God and the Mathematics of Infinity

What Irreducible Mathematics Says about Godhood

H. Chris Ransford

• Publisher: Ibidem Press
• Year: 2017
• ISBN:
  9783838270197
• Topic: Theology & Religion
• Subtopic: Religion & Science

Drawing on the science and mathematics of infinity, H. Chris Ransford analyzes the traditional concept of godhood and reaches surprising conclusions. He addresses humankind's abiding core debate on the meaning of spirituality and God. Using mathematics, he explores key questions within this debate: for instance, why does evil exist if there is a God? The book fastidiously does not take sides nor proffers opinions, it only follows allowable mathematics wherever it leads. By doing so, it makes a major contribution to an understanding of the nature of reality.

PART 1

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.

Using Mathematics

Whenever we fail to use a purely mathematical approach, whether from a believing or disbelieving or open standpoint, our views of the Godhead are bound to be almost equally naive. For instance, we may broadly agree on a definition of the Godhead as being infinite and disincarnate. But there are many quite different infinities, so which one is it exactly that we are favoring? We will probably naturally opt for some apex infinity—but, unless we do the math, we won't achieve anywhere near a full understanding or appreciation of the inescapable consequences that must flow from such a definition. Whenever or wherever infinities are involved, an astonishing degree of complexity kicks in and the plots thicken immeasurably, most often well beyond and differently from what we would naturally expect ([8]).

A math-based approach will also prove especially helpful in objectively analyzing received dogmas, i.e. the established foundational doctrines of many organized belief systems. Long-established dogmas are mostly accepted as are today, and reputed to spring from revealed truths. Any request for further justification is often staved off by a demand for a 'leap of faith', or some other demand for unquestioning obedience or acquiescence, often on the basis of the say-so of some (relatively) ancient texts deemed to be unassailable bearers of truth. As we shall however shortly see, dogmas are on the whole fully amenable to mathematical analysis.

As mentioned, math will also prove instrumental in analyzing the consequences of infinity—where the presence of infinity within certain contexts ineluctably leads. We cannot on the one hand accept or assert that a Godhead is infinite, and then blithely attribute traits, thoughts, or properties to that Godhead that would mathematically belie such infinity. We can hardly claim that we believe or for that matter disbelieve in something unless we fully know what it is that we believe or disbelieve in, including the flow-on consequences of such beliefs or disbeliefs. Whenever a contradiction that would belie infinity is uncovered, if we are to retain Its infiniteness there will be no choice but to abandon the corresponding purported trait of the Godhead.

Math will also be used to endow even everyday terms with precise meanings, and we shall endeavor to make such word definitions as broadly consensual as possible. Many concepts and/or realities tend to be loosely defined, and despite the widespread use of the same words, consensual meanings remain elusive, and different people may associate very different meanings to a same term or phrase. Illustrating the point, the age-old question 'Do you believe in God?' is utterly meaningless unless both words 'believe' and 'God' are very precisely defined, yet loose variations of this very question have historically led to all manner of strife, lethal and otherwise. We will therefore adopt here below a first definition for 'Godhead', which may then become further refined as logical analysis proceeds. Likewise, should we for instance say that a Godhead is infinitely intelligent, we must then find some way of defining intelligence appropriately accurately, so that its meaning both meets with consensus and becomes amenable to objective mathematical analysis. Most often, it will be easy to do so: for instance, the definition of 'infinitely intelligent' here would be, simply put, someone with an infinite IQ or IQ equivalent (irrespective of whatever particular IQ measure would be used, see note ([9]). Whenever some quality or attribute seems fuzzy, we will endeavour to find a way of defining it, even if provisionally so, bearing in mind that we could later on be led to a further refining of definition.

Last but not least, numbers will provide a handy way to illustrate certain concepts which would otherwise prove harder to apprehend—first and foremost, infinity itself. For instance, we can quite easily conceptualize how the unbounded series of whole numbers 1,2,3, 4,5,6...... goes on forever and never ends, and as such is infinite. Trying to conceptualize infinity, and infinities, by any other means or within contexts different from mere numbers may sometimes prove less straightforward, so that picturing infinities through the simple expedient of numbers shall provide a helpful shortcut towards visualizing infinity in a variety of other contexts.

What constitutes mathematical proof?

Two separate kinds of proof will be used in this book: first, the purely mathematical variety—proof that requires no further input from any other science to be able to stand on its own—be it physics, chemistry, psychiatry, psychology, or any other scientific discipline. A basic example of such a proof would be the statement that, "if A minus 1 is equal to 0, then necessarily, A is equal to 1." That's it—no further ado, no further discussion nor proof is needed to truthfully and incontrovertibly state that in this above case the value of A is unequivocally equal to 1, period. We'll call such proofs A-type proofs.

Examples of such A-type proofs include, for instance, the above-cited fact that if one believes in a Godhead, then one has no choice but to believe in the existence of some more complex universe or multiverse beyond our simple material universe—i.e., an outside reality that goes beyond the currently known material universe. Should our particular universe be finite (a question which we will revisit below), then the simple statement that a Godhead exists and has some infinite traits proves the existence of something else beyond that universe—because there is simply no mathematical scope or room to accommodate infinity within a finite universe. There are of course ways, as we shall see, whereby our very own universe could harbor infinity, so that a whole new separate universe or a metaverse or multiverse is not needed to accommodate infinity, but at the very least the existence of some infinity within the universe is in some way required if we are to accommodate divinity.

But the plot thickens, because there are many different infinities. Therefore, should we say and accept that a Godhead is infinite, or, say, infinitely good, what does this statement exactly mean? Are we content with some lower-ranking infinity, or shall we insist on a higher ranking, or, if such exists, on a or the apex infinity? We will examine these issues at some length in these pages.

Time is also involved: mathematically, we will see that if a Godhead exists, It by definition is the master of time and space. The only way to do so is to exist out of time, so that both the past and, crucially, the future do not hold sway ...

Table of contents

• Table of Contents
• Introduction
• PART 1
• PART 2
• PART 3
• Epilogue
• Further Reading
• End Notes