The Absurdity of Infinity: Astrophysicist Janna Levin Explains Whether the Universe Is Infinite or Finite in Letters to Her Mother

“The simpler the insight, the more profound the conclusion.”

In 1998, while on the cusp of becoming one of the most significant theoretical cosmologists of our time, mathematician-turned-astrophysicist Janna Levin left her post at Berkeley and moved across the Atlantic for a prestigious position at Cambridge University. During the year and a half there, she had the time and space to contemplate the question that would eventually become the epicenter of her career — whether the universe is infinite or finite. What began as a series of letters to her mother, Sandy, eventually became an unusual diary of Levin’s “social exile as a roaming scientist,” and was finally published as How the Universe Got Its Spots: Diary of a Finite Time in a Finite Space (public library) — a most unusual and absorbing account of the paradoxes of finitude.

“I’m writing to you because I know you’re curious but afraid to ask,” Levin offers in the opening letter — a “you” that instantly becomes as much her mother as the person Virginia Woolf memorably termed “the common reader.” From there, she springboards into remarkably intelligent yet inviting explorations of some of the biggest questions that the universe poses — questions most of us contemplate, sometimes consciously but mostly not, just by virtue of being sentient participants in the chaos and enchantment of existence.

A 1617 depiction of the notion of non-space, long before the concept of vacuum existed, found in Michael Benson's book 'Cosmigraphics'—a visual history of understanding the universe. Click image for more.

In an entry from September 3, 1998, Levin fleshes out her ideas on infinity and writes with exquisite Saganesque sensitivity to the poetics of science:

For a long time I believed the universe was infinite. Which is to say, I just never questioned this assumption that the universe was infinite. But if I had given the question more attention, maybe I would have realized sooner. The universe is the three-dimensional space we live in and the time we watch pass on our clocks. It is our north and south, our east and west, our up and down. Our past and future. As far as the eye can see there appears to be no bound to our three spatial dimensions and we have no expectation for an end to time. The universe is inhabited by giant clusters of galaxies, each galaxy a conglomerate of a billion or a trillion stars. The Milky Way, our galaxy, has an unfathomably dense core of millions of stars with beautiful arms, a skeleton of stars, spiraling out from this core. The earth lives out in the sparsely populated arms orbiting the sun, an ordinary star, with our planetary companions. Our humble solar system. Here we are. A small planet, an ordinary star, a huge cosmos. But we’re alive and we’re sentient. Pooling our efforts and passing our secrets from generation to generation, we’ve lifted ourselves off this blue and green water-soaked rock to throw our vision far beyond the limitations of our eyes.

The universe is full of galaxies and their stars. Probably, hopefully, there is other life out there and background light and maybe some ripples in space. There are bright objects and dark objects. Things we can see and things we can’t. Things we know about and things we don’t. All of it. This glut of ingredients could carry on in every direction forever. Never ending. Just when you think you’ve seen the last of them, there’s another galaxy and beyond that one another infinite number of galaxies.

Illustration from Thomas Wright’s visionary 1750 treatise 'An Original Theory,' found in Michael Benson's book 'Cosmigraphics'—a visual history of understanding the universe. Click image for more.

But having painted this bewitching backdrop for our intuitive beliefs, Levin sublimates the poet to the scientist, pointing out that however alluring these intuitions may feel, they are nonetheless ungrounded in empirical fact:

No infinity has ever been observed in nature. Nor is infinity tolerated in a scientific theory — except we keep assuming the universe itself is infinite.

It wouldn’t be so bad if Einstein hadn’t taught us better. And here the ideas collide so I’ll just pour them out unfiltered. Space is not just an abstract notion but a mutable, evolving field. It can begin and end, be born and die. Space is curved, it is a geometry, and our experience of gravity, the pull of the earth and our orbit around the sun, is just a free fall along the curves in space. From this huge insight people realized the universe must be expanding. The space between the galaxies is actually stretching even if the galaxies themselves were otherwise to stay put. The universe is growing, aging. And if it’s expanding today, it must have been smaller once, in the sense that everything was once closer together, so close that everything was on top of each other, essentially in the same place, and before that it must not have been at all. The universe had a beginning. There was once nothing and now there is something. What sways me even more, if an ultimate theory of everything is found, a theory beyond Einstein’s, then gravity and matter and energy are all ultimately different expressions of the same thing. We’re all intrinsically of the same substance. The fabric of the universe is just a coherent weave from the same threads that make our bodies. How much more absurd it becomes to believe that the universe, space and time could possibly be infinite when all of us are finite.

