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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.Although the layperson may think that mathematical logic is the logic of mathematics, the truth is rather that it more closely resembles the mathematics of logic. It comprises those parts of logic that can be modelled mathematically. Earlier appellations were symbolic logic (as opposed to philosophical logic); and metamathematics, which is now restricted as a term to some aspects of proof theory.
1 History
Mathematical logic was the name given by 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.
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 BooleThis article is not about George Boolos, another mathematical logician. George Boole [bul], ( November 2, 1815 December 8, 1864) was a mathematician and philosopher. As the inventor of Boolean algebra, the basis of all modern computer arithmetic, Boole is and then Augustus De MorganAugustus De Morgan ( June 27, 1806 March 18, 1871) was an Indian-born British mathematician and logician. He formulated De Morgan's laws and was the first to introduce the term, and make rigorous the idea of mathematical induction 1. Biography Childhood A, in the middle of the nineteenth century, who presented a systematic mathematical (of course non- quantitativeA quantitative property can be meaningfully measured using numbers; properties which aren't quantitative are called qualitative . Examples of quantitative properties include: the number of grains of sand on a beach, the width of a hair, and the time for a) 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 mathematicsPhilosophy of mathematics is that branch of philosophy which attempts to answer questions such as: "why is mathematics useful in describing nature?", "in which sense, if any, do mathematical entities such as numbers exist?" and "why and how are mathematic was greatly clarified by the 'new' logic.
While the traditional development of logic (see list of topics in logicThis is a list of topics in logic . Alphabetical list A Abacus logic Abduction (logic) Abductive validation Affine logic Affirming the antecedent Affirming the consequent Antecedent Antinomy Argument form Aristotelian logic Axiom Axiomatic system Axiomati) 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 languageIn mathematics, logic and computer science, a formal language is a set of finite-length words (i. character strings) drawn from some finite alphabet, and the scientific theory that deals with these entities is known as formal language theory''. Note that to a compilerA compiler is a computer program that translates a computer program written in one computer language (called the source language into an equivalent program written in another computer language (called the output or the target language . Introduction and h program to write it as sequence of machine instructions), and the semantic (constructing specific models or whole sets of them, in model theoryIn mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. It assumes that there are some pre-existing mathematical objects out there, an).
Some landmark publications were the Begriffsschrift and the
Principia Mathematica.
