Recently I’ve been reading Nicholas Orme’s *Medieval Children* with an eye toward blogging about my study. I have been making progress but for better or worse have encountered countless interruptions (and, lately, mishaps) from a modern child who at various points in his life has been referred to as having no “off” button, a “blonde blur” and the “tornado who lives in my house.” But I also affectionately refer to him as “Turtle” because he is the kindest and sweetest child I have ever met. The mother of a friend in our cul-de-sac told me once, “Well, sometimes he is a little too much boy for [her daughter], but they get on really well.”

He also is an engaging conversationalist and, especially in maths, way smarter than I ever will be–which is as it ought to arrive. So recently when a conversation that actually commenced several months ago and focused on infinity was re-ignited, I thought once more of a uni paper I wrote on the topic. We explored it a bit and while our conversation branched off in many different directions, it tended to come back to one of his favorite topics: numbers.

“How many numbers in between one and two?” I’d once asked him. He shrugged his shoulders and said something like, “Probably five or six, and some fractions into the bargain.” He was astounded and awed when I told him there were many, many more than that. Then came talk of fleas on fleas on fleas (a backwards travel he hadn’t anticipated), unending space. the speed of light and passing through it to end up in a different time.

One thing I’ve learned as a mother and a teacher is that children will never fail us when we ask them for ideas. I’ve lost count as to how many times I’ve presented Turtle with a dilemma and he pitched back a solution I hadn’t even explored. It makes me wonder how medieval mothers–and fathers, for that matter–approached and conducted conversations with their children, especially bright ones. I will learn more about this as I read on, certainly, perhaps learning from medieval children as I do from those who can actually talk to me now.

The paper re-printed below was written for *Communicating Math Ideas*, a class for teachers taught by one of the most magnificent professors I have ever had the great fortune to meet. In his classroom I learned a lesson similar to one I’d done several years before in an undergrad biology lecture hall. Because it was a GER *and* I considered myself “an English person” (in this world divided by language arts on one side and mathematics on the other), my heart wasn’t in it. Moreover it being packed full of students, individual attention was a pipe dream.

Both professors, however, have a gift in that they were able to singularly capture the attention of those whose walls were otherwise too thick to breach, and it is worth noting that humor had a privileged place in their respective techniques. Dr. Gary Davies (biology) persuaded me to realize the role played by affirmation, for better or worse: all our lives we were told or negatively supported in our own belief that we are not science people or not good at it–and that this was mere conditioning. Dr. Larry Foster (math) used less sentiment but possessed a wide repertoire of ways to fascinate us with mathematical magic. I still struggled with lots of the work and ended up with a C in both classes, but today feel better about those two Cs and the knowledge and interest I continue to carry, more than any A I earned regarding information I dropped off when I graduated.

I still find some maths challenging and didn’t miraculously become a whiz; in fact my ten-year-old child as mentioned above is way better at it than I am. But I remain interested and occasionally peruse the same library shelves (especially physics and astronomy) I did when in Dr. Foster’s class. Twice in recent months I entered into other conversations about infinity and time travel, bringing me back to those shelves and the paper referenced above. In some instances I haven’t a clue what I am reading but tend to follow the same pattern I did when reading *Tears of the Cheetah*: read carefully, write notes, find the links and be happily astounded. (It helped me, by the way, to create an on-the-spot lesson plan when once asked to substitute for a middle -school science teacher who had just gone home sick.) I try not to dismiss information that is above my head, but rather piece it together as best I can, finding at least one element I can understand.

There’s a small bit of background for the paper below, which is just now four years written. Individually it also didn’t earn me a fantabulous grade (B) and the professor told me I didn’t exactly address the question as he asked it. But he did enjoy the paper, including the weaving in of concepts as they relate to early childhood education. As it continues to carry me through ongoing and changing ideas and conversations, I enjoy it as well, even if I did, upon re-reading, have to stop and do some re-absorbing. There is still so much to know that I don’t, and this infinity is perhaps the best part.