A decade and a half later, Alan Lightman would come to write with a similar scientific poeticism about why we long for permanence in a universe defined by constant change. But however poetic the premise, Levin brings a mathematician’s precision to her “reasons for believing the universe is finite, unpopular as they are in some scientific crowds.” In another entry twelve days later, she writes:

Infinity is a demented concept…

Infinity is a limit and is not a proper number. No matter how big a number you think of, I can add 1 to it and make it that much bigger. The number of numbers is infinite. I could never recite the infinite numbers, since I only have a finite lifetime. But I can imagine it as a hypothetical possibility, as the inevitable limit of a never-ending sequence. The limit goes the other way, too, since I can consider the infinitely small, the infinitesimal. No matter how small you try to divide the number 1, I can divide it smaller still. While I could again imagine doing this forever, I can never do this in practice. But I can understand infinity abstractly and so accept it for what it is.

Pointing out that all titans of science — including Galileo, Aristotle, and Cantor — were besotted with the notion of infinity at some point, “each visiting the idea for a time and then abandoning the pursuit,” Levin notes that we can neither accept nor dismiss infinity on the basis of popular opinion alone. In early October, she writes:

Where in the hierarchy of infinity would an infinite universe lie? An infinite universe can host an infinite amount of stuff and an infinite number of events. An infinite number of planets. An infinite number of people on those planets. Surely there must be another planet so very nearly like the earth as to be indistinguishable, in fact an infinite number of them, each with a variety of inhabitants, an infinite number of which must be infinitely close to this set of inhabitants. Another you, another me. Or there’d be another you out there with a slightly different life and a different set of siblings, parents, offspring. This is hard to believe. Is it arrogance or logic that makes me believe this is wrong? There’s just one me, one you. The universe cannot be infinite.

[…]

I welcome the infinite in mathematics, where … it is not absurd nor demented. But I’d be pretty shaken to find the infinite in nature. I don’t feel robbed living my days in the physical with its tender admission of the finite. I still get to live with the infinite possibilities of mathematics, if only in my head.

Illustration by Lisbeth Zwerger for 'Alice in Wonderland.' Click image for more.

Understanding the infinite — both as a mathematical possibility and an impossibility of the physical universe — might be more a matter of coming to terms with infinite simplicity than with infinite complexity. In an entry from early February of 1999, Levin echoes Margaret Mead’s famous proclamation about clarity and writes:

I try to find a simple expression for my ideas. I figure if there is none, the ideas must be wrong. When I first started to work on topology I wondered about complex properties of spaces and didn’t take my own suggestions seriously until I realized the simple way to ask the question: is the universe infinite? Einstein’s simplest insights were profound. The simpler the insight, the more profound the conclusion.

In the remainder of How the Universe Got Its Spots, which is unbearably beautiful in both intellectual elegance and stylistic splendor, Levin goes on to explore questions of quantum relativity and freewill, death and black holes, spacetime and Wonderland, and more. Complement it with Levin on science, free will, and the human spirit, then revisit Alan Lightman on how dark energy explains why we exist and treat yourself to this poetic primer on the universe written in the 1,000 most common words in the English language.

Donating = Loving

Bringing you (ad-free) Brain Pickings takes hundreds of hours each month. If you find any joy and stimulation here, please consider becoming a Supporting Member with a recurring monthly donation of your choosing, between a cup of tea and a good dinner.

♥ $7 / month♥ $3 / month♥ $10 / month♥ $25 / month

You can also become a one-time patron with a single donation in any amount.

Brain Pickings has a free weekly newsletter. It comes out on Sundays and offers the week’s best articles. Here’s what to expect. Like? Sign up.

Einstein’s God: Krista Tippett and Theoretical Cosmologist Janna Levin on Free Will, Science, and the Human Spirit

“How we ask our questions affects the answers we arrive at… Science and religion… ask different kinds of questions altogether, probing and illuminating in ways neither could alone.”

Seven decades after a little girl asked Einstein whether scientists pray, Peabody Award-winning journalist Krista Tippett began interviewing some of the world’s most remarkable scientists, philosophers, and theologians about the relationship between science and spirituality in her superb public radio program On Being — the same trove of wisdom that gave us Sherwin Nuland on what everybody needs and Joanna Macy on how Rilke can help us live more fully. Tippett, who was awarded the National Humanities Medal for her ennobling work, collected the best of these dialogues in Einstein’s God: Conversations About Science and the Human Spirit (public library) — an immeasurably rewarding compendium featuring such contemporary luminaries as Parker Palmer, Freeman Dyson, Andrew Solomon, and Sherwin Nuland.

Lamenting that we have “lost a robust vocabulary for spiritual ethics and theological thinking” in the “polite, erudite, public-radio-loving circles” of public life, Tippett writes in the introduction:

The science-religion “debate” is unwinnable, and it has led us astray. To insist that science and religion speak the same language, or draw the same conclusions, is to miss the point of both of these pursuits of cohesive knowledge and underlying truth. To create a competition between them, in terms of relevance or rightness, is self-defeating. Both science and religion are set to animate the twenty-first century with new vigor. This will happen whether their practitioners are in dialogue or not. But the dialogue that is possible — and that has developed organically, below the journalistic and political radar — is mutually illuminating and lush with promise.

