God and the Mathematics of Infinity

Which Nature of Reality?

Theologians and, more recently, scientists have traditionally taken on the role of answering the question of what it all means. Their day job is to probe the ultimate nature of reality—to understand what it is that makes the world tick. These two communities ([2]) approach the issue from vastly different angles and with totally different tools, yet they share a common purpose of understanding and describing reality.

Theologians

The community of theologians goes back thousands of years, and still strongly endures. Many claim to hold special knowledge of Godhood, imparted to them through a variety of ways—meditation, divine revelation past or present, ancient scriptures, and the like. But different theologians routinely offer starkly different and, despite areas of overlap, often mutually contradictory visions of who or what the Godhead may be. Since irreconcilable views of divinity have historically led to severe social disharmony, to crimes and wars, both civil and foreign, and still do so today, some objective means of telling what may possibly be true or at least harbor a measure of truth from what is likely patently wrong is long overdue. Much adding to the confusion, academics with impeccable credentials, from a wide range of reputedly objective disciplines—Richard M. Gale, Michael Martin, Richard Swinburne, Victor Stenger, Peter Russell, and many others, have approached the subject from a variety of supposedly rigorously impartial angles over the years, and yet have still reached opposite conclusions with seemingly metronomic regularity—which further underscores the need for an absolutely objective tool of analysis. Could it be that, much like the faint flapping of butterfly wings may bring about inordinately big effects on distant, virtually unrelated related events—a phenomenon known as the 'butterfly effect'—the slightest subconscious bias may be stealthily determining the eventual outcome of analyses not incontrovertibly fully rooted in pure calculation-driven objectivity?

So whom, and what, are we to believe? And why and how different and incompatible views of Godhood can arise in the first place? Historically, the use of some mathematics and/or logic has been sometimes attempted: so-called ontological arguments were made by some theologians to demonstrate the existence of a Godhead. Such arguments, however, seem flawed ([3]). A far more compelling case for the possible existence of a Godhead has come from a far unlikelier source—a mathematician who was not in the business of seeking answers to queries of a spiritual or theological nature, but who appeared to stumble onto one: Georg Cantor, the pioneer of the formalized study of infinities, demonstrated that the mere existence of infinities ultimately leads to a stark mathematical contradiction, a full-blown breakdown of mathematics and of logic itself. He could only resolve the contradiction—which he called an antinomy—by positing the existence of a super-infinity, something much too vast to ever be approachable through the mathematics of infinity alone, but which required the deployment of much more than mathematics to be even remotely fathomed—an infinity approachable by us, however dimly, only if we use both our left (logical) and right (intuitive) brain. He made an argument that this super-infinity is the Godhead itself. Again, contrarily to the ontologists' approach, Cantor did not set out to find a mathematical definition or proof of Godhead, but, as he saw it, had to invoke the Godhead in order to resolve the intractable contradiction he discovered in the mathematics of infinity.

We will look more in depth at Cantor's arguments below. Generally speaking, everything about a Godhead is about infinities and infinite attributes ([4]), although, surprisingly, a few theologians disagree. The theologian Harold Kushner, for instance, argues in his book 'When Bad Things Happen to Good People' that the Godhead is not infinite. He is led to this puzzling conclusion by his analysis of the question of why a Divinity would see fit to allow 'good people' to undergo 'bad things', from his standpoint as someone who believes in a specific Godhead with precisely defined qualities and attributes, set forth in the narrowly defined framework of a rigidly established dogma. Building on lines of thought first put forward by Gersonides in the fourteenth century and more recently by others, such as Levi Olan, Kushner rather extraordinarily concludes that Godhead is in fact powerless to stop 'bad things' from happening. Under his view, the Godhead is neither infinite nor almighty. We shall keep here with the majority view that any Godhead must be infinite, and that infinity is the very quality that ultimately gives rise to the disruptive, extraordinary phenomenon of divinity. A more in-depth analysis of the question is presented in note ([5]).

The issue of why and how it is legitimate to use simple numbers-based mathematics in this context is a valid question, dealt with under ([6]). The bottom line is that some math is at the very least valid within certain areas and domains relevant to the questions at hand, and we shall restrict ourselves to such domains. By demonstrating incontrovertible facts, math will enable us to tell apart what can possibly be, and what most definitely cannot be. It will enable us to come closer to an understanding of who or what a Godhead could possibly be, if there is such a thing as a Godhead. It will also help in circumscribing the existential question itself—can there possibly be a Godhead to begin with?

A few guidelines on how mathematics can and should be used is appropriate here, so please bear with me as I briefly set them forth here. Broadly speaking, math can only deal with precisely defined words describing sharply delineated concepts. For instance, should we say that Godhead is love, or compassion, these somewhat fuzzy concepts cannot be readily probed or analyzed by numbers or by math. But should we say that Godhead is infinite love, infinite compassion, then the 'infinite' part of the statement is directly amenable to mathematical analysis: indeed, math is the only tool available in the box that objectively deals, or even can deal, with infinities.

Within the framework of a number of possible limitations and provisos, math will thus allow for deploying non-subjective, logical, 'left-brain' approaches, all the more indispensable because the wonted subjective, right-brain approaches seem to unfailingly lead to contradictions. Somewhat unexpectedly, math will also turn out to be helpful in analyzing emotional and right-brain approaches, and it will demonstrate why contradictions inevitably arise when exclusively right-brain approaches are used.

Scientists

Scientists, especially physicists, constitute the second community whose job it is to probe and understand reality, and therefrom to explain it to the rest of us. That there happens to be no more consensus as to what it all means amongst physicists than there is amongst theologians will surprise no one, and only underscores anew the acute need for using the most objective tool of analysis bar none.

A key ongoing debate among physicists today is between the so-called Aristotelian view of reality, and the so-called Platonic view. At its core, the Aristotelian view is a materialistic view ([7]), whereas the Platonic view holds that the ultimate nature of reality is not materialistic, but that abstract concepts, such as, first and foremost, an underlying abstract mathematical structure, play a or indeed the determinant role in weaving the fabric of our reality. In different shadings, this latter view has become more popular of late, in part thanks to published works by the likes of Max Tegmark, Lawrence Krauss, and others. Of course, if a person believes in any deity, that person then necessarily takes the Platonic view of reality, because he or she believes in a Universe which, at the very least, simultaneously accommodates both material reality itself plus some spiritual, immaterial transcendent being. Indeed, as we shall see, not only modern common sense but also straightforward mathematics proves that a Godhead, if It exists, cannot possibly be material.

This modern view may seem self-evident today, but not so long ago the image of Godhead as some avuncular man in the sky was not uncommon. In 1961, the Soviet cosmonaut Yuri Gagarin became the first human to travel in space. The then Soviet leader Nikita Khrushchev was quoted afterwards as saying, in all seriousness, in a speech in support of the USSR's secularist policies, "Gagarin flew into space, but didn't see any God there" (a quote that was later falsely attributed to Gagarin himself.) As recently as 1971, even John Lennon saw fit to write the lyrics: 'Imagine there's no heaven, above us only sky', in apparent reference to the then still surprisingly commonly held view of a material, three-dimensional Godhead resident somewhere in space.