Anthony Mistretta
Discrete Mathematics
1
1) Describe the origins of mathematical induction. Who were the first people to use it and to which problems did they apply it?
Mathematical Induction Paper
If I were to take a guess at this, I would venture to say that the origins of mathematical induction began with the indigenous Eskimos of the Adirondacks as they trekked over frozen rivers, “Well Sesi, we have made it this far, maybe we can go a little more. … Sesi?”
The modern version of mathematical induction appears to be the result of setting up a rigorous means of defining the phrase “and so on and so forth” from earlier versions. One of the more recent people that I came across in my research was Giuseppe Peano. Peano was one of those guys who took to the habit of creating a theory of numbers and devised a set of axioms to define it. The theory of numbers helps to establish properties that are useful in applying a consistent system of arithmetic. To avoid circular reasoning, the axioms omit any allusion to arithmetic and speak of (x+1) as the successor to x, and denote it as x’. The fifth axiom is denoted as the “Principle of Mathematical Induction” and states to the effect: Let Mbe a set of natural numbers N. 1 is an element of M. If x is an element of M, then x’ is an element of M. Therefore M = N. (Goodman,102). Using his 5 axioms, Peano then demonstrates the properties of addition on the set of natural numbers. There are those who would claim that Peano borrowed from Richard Dedekind’s axioms, but they appear to be an independent construction.(Kennedy, 135).
While Peano’s use of mathematical induction is axiomatic, Dedekind provesmathematical induction. (Russell, 248). Dedekind and Cantor are contemporaries to the development of set theory. Dedekind uses a notion of chains as a means to prove the concept of mathematical induction. From what I can tell, a chain is regarded by Dedekind to be a succession of elements that link together from a basis point, and that the properties associated with each element are hereditary from one to the next.
I suppose a paper on mathematical induction would not be complete without mentioning Blaise Pascal. Pascal wrote a paper in 1654on the Arithmetic Triangle, a triangle which had been developed in China prior to 1000AD. In this paper, Pascal says:
Although this proposition has an infinite number of cases, I shall give a very short demonstration of it based on 2 lemmata. The first [lemma], which is self-evident [a basis step], is that this proportion holds in the second base [hypotenuse of the second smallest triangle]; for it is quite clear that is to 0 as 1 to 1. The second [lemma] is that if this holds in some arbitrary base then it necessarily holds in the following base.(Crossely, 44).
And it is the wrapping up of Pascal’s use of the phrase: “and so on to infinity” which is about the only caveat to this being defined as rigorous.
Other notables in the history of the mathematical induction are Jacob Bernoulli, Euler, Pierre de Fermat, Rabbi Levi ben Gershon, and even Archimedes.
Works Cited/Referenced
Crossely, J. (1987). The Emergence of Number. Singapore: World Scientific.
Goodman, A. Algebra from A to Z. Singapore: World Scientific.
Kennedy, H. (1972). The Origins of Modern Axiomatics: Pasch to Peano. The American Mathematical Monthly, 133-136.
Russell, B. (1903). The Principles of Mathematics. London: Cambridge, at the University Press.