The Axiom of Choice

This post is in response to a video by Veritasium "The Man Who Almost Broke Math (And Himself...)"

Having passed trigonometry by one point after failing twice and being sent to summer school, it might be understandable why I decided years ago to lock my enthusiasm for mathematics in a glass jar - sealed, covered with a dark cloth, and hidden in an abandoned storage room in an isolated village known as отказ (failure). It takes about 2-10 days for homemade pickles to be properly fermented depending on the level of sourness one desires. I expected that by the time I would come across this jar when it came time to move onto other endeavors in life that I would find only the remains of mold spores left in the jar.

Today, I put on my respirator and winded up staring into this jar for 33 minutes as opening the jar led to me watching a mini-documentary about the story of a man who had a natural math sense, almost went insane for being unable to create a tangible proof of his theory, only to find support after being humiliated in front of his family by the math community that was once his home in a complete stranger who not only helped this man ostracized by the math world but even had his work accepted as an axiom of math. Eventually, the work to prove and disprove this axiom would lead to a stalemate which would culminate into a mathematician receiving a Fields medal in 1966 for creating a truce between those who oppose or accept the Axiom of Choice.

Funny enough, the compromise was that you may choose accept of axiom of choice depending on the purpose for which you intend to use it. Therefore, you have a choice to accept the axiom of choice.

The Axioms of Set Theory are as follows:

1. Axiom of Extension
2. Axiom of Union
3. Axiom of Power Set
4. Axiom of Infinity
5. Axiom of Replacement
6. Axiom of Regularity
7. Axiom of Choice*

Ultimately, mathematicians mostly accepted the last axiom; however, some prefer to have more tangible proofs. The implementation of Axiom of Choice in mathematicians has even led to more concise proofs and a wider, further reach into the infinite (haha) possibilities.

Perhaps another day, I would like to learn more about the axioms in detail but I remember that the first time I learned about sets was around the same time everyone learned about sets - my first time watching someone put something into a bag, box, a corner, or a pocket. In our daily lives, we limit the number of items in a given set to a range based on the area of the space provided. Now, I might be butchering this math being unable to really solve much due to my previous failures; however, I have always found the placement of things, their organization, their reorganization, drawing from them, to be an amazing concept to play around in one's mind.

For example, I have recently had to make a decision and both major aspects of my personality - one that values immediate, extrinsic results and the other that values intrinsic, delayed gratification which I could only accomplish by essentially using a sort of metaphorical sets of possible positive and negative outcomes. Both sides reached a stalemate as an argument could be made for both depending on which choice we were more focused on making: the immediate gain or the potential for an even grander gain that may only be intrinsic rather than much extrinsic gain.

A common issue as a Muslim is believing that one must either choose the dunya (the temporal world) or the akhirah (the eternal world) but this belief is actually fundamentally wrong. At no point did Allah require for us to make the choice to go against our nature, our nafs. Our nafs naturally is inclined towards good but it also at times prefers immediate gratification and can become like an unruly, undisciplined child that grows up into a leech in society or even a criminal.

Rather, Allah created a realm in which there are many ranks that one can attain success in the akhirah and there are different paths as to how one can attain paradise or jannah. In fact, there are eight gates of jannah and each of those gates are divided into potentially one hundred or more degrees. According to authentic narrations, the Prophet (peace be upon him) has shared how different good deeds lead to being promoted a degree in jannah. Some say it could be 6200 degrees due to the hadith about being raised a degree for every verse one recites. Other narrations support that it could be one hundred. But the majority agree that just as Allah has 99 names in one hadith but He also states that some He has not revealed proves that it is but a parable for us to be able to understand.

Georg Cantor, the man who is described as almost having broken math, almost went insane while trying to prove his theorem as he could not find a point in which to begin his "well-order" of real numbers that could exist between two counting numbers (or more). The human mind cannot handle infinity; hence, from the wisdom of Allah is to provide analogies for which we might be able to comprehend His vast knowledge. Likewise, Allah has provided both clear evidences and deeper knowledge for those who wish to gain a deeper understanding. As for those who try to reconcile the varying opinions about the degrees of paradise as mentioned previously, it returns to the conclusion that we would struggle to understand a neat system as to how the human race should be rewarded and that is why we leave it to Allah, exalted is He.

Again, the axiom of choice comes to mind because one can choose to try to find a stronger set of evidences and ultimately content themselves with the same fact that whether you take one opinion or another, we arrive at the same truth of the subject matter when it comes to Islam. An agreed upon axiom which is that Allah knows best.

It has been so difficult for me to make a decision because I feared I would make the "wrong" choice and truthfully, it is difficult to even clearly say if something is a wrong or right choice when there is no clear cut aspect of the choice containing forbidden matters. The situation I was in contained many factors that I found difficult to put into words as at one point, I believe I was juggling almost 20+ factors in making the decision and the numbers only began to increase as I took into consideration long-term effects of my decision making.

Then, it became clear to me that I would drive myself insane deliberating upon what is the "best" choice for me as it depended on a lot of other unknown variables. In math, there is this idea that there is no "random" and that even when we pick things randomly it is due to a sort of algorithm that we may contain internally. I agree with this as Allah mentions that not even a leaf falls without his understanding.

Therefore, it was not random that I came across this video and hesitated to watch it the other day as I feared if it would trigger my current fears about going insane. But then again, I found myself deeply inclined to watch it and found myself enthralled by what I was watching and I could not understand why it was so interesting to me from both a passive (failed) math enthusiast and a (failed) spiritual type.

Alhamdulillah, by the end of the video and having written this reflection, I feel an increased appreciation for Allah's knowledge and wisdom that I would never be able to understand. I am grateful that He allowed us to make choices and even inclines us towards what may seem to be random people and ideas. Just as Ernst Zarmelo happened to attend the 1904 International Mathematics conference in which Cantor's work was being disparaged by Julius Kunig. It was that day that Zarmelo had a major lightbulb moment, retreated into the studies he had done from Cantor's work and was able to find a pattern among other mathematicians' work for centuries. Zarmelo did not just so happen to be there that day nor did Cantor just so happen to be obsessed with his idea that there are countable and uncountable infinite sets. Zarmelo may have spared Cantor from ultimately killing himself and having his family facing severe public humiliation. Likewise, the work done by Zarmelo would later lead to a major mathematics breakthrough which made the world of possibilities - almost a bit more attainable. Well, if you choose to believe the set of infinite possibilities can be countable or uncountable.

Everything happens for a reason and though we are allowed to make choices, Allah knows best.