Origin of Inductive ProofsDate: 12/03/97 at 12:32:55 From: Jeffrey Nietzold Subject: Origin of inductive proofs. What is the origin of the method of "inductive proof"? Why was it developed and how was it used? Date: 12/03/97 at 13:17:33 From: Doctor Rob Subject: Re: Origin of inductive proofs. It is believed that the first person to explicitly state and use the Principle of Mathematical Induction was the French mathematician Joseph Louis Lagrange (1736-1813). He used it to prove the Binomial Theorem, which had been known earlier to Sir Isaac Newton. It was formalized and further developed by Augustus DeMorgan (1806- 1871). For details, see http://www-history.mcs.st-and.ac.uk/~history/Mathematicians/De_Morgan.html and http://www.shu.edu/academic/arts_sci/Undergraduate/math_cs/sites/math/reals/history/demorgan.html There are many equivalent forms of this idea. Probably the first to use an equivalent form was Pierre de Fermat (1601-1665), who used what he called the Method of Descent. The idea was to show that if a value of n exists for which a statement is false, then a smaller value also exists for which it is false. Using this, a still smaller false value could be found, and so on. This would produce an infinite chain of descending values of n for which the statement is false. This situation is impossible, however, since there are only a finite number of whole numbers between the starting value and zero. Thus no such value of n could possibly be found, and the statement must be true for all n. Do you see why this is equivalent to the Principle of Mathematical Induction? Fermat used his method of descent to prove that x^4 + y^4 = n^2 has no solution in positive integer values of x, y, and n, by showing that if there were a solution, there would be another with a smaller value of n. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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