Infinity and Me

Infinity and Me

By Russell Marcus, Professor of Philosophy, Hamilton College, USA

“You’re eight years old,” my father said. We were in the basement, gathering laundry. “Your sister is four. So you are twice her age. Two years ago, you were six and she was two, so you were three times her age. In four years, you will be twelve and she will be eight. So, you will only be one-and-a-half times her age. Four years after that, you’ll be sixteen and she’ll be twelve. You’ll be only one-and-a-third times her age.

            “When will she catch up?”

I wasn’t flummoxed, as he might have expected. I would always be four years older than Ilene. We could live forever and she would always be my little sister. But I was enthralled by my first lesson in asymptotes, by the mathematical insight that allowed us to reason with the infinite. It was, at the same time, a lesson in perspectives, the arithmetic and the geometric. From the one perspective, the difference in our ages is constant. From the other, it’s shrinking.

A few years later, I stumbled on Zeno’s racetrack. A runner must complete half the track before the full track, and then 3/4 and then 7/8, and then 15/16. There are an infinite number of these almost fractions, a convergent sequence amusingly easy to construct. The world doesn’t work like this. The runner always finishes. But the sequence of fractions never ends. The mathematics never finishes. The world’s perspective is simple and uninteresting. The mathematical perspective is puzzling. The proofs are unassailable but the possibilities are endless. I was not the kind of kid who finished first in an actual race. I was standing on the sidelines thinking about the potential of infinity.

I became obsessed with mathematical puzzles, not all of which involved infinity. I memorized a little novelty book called How Many 3-Cent Stamps in a Dozen? or how logical are you? (The answer is not four!) The amount of water flowing into a tank doubles every minute. The tank is full in an hour. When was the tank half-full? It’s like the inverse of Zeno’s paradox. The titillating answer: at fifty-nine minutes.

Eighth grade algebra, with Mrs. Zarchy. I am not happy. My family had moved the previous August and the new environment was a definite disappointment. When we registered for school over the summer, I was offered the opportunity to join the honors class if I would complete a forty page packet of math problems. I balked. My accomplishments to date were insufficient? No, thank you. I loved thinking about math, but I did not love either the drudgery or the danger of school. I was not about to complete forty pages of dull worksheets during summer vacation after being wrenched away from home and friends.

I was safer in the new school, not getting pushed around, not having my pockets emptied daily. No one was throwing pennies down in front of me to watch the Jew pick them up. My new community had lots of Jews. But they were no more friendly or welcoming. I was still largely alone with my thoughts.

And then Mrs. Zarchy was talking about infinity! The class was dull and slow, but here was perhaps an opportunity to think about something real. Imagine my disappointment when she said that the number of grains of sand on the beaches of Earth is infinite.

The next thing I know, I’m in the principal’s office, receiving stern instruction on how I was expected to show respect for teachers in this school. That was not the last time Mrs. Zarchy had to send me to the office. I was not forgiving. But I didn’t mind getting thrown out of class, actually. It was a relief from the other students and teachers.

I didn’t have no friends. In tenth grade, Marshall shared a copy of a big book about mathematics, art, and music: Douglas Hofstadter’s Gödel, Escher, Bach: an Eternal Golden Braid. These mathematical questions and puzzles I had been thinking about were more fertile than I had realized, with long tails extending like infinite series in directions I hadn’t imagined. There were funny directions, artsy directions, literary directions, even in the parts of the book I could understand. I was not ready for the whole thing, but the world had begun to open.

One Sunday night that year, my dad gave me a copy of a little book. “I think you’re ready for this.” Camus’ The Stranger. I fell asleep reading it. The next morning, I went to the bus stop in the morning, then hung around until my parents and sister left the house. I cut school and returned home to read. Over the next year or so, doors kept opening. Sartre. Beckett. Malcolm. Burroughs. Borges. I found myself empowered by seeing various perspectives. Would infinite life make me a happy God or a miserable prisoner? Would an infinite library undermine the possibility of meaning and meaningfulness? The literary philosophy of infinity helped open the world to me.

I’m not sure that I’m capturing well enough the ways in which thoughts about infinitary mathematics led me to philosophy, but the line is clear to me. The real world was miserable and unjust. I was seeking the transcendent. Religion was not an option. I saw arbitrary, inexplicable, unjustifiable rules and our rabbi would not engage my questions. Mathematical proof and insight allowed my thoughts to soar above my unhappy life.

