I remember the first time I saw the reductio ad absurdum in a class of mathematics. Also known as argument to absurdity, that elegant way that tries to establish something by showing the opposite situation, after some development it eventually ends up to absurdity or contradiction.
This method has been applied in mathematics, the classical example relates to the discovery by Pythagoras – disclosed to the chagrin of his associates by Hippasus of Metapontum in the fifth century BC – of the incommensurability of the diagonal of a square with its sides. The reasoning is as follows:
Let d be the length of the diagonal of a square and s the length of its sides. Then by the Pythagorean theorem we have it that d² = 2s². Now suppose (by way of a reductio assumption) that d and s were commensurable in terms of a common unit u, so that d = n x u and s = m x u, where m and n are whole numbers (integers) that have no common divisor. (If there were a common divisor, we could simply shift it into u.) Now we know that (n x u)² = 2(m x u)²
We then have it that n² = 2m². This means that n must be even, since only even integers have even squares. So n = 2k. But now n² = (2k)² = 4k² = 2m², so that 2k² = m². But this means that m must be even (by the same reasoning as before). And this means that m and n, both being even, will have common divisors (namely 2), contrary to the hypothesis that they do not. Accordingly, since that initial commensurability assumption engendered a contradiction, we have no alternative but to reject it. The incommensurability thesis is accordingly established.
What about philosophy? I personally consider it as a fascinating way to take your thoughts in a different path and see the types of results you can get, it is like watching how the solution is emerging in front of you. Defining the absurdity as the conflict the humankind has between the tendency to find meaning or the value of life, and the inability to find it within any certain, a fight between curiosity to know and the incapacity to obtain it. Albert Camus, an absurdist philosopher stated that individuals should embrace the absurd condition of human existence. Looking at it in more general terms, the absurdity has so many expressions, so many shapes, reaching areas like art: theater, music, literature, painting...
This method has been applied in mathematics, the classical example relates to the discovery by Pythagoras – disclosed to the chagrin of his associates by Hippasus of Metapontum in the fifth century BC – of the incommensurability of the diagonal of a square with its sides. The reasoning is as follows:
Let d be the length of the diagonal of a square and s the length of its sides. Then by the Pythagorean theorem we have it that d² = 2s². Now suppose (by way of a reductio assumption) that d and s were commensurable in terms of a common unit u, so that d = n x u and s = m x u, where m and n are whole numbers (integers) that have no common divisor. (If there were a common divisor, we could simply shift it into u.) Now we know that (n x u)² = 2(m x u)²
We then have it that n² = 2m². This means that n must be even, since only even integers have even squares. So n = 2k. But now n² = (2k)² = 4k² = 2m², so that 2k² = m². But this means that m must be even (by the same reasoning as before). And this means that m and n, both being even, will have common divisors (namely 2), contrary to the hypothesis that they do not. Accordingly, since that initial commensurability assumption engendered a contradiction, we have no alternative but to reject it. The incommensurability thesis is accordingly established.
What about philosophy? I personally consider it as a fascinating way to take your thoughts in a different path and see the types of results you can get, it is like watching how the solution is emerging in front of you. Defining the absurdity as the conflict the humankind has between the tendency to find meaning or the value of life, and the inability to find it within any certain, a fight between curiosity to know and the incapacity to obtain it. Albert Camus, an absurdist philosopher stated that individuals should embrace the absurd condition of human existence. Looking at it in more general terms, the absurdity has so many expressions, so many shapes, reaching areas like art: theater, music, literature, painting...
But going back to what we see everyday, how is something categorized as absurd?, what or who declares that something is indeed absurd. The critic?, the expert? I have no idea... In art my first thought is the Fountain by Marcel Duchamp (see the picture on the right). Is that Absurd?, art?, both?, none? Is time to end this writing now?, I think so... Let's talk more about all this in another time.
See you around,
Jesús
See you around,
Jesús