Origin of Inductive Proofs

Date: 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/