An Aesthetic Representation of Alfred North Whitehead's Philosophy of Mathematics Education

A STORY

Ali’s in the World of Heavens

Ali was beginning to tire of sitting by his brother on the bank now that he had finished his mathematics homework. He had excelled at today’s work to such a degree that he could do the problems without even thinking about them.[^1] Deeply bored, Ali peeked at the mathematics book his brother was reading. The book contained no numbers; Ali only saw meaningless symbols.[^2] “What is the use of a mathematics book”, Ali thought, “without numbers or practical problems to solve?” But as he studied the book, Ali realized the symbols were not stable. Instead, as his brother read, the symbols were mysteriously changing and transforming into a different set of symbols. Ali’s brother did not seem to notice the changing symbols for it was his thoughts that were defining the meaning of the symbols. Ali imagined his brother's brain expanding, getting bigger and bigger as it was filled with all the things he has been reading.

The hot sun made Ali feel very sleepy and stupid, and he turned his attention from the perplexing book to the small pleasure of making a daisy chain. Ali wondered if it would be possible to make a chain with an infinite radius and whether it would be worth the trouble of getting up and picking an infinite number of daisies. He also wondered whether the diameter consisting of all the real numbers would be larger than the total number of daisies in the circumference?[^3]

Pondering this question, Ali was saddened to realize that no daisy could live long enough to finish this infinite chain; it would either shrivel and die or be brutally consumed by a rabbit - a rabbit like the one with a white head and pink eyes that had just run by him. This idea of the daisies and a selfish rabbit seemed nothing more than sad[^4] until Ali heard the white-headed rabbit (or just Whitehead for the sake of simplicity) exclaim, “Oh dear! Oh dear!  Not again! Why do I have to lose my ability to talk every time I am here?”[^5] This was no ordinary rabbit.

Ali wondered whether Whitehead was real at all,[^6] or whether he might have imagined this impossible creature. “I think I should simply ignore it and continue to act according to my interest[^7] , which is to make this infinitely large chain of daisies,” Ali thought. Then, something even stranger happened.

The rabbit did not appear to be speaking because his lips weren’t moving, but Ali could still hear Whitehead saying, “Importance is derived from the immanence of infinitude in the finite”.[^8] Ali was confused about how to respond - whether in words or through his thoughts -  since Whitehead seemed to magically communicate through thought rather than speech. Finally, Ali made up his mind. Rather than putting his thoughts into words, he simply thought, “To make a daisy chain is my desire, which is a matter of fact that is so important to me that I should not waste my time talking to you”. Despite his desire to build an infinitely large chain of daisies, and its relative importance to him, Ali’s curiosity burned. He wondered why the rabbit was complaining that it can’t talk if it didn’t need to talk? Once again, Ali heard the rabbit’s voice in his head, “Expression is founded on the finite occasion[^9] and only animals express themselves unconsciously. I am not an animal, so stop referring to me asit !” Ali was stunned upon this outburst. This was such an unusual occurrence in an ordinary day. Was it really an ordinary day?

At this point, Ali found himself thinking about the book that his brother was reading, and he wondered about the mode of communication between his brother and the author of the book. Were the numbers elements of a finite world; were the symbols elements of an infinite world? Perhaps the relationship between numbers and mathematical symbols paralleled the relationship between language and thought. Perhaps these concepts were related to each other just as any reflection is related to the original object. It is never a perfect translation. There is always some distortion. Which element is more important? Which element reflects the infinite?

Ali shouted angrily, “Who are you? What are you?”

Through his thoughts Whitehead explained, “I was once a nebula and later united with some dust particles from the Moon. Finally I arrived on Earth in the form of a white-headed rabbit”.[^10] Without thinking another word, Whitehead started running. He was running purposefully to reach a destination rather than trying to escape from Ali. He seemed to know that Ali was going to follow him wherever he went, even down a dark, unmarked hole in the ground. Following his curiosity and the intriguing rabbit, Ali found himself in a new alternate reality.

