All posts by hyunmoon

Hyunmoon graduated from Princeton in 2013 with a degree in mathematics. He is interested in modern cosmic ideologies and is now at Seoul National University trying to understand the structure of empty space through the mathematics of Lagrangian Floer Homology.

The Tree of Pythagoras

How many penguins are there on the ice?
One, two, three, four, five. There are five penguins.

How many fingers are on a baby’s hand?
One, two, three, four, five. There are five fingers.

It is quite odd that, despite there being quite little in common between penguins and baby fingers, we can still proceed to count them in exactly the same manner. This is because counting is not so much about things as it is about us–about how our minds work with distinguishable volumes.

We take a mirror and observe ourselves amidst the counting process, and we find a lot of things going on! When we count, we recognize a large mass of interest, identify pieces that are similar and pieces that are different, associate a different number to each piece, and separate the unidentified mass with the mass we have previously identified.

Counting has three traits that make it very interesting. Firstly, it is a basic ability of (roughly) all human minds; secondly, it is consistent, producing the same result regardless of who is counting; and thirdly, it is intangible.

Although these are very crude distinctions, not many things satisfy the three traits simultaneously. For instance, desires, aesthetics, and opinions are intangible, and are capacities common to everyone, but they are not consistent, whereas  limbs, organs, and DNA are (roughly) consistent and common to everyone, but are not intangible. Even still, tradition, culture, and expertise are consistent and intangible, but are not common to everyone.

But to name a few other things that do (or could) share the three traits of counting are reason, justice, morality, and harmony.

A natural question to ask here then would be: is the process of counting at all more intimately related to any of reason, justice, morality, and harmony? Can we create an ideal world of justice through reason alone? Can we prove what is good and righteous with the tools of mathematics?

To begin with, the relation between counting and harmony had been known since the ancient times; its discovery is accredited to Pythagoras. Pythagoras discovered that every harmonious sound can be expressed by comparing two ways of counting, and it was striking, because well, there seems to be no immediately obvious reason for them to be related at all. Excited with this result, Pythagoras went on to establish an exclusive cult-academy of Apollo, where he preached the divinity of numbers and  taboos against beans as the ultimate truth.

One could say Pythagoras went overboard with a cult; but I would suppose he was only very optimistic that the aforementioned similarities between mathematics, harmony, justice, reason, and ethics would ultimately all point towards an essence that would unite them all. Pythagoras was just the first to carry the hopes that would later fascinate the dreams of Plato and the Enlightenment philosophers as well, of utopias governed by perfect morality and perfect faculties of reason.

Driven by the success in the theory of rational numbers, one of the things Pythagoras taught was that every existing number was in fact, rational, and was treated by his followers as a demigod with exclusive access to divine knowledge. However, his teachings turned out to be incorrect, as it did not acknowledge the existence of irrational numbers. According to folklore, Pythagoras is accused of murdering the student who approached him with the length of the diagonal of a unit square.

The Pythagoreans’ murder of this student seems to be only one of the many instances of persecution towards those with foreign knowledge, by those who defend a knowledge system already established in its place. In these instances, it seems as if there is a curious underlying connection between what the knowledge system would claim as right and what the moral system would pronounce as evil; is it natural to assume our knowledge systems capable of distinguishing even moral goods and evils as well?

But before we incriminate Pythagoras for the murder, I think we can at least first understand his desire to assume perfection and mastery, as also one that we might as well find in ourselves–to wish that every number must be one we have known very well already, to wish that everything that could be would already be something we were good at, seems a very common wish that anyone could naively hope for.

In fact, I think this desire can also be found in the story of Adam and Eve, the first created man and woman in Genesis. Adam and Eve also, I presume, knew of the same marvels of knowledge, the power of that godly point of view around which things seem to fall to complete order and harmony. They desired it so much that they were tempted to eat the fruit from the Tree of the Knowledge of Good and Evil that God had expressly forbidden them from eating. They assumed their knowledge was enough to tell them what was Good for them; but their knowledge could only foresee the forbidden fruit, as another harmless, nourishing fruit. With great excitement would they then have been tempted to believe, that the fruit would in fact, make them even more like God.

Instead, as the narrative turns out, taking the matters of Good and Evil into their hands resulted in a curse of humanity they could not have foreseen. The knowledge of Adam and Eve was not robust enough to predict the outcomes of good and evil as they had supposed. The story of Adam and Eve makes me wonder then, whether pretending our knowledge is all, that we can determine what is good by ourselves, that it would make us godly, will one day only make us realize how bare we really are, and our “perfect knowledge” compared to the light of God, or a spark of truth from an other, instead end up only creating space for fear, lies, and sin.

Reading Genesis as explaining the cause (rather than the origin) of evil, the story might be telling us that when we mistake our knowledge as godliness, it follows as a natural consequence to be banished from the lifestyle of Eden, of the joy in observing and cultivating the beautiful things of the world, wishing only the best of things to come.

