We Live in a Mathematically
Enchanted Universe
| Mathematical Idea | Description |
|---|---|
| Golden Ratio (φ) | Approximately 1.618. A number associated with proportions found in art, architecture, plant growth patterns, and geometry. |
| Pi (π) | The ratio of a circle's circumference to its diameter (approximately 3.14159), foundational for geometry and physics. |
| Euler's Number (e) | Approximately 2.718. Central to calculus, exponential growth, probability, and complex analysis. |
| Imaginary Unit (i) | Defined as √−1. Essential to complex number theory and many areas of modern physics. |
| Fibonacci Sequence | A sequence in which each number is the sum of the previous two, appearing in many natural patterns. |
| Perfect Numbers | Numbers equal to the sum of their proper divisors, such as 6 and 28. |
| Prime Numbers | Natural numbers greater than 1 divisible only by themselves and 1. |
| Catalan's Conjecture | A famous problem in number theory concerning powers of integers; now proved as Mihăilescu's Theorem. |
| Transcendental Numbers | Numbers that are not roots of polynomial equations with rational coefficients, including π and e. |
| Magic Squares | Arrangements of numbers in square grids where rows, columns, and diagonals sum to the same value. |
A Mathematically Enchanted Universe
We live in a mathematically enchanted universe. Mathematics is often treated as a practical instrument for calculation and prediction, but it can also function as a vehicle for awakening awe. It does so for most if not all mathematicians, and it can also do so for novices who appreciate mathematics but are not necessarily "good at it." Mathematics is not about numbers alone; indeed "number theory" is but one of many forms of mathematics, Still, numbers have a special kind of majesty. Certain numbers stretch imagination and enlarge our sense of reality itself. They invite us into dimensions of mystery that are real but exceed ordinary experience.
Indeed, our universe is filled with what the physicist and mathematician Antonio Padilla calls "fantastic numbers." His book Fantastic Numbers and Where to Find Them: A Journey to the Edge of Physics is presented below. Through astonishing scales of magnitude, infinities, and strange mathematical realities emerging from modern physics, Padilla opens doors into a universe that is at once intellectually challenging and deeply wondrous.
A simple awareness of these numbers, even on the part of non-mathematicians who may not understand their mathematical significance, reveal the mathematical enchantment of the universe. Consider:
From a Whiteheadian perspective, these "fantastic numbers" and related concepts are not simply inventions of the human mind, nor are they material objects existing independently in space and time. Rather, they are symbolic representations of potential patterns which, if actualized, can give shape and form to the universe, Some of these "numbers" are embodied in the patterns we disover in the actual universe, and some are "numbers" that could be embodied in our universe or other universes, by may or may not be. The key is that they are potential patterns and can be considered in abstraction from their embodiment in the universe, whether or not they are actualized.
More specifically these numbers and the ideas behind them can be understood in at least two ways.
On the one hand, they may be pure potentialities beyond time and space—Whitehead calls them eternal objects—that can be conceptually entertained but may or may not find realization in the actual universe. Examples might include perfectly symmetrical multidimensional geometries, mathematical structures involving higher dimensions, or exotic numerical relations that can be be imagined even if they are never embodied in our cosmos.
On the other hand, they may function as relevant and available fields of real potentiality—Whitehead calls them propositions—that are pertinent to the actual world and that serve as guiding principles available for particular forms of realization. Examples here might include mathematical possibilities embodied in gravitational relationships, quantum probabilities, or ratios and numerical patterns that shape the formation of galaxies, stars, and living systems. As eternal objects fantastic numbers symbolic representations of patterns of definiteness in themselves; as propositions they are possibilities relevant to actual entities for actual embodiment.
Neither eternal objects nor propositions possess agency or feeling in themselves, but through their actualization in feeling subjects they can become embodied in the evolving patterns of the world.