Down, down, down he tumbled.  Suddenly, Ali landed on a heap of sticks and dry leaves with a loud thump! Looking around, he saw he had landed next to a man in a suit. With a twitch of his nose, the man introduced himself saying, “Hello, I am Whitehead. I’m glad you have arrived in the World of Heavens. This passage between our worlds is called Gabriel’s Horn.[^11] We tried to make the passage nicer, but we could never finish painting the inside since it is infinitely large. The hole is too small and cumbersome for two people at once. I’m doing entirely too much talking! This is what you need to know. This world is very different from your world. Here, everyone and everything can talk. Take care, as all your thoughts are automatically translated into words that others will hear.[^12] Like raindrops on a pond, your words will constantly fall and the ripples of consequence they create may be hard to control. Welcome, and good luck”. At this, the man ran away, leaving Ali alone in his quest for eternal knowledge.

With Whitehead’s warning weighing on his mind, Ali wandered around a bit, trying to figure out what this whole new world was about. After a bit, he came across someone who was shaped like a symbol. This person looked exactly like the letter “U” and “U” was wearing a black tie. He or she  - it was hard to know  - introduced itself in the fashion of James Bond saying, “Hello. I am Stan - Under Stan. I’ll be your guide in this world, and I shall do my best to help you understand everything as it was, as it is, and as it shall be”.

Hearing the word “everything” was enough to scare Ali. He knew he would never have enough time to understand all the weird things in this place, and he was already yearning to return home and feed his cat. “Forget about your cat.  Let's concentrate on the daisies,” U said. U continued excitedly, “Wasn’t it you who wanted to have an infinitely large chain of daisies?” U asked the question in a very Aristotelian way, so Ali nodded his head in agreement. “What would you need for such a chain besides infinitely many daisies?” U continued.

“What a stupid question!” Ali thought, but he tried to dismiss this idea so as not to offend U. Instead he responded, “I would need time, an infinite amount of time”. U was happy to hear this answer and calmly said, “Well, then my Master shall give you what you wish for since he has infinite patience”.[^13] U pointed out a huge green sign with the following words written on it:

I am Master of this World;

And what I know not, is not knowledge.[^14]

Ali rebelled against this idea, once again falling back on the constraints of time and place found in his world, rather than exploring the logic of this new world. Ali wondered how this statement could make any sense when even human knowledge was constrained by time and place. Moreover, Ali had his own experiences and knowledge to consider and perhaps that would be outside the master’s domain. Directing his thoughts to U, Ali asked “What about my own perspective? What about my imagination?”

U had not been chosen to initiate newcomers without reason, and he had been in this business for years. This was perhaps the thousandth time that he had been forced to answer such an arrogant question. However, instead of getting angry he chose to stay calm and intelligently answered, “You are forgetting the fact that our Master has infinite knowledge. Your imagination is actually part of His imagination. He is giving you what you need in a non-changing world so you can grasp the meaning of everything, and perhaps even build your chain of daisies.[^15] His gift is infinite time. Now that you have infinite time, and because the daisies will not die, you may have a chance to succeed in your quest. You may also be interested to know we have some rational daisies, as well as irrational ones. They are all real, though. You can build a chain in which the diameter can be smaller or greater than the circumference. It is all up to you”.

Ali wondered how this was possible. How could one's knowledge not be limited by the present parameters of the universe?[^16] Ali could only imagine an answer if he believed that the master had existed in both realities since the beginning of time. But that would also mean Ali was going to be forced to stay here with the master for the rest of time in order to understand him. Ali was upset, thinking he would not be able to see his brother anymore. “Perhaps,” he wondered, “the best way to proceed is to just try and enjoy the present, exactly as it is right now”. Ali maintained hope that he would be able to change the future through his actions in the present and hopefully return back to his own world. However, the question remained. Were these two worlds somehow consistent with each other, or were they contradictory?

U told Ali to follow him and it showed him an apple. Admiring the apple, Ali noticed how wonderful this apple was, how perfect and delicious. Ali also remembered the rotten apple that he had given to his cat the day before. That is, he tried to remember the rotten apple. He pushed himself, trying to visualize the apple, but he just couldn't imagine the rotten apple, because the image of the perfect apple defeated all the other thoughts in his mind. “How could this happen?” Ali asked.