Had I been a better student, I would have been studying multivariable calculus in my first college semester instead of Calc II. Prof. Rosen, a year from retirement, was more mathematically adept than Mrs. Zarchy, but still I couldn’t pay much attention. I was, like most curious young people, a grotesque: comically unbalanced. I was full of intellectual ambition and bravado and emotionally and otherwise underformed. I loved aspects of the math, thinking about it, but lacked the grit really to learn the skills. The mathematics of analysis was too hard for me even to get to the compelling questions about density and continuity and the weirdness of infinitesimals. I was missing opportunities.

I fared better with discrete mathematics and its intuitive proofs by mathematical induction. Generalizing inferences from n to n +1 and thus to all numbers still captured my fancy. But philosophy was more welcoming, connecting in a different way to the eternal truths. None of the philosophy professors handed back a final essay saying, as my adviser and history teacher, Professor Bannister, did, “I expected more from you.” And I learned to write a little in Ethics: Hans Oberdeik gave me a C– on my first essay, a B– on my second, and an A– on my third, with a comment: “I can see that you’re getting it.”

Junior year, confused and adrift, I found mathematical logic. Studying this new formal logic, developed in an historically fecund research period of about fifty years starting near the end of the nineteenth century, was like getting credit for eating candy. I realized that I could return to Gödel, Escher, Bach and get through it this time. And then there was the Cantor proof in Chapter 13, so clear and decisive and compelling, opening my eyes to the different levels of infinity.

My father had already taught me that there were the same number of even numbers as there are natural numbers, another one of those perspective-broadening mathematical results. From one perspective, there were clearly more natural numbers than even numbers, since the natural numbers contain both odds and evens. From another perspective, the two sets are the same size. For each natural number, there is an even number and for each even number a natural number. In short: you can, in principle, make a complete list of the even numbers. Any set whose members you can list is the same size, even if the list is infinitely long. Both the list and the natural numbers go on forever and never run out.

The Cantor proof, I found, took this reasoning further. There are some sets that are so big that you can’t even list them. Such sets are even larger than infinity! Or, rather, they are larger than the smallest infinity. There are different sizes of infinity. My mind was blown.

My first encounter with the Cantor proof was in perhaps its most accessible form, on the decimal expansions of the real numbers. It’s a reductio proof, a proof by contradiction. Imagine that you could list all of the real numbers. We can construct a number, a diagonal number, that provably does not appear on the list. Add it to the list and try again. We can construct a new number that provably does not appear on the new list. Indeed, there are too many real numbers even just between 0 and 1 to list.

Consider a possible list of all the real numbers between 0 and 1. We can represent that list abstractly, using a concatenation of variables each of which stands for a digit of the decimal expansion of some real number, the digits that come after the zero and the decimal point.

            λ          . a1 a2 a3 a4 a5 a6 a7...

                        . b1 b2 b3 b4 b5 b6 b7...

                        . c1 c2 c3 c4 c5 c6 c7...

                        . d1 d2 d3 d4 d5 d6 d7...

                        . . . .

Note that λ is an infinitely long list of infinitely long concatenations of digits. We’re assuming, hypothesizing, that λ contains, at some point somewhere on the list, the digits that come after ‘0.’ in every real number between 0 and 1.

We can, instead, show that there has to be a number that does not appear on the list. Consider a number, we can call it N, defined by concatenating one term from each of the numbers in the list λ. We select the number N by taking the first term from the first number, the second term from the second number, and so on.

             N = .a1 b2 c3 d4 e5 f6 g7...

The number N could be on our list λ. It could be the same as the first number in λ, if b2 were the same as a2 and c3 were the same as a3 and so on. It could be the same as the second number in λ if a1 were the same as b1 and c3 were the same as b3 and so on.

But we can change each digit in N to create a new number N* which cannot be in λ. To construct N*, add one to each digit of N other than nine, and replace all nines in N with zeroes. N* is certainly not in λ. For, N* is different from the first number in λ in its first digit, different from the second number in λ in its second digit, and so on, for all numbers on the list.

In a quixotic attempt to complete the list, we could add N* to λ, to make a new list, λ*. But the same procedure allows us to form a new number, say N**, that’s not in λ*. However complete we make our list, we can always find a number that is not in it. All possible lists of real numbers are necessarily incomplete. There are strictly more real numbers than natural numbers.

Majestic.

