The Beauty of Very Large Numbers: Where Religion and Mathematics Meet

The Beauty of Very Large Numbers

Where Religion and Mathematics Meet

Religion and mathematics meet in many places. They meet in the search for order, in the cultivation of wonder, in the appreciation of beauty, and in the recognition that reality exceeds the limits of ordinary human understanding. Certainly they meet in the stars, and in our sense that we are small but included in a greater, ungraspable whole revealed in the night sky. This is why religion begins, in so many settings, with navigation of the seas and the darkness of the night sky in deserts. Perhaps also on mountains and by lakes. It is in such settings that we sense the "something more."

One of their most intriguing meeting places, however, is not so much environmental as it is intellectul, It is related to night skies but not reducible to them, It is the sheer beauty of very large numbers. Whether expressed through the cosmological imagination of the world's religions or through the symbolic language of mathematics, such numbers stretch the mind beyond the familiar. They become invitations to spiritual awakening.

We are awakened to a contrast between the possible and the actual. The numbers are quantifiable, and yet yet existentially uncountable, even if we or our computers have all the time in the world. They are sublime in their unique combination of quantifiability and uncountability. It is the contrast that makes them so beautiful, And so spiritual.

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There are many kinds of healthy religious feeling: attentive presence, a sense of beauty, compassion, connection, devotion, enthusiasm, faith, forgiveness, gratitude, listening, love, playfulness, silence, wonder, and a sense of mystery, among others.

Very large numbers, as contemplated in the world's religions and by mathematicians, have a remarkable capacity to evoke several of these at once. They can awaken a sense of beauty in their elegance, playfulness in their imaginative construction, wonder before their almost inconceivable magnitude, mystery before realities that exceed the limits of ordinary human understanding, and even love—a love of truth, of discovery, of the universe, or of the immeasurable itself. In this respect, they function not merely as mathematical objects but as invitations to contemplation.

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By "very large numbers," I do not mean infinities, which possess their own kind of beauty. I mean extraordinarily large but finite numbers, such as those explored by Richard Elwes in Huge Numbers: A Story of Counting Ambitiously, from 4½ to Fish 7 (Basic Books, 2026). You will find an interview with him at the bottom of this page.

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Some of these numbers can be described with perfect precision by mathematicians but can never be practically counted. They can be symbolized but never fully written out; named but never enumerated digit by digit. In this sense, they are immeasurable in practice, even though they are perfectly finite and therefore measurable in principle. They occupy a fascinating middle ground between the ordinary numbers of everyday life and the infinities of higher mathematics, inviting us to contemplate magnitudes that can be defined exactly but never fully realized in experience.

Part of what is attractive about them is that we can feel their presence, even as we cannot count them. In this sense, they are a little like God as felt by many Jews, Muslims, and Christians, and perhaps also, in a different way, like the Dao or Brahman or

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What are these numbers ontologically? In the spirit of the philosophy of Alfred North Whitehead, I suggest that they are what Whitehead calls eternal objects or timeless possibilities which may or may not be actual in some universe or dimension of the space-time continuum, but which are real in the sense that we can feel their presence intellectually and experientially, even as they exceed ordinary human comprehension. As Whitehead puts it, we can "entertain" them through conceptual prehensions. We feel them intellectually and experience wonder along the way.

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In the historical religions they are often expressed linguistically, in terms of stories.

Consider the Buddhist idea that the duration of a kalpa, a cosmic epoch, is like the time it would take to wear away a mountain a mile high, a mile wide, and a mile long by brushing it gently with a silk cloth once every hundred years. Or consider the Jain image of a palya: an enormous pit, roughly eight miles deep, eight miles wide, and eight miles long, filled with the fine hairs of a seven-day-old lamb. Remove one hair every hundred years, and the time required to empty the pit is a single palya. Or consider the Hindu vision of a single day in the life of Brahmā lasting 4.32 billion human years, with Brahmā himself living for more than 311 trillion years before the cycle begins anew. Each image invites us to imagine spans of time so vast that ordinary intuition begins to fail.

But they can also be expressed in terms of mathematical symbols.

Consider also the mathematical idea of a googolplex, the number 1 followed by a googol zeros. It is so unimaginably large that there are not enough atoms in the observable universe to write out all of its digits. Or consider Graham's number, one of the largest numbers ever to appear in a serious mathematical proof. It is so immense that mathematicians cannot even describe it using ordinary exponents but must employ a special recursive notation that generates numbers far beyond everyday comprehension. Or consider TREE(3), a number arising from a problem in graph theory that is so unimaginably vast that even Graham's number is insignificant by comparison. These, too, are finite mathematical realities that stretch the imagination almost to its breaking point.