I follow a Christian tradition that teaches that we are not to assume any righteousness of ourselves apart from the one given to us by grace, and that we are not to think of anybody as stupid, likening this very thought to the equivalent of murder. Though humanity has long been banished from the Garden of Eden, I think it may well be the case that we are still forbidden from writing the knowledge of Good and Evil ourselves. Rather, we are to obey what has been revealed as Good and Evil by the laws of God.

My own experiences of learning was as much a revelation of my ignorance as it has been an acquiring of knowledge; and it makes me doubt whether there can be ultimate knowledge of any kind. Even in matters of doctrine and theology, so deeply intertwined with the revelation that we hold ultimate, I think that what we know now has always been very little compared to what we have not yet known; and thus, that our growing knowledge should bring us to humility, instead of bringing others to our judgment.

In our current body of knowledge, there are two large portions of “facts” (statements considered consistent and universal) that constitute our ultimate notions of reality. One consists of historical facts, and the other consists of scientific facts. I think the reason they are so deeply trusted is because both kinds withstand the passage of time.

Historical facts withstand the passage of time by the way we understand the system of causality. Since an event cannot be influenced by any event occurring after it, a historical fact remains true for all moments after it occurs.

On the other hand, scientific facts withstand the passage of time by making average statements of time. Since we have taken the average  behavior of all time, the resulting phenomena are essentially timeless; ready to be imagined to reoccur at any arbitrary point in time, given the right preconditions. Scientific facts are produced after assuming the inductive hypothesis–that what has occurred today will occur tomorrow–which seems bizarre in the time scales of our schedules, but holds for much material behavior.

Both systems are powerful, but I think it deserves noting how the initial assumptions of the two types of facts are in fact, mutually exclusive. One assumes the fundamental particularity of every moment in time, viewing reality as consisting of events, while the other assumes the fundamental homogeneity of every moment in time, viewing reality to consist of timeless substances. Thus it is inevitable that these portions by themselves can only illuminate a portion of the reality that we believe is ultimate; what is really real, really important, really true.

As such, I think that even facts can only be as ultimate as the rational numbers had been to the Pythagoreans, and there will yet to be an infinitude of truths of faith that do not yet pertain to such facts, among the miraculous, the mysterious, and the cosmic imprints that have been revealed.

It sure must be exciting to share!

Hyunmoon graduated from Princeton in 2013 with a degree in mathematics. He is interested in modern cosmic ideologies and is now at Seoul National University trying to understand the structure of empty space through the mathematics of Lagrangian Floer Homology.

A Babel of Fractions

Once upon a time there was a mathematician who knew that One could be split into two halves. He was asked to participate in the construction of the Tower of Babel. He began teaching people how to split one thing into two equal pieces, and as more and more people learned, the Tower grew higher and sturdier.

One day, he awoke from his bed to see a swarm of people waiting outside his door. People were confused the halves that they had no longer were the same.

“I swear, 256/512 is the right way to divide one into two equal pieces!”
“No, I can bet my life that 3/6 is the right way!”
“It’s definitely 10/20 that is fundamental! Check the number of fingers on your hands!”

When he came out the door, he asked for everyone to give him a sample of every one half that they considered to be true. Then he told them, “I will take a look at these” and set out to examine them carefully.

He spent his days looking, looking, looking, looking, and there were so many of them that he died before he could find an answer.

Meanwhile, the constructors of Babel split and went their ways each thinking they had the correct number and everyone else was wrong.

One day two grandchildren of the constructors of Babel met and were sharing their family heirlooms, a/b and c/d. They were young and innocent and let each other play with their numbers, and they discovered that ad and bc were the same!

News spread to their parents and people everywhere started matching their fractions together. Family feuds were reconciled and the king decreed that  henceforth all fractions that can be reduced to the same lowest terms would be one and the same.

And the people lived happily ever after.

I wrote this story wondering to what extent the notion of a half could model  the notion of truth. It does explain how an idea can be one and infinitely many at the same time, without there being any logical contradiction.

Moreover, it illustrates how not obvious and critical the knowledge of ad = bc is to establish a single, united system of numbers that we call a half.

To understand a half we must not only understand how a fraction is different from another, but also how different fractions can be one and the same. Though we may believe we understand 1/2, we might really not have understood a half without figuring out all the ad = bc s. The task of translating many different expressions of the same idea is essential for mutual understanding, especially in our time when modernity has left us with many separated towers of traditions and expertise, inside each of which we can all lose ourselves forever.

It is funny how the notion of fractions can not only tell us how to share things but also how to share truths. Though for the battlefield of all the intricately intertwined truths out in the world, the task of establishing translations will be immensely more difficult, yet still I hope that perhaps at the end of it all, we will have a very handsome shared truth for everybody.

Hyunmoon graduated from Princeton in 2013 with a degree in mathematics. He is interested in modern cosmic ideologies and is now at Seoul National University trying to understand the structure of empty space through the mathematics of Lagrangian Floer Homology.