**********

The Nature of Infinity: A Multi-Discipline Perspective Study

When I began to discuss this topic with someone more learned than I, he warned me to stay away from it: “It has driven men insane.” His comment summoned memories of driving through North Dakota and its immeasurable flatness, an experience I never wish to repeat, despite prior romanticism of all things prairie. The flatness as I drove went on and on, forever and a day, it seemed, except that at the time implied there would be an end to this drive. There were no indications to dispel the notion I began to entertain that I was not actually moving at all. No exits, no other cars, no mile markers, no wandering buffalo, nothing. I did not understand then they don’t roam around the streets like our moose do, and I cursed this endless country. Even the lone structure I thought I saw in the distance never got any closer. I started to worry about Zeno’s paradox (absorbed courtesy a childhood obsession with Lewis Carroll and discussions with my father) and that I might never get back home. I remember thinking, “This must be what it is like to go mad.” It enveloped me like a dark cloak that was only lifted when, outside Edmonton, I saw a green sign with an arrow and the printed word, “Alaska.” It was still three days and a drive through the Canadian Rockies away, but that cloak was off my shoulders.

Occasionally subsequent listeners of my yarn would talk about perception and how some people are used to such a land—even that they would have recognized natural distance markers—and I would respond dramatically, “You’ve not looked into the eyes of infinity.” They would laugh at the absurdity in my memory of driving towards Montana and the comfort and relief promised by its name. But even the learned Greeks had a word to represent fear of what Marcus Aerelius referred to as a “fathomless gulf, into which all things vanish”: *apeirophobia* (Tyler). I had not been alone with my feelings of displacement.

Infinity actually comes in two sizes, Tyson continues, the great and the small. The flea on a dog has pests on him, and they have pests on them, and so on down to the uncomprehendingly small, as he quotes Jonathan Swift. The largeness of infinity, on the other hand, can be contemplated in terms of numbers as well as in the forward moving of time that never ends, as in Van Loon’s 1921 musings on the topic in *The Story of Mankind* (itself a heady title):

High up in the North in the land called Svithjod, there stands a rock. It is 100 miles high and 100 miles wide. Once every thousand years a little bird comes to the rock to sharpen its beak. When the rock has thus been worn away, then a single day of eternity will have gone by.

If this bird, who for his size must have a rather small beak, comes *once every thousand years *to sharpen it, how many thousands will it take to wear the enormous rock down? On the one hand, our lives—roughly *70*–*75* years for each of us—is what part of that rock? Is the chip that represents each of us visible to the naked eye? How many of us can fit onto the tip of his beak? On the other hand, tedious as it may be, perhaps we could come up with some number and see that, indeed, at least infinity has a defined end. (Does that by definition disqualify it from being *infinity*?) But then again, can we really come up with a number of that size? Each time the rock is worn down, we will recall, is a *single* day. Children come up with words all the time to represent numbers they cannot yet grasp. How do we reach for such a tool? When we begin to have a grasp on the vastness, is the worry over smallness of ourselves in such a scenario mere conceit? And do we rarely encounter the type of fear the suicidal Buber (Tyler) did because we engage in lifelong avoidance of having to grapple with the terrifying enormity of it all?

In conversations about the unknown or frightening, sometimes we talk about how the process of naming something renders it understandable or better appreciated—or at least nerve wracking to a lesser degree. This seemed true as an experiment at the time of my journey, and helped me endure that difficult portion of it. For instance the numbers on my watch told me time was definitely passing, and the speedometer registered miles actually traversed. Bringing this idea to bear on infinity, rather than just imagining a vague sort of never-ending vastness, we can look at numbers to bring some concreteness to the idea. There is precedence for this line of thought: In ancient Greek cultural context, explains classics historian Reviel Netz, there existed a paradigm “that in order to understand things you should find the precise integer numbers that govern them” (Nova).

Unlike small children we as adults have internalized the *idea* of numbers, and don’t tend to understand them as something magical or unknown or vague; they mean more to us than to young children, who are busy cracking some kind of code by coming to understand these bizarre shapes. (If you don’t believe me, try writing in your checkbook register for a day using numerals *hindiyyah*: ٠,١,٢,٣,٤,٥,٦,٧,٨,٩., and you’ll get an idea of children’s perspective.) So, if we take the familiar numbers, say, *1* and *2*, we get a better idea of what lay in between without too much reeling; they keep us within a certain comfortable framework.

Even though there are infinitely many fractions between each pair of consecutive counting numbers, the arrows (Figure 1) show how to match the red counting numbers with the blue fractions (Bellevue).

Perhaps it is a bit like the difference between knowing you have a great distance to cover but you don’t know how much, and knowing you’ve 600 more miles to go before you reach the border.

One of the things I frequently contemplated on the drive was whether I should have taken a different route. Ever since I made my way by Chicago via (mistakenly) Gary, Indiana, I started second-guessing myself. I’m fairly skilled at reading a map, but sometimes I stopped and looked for the straightest line from where I was to the next stop where I wanted to be. Some alternatives looked attractive, and while the time may have been shorter, the points crossed wouldn’t have.

