DISCRETE MATH
Mathematical Database
MATHEMATICAL INDUCTION
1. Introduction
Mathematics distinguishes itself from the other sciences in that it is built upon a set of axioms and definitions, on which all subsequent theorems rely. All theorems can be derived, or proved, using the axioms and definitions, or using previously established theorems. By contrast, the theories in most other sciences, such as the Newtonian laws of motion in physics, are often built upon experimental evidence and can never be proved to be true.
It is therefore insufficient to argue that a mathematical statement is true simply by experiments and observations. For instance, Fermat (1601–1665) conjectured that when n is an integer greater than 2, the equation x n + y n = z n admits no solutions in positive integers. Many attempts by mathematicians in finding a counter-example (i.e. a set of positive integer solution) ended up in failure. Despite that, we cannot conclude that Fermat’s conjecture was true without a rigorous proof.
In fact, it took mathematicians more than three centuries to find the proof, which was finally completed by the English mathematician Andrew Wiles in 1994.
To conclude or even to conjecture that a statement is true merely by experimental evidence can be dangerous. For instance, one might conjecture that n 2 − n + 41 is prime for all natural numbers n.
One can easily verify this: when n = 1, n 2 − n + 41 = 41 is prime; when n = 2, n 2 − n + 41 = 43 is prime, and so on. Even if one continues the experiment until n = 10, or even n = 20, one would not be able to find a counter-example. However, it is easy to see that the statement is wrong, for when n
= 41 the expression is equal to 412 which definitely is not prime.
While experimental evidence is insufficient to guarantee the truthfulness of a statement, it is often not possible to verify the statement for all possible cases either. For instance, one might conjecture that 1 + 3 + 5 + + (2n − 1) = n 2 for
MATHEMATICAL INDUCTION
1. Introduction
Mathematics distinguishes itself from the other sciences in that it is built upon a set of axioms and definitions, on which all subsequent theorems rely. All theorems can be derived, or proved, using the axioms and definitions, or using previously established theorems. By contrast, the theories in most other sciences, such as the Newtonian laws of motion in physics, are often built upon experimental evidence and can never be proved to be true.
It is therefore insufficient to argue that a mathematical statement is true simply by experiments and observations. For instance, Fermat (1601–1665) conjectured that when n is an integer greater than 2, the equation x n + y n = z n admits no solutions in positive integers. Many attempts by mathematicians in finding a counter-example (i.e. a set of positive integer solution) ended up in failure. Despite that, we cannot conclude that Fermat’s conjecture was true without a rigorous proof.
In fact, it took mathematicians more than three centuries to find the proof, which was finally completed by the English mathematician Andrew Wiles in 1994.
To conclude or even to conjecture that a statement is true merely by experimental evidence can be dangerous. For instance, one might conjecture that n 2 − n + 41 is prime for all natural numbers n.
One can easily verify this: when n = 1, n 2 − n + 41 = 41 is prime; when n = 2, n 2 − n + 41 = 43 is prime, and so on. Even if one continues the experiment until n = 10, or even n = 20, one would not be able to find a counter-example. However, it is easy to see that the statement is wrong, for when n
= 41 the expression is equal to 412 which definitely is not prime.
While experimental evidence is insufficient to guarantee the truthfulness of a statement, it is often not possible to verify the statement for all possible cases either. For instance, one might conjecture that 1 + 3 + 5 + + (2n − 1) = n 2 for