When actualized by entities within the universe—entities that do possess agency and feeling, namely actual entities—these potentials and possibilities become embodied as actual patterns of nature, providing entities with forms of definiteness and unity they otherwise would not have. In this sense they function as formal causes in the natural world, complemented by material causes (the energetic activity of the universe), efficient causes (the causal efficacy of the past actual world), and final causes (the subjective aims or purposes within entities themselves).
Different cosmic epochs might involve different relevant patterns and different mathematical possibilities becoming actualized. The eternal objects themselves remain neutral with respect to embodiment; they do not decide where or how they are realized. What becomes relevant depends on the situations at hand and, in a Whiteheadian framework, on the ordering function of the universe itself. Thus the mathematical architecture of one cosmic epoch might differ dramatically from another, with different forms, relations, and even different "laws" becoming significant for the creative advance into novelty.
These "fantastic numbers," insofar as they ingress into the world, help give reality its patterns. Pure mathematics is the activity of discovering and working with such patterns in abstraction from their embodiment. Applied mathematics is the activity of recognizing and utilizing those patterns that are, or can become, actualized in the world.
Indeed, our universe is filled with what the physicist and mathematician Antonio Padilla calls "fantastic numbers." His book Fantastic Numbers and Where to Find Them: A Journey to the Edge of Physics is presented below. Through astonishing scales of magnitude, infinities, and strange mathematical realities emerging from modern physics, Padilla opens doors into a universe that is at once intellectually challenging and deeply wondrous.
A simple awareness of these numbers, even on the part of non-mathematicians who may not understand their mathematical significance, reveal the mathematical enchantment of the universe. Consider:
- a googol (1010010^{100}10100): a 1 followed by 100 zeros
- a googolplex (101010010^{10^{100}}1010100): so vast that there is not enough room in the observable universe to write it out in ordinary form
- Graham's Number: one of the largest numbers ever used in a serious mathematical proof
- TREE(3): a number so immense that it dwarfs even Graham's Number
- the Planck length (approximately 1.6×10−351.6 \times 10^{-35}1.6×10−35 meters): a scale at which ordinary understandings of space and time may begin to break down
- the extraordinarily small value 10−12010^{-120}10−120: associated with the cosmological constant problem in physics
- infinity: not simply a large number but a concept that stretches ordinary thinking beyond limits
- zero and imaginary numbers: mathematical entities that challenge intuitive notions of quantity and reality
From a Whiteheadian perspective, these "fantastic numbers" and related concepts are not simply inventions of the human mind, nor are they material objects existing independently in space and time. Rather, they are symbolic representations of potential patterns which, if actualized, can give shape and form to the universe, Some of these "numbers" are embodied in the patterns we disover in the actual universe, and some are "numbers" that could be embodied in our universe or other universes, by may or may not be. The key is that they are potential patterns and can be considered in abstraction from their embodiment in the universe, whether or not they are actualized.
More specifically these numbers and the ideas behind them can be understood in at least two ways.
On the one hand, they may be pure potentialities beyond time and space—Whitehead calls them eternal objects—that can be conceptually entertained but may or may not find realization in the actual universe. Examples might include perfectly symmetrical multidimensional geometries, mathematical structures involving higher dimensions, or exotic numerical relations that can be be imagined even if they are never embodied in our cosmos.
On the other hand, they may function as relevant and available fields of real potentiality—Whitehead calls them propositions—that are pertinent to the actual world and that serve as guiding principles available for particular forms of realization. Examples here might include mathematical possibilities embodied in gravitational relationships, quantum probabilities, or ratios and numerical patterns that shape the formation of galaxies, stars, and living systems. As eternal objects fantastic numbers symbolic representations of patterns of definiteness in themselves; as propositions they are possibilities relevant to actual entities for actual embodiment.
Neither eternal objects nor propositions possess agency or feeling in themselves, but through their actualization in feeling subjects they can become embodied in the evolving patterns of the world.
When actualized by entities within the universe—entities that do possess agency and feeling, namely actual entities—these potentials and possibilities become embodied as actual patterns of nature, providing entities with forms of definiteness and unity they otherwise would not have. In this sense they function as formal causes in the natural world, complemented by material causes (the energetic activity of the universe), efficient causes (the causal efficacy of the past actual world), and final causes (the subjective aims or purposes within entities themselves).