I was still not capable of being a good student, of reading what I found dull and of submitting to the demands of teachers I did not respect. But I was thirsty for what I found worthy and contemplated applying to graduate school despite my C-level GPA. My one true mentor, a professor of education named Ann Renninger, advised me to go study with Hofstadter, whose work had inspired me actually to work and learn. How impossible!, I thought. I might as well apply to study with God.

One of my deepest regrets is not respecting Ann enough to sublimate my ego and take her advice. There was that grotesqueness again, a balance of arrogance and humility. I thought I knew more than Ann about how to manage graduate education and I thought that I was unworthy of studying with the author of the book that inspired me to develop beyond my unbalanced youth into a more well-rounded person.

My incompetence as a student became apparent that first semester of junior year. I got an A in logic and finished none of my other courses. Swarthmore and I agreed to see other people for a while. I lived in West Philly for a term, delivering pizzas, making some music, trying to grow up.

When I returned to school in the fall, Oberdeik, the Philosophy chair, was on leave, so I met with Hugh Lacey instead, to discuss how to complete the requirements for graduation. I planned a few courses and what Hugh called, “a little qualifying paper.” I would take a philosophy of mathematics seminar with visiting professor Mary Tiles, and I would have an adviser for my qualifying paper, Dorothea Frede. The topic, of course, was Cantor and the development of transfinite arithmetic

In order to compensate for my semester away and graduate on time, I was taking extra courses, so I was busy. My time off had motivated me at least to try to learn how to work, to develop some fortitude, to sit in my seat and persist when things got tough. For the first time in my life, I sacrificed immediate pleasures for studying, writing essays on Friday nights and letting the noise of campus recede into the background.

I took my newfound logic skills into a course in Ancient Philosophy with Frede in the fall. I had been frenemies with Plato since high school, loving the kinds of questions he asked and hating pretty much everything about his answers, especially the authoritarianism of the Republic and the arguments for the eternality, the infinity, of the soul and knowledge as recollection, which I found insipid. Infinity was not for people. That was the realm of mathematics.

In Plato’s Meno, Socrates walks an uneducated child through a mathematical construction, learning how to double the area of a square and determining the length of a side of the larger square. The child makes mistakes. Socrates asks leading questions to help the child see the errors. Eventually the child arrives at a solution, which Socrates says had been buried in his soul, beneath the errors.

Socrates: What do you say of him, Meno? Were not all these answers given out of his own head?

Meno: Yes, they were all his own.

Socrates: And yet, as we were just now saying, he did not know?

Meno: True.

Socrates: But still he had in him those notions of his—had he not?

Meno: Yes.

Socrates: Then he who does not know may still have true notions of that which he does not know?

Meno: He has.

Socrates: And at present these notions have just been stirred up in him, as in a dream; but if he

were frequently asked the same questions, in different forms, he would know as well as any one at

last?

Meno: I dare say.

Socrates: Without any one teaching him he will recover his knowledge for himself, if he is only asked

questions?

Meno: Yes.

What utter bullshit!, I thought. You can frame your instruction with questions, Socrates, but that doesn’t mean that you’re not teaching. It looked to me as if Plato were making simple logical errors. And now I had mathematical logic in my pocket, a tool to uncover the mistakes. I was going to use it.

I spent the whole semester in Ancient translating Plato and Aristotle into first-order quantificational logic. Once again, mathematics helped me to see other perspectives. For one, it turns out that Plato and Aristotle had their logics straight. For another, that vaunted new logic was insufficient to capture sophisticated philosophical arguments at all but the most shallow level. I was a little humbled, though not quite enough to figure out how important and stubbornly persistent that Meno argument is. It took twenty-five years for me to see Plato’s point about the importance of the psychology of mathematical insight.

I also wasn’t humbled or persistent enough to get much out of Descartes’s Meditations in my spring Early Modern Philosophy class. I got stuck in the First Meditation, where Descartes wonders whether we can know whether we are dreaming or whether there is a demon injecting false ideas into our minds. It wasn’t until I started teaching the work that I realized that he doesn’t care about dreaming, though he says so clearly at the end. So, I missed the most important part, how our knowledge of infinity cannot come from sense experience, how we must have some non-empirical insight to explain our knowledge of mathematics.

