I just had a realization about good mathematicians. Often times in Mathematics, a successful proof may be cracked only if one considers all possible cases, and the exclusion of any case would skew the entire argument/proof. I have often noticed that the cases I do consider, I am able to analyze them with a certain depth, and think it through with advanced analysis. Nonetheless, often times, my proof goes amiss because I failed to think of a possible case, or a possible interpretation of the problem from the very start.
Now, I realize that this may be a very horrid characteristic in a mathematician. However, I now understand the factors (or the interaction of factors) that influence such behavior on my part. I hope that the identification of these factors can help my further analytical math. Read on for more clarity and stay with me here.
Luckily, I do think that this flaw in my mathematical thinking actually lends to an improvement in my general thinking, when it comes to life & such fun things. And this is how:
I always leave room for something that I do not know; perhaps a statistician would say, I always leave some room for error. I would like to connect this idea with another now. I do not know what I do not know- a teacher somewhere had taught me that, and it stuck to me. Let’s further connect this. My religious tendencies also speak so. I do not know what I do not know, so there is always a possibility that there is God or there isn’t. Moreover, there could even be OTHER cases that I have not considered still, and I leave room for that.
I am sure many of you will be able to relate to the next bit I am writing. Often times when we would not know what will happen later on, we would try to think of the possibilities; say A, B, and C, and the subsets of this set. And many many times, the resulting event would be of an entirely different nature (perhaps because you forgot to consider a D in the original set; so despite considering all subsets, you missed something).
I am sure many of you have also thought that A could potentially happen, “… but if I expect that A will happen, B would happen, and if I expect B would happen, then A would happen…I want A to happen so I will think that A will not happen (because this will make A happen), but then aren’t I really still thinking that A would happen..” and that cycle will keep going until you stop thinking. So, what was the solution to this? It’s a catch-22 type of thought, and probably, it is just something you cannot figure out because you do not have enough evidence. Primarily, this thought just portrays that no matter we expect, the result will be opposite or not as expected. But this sort of mechanism indicates that something I expect will not happen because there will be other factors (unknown to me) that will affect the ultimate result.
What I am trying to say here is that we almost always account for things we do not know. My personal opinion is that many mathematicians think highly of themselves and their academic skills. I like to think of myself as a humble mathematician. While I am proving, I am not sure that this is ABSOLUTELY correct.
Moreover, many good mathematicians say, “I got it, I got it” ten times before they actually have the solution. Despite the fact that they are wrong so many times in order to get the correct answer ultimately, they do not fail to be highly confident each time they claim they “got it.”
Even if I set up a proof correctly, I tend to think I may have missed something, but I do my best anyway. I like this approach in life. I stay humble. But darn it!! Those arrogant math-know-it-alls beat me at math.
Now that I understand the conflict between the different perspectives, will I be able to objectify myself from how I intrinsically have trained myself to think (as if towards a virtue), and do better mathematical analysis by changing the thinking cap a bit?