Different cosmic epochs might involve different relevant patterns and different mathematical possibilities becoming actualized. The eternal objects themselves remain neutral with respect to embodiment; they do not decide where or how they are realized. What becomes relevant depends on the situations at hand and, in a Whiteheadian framework, on the ordering function of the universe itself. Thus the mathematical architecture of one cosmic epoch might differ dramatically from another, with different forms, relations, and even different "laws" becoming significant for the creative advance into novelty.
These "fantastic numbers," insofar as they ingress into the world, help give reality its patterns. Pure mathematics is the activity of discovering and working with such patterns in abstraction from their embodiment. Applied mathematics is the activity of recognizing and utilizing those patterns that are, or can become, actualized in the world.
Fantastic Numbers and
Where to Find Them
Antonio Padilla is a leading theoretical physicist and cosmologist at the University of Nottingham. He is the Associate Director of the new Nottingham Centre of Gravity and has served as the chair of U.K. Cosmology for over a decade. In 2016, he and his collaborator shared the Buchalter Cosmology Prize for their work on the cosmological constant. He is also a star of the Numberphile YouTube network, where his most popular videos include a discussion of Ramanujan’s sum of all positive integers, which has been viewed more than seven million times.
"A fun, dazzling exploration of the strange numbers that illuminate the ultimate nature of reality.
For particularly brilliant theoretical physicists like James Clerk Maxwell, Paul Dirac, or Albert Einstein, the search for mathematical truths led to strange new understandings of the ultimate nature of reality. But what are these truths? What are the mysterious numbers that explain the universe?
In Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity (FSG,2022), the leading theoretical physicist and YouTube star Antonio Padilla takes us on an irreverent cosmic tour of nine of the most extraordinary numbers in physics, offering a startling picture of how the universe works. These strange numbers include Graham’s number, which is so large that if you thought about it in the wrong way, your head would collapse into a singularity; TREE(3), whose finite nature can never be definitively proved, because to do so would take so much time that the universe would experience a Poincaré Recurrence—resetting to precisely the state it currently holds, down to the arrangement of individual atoms; and 10^{-120}, measuring the desperately unlikely balance of energy needed to allow the universe to exist for more than just a moment, to extend beyond the size of a single atom—in other words, the mystery of our unexpected universe.
Leading us down the rabbit hole to a deeper understanding of reality, Padilla explains how these unusual numbers are the key to understanding such mind-boggling phenomena as black holes, relativity, and the problem of the cosmological constant—that the two best and most rigorously tested ways of understanding the universe contradict one another. Fantastic Numbers and Where to Find Them is a combination of popular and cutting-edge science—and a lively, entertaining, and even funny exploration of the most fundamental truths about the universe."
For particularly brilliant theoretical physicists like James Clerk Maxwell, Paul Dirac, or Albert Einstein, the search for mathematical truths led to strange new understandings of the ultimate nature of reality. But what are these truths? What are the mysterious numbers that explain the universe?
In Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity (FSG,2022), the leading theoretical physicist and YouTube star Antonio Padilla takes us on an irreverent cosmic tour of nine of the most extraordinary numbers in physics, offering a startling picture of how the universe works. These strange numbers include Graham’s number, which is so large that if you thought about it in the wrong way, your head would collapse into a singularity; TREE(3), whose finite nature can never be definitively proved, because to do so would take so much time that the universe would experience a Poincaré Recurrence—resetting to precisely the state it currently holds, down to the arrangement of individual atoms; and 10^{-120}, measuring the desperately unlikely balance of energy needed to allow the universe to exist for more than just a moment, to extend beyond the size of a single atom—in other words, the mystery of our unexpected universe.
Leading us down the rabbit hole to a deeper understanding of reality, Padilla explains how these unusual numbers are the key to understanding such mind-boggling phenomena as black holes, relativity, and the problem of the cosmological constant—that the two best and most rigorously tested ways of understanding the universe contradict one another. Fantastic Numbers and Where to Find Them is a combination of popular and cutting-edge science—and a lively, entertaining, and even funny exploration of the most fundamental truths about the universe."