I find within me countless ideas of things which even though they may not exist anywhere outside me still cannot be called nothing; for although in a sense they can be thought of at will, they are not my invention but have their own true and immutable natures. When, for example, I imagine a triangle, even if perhaps no such figure exists, or has ever existed, anywhere outside my thought, there is still a determinate nature, or essence, or form of the triangle which is immutable and eternal, and not invented by me or dependent on my mind. This is clear from the fact that various properties can be demonstrated of the triangle, for example that its three angles equal two right angles, that its greatest side subtends its greatest angle, and the like; and since these properties are ones which I now clearly recognize whether I want to or not, even if I never thought of them at all when I previously imagined the triangle, it follows that they cannot have been invented by me. (Meditation 5)

That’s that transcendence that I had been seeking. Mathematical objects, including infinite numbers, have a true and immutable and eternal nature, waiting to be discovered. Sense experience, on which empiricists base their entire world view, provides an anemic perspective on the truth. We need pure thought, mathematical insight. I missed it!

I did not miss, though, Mary Tiles’ observation that even something as proven as Cantor’s theorem had detractors. Yes, we can use the list method, one-one correspondence, to measure the sizes of sets, and infer that there are different sizes of infinity. But that other perspective, the one on which there are more natural numbers than even numbers, is also worth keeping in mind. We can even enshrine it in a principle: a whole is greater than its proper part. A different kind of mathematics was waiting to be discovered.

At Swarthmore, there were various ways to complete one’s degree. Some students went honors, taking double-credit seminars for two years that culminated in written and oral exams at the end of their senior year. Others went course. Some students were writing theses. Many of my entering cohort were on what we called the five year plan, taking more time away from school. Everyone was busy in their own way. Many seniors stressed about their upcoming honors exams or their theses. I had my head down, trying to work. And I wasn’t writing a THESIS, just a little qualifying paper in addition to my heavy course load. I read and wrote almost nothing on Cantor all year, aside from the work I did for Tiles’ seminar in the fall. But I did pretty well otherwise that year. While I had no plans for after May, it looked like I was actually going to graduate.

In April, Oberdeik, who had returned to campus and to chair, called me into his office. “You’ve been miscreant,” he told me.

 “Huh?” I didn’t know what ‘miscreant’ meant.

 “You neglected to register for the thesis.”

  “I’m not writing a thesis. Hugh told me that I’m just writing a little qualifying paper.”

  “Yes. We call that a thesis.”

Fuck.

 “Don’t worry about it,” Hans assured me, blithely unaware of the measure of my miscreance. “I cleared it up with the Registrar. Now, let’s schedule your defense.”

I had two weeks to do a year’s worth of work. Fortunately, I had been thinking about infinity since at least that day in the basement with my dad. Fortunately, there were structural pressures against failing me and preventing me from graduating. Fortunately, I had learned enough philosophy and writing and mathematics to stumble my way through. Fortunately, any sort of mathematical ability, even the little that I had mustered, was enough to look, well, at least not pathetic to philosophers. Unfortunately, I could read body language, for instance the look on Frede’s face during that oral exam. It was the same look of disappointment that Bannister had given me sophomore year. From one perspective, I was going to get a college degree. From another, I had a lot of growing up to do. And I was exhausted just trying.

On graduation day, I introduced Ann to my folks. She told them, “It may take him a while to find his way, but trust him. He has the tools. He’ll find it.” Those words were for my parents, but they were important to me. It took me a long time to start to believe them, and to believe in myself, to appreciate that Ann’s perspective on me was worth heeding. I found my way through teaching, high school mathematics at first, then various subjects and diverse kinds of students, and, eventually, college philosophy.

I’ve been teaching, now, for over 35 years. I’ve seen lots of other curious grotesques and I try to help them round out. In my infinity course, we read Zeno and Cantor and Hofstadter and Sartre. I teach formal logic and help students to collaborate so that they can hear new and various perspectives. I make my students suffer through Descartes’s work, and Plato’s, which they almost all despise, so that I can challenge their easy empiricism with the beauty of pure thought, of mathematical insight.

Earlier this term, a student in my Wittgenstein seminar stayed after class to tell me about a book he found in the library while searching for an essay topic. Had I heard of it? It’s called Gödel, Escher, Bach. He thought I might like it.

Russell Marcus is a philosopher specializing in philosophy of mathematics and the pedagogy of philosophy. He is Chair of Philosophy at Hamilton College and president of the American Association of Philosophy Teachers. At Hamilton College, he teaches a course on Infinity and Philosophy of Education, amongst others.

April 5, 2024