Disclaimer: This is not an expert view on Proofs. It’s just a view by someone who’s fascinated by them. Please read this article as a ‘completely true sharing of experience’.
I am always fascinated by Mathematical Proofs.
I know that most of us don’t understand what they even are, let alone understand what they are trying to prove. To the Math-loving and the Math-expert communities, proofs are a way to validate even the weirdest and the most non-nonsensical statements (presumably to the common man).
Wikipedia says that in Mathematics, a proof is an inferential argument for a Mathematical statement. It says that they are examples of exhaustive deductive reasoning or inductive reasoning; this is for the Math community. If I were to explain this to someone who has problems in Mathematics, I would say that Proofs are the way to answer the ‘How’ to every statement. They are more like an experiment, where we have an observation, reach to an inference and then state a conclusion. Don’t even think for a moment that all the blame game that happens between married couples are based on proofs. Most of them are based on ‘assumed expectations’. The probability of an assumed expectation to be true is itself a variable. No mathematician or Math geek will ever be able to explain this to them. Okay, back to business.
All proofs may look the same, but there are actually certain types of proofs, IN MY EYES. The first, which we all have dealt with at some point of time, is the LHS-RHS equality type. This is actually the first type of any proof we ever encounter in our lives, and perhaps the simplest of them all. Then there’s the fake type of proofs, or I should say, ‘missense’ type of proofs. These are the ones which seem extremely legitimate given the argument and the situation, but they aren’t. These are like most of the relationships today; they seem real at first, only to be realised later how missensical they actually are.
There is an other category of proofs, more like a long-term relationship, which is today extremely difficult to find. One of them in this category is the 129 page-long proof to Fermat’s Last theorem by Andrew Wiles, which took 350+ years to be found. Some of them in this category are the unlucky ones too, as they have been discarded after a counter-example was found to them.. Then there is a category of incomplete proofs, for example the incomplete proof for Riemann’s Hypothesis.
Then there’s this type of proof, which I want to talk about, and which I love the most – PROOF BY CONTRADICTION.
Proof by Contradiction is by far, I think, easily falling in the range between easiest and hardest types of proofs present today. Easiest because, they are introduced at as early as the 10th standard, and hardest, because it is often difficult to actually find out a contradiction sometimes, especially when we don’t know where to draw the contradiction from.
Let’s talk Mathematics.
In Proof by Contradiction, we show that a claim P is true by showing that its negation, ~P, leads to a contradiction. If ~P leads to a contradiction, ~P then can’t be true, and therefore P must be true. A contradiction can be any statement that is a well known set of false statement(s), which is/are obviously inconsistent with one another.
For the common man, Proof by Contradiction is equivalent to finding out how is walking in a straight line and in a circle at the same time not possible. You could also think that finding the way to show that you alone can’t catch two different birds on two different paths at the same time.
In all the years I have dealt with proof by contradiction, one thing that has perhaps always struck me is that it contains hidden rhetorical questions. Actually, it does not ask rhetorical questions; we are the ones who ask these questions ourselves. In a PDF I read to know more about proof by contradiction, perhaps it was best rephrased like this – In proof by contradiction, we build a world, and destroy it when we find a loophole in our world. I don’t know if you have observed or not and I am right or not, but I see that every time we sit down to validate a statement using Proof by Contradiction, we are actually trying to create a paradox in our assumed system. I believe that every paradox existing today is a contradiction to our assumed set of rules and beliefs.
I believe that the reason behind the high preference and use of Proof by Contradiction is that it subtly uses the process of cancellation, and highly optimises our efficiency in every way possible. I know that some people argue for the existence of a constructive proof to every proof by contradiction, but consider this: You have two roads, one long and one short to reach the same place. Unless one’s out of mind, the natural option would be to take the short road, isn’t it? The point is, most of the proofs by contradiction are short and referable, as negation needs only one example. Proof by Contradiction leaves us with no other choice but the one we assumed to be false.
Also, most of us aren’t always ready for a constructive proof, because often is the case that we get lost in a particular step or two, and thus we have restart all over again to get the gist of it. What’s more, in a constructive proof, we have to create everything from the scratch, and that is CUMBERSOME! This also reduces our efficiency. However, when we use proof by contradiction, we already have the negation of the given statement to start with, and the related concept to work with. This increases our efficiency and lessens our time consumption. I believe that all the work done in Mathematics is to optimise our work speed and reduce our difficulty in understanding the patterns of the universe. And if this is true, then most of the cumbersome constructive proofs are themselves contradictory to our cause (see, proof by contradiction!).
And also, allow me to say this: I am NOT against constructive proofs; to be more broad, I am not against any proof. If I find a constructive proof more efficient and explanatory, I will stand by it. If I find a Proof by Contradiction the same, I will defend it. The things is, I believe that the existence of Mathematics is always to be an instrument in the evolution of human civilisation, and therefore itself always improve in its efficiency and approach towards the common man, unlike Pythagoras, who kept his discoveries pretty much secret and hindrance to the natural evolution of Human Civilisation (that’s probably why he was murdered). I stand against anything which hinders evolution of any kind, either with a loud voice, or a silent rebellion, and I am not a stone-man, who won’t change sides when the time and situation demands.
To end this, I would say: I end this here, and I am lying.