Interview with Padilla on New Books Network
Book Review:
Kirkus Review
"A fascinating book that guides us through the labyrinth of numbers and what they mean.
Theoretical physicist Padilla, whose research focuses on the intersection of advanced mathematics, physics, and cosmology, sees the world as a collection of particles, waves, and forces described by a spectrum of numbers ranging from the vanishingly small to the unbelievably large. Dividing the book into “big numbers” and “little numbers,” the author traces the history of each of his fields of study, providing sketches of the people who made the key breakthroughs.
This is lively history, from the Newtonian world of movement and collisions to the mysteries of quantum mechanics, with Einstein and a host of others making appearances. Finding numbers large enough to make sense of the universe has been a constant problem. Graham’s number, usually written as a mixture of numerals and esoteric symbols, was the standard for a while; Padilla describes it as a “black hole head death,” too big for the human brain to handle as anything but an abstraction. Graham’s number was eventually surpassed by TREE(3), the product of a self-replicating series.
At the other end of the numerical scale, Padilla stares into the abyss of supertiny particles and components of atoms, confronting the randomness of the way they stick together, or don’t. “It worries me to think that everything I know should never have existed: me, my family, my closest friends,” he writes. “This book should never have existed and yet, somehow, you’re reading it, right now, in a moment that might never have arrived.”
Padilla has a knack for effectively deconstructing difficult concepts, using explanations that include Usain Bolt, Lego, and Squid Game. Though parts of the book are extremely challenging, like James Gleick’s Chaos and Stephen Hawking’s A Brief History of Time, it is a remarkable piece of work that is well worth the effort."
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Fantastic Numbers & Mathematical Enchantment: A Whiteheadian Approach
Mathematics need not belong exclusively to a mechanistic image of the universe. Modernity often inherited a picture of reality in which matter consists of inert stuff moving through empty space, all relations are entirely determined by the past, and experience or inwardness belongs only to human beings, if it exists at all. In such a worldview, mathematics easily becomes a language of prediction and control—a tool for measuring and managing a lifeless world.
Yet mathematics itself need not be confined to such assumptions. In the spirit of Process and Reality and Science and the Modern World, mathematics can be liberated into a more organic vision of reality: one in which the world is composed not of dead substances but of events and relationships, not of isolated entities but of interdependent processes, and not of empty mechanisms but of experiential life at many levels.
Within such a perspective, mathematics becomes not merely an instrument of calculation but a pathway into wonder, imagination, and the discovery of hidden patterns of relation. It can become one of the ways modernity awakens again to mystery—not replacing artistic, emotional, bodily, musical, or spiritual ways of knowing, but standing alongside them as one among many forms of participation in a living universe.
The use of mathematics and numbers as languages of wonder
Mathematics is often treated as a practical instrument for calculation and prediction, but it can also function as a vehicle for awakening awe. Certain numbers stretch imagination and enlarge our sense of reality itself. They invite us into dimensions of mystery that exceed ordinary experience.
The role of numerical imagination
Human beings can inhabit realities far beyond direct sensory perception through mathematical thought. We can imagine scales of space and time, dimensions, infinities, and possibilities unavailable to immediate experience. Numbers become invitations into hidden worlds.
The role of mathematics in making sense of the world
Mathematics not only describes patterns within nature but also reveals structures beneath appearances. Whether mathematics is discovered, invented, or some combination of both, it remains one of humanity's most remarkable ways of participating in the intelligibility of the cosmos.
Mathematical and non-mathematical ways of knowing: their complementarity
Mathematics is not the only way of knowing reality. Human life unfolds through multiple forms of understanding that complement rather than compete with one another:
- Artistic knowing: grasping reality through images, symbols, stories, color, and creative expression.
- Emotional knowing: recognizing meaning through feelings such as joy, grief, fear, longing, and love.
