MATHEMATICAL LOGIC - HISTORY

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In the history of ideas the 19th century is marked by an extraordinary development of logic/ A discipline which had remained for more than twenty centuries in approximately the state to which the mind of Aristotle reduced it, suddenly entered upon a period of rapid growth and systematic development.  While the essential elements of the Aristotelian logic have been overthrown  or shaken, the labors of Boole, Russel, Whitehaed and a host of fellow workers have produced a calculus of classes and a calculus of propositions in which Aristotelian theory of the syllogism is seen to occupy only a tiny corner. The potentialities of the new logic as a scientific instrument have already been indicated in the illumination which the application of modern logic has brought to the foundation of maths. The field of logic has been traditionally restricted to propositions true or false in themselves: the realm of fiction and fictions has appeared to lie beyond the domain of logic. But the development of the concepts of system and order in the new logic shows this limitation to be unjustified. Sentences which have a variable truth value relative to a defining set of postulates or hypotheses are as susceptible to logical analysis as any of the sentences about the mortality of Socrates that filled the older textbooks.

Mathematical Logic is the outcome of the joining together of four different lines of thought. There are 1) old logic, the invention Aristotle; 2) the idea of a complete and automatic language for reasoning; 3) the new developments in algebra and geometry which took place after 1825 and 4) the idea of the parts of maths as being systems of deductions, that is of chains of reasoning in agreement with rules of logic. These rules give one the power to go from a statement s₁ to another statement s₂ when s₂ is necessarily true if s₁ is taken to be true.

1) Aristotle syllogistic is a theory of syllogistic implication, i.e. implications with two and only two conditions (premises) and one outcome (conclusions). What Aristotle made the attempt to do was to give a complete account of the different possible detailed forms of syllogistic implications and a complete body of rules as tests of the validity of the validity of any given syllogistic implication. However, both Aristotle's and his successors' logic was one of the rules whose statement was in everyday language (Latin); no special signs for the operations of reasoning were used and they seemed to have no idea that it was possible for logic to be turned into a sort of maths. So they took no step forward in the direction of turning logic into mathematical logic.

2) Descartes appeared to have been the first person to have the idea of a general language - "universal" - as  sort of arithmetic, though he made no attempt to suggest such a language. A little later Leibnitz had designs for a new and general language that were not unlike Descartes', but he took then some stages further. But though he frequently gave attention to the idea of such language, he did little with it. His designs go no further then being designs - and rough ones.

3) Between 1825 and 1900 algebra and geometry underwent great changes which had a strong effect upon the growth of mathematical logic. Abel's and Galoi's group theory came quickly into being as the first new branch of abstract algebra. The systems of vector algebra and matrix algebra were worked out not long after the death of Galois in 1832. These systems were company for group theory as new branches of abstract algebra.

4) The idea of the parts of maths as system of deductions goes back to Euclid's Elements. What a system of deduction is. This question was viewed in the 1890s as interesting and important by that Italian school of maths that was headed by Peano. Peano's work followed by that one by D.Hilbert  had a great effect of later development of logic. From their work came a new branch of mathematical logic: metamathematics. The birth of mathematical logic  was in the 1840s.

Boole's Algebra of Logic

What was the start of mathematical logic? The shortest and simplest answer is George Boole's Mathematical Analysis of Logic of 1847. The earlier teaching in logic of which Boole had a knowledge and which had an effect on him were, on the one hand, those of the old logic and, on the other hand, those of W.Hamilton and De Morgan.

The name used in logic and maths for a group of all the things that have a certain simple or complex property is class and the things that have the property are said to be the element of the class. The ideas of class and lass elements are root ideas in all present-day maths. The outcome of the Hamilton - De Morgan theory was to make possible a view of logic as being at least in one of its branches an algebra of classes. Boole was the first man to have this view clearly and where others had been completely at a loss, he was able to give a good theory of it. Moreover, Boole was the first man to give a united theory of logic in his second work The Law of Thought.

Much of the work in mathematical logic in the 100 years and more that have gone by after The Laws of Thought has been given to taking out the errors from Boole's ideas, making some of the parts of his theory stronger, putting his algebra in the form of a system of deductions and, lastly moving from it towards more general theories in abstract algebra, for example, towards the theory of those "partordered" classes every two elements of which have a greatest lower limit and a smallest higher limit named lattices.