For practical purposes that would have been good enough, but looking back as well as at some of the information at my fingertips now makes me contemplate the idea of superimposing a great triangle somewhere around Dickinson, North Dakota. If I began on one side to bisect the triangle, back then I might have thought that being near the apex would make the drive shorter than had I began at the wide angle of the triangle. Well, it would have, but in mathematical terms it is interesting to note that I still cross the same number of lines. So infinity in one sense, with a perspective towards time, can be much shorter or smaller depending on where you start and stop. But in terms of numbers, there are no shortcuts. Figure 2 below illustrates this.

(Fig. 2) [Image used in original paper no longer available; please see bijection, page nine.]*

Line segments with different lengths have the same number of points. (Schechter)

So, no shortcuts.

Then again, maybe there are. If you consider the numbers used to measure time, you’ve got it made. Closer to the apex it takes only two hours to get to Dickinson, while from the other end it takes eight. But, again, time is an abstract concept. Is it more real than something that can be mapped out concretely? What if we measured distance in a non-concrete fashion? Is that possible? Well, as Vilenkin writes, “It is hard to reconcile oneself to the thought that a path a million light years long has only as many points as the radius of an atomic nucleus!” (63). As a grown up even I don’t have a firm grasp on the exactness of those distances, but I do have an awareness of what they imply, and I can understand that paradox in Vilenkin’s statement. Not to mention the time it takes to cover such distances (and the differences between space and Earth times). How could you ever explain such concepts to a child?

John Monaghan researched young people’s ideas about infinity and reviewed Piagetian studies on children’s understanding of the concept. In 1954 Piaget and Inhelder tasked children with subdividing geometric shapes, such as a line. They found that “[i]n the concrete operational stage children could continue a large but finite number of divisions” (241). The problem, however, was that the Piagetian framework, in which children develop, progressing from stage to stage, did not account for the internally contradictory nature of many of the children’s responses. Later, Fischbein et al. expanded the study with older children (10-15 at various levels of attainment) by taking intuition into account. They determined that “our intuition of infinity is *intrinsically* [emphasis mine] contradictory because our logical schemes are naturally adapted to finite objects and events” (243).

How contradictory is infinity itself? In terms of numbers, my own musings of the comparison between positive and even integers still leaves me in awe: they are equal in amount. Richard Morris backs me up in this by affirming, “intuition would tell us there are twice as many positive integers.” Galileo came up with the same conclusion but referred to infinity as “inherently incomprehensible.” His conclusion that it is best avoided (4) didn’t help Giordano Bruno, who likely heard it from someone if not Galileo and was burned at the stake during the Inquisition for, it is believed, his ideas about an infinite universe (40).

Summoning images to my mind as I type brings ideas I am fairly certain I never contemplated while on the road. The geometry of the angles (and sometimes curvatures) produced when looking off into the distance, the great opening that lay before me through North Dakota, the Big Sky of Montana when I finally reached it—all of these are ideas that touch on multiple perspectives that can be repeatedly questioned or played with to look for different ideas. For instance, if I saw the above diagram, or this stretch of North Dakota as a work of art, but didn’t have any idea about its measurability, how would I gauge it? Geometry in art can be a bit like statistics—the outcome can change depending on who looks at the figures. This brings up the entire concept of visual literacy and how the eyes and sight work. Is eyesight passive—does it simply register based on things like shape or condition—or does what is recorded engage and process based on what the person has stored in his or her brain? Is the structure in the distance on the other side of the field or many miles away? Is the sky in Montana actually big, or did it appear that way simply because I’d long heard that expression? Looking across a Montana prairie one sees, without being aware they are seeing, as if from a mountaintop. However, from this perspective of flat land where one stands, there actually exists a drop, or curvature of the earth, on either side so subtle as to create the image of a sky larger than it is. Moreover, one could turn 360° and the effect remains constant; the perspective is not changed by direction. The mind of this observer perceives the horizon as being infinitely far away, when in fact the point they have focused on is closer than it appears.

Camerawork can be just as tricky with the way lenses make objects appear closer than they are and I’ve often wondered how that can be. Geometrical configurations are not manageable in the way numbers can be (*Nova*), and this may be why the subject unnerves many people. For the same reasons, perhaps oddly enough, this may also be why people revere art as they do: they are repeatedly trying to conceive of and interpret what they see.