- Musical knowing: sensing rhythms, harmonies, tensions, and relationships through sound and movement.
- Bodily knowing: understanding through touch, movement, habits, gestures, and embodied experience.
- Spiritual knowing: experiencing connection, reverence, transcendence, depth, and the sacred dimensions of life.
Mathematics reveals some truths of the universe; poetry, music, love, and bodily life reveal others. A fully human understanding of reality may require all of these modes of participation.
Two Kinds of Fantastic Numbers: Eternal objects and Propositions
From a Whiteheadian perspective, fantastic numbers can be understood in more than one way. In some contexts they may be interpreted as representations of eternal objects: pure potentials for definiteness, patterns or forms that can be realized in actual experience. Mathematical relationships such as π, the Golden Ratio, prime number structures, or geometric symmetries may fit this understanding. They need not depend on our universe for their reality. They can be thought of as possibilities available for realization in any universe—or perhaps in many universes—whether or not they happen to be embodied in a particular world.
In other contexts, fantastic numbers may function as propositions: possibilities emerging from our encounter with the actual world that beckon us toward further understanding. Whitehead describes propositions as lures for feeling. Numbers associated with black holes, cosmic inflation, quantum fields, entropy, or unimaginably large scales such as Graham's Number may operate in this way. They arise not simply as timeless abstractions but as imaginative invitations generated by our engagement with reality itself. They encourage us to think beyond the familiar and to entertain worlds and structures we have not directly experienced.
At the same time, these numbers are not merely subjective constructions or imaginative exercises. They help us understand the objective world itself. They allow us to describe the structure of galaxies, estimate the age of the universe, model quantum interactions, understand spacetime, and identify hidden patterns in nature. In Whitehead's terms, eternal objects and propositions are not detached from reality; they are ingredients in the way reality becomes intelligible. Mathematical ideas become lenses through which the world discloses itself.
A number can thus become more than an abstract quantity. It can become an invitation—a lure drawing thought toward unexplored realities while also helping us grasp the world we inhabit. In this sense, fantastic numbers are not merely objects of calculation. They are pathways of imagination, understanding, and wonder.
Cosmic epochs and the possibility of other mathematics
Whitehead speaks of cosmic epochs: large-scale orders of existence governed by particular patterns of relationship and regularity. The mathematical structures familiar to us may characterize our cosmic epoch, but other epochs could embody different forms of order altogether. Perhaps entirely different mathematical systems might obtain under different cosmic conditions.
And there is a still more speculative possibility: perhaps there could be cosmic epochs in which mathematics, at least as we understand it, does not play a central role at all. Such worlds may be unimaginable to us. Yet the very act of imagining them reminds us that reality may be more expansive than any conceptual system we presently possess.
Mathematics and the extensive continuum
Whitehead proposes the idea of an extensive continuum: a field of potential relatedness underlying the world of actual events. It is not itself made up of actual things or objects but is instead a realm of possible relationships of extension—possible ways that entities can stand in relations of nearness and distance, inclusion and separation, overlap and succession.
Mathematics often gives us remarkably powerful ways of describing these relations. Geometry, topology, and modern physics all reveal aspects of the structures of extension that characterize our universe. In our cosmic epoch many patterns of extension appear deeply mathematizable. Mathematical descriptions have proven extraordinarily successful in uncovering hidden regularities in nature.
Yet from a Whiteheadian perspective, the extensive continuum itself should not necessarily be identified with mathematics. Mathematics may describe some of its structures without exhausting its reality. There may be dimensions of potential relatedness that cannot be fully rendered in equations or quantitative models. The continuum could contain possibilities of relation that exceed present mathematical frameworks—or perhaps exceed mathematization altogether.
This raises an interesting possibility: mathematics may be one way of mapping the terrain of possibility, but not the terrain itself. Reality may be richer than any map. The extensive continuum might remain, at its depths, more spacious than the mathematical forms through which we currently approach it.