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Whether presented verbally or in terms of mathematical symbols, these numbers are abstract, but in their very abstractness they bristle with a certain kind of aliveness. They have their own identities - awakening the imagination, enlarging our sense of reality, and inviting forms of contemplation that transcend calculation. They are not lifeless symbols detached from experience but windows of possibility. Almost, dare I say, like angels.

It is not clear to us whether we discover them, as if they already exist; or if we create them, because they emerge out of our own imaginations; or both. They may emerge in both ways, as discovered and created. There is no need to decide this. What is important is to know that they can be associated with the very big, the very small, and the immeasurably beautiful.

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We usually associate these numbers with realities that are unimaginably large: a universe without beginning or end containing trillions upon trillions of galaxies, each with trillions upon trillions of stars, each with trillions upon trillions of planets, and more. We might well say that ordinary arithmetic cannot adequately express the magnitude of it all.

But they can also be associated with realities that are unimaginably small. Imagine the trillions upon trillions of atoms in a grain of sand, each containing a subatomic world of extraordinary complexity. Here, too, numbers stretch the imagination beyond the limits of ordinary perception, inviting us to consider depths hidden within the seemingly ordinary.

They can also serve as metaphors for qualities rather than quantities. Imagine a tenderness so deep that it is like a kalpa —not because love can be measured, but because its depth exceeds the ordinary scales of human imagination. Or imagine a compassion so vast that no finite number could contain it, or a creativity so inexhaustible that every act of creation opens the possibility of countless more. The point is not arithmetic but feeling. If immense numbers can function as lures for contemplating cosmic time, perhaps they can also become lures for contemplating immeasurable compassion, inexhaustible forgiveness, boundless creativity, or the infinite depths of divine love.

In this way really big numbers have a kind of grace. They evoke a sense of beauty, wonder, playfulness, and love.

In this way, really big numbers have a kind of grace. They evoke beauty, wonder, playfulness, and even love. They remind us that the human mind is capable not only of solving problems but also of delighting in mysteries that forever exceed its grasp.

It is tempting to dismiss their relevance in light of the pressing needs of the world. What good is a palya, a googolplex, or Graham's number when people are hungry, lonely, frightened, or at war? Yet perhaps, precisely because those needs are so pressing, we also need marvelous abstractions that, for a moment, lift us beyond the urgencies of everyday life. They are like the sabbaths of Judaism. They refresh the imagination, enlarge the heart, and renew our capacity for hope. Then we can return to the world, like a Buddhist bodhisattva, with bliss-bestowing hands—bringing hope that there is so much more to what is possible, than what is actual,

Jay McDaniel

Jainism — The palya and sāgaropama use vivid images—a pit filled with lamb's wool or an ocean—to evoke durations so vast they defy ordinary imagination.
Buddhism — A kalpa is illustrated by images such as a mountain worn away by a silk cloth brushed against it once every hundred years, or by comparing eons to the grains of sand in the Ganges.
Hinduism — A single day of Brahmā lasts 4.32 billion years, and Brahmā's lifetime extends for over 311 trillion years, placing human history within immense cosmic cycles.
Mahāyāna Buddhism — Sutras frequently speak of countless Buddha-fields, innumerable bodhisattvas, and "incalculable" eons to evoke the boundlessness of awakening and compassion.
Pure Land Buddhism — Amitābha Buddha is described as possessing immeasurable light and immeasurable life, emphasizing the inexhaustibility of compassion rather than measurable quantity.
Daoism — Daoist cosmology imagines vast cosmic cycles and unimaginably long-lived immortals, inviting contemplation of nature's rhythms beyond ordinary human lifespans.
Yoruba Religion — While less focused on explicit large numbers, Yoruba cosmology speaks of innumerable ancestors, spirits, and generations participating in an ongoing sacred community extending far beyond living memory.
Ancient Egyptian Religion — Egyptian religion imagines the eternal cyclical renewal of the cosmos through the daily journey of the sun and the recurring rhythms of cosmic order (ma'at), inviting people to see human life within an enduring cosmic pattern.
Dogon Tradition — Dogon cosmology describes layered symbolic orders extending far beyond ordinary human perception, using large-scale cosmological imagination to cultivate awe before creation.
Islamic Mysticism — The Qur'an and later Sufi writings employ immense numbers symbolically—for example, "a day with your Lord is like a thousand years," and elsewhere "fifty thousand years"—to suggest that divine time transcends ordinary human reckoning.
Mayan Traditions — The Maya developed calendrical systems capable of measuring vast cycles extending millions, and in some inscriptions even billions, of years into the past and future, locating human life within immense cosmic rhythms.
Modern Mathematics — Contemporary mathematics offers its own lures for feeling through numbers such as the googol, googolplex, Graham's number, and the Busy Beaver function, each stretching the imagination beyond ordinary numerical comprehension.
Artificial Intelligence and Higher-Dimensional Mathematics — AI is beginning to help mathematicians visualize and explore mathematical structures in hundreds or even thousands of dimensions, revealing patterns that exceed unaided human imagination. Like the great cosmological images of the world's wisdom traditions, these AI-assisted visualizations invite wonder before realities that can be investigated mathematically but never fully pictured in the mind.