As geometry seems linked to art, so too does it to astronomy. We’ve all seen the diagrams drawn around pictures of sparkling skies; how many of us understand them? Is the vastness part of, or an addition to, what baffles us? There are numbers to work with, although in this context they seem so unreal, unbelievable. Perhaps a bit more “down to earth” would be our modern understanding of how astronomy and space exploration is related to national defense. Galileo, after all, understood the implications of the telescope when it was still being sold as a novelty item (Couper, 136). When he looked through his he could see ships so far out to sea that the lookouts posted wouldn’t see them for two more hours (139). Galileo brought a seemingly infinite distance down to a manageable size for the safety of the city.

But what of those who watched the stars before the telescope? People have been doing this for millennia; the distances they marveled at left them perhaps as awestruck as my driving distance left me. Certainly, some may say, the drive across one U.S. state is no match for the distance between Earth and stars billions of miles away. Too right they would be, although I still might be able to claim an infinitive drive when the question of sizes of infinity is raised. Is an ancient star gazer’s infinity bigger than mine? Stephen Battersby writes that the infinity behind the idea of endless time and space is a “trifle.” Not only that, “multiplying infinitely many infinities together, the result is smaller than another infinity.” Does this mean all the galaxies together are smaller than North Dakota?

The numbers might be fun to play with on that, and it is interesting to note that beliefs on the universe’s infinity swung like a pendulum as time went on. Aristotle taught that it was quite finite, and his authority (Greek fear?) influenced Church doctrine. We will remember that Bruno lost his life over disagreement with contemporary authorities. But by the seventeenth century his idea no longer was so controversial (Morris, 158). The pendulum has swung back and forth and continues to do today: compelling evidence once pointed to an infinite universe, but many physicists and astronomers now believe we will never know because it is so close to the borderline (159). Perhaps most fascinating to contemplate, in opposition to the idea that can barely conceive of infinity, especially given Monaghan’s discussion on the finite nature of the human mind, is that we reject the idea of a finite universe. For it to be so, there is an indication that somewhere along the way it stops. And then what?

Works Cited

Battersby, Stephen. “To Infinity and Beyond.” *New Scientist* 179.2414 (Sept. 27, 2003): 31.

Bellevue College Science Division. “Counting to Infinity.” Bellevue College Science

Division. N.D. Bellevue College. 15 April 2009.

<http://scidiv.bcc.ctc.edu/math/Infinity.html>.

*ibid*. 6 Apr. 2013.

Couper, Heather, and Nigel Henbest. *The History of Astronomy*. Buffalo: Firefly, 2007.

Monaghan, John. “Young Peoples’ Ideas of Infinity.” *Educational Studies in Mathematics*

48.2/3 (Oct. 2001): 239-257.

Morris, Richard.* Achilles in the Quantum Universe: The Definitive History of Infinity.*

New York: Holt, 1995.

Nova, Interview with Reviel Netz, *Working with Infinity: A Mathematical Perspective*,

Nova, PBS, Boston, Sept. 2003. 15 Apr. 2009

< http://www.pbs.org/wgbh/nova/archimedes/infinity.html>.

Nova, Interview with Reviel Netz, *Working with Infinity: A Mathematical Perspective*,

Nova, PBS, Boston, Sept. 2003. 6 Apr. 2013

http://www.pbs.org/wgbh/nova/physics/working-with-infinity.html

Schechter, Eric. “Georg Cantor (1845-1918): The Man Who Tamed Infinity.” Vanderbilt

University Department of Mathematics. 15 April 2009

<http://www.math.vanderbilt.edu/schectex/>.

Schechter, Eric. “Georg Cantor (1845-1918): The Man Who Tamed Infinity.” Vanderbilt

University Department of Mathematics. 6 Apr. 2013

http://www.math.vanderbilt.edu/~schectex/courses/infinity.pdf.

Tyson, Peter. “Contemplating Infinity: A Philosophical Perspective.” Nova, PBS, Sept.

2003. 15 April 2009

<http://www.pbs.org/wgbh/nova/archimedes/contemplating.html>.

Tyson, Peter. “Contemplating Infinity: A Philosophical Perspective.” Nova, PBS, Sept.

2003. 6 Apr. 2013

<http://www.pbs.org/wgbh/nova/archimedes/contemplating.html>.

Vilenkin, N. Ya. *In Search of Infinity*. Trans. Abe Shenitzer. Boston: Birkhaüser, 1995.

*****For an interactive illustration of this point (no pun intended), click here.