developed an 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.
While the Greek development of 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 and the semantic dimensions. Syntactic has to do with the correct or formal structure of a string of symbols in a formal language, as, for example, sending a string from a formal language to a compiler program to write it as sequence of machine instructions. Semantic has to do with interpretation or use of a string of symbols, as, for example, constructing specific models or whole sets of them, in model theory. This study of mathematics from the outside is known as metamathematics.
Some landmark publications were the Begriffsschrift by Gottlob Frege, Studies in Logic by Charles Peirce, Principia Mathematica by Bertrand Russell and Alfred North Whitehead, and On Formally Undecidable Propositions of Principia Mathematica and Related Systems by Kurt Gödel.
At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. The system of first-order logic is the most widely studied because of its applicability to foundations of mathematics and because of its desirable properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with nonclassical logics such as intuitionistic logic.
Barwise's "Handbook of Mathematical Logic" (1977) divides mathematical logic into four parts:
The border lines between these fields, and also between mathematical logic and other fields of mathematics, are not always sharp; for example, Gödel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Loeb's theorem, which is important in modal logic. The mathematical field of category theory uses many formal axiomatic methods resembling those used in mathematical logic, but category theory is not ordinarily considered a subfield of mathematical logic.
The study of computability theory in computer science is closely related to the study of computability in mathematical logic. There is a difference of emphasis, however. Computer scientists often focus on concrete programming languages and feasible computability, while researchers in mathematical logic often focus on computability as a theoretical concept and on noncomputability.
The study of programming language semantics is related to model theory, as is program verification (in particular, model checking). The Curry-Howard isomorphism between proofs and programs relates to proof theory; intuitionistic logic and linear logic are significant here. Calculi such as the lambda calculus and combinatory logic are nowadays studied mainly as idealized programming languages.
Computer science also contributes to mathematics by developing techniques for the automatic checking or even finding of proofs, such as automated theorem proving and logic programming.
All links retrieved January 27, 2017.
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