To infinity and beyond? ∞ for effort.
A word of warning for you, dear reader: if you start reading this week’s Column, you will never be able to reach the end of it. Try as you might, there will always be more paltry words to wade through. Indeed, if you find yourself reading these words and have got this far, it’s already too late. I think that was dear old Zeno’s point with his famous tales about arrows never reaching their targets and Achilles never catching tortoises, and the like.
Your Director’s thoughts this week have been beset with images of infinity, for reasons which are not altogether clear to me but might, as these things often do, have their roots in a certain kind of Monday morning malaise. You can picture the scene, I’m sure, your aged Director staring at his email inbox on Monday morning; the usual three-digit number that always feels unmanageable but somehow gets managed. I recalled an old Head of Department opining that Hell for him would be an infinite pile of books to be marked. One unmarked book added for every book marked and so on, forever. Staring at my emails, I had the thought: what if my inbox were infinite? What if, every time I cleared a message, another appeared? A sobering prospect, though not that far from reality on a Monday morning.
But of course, infinity doesn’t really work that way. It cannot be approached incrementally. Galileo back in the day recognised that the infinite cannot be reached by simply adding more and more bits and bobs together. Think of a line extended for billions and billions of light-years. Such a line is no doubt ‘long’ in one sense but it is no closer to infinity than it was at the millimetre mark. Galileo wrote the infinite cannot be reached by a successive progression of steps but only at a single stroke. Zeno, with his arrow that never reaches its target and Achilles never outrunning the tortoise, anticipated the strangeness of such leaps. Hilbert, imagining his infinite hotel, also reminded us that infinity is not just really really big: it behaves according to rules utterly unlike our everyday experience. Even a fully occupied hotel with an infinite number of rooms can still accommodate more guests, even an infinite number of them. Briefly his lovely thought experiment goes like this: an infinite hotel with every room full can always accommodate a new guest simply by having every current guest move to the next room, demonstrating that infinity plus one is still infinity. As well of course also demonstrating that staying for more than one night in an infinite hotel would be annoying. I was reminded of Hilbert’s hotel at the start of term as I navigated the once familiar corridors that now housed new Departmental guests. But never fear, your Director has grown adept at the odd counter intuitive leap or three.
There’s also a kind of counter-intuitive leap going on here with infinity which invites us to shift from a Newtonian, left-hemisphere universe which is linear, sequential, computational, to a right-hemisphere perspective which is holistic, intuitive, and capable of grasping the seamless flow of existence. As philosopher Henri Bergson observed, the logician, by following a straight line of reasoning, becomes exterior to himself. He only returns to true understanding when he gets back to intuition, tracing the curve of thought rather than a rigid, straight path. To truly know infinity, we must trace the curve. Straight lines just won’t do. It is no coincidence that there are no straight lines in ‘Nature’.
And now for the literature section…
Writers and poets have long sought to navigate this curved understanding of infinity, each offering a different window on it. Shakespeare, for example, in Sonnet 18, offers a glimpse of a static, frozen infinity: So long as men can breathe or eyes can see, / So long lives this, and this gives life to thee. Here, poetry seeks the actualised infinite, understood as a finished perfection that exists outside of time, immortalised in art. In Hamlet, the prince reflects What a piece of work is man! How noble in reason…how infinite in faculties! capturing humanity’s boundless potential. And in King Lear, human folly, grief, and cruelty spiral outward, unbounded, echoing the terrifying and tragic breadth of a limitless cosmos.
In contrast, the Romantics embraced a potential, processual infinity. Wordsworth, in Tintern Abbey, writes of a sense sublime / Of something far more deeply interfused, sensing in the landscape a continuity beyond individual life. Coleridge’s Ancient Mariner drifts across seas whose vastness is both literal and metaphysical, famously noting water, water, everywhere, / Nor any drop to drink. This vastness isn’t static; it is an ongoing, terrifying process. Shelley, in Mont Blanc, marvels at mountains that inspire both fear and exhilaration, prompting the mind to expand without ever reaching completion. Keats, in Endymion, muses that a thing of beauty is a joy forever, embracing the infinite potential in a single experience. Byron, too, when he wasn’t gallivanting around Europe, exults in infinite skies and oceans, recognising human finitude against nature’s boundless canvas.