Jainism and the Pit

Imagine a pit roughly eight miles deep, eight miles wide, and eight miles long. Fill it with the fine hairs of a seven-day-old lamb. Then remove one hair every hundred years. How long would it take to empty the pit? In Jain cosmology, the answer is one palya. Translated very roughly into modern arithmetic, this would be somewhere between 10²⁴ and 10²⁵ years—that is, between one septillion and ten septillion years, many trillions of times older than the approximately 13.8-billion-year age of the observable universe.

The context is Jain cosmology, especially its effort to convey the immense age and scale of the universe and the extraordinarily long processes of karmic and spiritual development.

More specifically, the palya appears in Jain scriptures as one of several enormous units of time used to describe:

the age of the cosmos (which has no beginning or end),
the duration of cosmic cycles,
the lifespans of heavenly and infernal beings,
the length of time souls may remain in various states of existence, and
the immense span over which karma accumulates and is eventually exhausted.

Unlike the Abrahamic traditions, Jainism has no doctrine of creation. The universe is eternal: it was never created and will never cease to exist. Time itself moves through endless ascending and descending cycles (utsarpiṇī and avasarpiṇī), each lasting innumerable palyas and even larger units called sāgaropamas.

The image of the pit is therefore not an isolated curiosity. It serves a theological and spiritual purpose. It reminds practitioners that liberation (moksha) is not simply a matter of a single lifetime but unfolds against a cosmic backdrop of almost unimaginable duration. At the same time, it places ordinary human concerns in perspective, cultivating patience, humility, and perseverance.

A googol is the number 1 followed by 100 zeros (10¹⁰⁰). It was named in 1938 by nine-year-old Milton Sirotta, the nephew of mathematician Edward Kasner. A googol is vastly larger than the estimated number of atoms in the observable universe (about 10⁸⁰), yet it is still a perfectly ordinary, finite number. It reminds us that mathematics can describe quantities far beyond everyday experience.
Googolplex

A googolplex is 1 followed by a googol zeros (10^(10¹⁰⁰)). It is so large that there are far too few atoms in the observable universe to write all of its digits. Although finite, it lies beyond any practical way of being represented physically, stretching the imagination far beyond ordinary numbers.

Graham's number is one of the largest numbers ever used in a serious mathematical proof. It is incomparably larger than a googolplex and cannot be written in ordinary decimal notation. Instead, mathematicians define it using Knuth's up-arrow notation, a shorthand that generates numbers of astonishing size. It shows how mathematical reasoning can lead naturally to quantities that exceed human intuition.
The Busy Beaver Function

The Busy Beaver function is not one number but a rule that produces numbers of extraordinary size. It asks for the largest number of steps a very simple computer program can take before stopping. As the programs become only slightly more complicated, the answers grow faster than almost any numbers mathematicians have ever imagined. It reveals that mathematics contains well-defined quantities that no general algorithm can calculate.

Artificial intelligence is helping mathematicians explore mathematical structures in hundreds or even thousands of dimensions—far beyond what people can picture. AI can discover patterns, suggest conjectures, and create visualizations that reveal relationships hidden from ordinary intuition. Rather than replacing mathematics, AI expands our ability to explore mathematical worlds that lie beyond the reach of the unaided human mind.

Huge Numbers: A Story of Counting Ambitiously, from 4 1/2 to Fish 7

"What if, every time you wanted to write down 1,000,000, you had to draw a picture of a god? And what if that number were the biggest you had a symbol for? If you were doing math in ancient Egypt, those were the rules: anything bigger broke math.

As mathematician Richard Elwes shows in Huge Numbers: A Story of Counting Ambitiously, from 4 1/2 to Fish 7 (Basic Books, 2026) this is the strange story of math. Even today, writing down some numbers is beyond us: try it with all the zeroes in a googolplex, or an outrageous alien number like TREE(3). Safer not to try: even harnessing every particle in the universe, you wouldn’t come close. But this book is no mere bestiary of numerical monsters. It shows how, by hunting down and studying ever-bigger numbers, arithmetic has reshaped human thought and made our modern era of science and computation possible.

Where many math books celebrate abstract algebra or ineffable infinities, Huge Numbers is both more practical and far weirder. It reveals a world where most numbers remain out of reach until we discover how to chase them down and tame them, and so remake our world again."