Illustration from Thomas Wright’s visionary 1750 treatise 'An Original Theory,' found in Michael Benson's book 'Cosmigraphics'—a visual history of understanding the universe. Click image for more.

Tippett invokes her grandfather, a “preacher of hellfire and brimstone” with a “large, unexcavated mind that frightened him” and “sharp wit, a searching attentiveness, a mysterious ability to perform mathematical feats in his head”:

People like him became the object of erudite parody, straw men easily blown down by prophets of reason. His kind of religiosity was small-minded at best, delusional at worst, and, most damnably, the enemy of science.

The mundane truth is this: my grandfather did not know enough about science to be against it. I summon his memory by way of tracing, for myself, why I’ve found my conversations with scientists to be so profoundly sustaining. It is not just that they are intellectually and spiritually evocative beyond compare. Cumulatively they dispel the myth of the clash of civilizations between science and religion, indeed between spirit and reason, that we’ve accepted as the backdrop for so many tensions of the modern West.

[…]

How we ask our questions affects the answers we arrive at. Light appears as a wave if you ask it “a wavelike question” and it appears as a particle if you ask it “a particle-like question.” This is a template for understanding how contradictory explanations of reality can simultaneously be true.

And it’s not so much true, as our cultural debates presume, that science and religion reach contradictory answers to the same particular questions of human life. Far more often, they simply ask different kinds of questions altogether, probing and illuminating in ways neither could alone.

Hardly anything illustrates this notion more crisply than a line from the bewitching novel A Madman Dreams of Turing Machines — “To see some truths you must stand outside and look in.” — by astrophysicist and theoretical cosmologist Janna Levin, one of Tippett’s interviewees, who studies the shape and finitude of the universe. In her conversation with Tippett, Levin reflects on the relationship between mathematics and truth, central to both her novel — which explores the parallels between the extraordinary minds of computing pioneer Alan Turing and mathematician Kurt Gödel — and her life:

I would absolutely say I am also besotted with mathematics. I don’t worry about what’s real and not real in the way that maybe Gödel did. I think what Turing did, which was so beautiful, was to have a very practical approach. He believed that life was, in a way, simple. You could relate to mathematics in a concrete and practical way. It wasn’t about surreal, abstract theories. And that’s why Turing is the one who invents the computer, because he thinks so practically. He can imagine a machine that adds and subtracts, a machine that performs the mathematical operations that the mind performs. The modern computers that we have now are these very practical machines that are built on those ideas. So I would say that like Turing, I am absolutely struck with the power of mathematics, and that’s why I’m a theoretical physicist… I love that we can all share the mathematical answers. It’s not about me trying to convince you of what I believe or of my perspective or of my assumptions. We can all agree that one plus one is two, and we can all make calculations that come out to be the same, whether you’re from India or Pakistan or Oklahoma, we all have that in common. There’s something about that that’s deeply moving to me and that makes mathematics pure and special. And yet I’m able to have a more practical attitude about it, which is that, well, we can build machines this way. There is a physical reality that we can relate to using mathematics.

A 1573 painting by Portuguese artist, historian, and philosopher Francisco de Holanda, a student of Michelangelo's, from Michael Benson's book 'Cosmigraphics'—a visual history of understanding the universe. Click image for more.

When Tippett stretches this into the difficult question of whether “the fact that one plus one equals two [has] anything to do with God,” Levin — a self-described atheist — echoes Tolstoy’s quest for meaning and answers with remarkable poetry and poise:

If I were to ever lean towards spiritual thinking or religious thinking, it would be in that way. It would be, why is it that there is this abstract mathematics that guides the universe? The universe is remarkable because we can understand it. That’s what’s remarkable. All the other things are remarkable, too. It’s really, really astounding that these little creatures on this little planet that seem totally insignificant in the middle of nowhere can look back over the fourteen-billion-year history of the universe and understand so much and in such a short time.

So that is where I would get a sense, again, of meaning and of purpose and of beauty and of being integrated with the universe so that it doesn’t feel hopeless and meaningless. Now, I don’t personally invoke a God to do that, but I can’t say that mathematics would disprove the existence of God either. It’s just one of those things where over and over again, you come to that point where some people will make that leap and say, “I believe that God initiated this and then stepped away, and the rest was this beautiful mathematical unfolding.” And others will say, “Well, as far back as it goes, there seem to be these mathematical structures. And I don’t feel the need to conjure up any other entity.” And I fall into that camp, and without feeling despair or dissatisfaction.