Eternal Objects, Propositions, and the Divine Prehension of Possibility
There is an important qualification in Whitehead's philosophy. Eternal objects and propositions do not simply float in a Platonic realm independently existing in isolation. They become relevant only insofar as they are felt, entertained, or prehended by subjects. Eternal objects themselves are neutral with respect to their realization. They do not make decisions, exert causal power on their own, or seek embodiment. The possibility of circularity does not decide to become a planet; numerical patterns do not decide to become galaxies; mathematical relationships do not actively seek expression in the world.
For Whitehead, this is where the role of God becomes important. God, understood as the primordial mind of the universe, prehends the realm of eternal objects and orders them according to their potential relevance for the world. In this primordial valuation, possibilities become graded according to their capacity for intensity, harmony, beauty, and richness of experience. Through this valuation, eternal objects become available as meaningful possibilities for actual entities. God does not force their realization, but offers them as possibilities—as lures toward forms of becoming.
Whitehead also leaves open the possibility of different cosmic epochs. A cosmic epoch is a large-scale ordering of reality in which certain kinds of relationships, patterns, and laws become dominant. We ordinarily assume that the mathematical structures and physical laws familiar to us—those governing spacetime, matter, energy, and number—are universal and eternal. But Whitehead's philosophy suggests another possibility: there could be cosmic epochs structured by different patterns of order altogether.
In such epochs, different eternal objects might become relevant to actualization. Entirely different mathematical structures—or what would appear to us as different mathematical "laws"—could govern reality. Possibilities now dormant or unimaginable to us might become active and central, while some mathematical forms fundamental to our universe might lose their relevance altogether. The extensive continuum associated with a given cosmic epoch would provide the field of potentiality appropriate to that epoch's mode of becoming.
From this perspective, the fantastic numbers explored by contemporary physics may represent only a small subset of a much larger field of possibility. They may belong to the mathematical architecture of our cosmic epoch rather than to reality as such. If so, the universe—or perhaps reality itself—may be even stranger, richer, and more imaginative than we currently suppose.
Mathematics and the Good
Alfred North Whitehead
"Whitehead is well known for saying that much of Western philosophy is a series of footnotes to Plato. In "Mathematics and the Good," published in 1941,* he is explicitly inspired by Plato as he discusses the connections between mathematics and the good.
The essay begins: "About two thousand three hundred years ago a famous lecture was delivered. The audience was distinguished: among others it included Aristotle and Xenophon. The topic of the lecture was The Notion of The Good. The lecturer was competent: he was Plato." Plato's subject, he adds, was mathematics, and Plato's general idea, too often ignored by his successors, is that mathematics and the Good are inseparable. Whitehead agrees.
Whitehead proposes that mathematics is the exploration of patterns abstracted from their exemplifications in nature and human life and argues that these patterns have their value and place as embodied in the experience of the world. When embodied in experience, he adds, they can be considered "good" or, in his words, "bad." Therefore, the good he speaks of is not moral goodness, as exemplified in Greek virtues (temperance, courage, wisdom, and justice) or Christian virtues (faith, hope, and love), but rather the goodness of experience as intense and harmonious, or as, in the case of triviality, maladaptive. Value is, in Whitehead's philosophy, richness of experience as enjoyed by human beings, other animals, and actual entities of any size and scale.
One of his key points, in support of Plato who believed the same, is that mathematics and the concept of the good, as understood in this context, are inseparable, and that the subsequent separation of the two in Western history was a mistake. In discussing geometry, arithmetic, and algebra, he highlights that the activity of abstraction itself, as exemplified in mathematics, is a remarkable achievement of the human imagination. It gives rise to consciousness.
However, he also emphasizes that patterns themselves, in abstraction, are not where value or goodness lies; instead, goodness lies in the activity of experience itself, as it both exemplifies and discerns patterns. Additionally, he argues that the universe revealed in abstraction and in experience is nested within an "unbound" infinity of actuality and potentiality. He also suggests that this unbound infinity, in which everything is connected to everything else in myriad ways, is "awakened" by the creativity of the actual world itself.
