Friends, in today’s VERITAS we will try to understand a technique of mathematical proof. This is a very powerful technique and helps prove some very complicated mathematical theorems. The technique is called “Reductio ad absurdum” which means reduction to an absurdity. This technique is also called proof by contradiction.
This technique is very old. It has been there since the time of Euclid(330 – 275 BC). Euclid used this very effectively in the proof of several theorems. This is how it works: suppose we want to prove a theorem which can be stated by a statement S. We assume that S is not true i.e we assume that our theorem is false. Then we show that the consequences of such an assumption is something absurd which can never be true. So this means that our assumption that S is false is not valid. Thus S is a true statement.
The mathematician Hardy( Ramanujan’s friend) wrote in his book, A Mathematician’s apology: “Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game”
Notice that our technique makes use of the law of the “excluded middle”. According to this law a statement can either be true or false.
And if a statement is not true then it MUST be false. Not every school of mathematical thought accepts this technique as a valid method of mathematical proof. The school of intuitionism does not take the law of the excluded middle to be universally valid and thus does not agree that Reductio ad absurdum should be a valid method of mathematical proof.
Lets look at an example of a proof by Reductio ad absurdum. This is Euler’s proof that the number of prime numbers are infinite i.e there is no largest prime number. But before we start our proof we should remember that every non prime natural number can be written as a product of primes in essentially one way. This is called the fundamental theorem of arithmetic. We will use this theorem in our proof that the number of prime numbers is infinite:
proof: Lets assume that our theorem is false i.e there are a finite number of prime numbers. That means that there is a largest prime. Now lets multiple all the primes together and add one to the product. This new number leaves a remainder of 1 when divided by ANY prime. Therefore it cannot be divided by any prime ie it is not divisible by ANY prime.
So it HAS to be another prime. But that is absurd because we had already taken all the primes and multiplied them . And now we have found another prime! SO this contradicts the statement with which we started. So our statenment that the number of primes is finite is wrong. So we have proved that the number of primes is infinite!!
See what a powerful technique this is! But it is also a very dangerous one. Euclid made a mistake in his proof of the fifth postulate(the parallel postulate). The problem is caused when a statement looks false but is not really PROVEN false.
A question to you: do you think that God listens and acts upon prayers?? Should be apply Reductio ad absurdum here? Ivan Turgenev once said ” Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: `Great God, grant that twice two be not four'” !!!!!
How very absurd ! 🙂
Go, wondrous creature! mount where Science guides:
Go, measure earth, weigh air, and state the tides:
Instruct the planets in what orbs to run,
Correct old time and regulate the Sun;