What is the mathematical idea behind recursion?

Question

Accepted Answer

What is the mathematical idea behind recursion?
    
        
            In mathematics, many problems can be written in recursive form. For example,
            the sum of natural numbers can be expressed using a smaller version of the same problem.
        

        
            Sum of n natural numbers = n + Sum of (n - 1) natural numbers
        

        
            
                Expression
                Recursive Meaning
            
            
                Sum(n)
                n + Sum(n - 1)
            
            
                Sum(n - 1)
                (n - 1) + Sum(n - 2)
            
            
                Sum(n - 2)
                (n - 2) + Sum(n - 3)

Expression	Recursive Meaning
Sum(n)	n + Sum(n - 1)
Sum(n - 1)	(n - 1) + Sum(n - 2)
Sum(n - 2)	(n - 2) + Sum(n - 3)

What is the mathematical idea behind recursion?

Answer with Explanation

What is the mathematical idea behind recursion?