Borges’ Library of Babel dramatises the tension between potential and actual infinity: a universe containing every possible combination of letters, where knowledge is simultaneously attainable and futile. Milton’s Paradise Lost stretches across heaven, hell, and chaos unformed, a cosmic process rather than a finished tableau. Lewis Carroll’s Alice tumbles endlessly through mirrored worlds, each reflection spawning yet another, an endless regress of curiosity and imagination.
Not all infinities, however, are creative or life-giving. Multiverse theories, positing an infinite number of universes to explain fine-tuning, risk a dead infinity: a cosmos in which everything always goes on infinitely often, as astrophysicist Marco Bersanelli warns, and where nothing novel ever emerges. Such ‘actualised’ infinities are static, sterile, and uncreative, lacking the dynamism necessary for novelty, engagement, or purpose. They are a library of books that have already been written, or a hotel with no new guests to welcome.
To my mind, a better way of understanding the notion of infinity is that it is potential, processual and emergent. Wagner, for example, did not find a pre-existing Ring Cycle hovering in the cosmos already created, determined by the idea that in an infinite universe everything already exists somewhere. He brought something genuinely new into being. In the Kabbalistic tale of creation, humanity’s task is to mend shattered vessels and restore them to something more beautiful than before; an act of ongoing creation. The imagination, properly understood, participates in reality’s ongoing creation rather than merely observing pre-formed structures.
A living cell is not a static entity; its membrane flows, matter enters and exits, and its metabolism transforms it in thousands of ways that scientists in their white coats are still trying to understand. Life persists precisely through this constant change. As the pre-Socratic philosopher Heraclitus reminds us: All things flow; yet by changing, a thing remains the same. Blake unsurprisingly put it beautifully: Hours of folly are measured by the clock; but of wisdom, no clock can measure time. Eternity, then, is adverbial, a manner of being, not a thing, it’s a process rather than a noun. We have discussed this before, as you know.
Infinity and time are continuous, relational, and flowing. Bergson emphasised that time cannot be dissected into static points; it is seamless, like a melody. To attempt otherwise is to enter a devitalised world. And nobody wants to live there…
Science, mathematics, and art converge on this theme. Fractals repeat endlessly at new scales; the cosmos may be infinite in extent, yet comprehensibility demands a leap of imagination; Hawking and others remind us that mystery and structure coexist. Process philosophers such as Whitehead and physicists such as Edward Nelson distinguish between sterile, completed infinity and the generative, ongoing infinity of becoming. In human experience, creativity, attention, and imagination participate in this emergent flow.
And so, despite my earlier warning, here we are at the end of infinity, or at least this week’s Column. Though, if Zeno is to be believed, you’re still not quite there. So, dear reader, as you wade, perhaps endlessly, through the river of these paltry words, consider the infinite in your own life: the lessons yet to be learned, the questions yet to be asked, the acts of kindness yet to be performed. We are finite beings in an infinite web of connection, thought, and potential. Our choices matter because we participate in the creative, processual infinity of existence; our joys and sorrows are threads in a river that is always moving, never the same, yet recognisably continuous.
The best kinds of infinity infinity are simply the endless potential contained in a small, ordinary moment, mirroring Blake’s famous words: A world in a grain of sand, / And a heaven in a wild flower, / Hold infinity in the palm of your hand / And eternity in an hour.
Until next time, Happy Reading/Being Infinite
Fun problems to solve #1
A librarian has books numbered 1, 2, 3, … forever. She removes every book with a prime number; then every book divisible by 2; then every book divisible by 3; and so on, performing the step for every natural number. Which books remain at the end?