Whitehead is impressed with the fact that the Good itself (with an upper case G) can be considered in abstraction from its instances, just as mathematical patterns can be thus considered. The question emerges: Does the Good exist even apart from its instances? Does it exist, for example, as an ideal in the mind of the universe, by which human beings are somehow lures or beckoned? Does it exist in God?
He does not say.
Another related question is: Do the mathematical patterns themselves, when considered in abstraction from their instances, also exist apart from those instances?
Here, too, he does not say.
But he is clear that the mathematical patterns are neither good nor evil. "Every abstraction derives its importance from its reference to some background of feeling, which is seeking its unity as one individual complex fact in its immediate present. In itself, a pattern is neither good nor bad."
He is most interested in the process of abstraction itself, as a supreme achievement of the human mind, but also as incomplete without other dimensions of life, including feeling. It is when the mathematical patterns are conjoined with feeling, in the actual process of experience, that they have value. When embodied or actualized, they find their place in the creativity of the universe, a creativity which surpasses even the patterns. In their very finitude, creative entities in the universe awaken the infinite."
Nagel E. Alfred North Whitehead. Mathematics and the good. The philosophy of Alfred North Whitehead, edited by Paul Arthur Schilpp, Northwestern University, Evanston and Chicago1941, pp. 666–681. Journal of Symbolic Logic. 1942;7(2):101-101. doi:10.2307/2266317
Here are some excerpts.
"The notion of the importance of pattern is as old as civilization. Every art is founded on the study of pattern. Also, the cohesion of social systems depends on the maintenance of patterns of behavior, and advances in civilization depend on the fortunate modification of such behavior patterns. Thus, the infusion of pattern into natural occurrences, the stability of such patterns, and the modification of these patterns are the necessary conditions for the realization of the Good. Mathematics, being the most powerful technique for understanding patterns and analyzing their relationships, plays a fundamental role in this process. This understanding of patterns is the fundamental justification for Plato's lecture topic."
*
"But this notion [of triviality in experience] is meaningless except as a reference to the background of feeling—namely emotional and analytic experience—within which that total pattern arises. Every abstraction derives its importance from its reference to some background of feeling, which is seeking its unity as one individual complex fact in its immediate present. In itself, a pattern is neither good nor bad. But every pattern can only exist in virtue of the doom of realization, actual or conceptual. And this doom consigns the pattern to play its part in an uprush of feeling, which is the awakening of infinitude to finite activity."
*
"Abstraction involves emphasis, and emphasis vivifies experience, for good, or for evil. All characteristics peculiar to actualities are modes of emphasis whereby finitude vivifies the infinite. In this way Creativity involves the production of value-experience, by the inflow from the infinite into the finite, deriving special character from the details and the totality ofthe finite pattern. This is the abstraction involved in the creation of any actuality, with its union of finitude with infinity. But consciousness proceeds to a second order of abstraction whereby finite constituents of the actual thing are abstracted from that thing. This procedure is necessary for finite thought, though it weakens the sense of reality. It is the basis of science. The task of philosophy is to reverse this process and thus to exhibit the fusion of analysis with actuality. It follows that Philosophy is not a science."
The essay begins: "About two thousand three hundred years ago a famous lecture was delivered. The audience was distinguished: among others it included Aristotle and Xenophon. The topic of the lecture was The Notion of The Good. The lecturer was competent: he was Plato." Plato's subject, he adds, was mathematics, and Plato's general idea, too often ignored by his successors, is that mathematics and the Good are inseparable. Whitehead agrees.
Whitehead proposes that mathematics is the exploration of patterns abstracted from their exemplifications in nature and human life and argues that these patterns have their value and place as embodied in the experience of the world. When embodied in experience, he adds, they can be considered "good" or, in his words, "bad." Therefore, the good he speaks of is not moral goodness, as exemplified in Greek virtues (temperance, courage, wisdom, and justice) or Christian virtues (faith, hope, and love), but rather the goodness of experience as intense and harmonious, or as, in the case of triviality, maladaptive. Value is, in Whitehead's philosophy, richness of experience as enjoyed by human beings, other animals, and actual entities of any size and scale.