“Nothing ever changes in this place, neither for better or worse, even in my mind,” U replied in a philosophical manner. It added that, “Our universe escapes from the limitations of the finite. Things that you find contradictory in your own world are simply consistent in this world. This is because our Master can presently see the past and future,[^17] and any contradictory things work in harmony in our world”. U tried to explain this unexpected reality as the two walked along a long path through a garden. Like everything here, the garden was unusual, filled with animals and flowers chatting with each other. Not only did the animals and plants seem to be communicating, the stones, and even the dust particles floating in the air seemed to be communicating. Ali even saw the numberpi in the air flirting with a circle.

“This is too chaotic. I can’t understand a word when so many things are talking at once,” Ali complained. He was burning to know more about this world that was pulsating with consciousness, and wished he could interview each element of the world to learn about its experiences. During his journey, Ali met with a wide variety of things, and each seemed to be living and talking. They were all working in harmony, without conflict. Ali saw a tree producing apples and serving them in a respectful way to others. As its presence thrived in the environment, Ali heard the tree saying, “Thank you for bringing me into existence, Master. You are beautiful and perfect; you give life to me without expecting anything back but my appreciation”. If only he had the time, Ali would ask the trees how photosynthesis felt inside their bodies, and what they felt when chloroplasts transformed solar energy into ATP. He longed to talk to the number pi and learn about the beautiful circles around him. Everything here seemed to be beautifully connected.

Sadly, there was no time to explore. Instead, someone called Anaxagoras approached Ali and told of his adventures trying to square the circles.[^18] Ali wondered if “squaring” was slang only when the guy called Tangent said he used to live next to the most beautiful circle in the world, but could never get inside its heart.

Ali explored the World of Heavens, never feeling as though he had enough time. For days on end he observed the elements of the land living in harmony witnessing their contributions to the environment, and trying to learn about the experiences of each element.[^19] At the end of each day, Whitehead visited Ali and tried to explain the ways of this world, naming all the elements Ali had encountered. Whitehead only introduced the name of something after Ali had observed it for himself. When Ali asked why Whitehead wouldn’t tell him the name of things before encountering them, Whitehead explained, “In the World of Heavens, we must first let you see everything before you learn its name. In fact, you were supposed to see these things in the other world and then learn the names here, but for some reason you were rushed and born prematurely into this world”.

Grappling with learning so many new concepts, Ali felt as though he was reading a book in which the chapters included everything ever written by the master - this master who claimed to have knowledge of all unchanging things, and represented them in the book with their true value and meaning. After spending time in this place, Ali found himself thinking, “This should be the true world, a world which is both complete and consistent.” Ali threw himself into learning the ways of the new world so that he could gain an intuitive understanding of how this foreign world operated. Ali learned about tangents, and circles; he learned about pi. He even had homework and tests in which the answers always referenced the lessons he had learned that day. There were never any trick questions[^20] and by the end, Ali was able to answer many questions without much thinking.[^21]

Suddenly, Ali awoke from his long sleep to find his brother shaking him. Ali, understanding he had returned to his native world, saw that with his visit to the World of Heavens his present world had changed because his views and interpretation of this changing world had changed. Ali asked to see his brother's book and studied it, looking for clues that might show him how to understand the universe from a new point of view that would combine all three perspectives: the book’s perspective, the universe’s perspective, and Ali’s own perspective. Ali found that although the book, Ali, and the universe appeared to represent three different perspectives, they were basically telling the same truth. He discovered that as he knew more about himself, he understood the book better, and as he better understood the book, he better understood the universe. Thus, Ali realized that he himself was a small example of the entire universe. From his experience in the World of Heavens, Ali had learned infinity was not possible in this world. All the forms of this world were essentially referent beyond themselves, even the numbers and symbols found in mathematics. Ali now understood that in this world he would learn and grow until he was ready to reenter the World of Heavens and its harmonious, coherent existence. Ali had come to see the trees on the bank as serving apples to people, and being replaced by new trees when the time came. Ali now understood that consciousness infused every particle of dust, every breath of life, and every concept in the universe. Ali saw the master as alive and charged with supervising and controlling the universe in order to prepare all his subjects for the unchanging world of eternity.

As for the rabbit, Whitehead was in the fields looking for another human to take to the World of Heavens for a taste of eternity.