Infinity and Infinitesimals
Infinity and Infinitesimals
By Rocco Gangle, Professor of Philosophy, Endicott College, USA
The concept of infinity has traditionally been associated with the divine. The divine in turn, particularly in Western culture, has typically been associated with transcendence, that is, with distance, hierarchy and disassociation. Indeed, in the Aristotelian-Ptolemaic cosmos that held sway in the West for approximately two millennia, the various cosmic parts were essentially ordered according to their properly layered places, and the reigning place of the Prime Mover, or God, was that of the greatest distance from the humble center of things, our fourfold elemental and sublunary Earth. For this cosmology, God was greatest and most divine because furthest away from us.
It may perhaps be surprising, then, to find that the predicate infinite was for centuries, at least in Christian metaphysics, attributed to God only in a negative sense. From Gregory of Nyssa to Thomas Aquinas, to call God infinite was simply to reject any limitation, any qualitative “finitude”, to God’s divinity. Only with the late Scholastic philosophy of Duns Scotus does infinite become a strictly positive attribute of God and the divine. This is because for Scotus the transcendental attribute of “being” must be predicated in a univocal manner both to God and to creatures. That is to say, roughly speaking, the “concept” of being must have exactly the same sense when one affirms, “God exists,” as when one says, “This meadow flower exists”. What then distinguishes, first and foremost, the being of God from the being of the flower is, for Scotus, an ultimately quantitative qualification: God enjoys being in infinite measure, whereas all other beings, such as the spring flower in the mountain meadow, are understood to be essentially finite, to possess immeasurably less being than God.
An interesting point of contrast with this distinction between creaturely finitude and divine infinity may be found in traditional Hindu and Buddhist cosmology, where massively large – but still finite – numbers are associated with vast ages of time and enormously long cosmic cycles. Billions upon billions of kalpas mark the lifespans of certain epochs, where a single kalpa lasts somewhere in the neighborhood of 17 million years (or, in alternate interpretations, 17 billion years). In the context of Buddhist cosmology, then, even the devas and gods become necessarily subject to the inexorable law of impermanence. Like all things, they too shall eventually decay and die. It will simply take a practically immeasurable duration for them to do so. Even if existing at vastly different scales, we are ultimately like them and they are ultimately like us, in finitude.
But rather than being distinguished by infinite distance or by vast yet finite difference, might instead divine and human being find common ground in some type of infinity shared by both? What if the infinity of the divine were not primarily associated with distance, largeness and transcendence? What if divine infinity were instead linked to experiences of proximity, smallness and immanence with which we may be intimately familiar? Is such a notion of the infinitely small and near even comprehensible? At roughly the same time that the “subtle doctor” Duns Scotus developed his theory of the infinite being of God, the Dominican popular preacher Meister Eckhart gave sermons to the simple laypersons of the Holy Roman Empire telling of the unity of God and the Soul and explaining the seemingly abstract notion of the birth of the Son from the Father in the divine Trinity in terms of an incessant production of global intelligibility – the logos, or Word – at the heart of ordinary, individual being. God is within you, Eckhart teaches, and the infinity of God is at home in the infinity of your own soul. Eckhart’s metaphysics thus overturns the static hierarchy of the Aristotelian-Ptolemaic cosmos. For Eckhart, God is no longer the most distant, but rather the most proximate being. In the infinite soul, all things are immeasurably near. Just think for a moment of the cosmic event of the Big Bang and some 14 billion years of time collapse in your human soul to something like direct intellectual contact with that event, a distance less than any measurable quantity. What greater distance could a god hope to overcome than this?
How can we think such infinite proximity? Here, the mathematics of infinitesimals comes to our aid. The core idea of an infinitesimal is that it is a quantity that is less than any measurable real quantity and yet, for all that, is not equal to zero. This concept was introduced in the late seventeenth century by the philosopher Gottfried Leibniz in his development – parallel to that of Isaac Newton – of the differential calculus. For Leibniz, infinitesimals remained mathematical “fictions” that nonetheless served highly useful purposes. They enabled straightforward calculations, for instance, involving the derivatives of many standard mathematical functions.
The infinitesimals of the early modern era were not merely mathematical instruments, however; they were equally metaphysical speculations. Leibniz himself was strongly influenced in this regard by the English Quaker philosopher Anne Finch Conway, whose brilliant and underappreciated masterpiece The Principles of the Most Ancient and Modern Philosophy emphasized the infinity of both divine and creaturely reality. Indeed, for Conway, precisely because God’s divinity is infinite in power, creatures must therefore be infinite in quantity: there are infinitely many of them, and they swarm throughout “infinite worlds”. But even more extraordinary for Conway’s metaphysics is the fact that in her view every individual and finite creature is itself internally infinite by virtue of its infinite divisibility. Not only are there infinitely many beings (reflecting the unlimitedness of the divine nature) but every being is itself composed of infinitely many, ultimately infinitesimal, parts. Reality is in this way infinite both “all the way up” and “all the way down”. Importantly, for Conway it is due to their infinite divisibility that all things are thus, in some respect, necessarily intertwined with all others at the smallest level. By way of the infinitely small and near, all things become, at root, One.
Infinitesimals were the scandal of their day. Theologian and philosopher George Berkeley famously mocked the very idea of infinitesimals as “ghosts of departed quantities”. It is, in retrospect, profoundly ironic that a philosopher committed both to the denial of matter and to a metaphysical system in which the unity of reality is guaranteed only by the infinite mind of a Supreme Being should have sneered at the notion of a “little infinite” that by its very nature would bind the diverse beings of ordinary experience into an ultimately continuous relational nexus. In any event, the later orientation of differential calculus around the method of limits as expressed in the “delta-epsilon” formalism of Weierstrass and others became the standard approach in the late nineteenth century and throughout the twentieth. Infinitesimals, it seems, had become no more than departed ghosts of departed ghosts.
It is remarkable, then, that in the 1960s mathematician Abraham Robinson developed an entirely rigorous theory of calculus using infinitesimals. His resulting “non-standard analysis” thus perhaps vindicates the original intuition of Leibniz, following Conway, that a metaphysics of continuity grounded in the infinitely small may be at once the most mathematically accurate available to us and the one modeling our greatest proximity to the infinite divine. Today, philosophers such as François Laruelle take inspiration from Robinson and his overturning of the long-standing consensus opposed to infinitesimals. Laruelle’s “non-standard philosophy” integrates Robinson’s mathematical vindication of the infinitely small with the phantasmagoric physics of quantum mechanics to envision an immanent “philo-fiction” conceived on analogy with “science fiction” that would empower a mode of thinking in which imaginative speculation would be one with a divine creative power that has become counter-authoritarian and unlimitedly open to difference in its infinite smallness. But such a type of philosophy would be no less divine for that. Perhaps in this discombobulated age, infinitesimals promise the best chance for our divinity.
Rocco Gangle is Professor of Philosophy at Endicott College where he has been teaching since 2007, having taught previously at Oberlin College and the University of California, Merced. His current research focuses on metaphysics, semiotics, diagrammatic logic, and category theory. He is also one of the foremost translators and expositors of the work of contemporary French thinker Francois Laruelle. He has published several books, including Diagrammatic Immanence: Category Theory and Philosophy (2015) and, with Gianluca Caterina, Iconicity and Abduction (2016). He is co-director of the Center for Diagrammatic and Computational Philosophy. At Endicott, Gangle teaches a variety of courses in philosophy, intellectual history, and religious studies.
March 7, 2024