One of his key points, in support of Plato who believed the same, is that mathematics and the concept of the good, as understood in this context, are inseparable, and that the subsequent separation of the two in Western history was a mistake. In discussing geometry, arithmetic, and algebra, he highlights that the activity of abstraction itself, as exemplified in mathematics, is a remarkable achievement of the human imagination. It gives rise to consciousness.
However, he also emphasizes that patterns themselves, in abstraction, are not where value or goodness lies; instead, goodness lies in the activity of experience itself, as it both exemplifies and discerns patterns. Additionally, he argues that the universe revealed in abstraction and in experience is nested within an "unbound" infinity of actuality and potentiality. He also suggests that this unbound infinity, in which everything is connected to everything else in myriad ways, is "awakened" by the creativity of the actual world itself.
Whitehead is impressed with the fact that the Good itself (with an upper case G) can be considered in abstraction from its instances, just as mathematical patterns can be thus considered. The question emerges: Does the Good exist even apart from its instances? Does it exist, for example, as an ideal in the mind of the universe, by which human beings are somehow lures or beckoned? Does it exist in God?
He does not say.
Another related question is: Do the mathematical patterns themselves, when considered in abstraction from their instances, also exist apart from those instances?
Here, too, he does not say.
But he is clear that the mathematical patterns are neither good nor evil. "Every abstraction derives its importance from its reference to some background of feeling, which is seeking its unity as one individual complex fact in its immediate present. In itself, a pattern is neither good nor bad."
He is most interested in the process of abstraction itself, as a supreme achievement of the human mind, but also as incomplete without other dimensions of life, including feeling. It is when the mathematical patterns are conjoined with feeling, in the actual process of experience, that they have value. When embodied or actualized, they find their place in the creativity of the universe, a creativity which surpasses even the patterns. In their very finitude, creative entities in the universe awaken the infinite."
Nagel E. Alfred North Whitehead. Mathematics and the good. The philosophy of Alfred North Whitehead, edited by Paul Arthur Schilpp, Northwestern University, Evanston and Chicago1941, pp. 666–681. Journal of Symbolic Logic. 1942;7(2):101-101. doi:10.2307/2266317
Here are some excerpts.
"The notion of the importance of pattern is as old as civilization. Every art is founded on the study of pattern. Also, the cohesion of social systems depends on the maintenance of patterns of behavior, and advances in civilization depend on the fortunate modification of such behavior patterns. Thus, the infusion of pattern into natural occurrences, the stability of such patterns, and the modification of these patterns are the necessary conditions for the realization of the Good. Mathematics, being the most powerful technique for understanding patterns and analyzing their relationships, plays a fundamental role in this process. This understanding of patterns is the fundamental justification for Plato's lecture topic."
*
"But this notion [of triviality in experience] is meaningless except as a reference to the background of feeling—namely emotional and analytic experience—within which that total pattern arises. Every abstraction derives its importance from its reference to some background of feeling, which is seeking its unity as one individual complex fact in its immediate present. In itself, a pattern is neither good nor bad. But every pattern can only exist in virtue of the doom of realization, actual or conceptual. And this doom consigns the pattern to play its part in an uprush of feeling, which is the awakening of infinitude to finite activity."
*
"Abstraction involves emphasis, and emphasis vivifies experience, for good, or for evil. All characteristics peculiar to actualities are modes of emphasis whereby finitude vivifies the infinite. In this way Creativity involves the production of value-experience, by the inflow from the infinite into the finite, deriving special character from the details and the totality ofthe finite pattern. This is the abstraction involved in the creation of any actuality, with its union of finitude with infinity. But consciousness proceeds to a second order of abstraction whereby finite constituents of the actual thing are abstracted from that thing. This procedure is necessary for finite thought, though it weakens the sense of reality. It is the basis of science. The task of philosophy is to reverse this process and thus to exhibit the fusion of analysis with actuality. It follows that Philosophy is not a science."