State University of New York at Old Westbury
This paper will explore the possibility of infinity structures philosophically, mathematically and physically. My recent research as an artist has led me to speculate on the existence of such unique structures, in which there is a stable exterior and an infinitely expanding interior, a wondrous yet seemingly impossible structure whose internal dimensions exceed its external ones.
The project began when I first asked myself a driving question: how can I make a structure in which the inside is continuously expanding and contracting, shifting, morphing, and folding back on itself with no effect on the outside? I began to look for theoretical, philosophical, and physical models, structures and explanations of anything similar. I pressed forward, working on a list of qualities that would be attributable to this structure: extension in space, a temporal dimension, and, most importantly, infinitely changing yet bounded. These were my starting points. The notion of the infinite fully captured me and I set about researching it from a historical and philosophical perspective. I will begin by grounding my question genealogically and sketching out a selective historical greatest hits. Then I will present what I hope to be some new possibilities drawn out of my research and speculations, and I will pose a few corresponding philosophical questions.
As early as the pre-Socratic Greeks, philosophers and mathematicians have taken up the fascinating and paradoxical concept of infinity and have struggled to comprehend it rationally. They developed a concept for a boundless, ever-changing infinity and called it Aperion. There have been disagreements as to the similarities between Aperion and Infinity. It was Anaximander of Miletus (mid-6th century BCE) who first referenced it. He referred to the Aperion as that which is unbounded, infinite, indefinite, or undefined. Aperion also was seen as an originary or fundamental substance which possesses no specific qualities and is unlimited.
According to Bertrand Russell, it was the pre-Socratic Zeno, famous for developing a series of paradoxes (the best known of which is the paradox of motion and infinite regress¬), who founded the philosophy of the infinite. In “Mathematics and the Metaphysicians” (1919), Russell states, “Zeno was concerned, as a matter of fact with three problems, each presented by motion but each more abstract than motion, and capable of purely arithmetical treatment. These are the problems of the infinitesimal, the infinite and continuity.”  These would become the mathematical problems for the future.
After the Pre-Socratics, Aristotle proposed that there are two types of infinity. He introduced the distinction between the potential and actual infinite. For him, infinity as potentiality is incomplete, always growing, and it has the absence of limit. In Metaphysics, Aristotle writes, “Infinity is not a permanent actuality, but consists in a process of coming to be, like time and the number of time. A line consists of two halves, but only potentially, for the actualization of the halves divides them from one another.”  For Aristotle, the existence of potential infinity is all that is necessary since it allows mathematicians to do their mathematics.
Aristotle rules out actual infinity for in his view our understanding of the world depends on our ability to comprehend substances in the finite world around us, which is real and actual (energeia) , in the fullest sense, and all actual things come to an end. In Physics, he states, “But we must not construe potential existence in the way we do when we say that it is possible for this to be a statue – this will be a statue, but something infinite will not be in actuality.” 
In medieval times, the church adopted this ontological division of potential and actual and merged Aristotelian metaphysics with church doctrine, instituting the law that only God is actual infinite and nothing else can be. Church authorities believed infinity was an attribute of God. This doctrine maintained a stranglehold on Western thought for several centuries.
The scientific revolution in the seventeenth century marked an explosion of philosophical and mathematical learning, and new mathematical theories of the infinite emerged. In 1609, Johannes Kepler wrote the Astronomia Nova, in which he develops the infinity of infinitesimals. While under house arrest during 1636–1638, Galileo wrote Two New Sciences, in which he gives an early preview of what would become the modern notion of infinity and of something reaching toward set theory that, anticipating Cantor, was later developed into the form of the infinite we have today. In Everything and More (2003) David Foster Wallace, states, “Two New Sciences is…the first truly modern attitude toward actual infinities as mathematical entities.” 
Gottfried Wilhelm von Leibniz, who was a midwife to the birth of modern science, was a rationalist and inventor of the system of calculus (alongside Newton). He also discovered the binary system, the foundation of modern computing. According to Leibniz, the smallest unit is the monad (think metaphysical subatomic particle). These monads have stable insides yet infinitely unfolding and expanding outsides – he calls them “insides without outsides.” The Monad, then, is the infinitesimal inverse of an infinity structure. Learning about the Monad was a profound revelation for my project. In my work, the infinity structures became expanding insides with stable outsides. With this I found my first promising model.