The emboldening poetics of Levin’s orientation to the universe and its meaning — at the heart of which is the same inquiry Alan Watts tussled with in probing what reality is — comes alive in this passage from her novel:

In the park, over the low wall, there are two girls playing in the grass. Giants looming over their toys, monstrously out of proportion. They’re holding hands and spinning, leaning farther and farther back until their fingers rope together, chubby flesh and bone enmeshed. What do I see? Angular momentum around their center. A principle of physics in their motion. A girlish memory of grass-stained knees.

I keep walking and recede from the girls’ easy confidence in the world’s mechanisms. I believe they exist, even if my knowledge of them can only be imperfect, a crude sketch of their billions of vibrating atoms. I believe this to be true… I am on an orbit through the universe that crosses the paths of some girls, a teenager, a dog, an old woman…

I could have written this book entirely differently, but then again, maybe this book is the only way it could be, and these are the only choices I could have made. This is me, an unreal composite, maybe part liar, maybe not free.

Another 16th-century painting by Francisco de Holanda from 'Cosmigraphics.'Click image for more.

Therein lies the obvious question — a question raised memorably and somewhat controversially by C.S. Lewis — of free will in a universe of fixed laws. Levin tells Tippett:

I think it’s a difficult question to understand what it means to have free will if we are completely determined by the laws of physics, and even if we’re not. Because there are things—for instance, in quantum mechanics, which is the theory of physics on the highest energy scales—which imply that there is some kind of quantum randomness so that we’re not completely determined. But randomness doesn’t really help me either.

[…]

There is no clear way of making sense of an idea of free will in a pinball game of strict determinism or in a game with elements of random chance thrown in. It doesn’t mean that there isn’t a free will. I’ve often said maybe someday we’ll just discover something. I mean, quantum mechanics was a surprise. General relativity was a surprise. The idea of curved space-time. All of these great discoveries were great surprises, and we shouldn’t decide ahead of time what is or isn’t true. So it might be that this convincing feeling I have, that I am executing free will, is actually because I’m observing something that is there. I just can’t understand how it’s there. Or it’s a total illusion. It’s a very, very convincing illusion, but it’s an illusion all the same.

In a sentiment that calls to mind Nobel-winning psychologist Daniel Kahneman’s revelatory work on intuition, exposing the lack of correlation between our confidence in our beliefs and the validity of the evidence behind them — something that often manifests as “the backfire effect” — Levin considers the nature of these convincing illusions to which human nature so easily succumbs:

Our convincing feeling is that time is absolute. Our convincing feeling is that there should be no limit to how fast you can travel. Our convincing feelings are based on our experiences because of the size that we are, literally, the speed at which we move, the fact that we evolved on a planet under a particular star. So our eyes, for instance, are at peak in their perception of yellow, which is the wave band the sun peaks at. It’s not an accident that our perceptions and our physical environment are connected. We’re limited, also, by that. That makes our intuitions excellent for ordinary things, for ordinary life. That’s how our brains evolved and our perceptions evolved, to respond to things like the Sun and the Earth and these scales. And if we were quantum particles, we would think quantum mechanics were totally intuitive. Things fluctuating in and out of existence, or not being certain of whether they’re particles or waves — these kinds of strange things that come out of quantum theory — would seem absolutely natural…

Our intuitions are based on our minds, our minds are based on our neural structures, our neural structures evolved on a planet, under a sun, with very specific conditions. We reflect the physical world that we evolved from. It’s not a miracle.

And yet, crucially, the lack of evidence for free will is by no means a license to abdicate personal responsibility in how we move through the world:

If I conclude that there is no free will, it doesn’t mean that I should go run amok in the streets. I’m no more free to make that choice than I am to make any other choice. There’s a practical notion of responsibility or civic free will that we uphold when we prosecute somebody, when we hold juries or when we pursue justice that I completely think is a practical notion that we should continue to pursue. It’s not like I can choose to be irresponsible or responsible because I’m confused about free will.

Six decades earlier, and long before the dawn of modern astrophysics, Anaïs Nin made a humanistic case for the same.

Einstein’s God is a spectacular read in its entirety, as is Levin’s novel. For more perspectives on the relationship between science and spirituality, step into the cultural time machine with Carl Sagan on science and religion, Flannery O’Connor on dogma, belief, and the difference between religion and faith, Alan Lightman on science and spirituality, Ada Lovelace on the interconnectedness of everything, Jane Goodall on science and spirit, and Sam Harris on spirituality without religion.

Donating = Loving

Bringing you (ad-free) Brain Pickings takes hundreds of hours each month. If you find any joy and stimulation here, please consider becoming a Supporting Member with a recurring monthly donation of your choosing, between a cup of tea and a good dinner.

♥ $7 / month♥ $3 / month♥ $10 / month♥ $25 / month

You can also become a one-time patron with a single donation in any amount.

Brain Pickings has a free weekly newsletter. It comes out on Sundays and offers the week’s best articles. Here’s what to expect. Like? Sign up.