God and the Mathematics of Infinity

Mathematics of Infinity

Throughout History, communities of virtually any size have organized cults and religions in one form or another, which as a rule have profoundly impacted culture and the way people think and live, and much contributed to shaping society. Historically, people have willingly followed their shamans, priests, and other holy men, probably in part because ‘inexplicable’ things and events do happen, above and beyond the earthquakes, bolts of lightning and other unusual natural events that were inexplicable in pre-scientific ages. In our enlightened times, the transcendent, the extraordinary and the inexplicable also sometimes put in an appearance at some point in ordinary people’s lives. As author David Mitchell put it in ‘The Bone Clocks’: ‘The paranormal is persuasive : why else does religion persist? ‘

Various texts telling of extraordinary people and events soon provided the foundation to lay down strict tenet systems and liturgies and rites, and many religions became rigidly codified and systematized. They grew in worldly power, and became the seats of political power and the repositories of all knowledge.

Then, some 300 years ago, the scientific age dawned and developed, oftentimes in opposition to the established churches. Science, technology and medicine were able to ever more impressive achievements, and soon scientists from a variety of disciplines began to deploy the same analytical skills and tools that serve them so well towards analyzing religion and faith, not least because of the strong sway they continue to hold over modern societies. Richard M. Gale, Michael Martin, Richard Swinburne, Victor Stenger, Peter Russell, George H. Smith, Paul Davies, David Eagleman and many others have commented on, and often tackled head-on the issue of the legitimacy of religion and belief.

The Mathematics of Infinity

Somewhat surprisingly, these scientists routinely reach totally different and contradictory conclusions. This very wide range of conclusions on the scientists’ part seems to feed a suspicion that whatever particular science discipline is used to justify a particular stance – biology, physics, neurology, biochemistry and so on – is perhaps little more than a subconscious amplifier for deep-seated cognitive bias. Should not science be flawlessly impartial? Despite a number of attempts, one discipline however has never been properly used in this context: elementary, fastidiously neutral, objective mathematics. Indeed, since math and theology are the only two human endeavors that deal in infinities, trying to use math in this context seems to make much sense, and we are justified in thinking that bringing these two fields together could just lead to interesting insights. The legitimacy of using mathematics in this context is of course not a given, suffice it to say here that within certain boundaries, a small subset of math has been shown to be a legitimate tool.

Most math is based on unproven axioms, i.e. assumptions which as such can hardly prove real-world hard realities without further ado. But a part of mathematics reduces instead to simple vocabulary definitions rather than hypotheses (such as the statement that 1+1 shall be called 2.) From these simple definitions, a whole unforeseen ménagerie of numbers springs, including unexpectedly exotic ones (imaginary, transcendental, transfinite, aleph, beth and other such numbers). Together, they can be drafted as tools to study the infinite and its consequences, and thus to analyze whether various qualities attributed to Godhood ultimately hold mathematical water, or reversely, lead to intractable contradictions.

Mining mathematics, we moreover soon find that more than just numbers can lead to unexpected insights (as a case in point, something called Tarski’s undefinability theorem logically justifies why some religions require a ‘leap of faith’: In essence, the theorem says that the truth of any statement made by means of some language cannot be proven from within the language that made the statement: if I say ‘this car is blue’, there is nothing within the English language that can prove that the car is actually blue. To establish veracity, corroboration from outside the language is needed.) Mathematical analysis can then often show whether some particular leap of faith is legitimate. To take an everyday example, math says that a lottery player who buys a ticket because she thinks she might win makes a legitimate ‘leap of faith’ that she’ll win: despite long odds, she might actually win, and there is nothing wrong with her leap of faith. The leap of faith, however, that God is transcendent rather than immanent―meaning present only at specific locations within space-time (including in possibly higher dimensional realms) rather than everywhere, is mathematically not legitimate: mathematical analysis shows that transcendence cannot be an attribute of God, but that immanence is.

That takes us to the very core of why mathematics can be so intriguingly powerful here: some theological questions, debated for centuries by theologians without finding their resolution, are incontrovertibly solved in a few minutes by mathematical analysis. Such questions include, surprisingly, why evil continues to exist in a Godlike universe, and more.

One question however, which mathematics itself says is in principle unanswerable, is that of existence itself. Does God exist? Mathematics proves that it cannot answer that one: it must firmly constrain itself to analyzing, if It exists, some of God’s purported attributes.

Chris Ransford
H. Chris Ransford, physicist and mathematician, earned advanced degrees in physics and engineering in France and Germany. He lectured at the University of Melbourne and was a research fellow at Monash University, Australia. He previously published The Far Horizons of Time (2015).