In “The Principles of Philosophy, or, the Monadology,” Leibniz describes the characteristics of the monad, which provides further support for the existence of infinite structures:
My topic here will be the monad, which is just a simple substance, by calling it ‘simple’ I mean that it has no parts [. . .] Monads are the true atoms of Nature—the elements out of which everything is made [. . .] I take it for granted that every created thing can change, and thus that created monads can change, I hold in fact that every monad changes continually [. . .] This particular series of changes should involve a multiplicity in the unit [. . .] And every momentary state of a simple substance is a natural consequence of its immediately preceding one, so that the present is pregnant with the future. 
The next major reinforcement for my project came through Spinoza, one of the most radical philosophers of the early modern period. His work was re-introduced to contemporary readers by Gilles Deleuze in the two popular books Spinoza: Practical Philosophy (2001) and Expressionism in Philosophy: Spinoza (2005). With significant new scholarship on Spinoza it is clear that he has significant resonance and relevance today. In his 12th letter, “The Letter on the Infinite,” he outlines a conception of “Infinities Contained within Limited Bounds,” and with this I had found a description of an infinity structure.
In this letter Spinoza lays out two forms of infinity: “Unlimited Infinities and Infinities Contained within Limited Bounds; one sort of infinity is called infinite because it is unlimited, and another sort is called infinite because its parts cannot be equated or explicated by any number, despite our knowing its maximum and minimum.”  He continues:
Some things are inﬁnite by their nature and cannot in any way be conceived to be ﬁnite, and that others [are inﬁnite] by the force of the cause in which they inhere, though when conceived abstractly, they can be divided into parts and regarded as ﬁnite, and that others, ﬁnally, are called inﬁnite, or if you prefer indeﬁnite, because they cannot be expressed in number, though they can be conceived as greater or lesser. For if things cannot be equated with a number, it does not follow that they must be equal. This is manifest enough from the example adduced and from many others. 
As is apparent, Spinoza directly counters Aristotle and the church in arguing that every infinite is actual.
It was at this point in my research that I felt I had found enough reinforcement and justification for my thesis, and that I was not simply in the realm of an imaginative projection or metaphysical speculation. I was now arriving at a more clear idea of how to define infinity structures: they are a sequence of spaces and structures, a sequence of structures opening up new spaces. A more extended definition could be that infinity structures are flowing states of matter, matter in movement without centers of reference or points of anchorage. Centers emerge but are constantly displaced or dissolved into the flux of matter.
After this examination of the philosophical, conceptual models and applications of the infinite, I proceeded in the critical light of mathematical results to ground my work moving from metaphysical fiction to the concrete. The mathematical genius Georg Cantor’s road to the theory of infinity develops through the invention of transfinite math/numbers in his paper on the Uniqueness Theorum leading him to then develop point sets and the abstract sets for which he is most famous. Arguably, Cantor invented set theory, the area of modern mathematics devoted to the study of infinity, in a single paper, “On a Characteristic Property of All Real Algebraic Numbers” (1874).
Cantor held that transfinite numbers and infinite sets actually exist and are reflected in actual real world infinities. In The Principles of Mathematics (1903), Bertrand Russell credits Cantor’s contribution to infinity with this statement: “The mathematical theory of infinity may almost be said to begin with Cantor.”  Cantor’s theorem implies the existence of “an infinity of infinities,” and proposes a theory of infinities of different sizes. He held to the reality of the actual infinite.
In just a few years following Cantor, Ernst Zermelo and Abraham Fraenkel formed a foundationally secure and rigorous version of set theory (ZFS). Kurt Gödel added to this conversation with his Incompleteness Theorem in 1931, and later others contributed to ZFS to become the present ZFC (nine collective axioms) system of contemporary set theory that in some part comprise the fundamental laws of mathematics.
The Cantorian conception of infinity (extensional discrete infinity) is a series of discrete points or elements that exist independent of each other. There is another contrasting conception of infinity namely that of the continuum (intentional continuous infinity), which is geometric and is conceived as a cohesive totality. This categorical duality between the two conceptions of infinity is nicely laid out by Yoshihiro Maruyama. 
A number of recent developments extend the frontiers of mathematics, such as expanding ZFC, Multiverse and Woodin’s V=ultimate L. These are hyper-abstract constructs that extend the infinite and yet some have bearing on actual real-world substances. Infinity has entered science and can be seen in real-world phenomena such as semi-infinite bodies of elastic material, infinitely deep liquids, thermodynamics, and the description of black holes and singularities, to name a few.
With this genealogy I now have the scaffolding necessary to proceed with this speculative project. I have found enough circumstantial evidence to support the claim that infinity may be a used as a malleable substance as well as a mathematical entity. I believe that infinity structures are a paradox that can give rise to conceptual advances, not unlike other paradoxes in the history of infinity. In moving forward with my project, I am addressing the structures not only as pieces of mathematics, but also as phenomenal substances.
