Philosophy & Mathematics share such an extraordinary rapport that they have shaped up civilizations, spurred up intellectual revolutions and at times together have surmounted the seemingly unsurmountable.
Despite arguments that Philosophy & Maths(Science in general) are at loggerheads with each other in terms of perspectives & interpretations, to a lot of us with interdisciplinary interests, they are different faces of the same coin. In the words of Will Durant - "Philosophy seems to stand still, perplexed, but only because she leaves the fruits of victory to her daughters the Sciences, and she herself passes on divinely discontent, to the uncertain & unexplored."
One such aspect of immense interest in both the disciplines is "Reductio ad absurdum"(Latin for 'reduction to the absurd'). Its a form of argument in which a proposition is disproven by following its implications logically to an absurd consequence. Familiar to students of Mathematics as "Proof by contradiction".
Now the beauty of this form of argument is two folded. One in the perspective of Math and the other in the light of Philosophy.
To illustrate the Math perspective, Hardy does it best in his famous statement - " Reductio ad absurdum, which Euclid loved so much, is one of Mathematician's finest weapons. It is a far finer gambit than any chess gambit; A chess player may offer the sacrifice of a pawn or even a piece, but a Mathematician offers the game." In the method of proof by contradiction, the very first statement would be to assume what is to be proved true as false!! Like if you had to prove the number of primes was infinite, you start by assuming the number of primes is finite(crazy!) and then prove the assumption to be false(crazier!). Maybe rather than being a method of proof, it is a method of disproving the false and thereby acting as an implied proof of the truth!! There lies the beauty for the beholder!!
In the light of Philosophy, the argument has rendered a lot of interesting twists in the tale. Consider Neils Bohr's statement: "The opposite of every great idea is another great idea." Carl Sagan used a reductio ad absurdum argument to counter this claim. If this statement is true, then it would certainly qualify as a great idea - it would automatically lead to a corresponding great idea for every great idea already in existence. But if the statement itself is a great idea, its opposite must also be a great idea. The original statement is disproven because it leads to an absurd conclusion: that an idea can be great regardless of whether it is true or false. Now what is true and what is false has been Philosopher's quest for over centuries. But how the approach of argument renders interpretation is interesting.
"If the windows of perception are cleansed, everything would appear as it is - infinite.
- William Blake"
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