The most troublesome concept of mathematics is Infinity, the unbounded and ever-growing totality, overpowering everything in its god-like immensity. It can be signified by a sleeping 8 sign ‘∞’. To even dare to name infinity is to commit blasphemy for only God can be ever-growing and without any limits. For most of its history, infinity has been banished from arithmetic and is still not permitted in Number Theory. But in mathematics, around 1870, Georg Cantor initiated the study of sets which opens up the science of infinity and transfinite arithmetic and reveals that there are so many different and distinct infinities that if we try to enumerate them their quantity exceeds what any number system allows, permits or enables us to count. There are more, ever increasing sized infinities than any infinity!
Cantor names two type of transfinite numbers the Alephs and the Omegas. He takes a Christian name for God (the ‘alpha and omega’), the first and last, drawn from the holy languages of Hebrew and Greek. It is no accident the Jorge Luis Borges named his infinite object the “Alpha”. He had obviously sneaked a peak at Cantor.
Cantor’s choice of names indicates that he is scaling the heavens, putting number’s claim on the names of the deity. The infinity of the set of counting numbers is breathtaking. But it is rapidly overpowered by the infinity of infinities that explodes beyond the universe and even beyond our powers of imagination. This adds a paradox to the normally regarded as super-rational science of arithmetic. In mathematics we dare to imagine the unimaginable, we celebrate and even arrogate to ourselves the power to touch the unreachable, while at the same time banishing the imaginings of philosophers, mystics, theologians and artists to the realm of the unreal.
Is infinity unreal? It is real, a cornucopia of excessive magnitudes in mathematics, an endless source of study from the denumerable natural numbers to inaccessible cardinals (very, very large infinite numbers) and beyond in the infinite space of set theory. In the world we only see the infinite in never-ending processes, such as a snail crossing half the remaining path width each day, or continuing forever. Even Archimedes knew that counting all the grains of sand on the beach or in the whole world would someday finish. But to this day some howl at the blasphemy of claiming that the infinite exists. Some claim to have tamed it and celebrate the banned division 1/0 which has in the past been equated to infinity ∞. Some, like Errett Bishop, have even reconstructed mathematics to exclude all possibility of it. Thus infinity is both a blessing and a curse in mathematics. It unleashes huge power but also great anger.
Of course the concept of Infinity already holds profound significance in religious and philosophical traditions. Precursors of infinity can be found in the two great ancient civilisations of Mesopotamia and Egypt. While Mesopotamian texts do not explicitly discuss infinity in recognisably mathematical terms, their cosmology reflects an understanding of unending cycles of time and the eternal nature of the gods. The Ancient Egyptian god Heh personifies infinity or eternity. According to Wikipedia “Heh was the personification of infinity or eternity in the … infinite realm of chaos as, in contrast with the finite created world.”
Infinity is a sacred concept. In Christian theology, infinity is divine because God is Infinite. God’s nature is described as infinite, because of His boundless attributes. ”His understanding is beyond measure.” says Psalm 147.
The Neoplatonist philosopher Plotinus conceptualized ‘The One’ as the infinite source of all existence. “The One is all things and no one of them; the source of all things is not all things, but their cause.”
Jain philosophy presents a sophisticated classification of infinities, distinguishing between enumerable, innumerable, and infinite, each with further subdivisions. “The Jains were the first to discard the idea that all infinities were the same or equal.” Says Singh, cited in Wikipedia.
In Western mystical traditions, particularly within the Kabbalah, Christian mysticism and Western esotericism, the concept of infinity is central to understanding the divine and the cosmos. In Kabbalistic cosmology, the Sephiroth, the Tree of Life is preceded by the Three Veils of Negative Existence. The first veil, Ain, represents nothingness. The second veil, Ain Sof, represents the infinite aspect of the divine, according to the International Order of Kabbalists.
Gregory of Nyssa, who lived c. 335–395 CE, was among the first theologians to articulate the idea of God’s infinity. His main argument for the infinity of God is that since God’s goodness is limitless, so too God is limitless. Nicholas of Cusa delved deeply into the nature of the infinite and discusses the paradox of attempting to comprehend the infinite. He claims that to define the infinite is to make it definite, and thus no longer infinite.
History reveals our love-hate relationship with infinity, but it is always troublesome. One illegitimate route to infinity comes by bring together the numbers zero and one. The equation 1/0=∞ is illegal for many reasons. First ∞ is not a normal part of counting, you never reach it no matter how far you continue 1, 2, 3, … . The unfinished infinity hides in the three dots, the ellipsis. So it is not a natural or counting number. But another problem is that 1/0, 2/0, 7/0, 139/0 would all have the same value, that is, ∞. So you could not reverse the division by multiplication by the divisor, as you should be able to, because you always get zero. It’s a dead end. It disrupts arithmetic.
Despite the seeming rationality of mathematics and arithmetic, it turns out the key concepts of number including zero, one and especially the unwelcome guest infinity, have strange powers and strong mystical associations and religious connotations. But contra to their sacred origins, a new profane order of Dataism has emerged, and now uses these numbers to regulate us and the world. The emergent technologies of control change the role of these numbers. We are now in a state where the power of zero and one, working together in zillionfold combinations, dictates every aspect of our lives; it is approaching totality, that is, infinity.
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Paul Ernest
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