My artwork explores what would be like to enter and experience such a structure phenomenologically. Walls continuously extended and folding back, floors shifting, then stable, until another rearranging. Some movements are infinitesimal, so tiny and evanescent that they seem to have no effect on the structure. Moving through at the times when it is seemingly stable one can sense its eminent variability as if asleep, dormant, an eruption of movement at any moment. There is a fun house horror as the space carves itself, the passages become transient and ephemeral, and the surfaces that seem single become endless. Ultimately the structure is un-navigable, periodically swallowing and reforming. This structure not only stretches our perception but our equilibrium, forcing body before brain, touching skin before our intellect.
To create an infinity structure artwork I begin with contextual architectural elements derived from the site of exhibition, for example, the floor plan of the gallery and other localized interior features. This becomes the core structure or object that will be added to, morphed, and modified in multiple iterations using 3D modeling software. The result is hundreds of iterations of these extensive transitive structures. For exhibition a select number of the forms are chosen to be physically produced using a combination of new technology tools including 3D modeling and printing, computer numerical control (CNC) milling, and projection mapping to create installations of these paradoxical structures. These works are preliminary movements toward manifesting infinity structures; they are immobile poses of these structures, attempting to catch movement in an instantaneous flash.
For Deleuze, wonder is a site of potential “outside of all representations.” Infinity structures are an example of such a wondrous site since they are a particular encounter with indeterminacy, forming, deforming, and undoing their own object-form. In A Thousand Plateaus, he writes, “It is a question of producing with the work a movement capable of affecting the mind outside of all representations.” 
Infinity structures are a confrontation with something that our mind cannot organize, contain, or fully make sense of. Our mind, our imagination, extends and stretches, but it can determine no internal boundaries for these structures that exist at the border of the unpresentable. Infinity structures are an evanescent multiplicity, a felt indeterminacy, and fully wondrous.
Infinity structures may seem like speculative fiction, but there are mathematical and scientific reasons to believe they could exist. With all the recent work that has been done on infinity in theoretical physics, cosmology, and mathematics, infinity structures according to our present understanding simply cannot be ruled out. And although there is no experimental evidence yet, these structures are quite plausible, so that as we advance in science and technology, it may be possible to construct an infinity structure in the future.
- Bertrand Russell, “Mathematics and the Metaphysicians” in Mysticism and Logic: And Other Essays (New York: Longmans, 1919), 74–96.
- Aristotle, “Metaphysics” in The Complete Works of Aristotle Vol. 2: The Revised Oxford Translation, ed. Jonathan Barnes (Princeton: Princeton University Press, 1984), 1552–1728.
- Aristotle, “Physics” in The Complete Works of Aristotle Vol. 1: The Revised Oxford Translation, ed. Jonathan Barnes (Princeton: Princeton University Press, 1984), 315–446.
- David Foster Wallace, Everything and More (New York: W.W. Norton and Company, 2003), 101.
- Gottfried Wilhelm Freiherr von Leibniz, “The Principles of Philosophy, or, the Monadology,” in Discourse on Metaphysics and Other Essays, trans. Daniel Garber and Roger Ariew (Indianapolis: Hackett, 1991), 68–81.
- Baruch Spinoza, The Letters, trans. Samuel Shirley (Indianapolis: Hackett, 1995), 102.
- Spinoza, The Letters, 101–107.
- Bertrand Russell, The Principles of Mathematics (London: Cambridge University Press, 1903), 304.
- Yoshihiro Maruyama, “Continuous vs. Discrete Infinity in Foundations of Mathematics and Physics,” History and Philosophy of Infinity Conference, University of Cambridge, 22 September 2013.
- Gilles Deleuze and Félix Guattari, A Thousand Plateaus, trans. Brian Massumi (Minneapolis: University of Minnesota Press, 1987), 144.
Robert Gero is an artist and philosopher. In recent years he has focused on an expanding notion of sculpture in theory and practice, pushing its interdisciplinary boundaries into design, temporary architecture, theoretical mathematics, experimental digital 3D, and collaborative works. As an artist he has exhibited nationally and internationally. Selected exhibitions include the Museum of Art and Design, New York; UCLA Art-Sci Gallery; 45th Venice Biennale; Artist Space, New York; Santa Monica Museum of Art. He was awarded an Art Matters grant, NY (2011). He completed his Ph.D. in Philosophy at the New School For Social Research and received an MFA from the California State University, Los Angeles. His research interest is grounded in the practical and theoretical intersection of art practice, theory and philosophy. Currently Gero is an Assistant Professor in the Department of Visual Arts at State University of New York at Old Westbury where he has been a faculty member since 2012.