On the Presumption of Theism 1.3.0

Abstract The question of the existence of G-d is one that many people are quick to weigh in on, but it is vexed and misunderstood. One of the principle misunderstandings involves the inclination to regard G-d as a material object, and to argue for or against his existence by reference to whether there is evidence for G-d in the sense that there is evidence for the CEO of a large corporation. Perhaps there is a rumour going around that this CEO secretly watches employees at work, and will reward any employee observed to be hard-working with a large annual bonus, but no employee has ever actually seen this CEO… Do they really exist, or are they a myth promulgated to keep employees in awe, and towing the cooperate line? Well, if we hire a skilled detective to conduct an investigation, and sift through the documentary and other empirical evidence, we can soon find out one way or another. But what if the CEO is defined as “immaterial”, and as both “immanent and transcendent”? Sherlock Holmes himself would be hard pressed to conduct an investigation into the existence of such an individual. More significantly, does it even make sense to say that something is “immaterial”, and “immanent and transcendent” at the same time? In this note it is explained why it does make sense to say this, and it is argued that G-d, so defined, is a condition of the existence of everything else. 

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If a debate arises concerning the question of the presence of a room of some item furniture, is the burden of proof on the one who claims the item is in the room, or is it on the one who claims that it isn’t? All things being equal, there is no clear cut burden of proof in such a case, and we must look to see if the room contains this item or not. If by contrast the debate concerns the presence in the world -the existence- of unicorns, we should feel sure that the burden of proof is on the one who claims that unicorns exist rather than the one who claims they do not. This is because, all things considered, we expect that unicorns don’t exist.

When it comes to the existence of G-d, do we presuppose existence or non-existence? Which is the more reasonable preliminary stance, so that the one who takes the opposite view must shoulder the burden of proof in an argument? I have found that there is no quicker way to diffuse a debate about G-d’s existence than to ask what would constitute evidence of G-d’s existence. What is absent such that if were present would satisfy the atheist that G-d exists? Or conversely what is present such that if it were absent would satisfy the theist that G-d does not exist? It is difficult to say what the signs of an all-knowing, all-powerful, omni-present, immanent yet transcendent being might be, so that we can say that we don’t see a sign of this being, or that we do. Certainly, G-d in the sense of the term as it is used in the Judeo-Christian tradition is quite unlike a chair or a unicorn. G-d is not merely a item or a entity in the world, but is the very ground of existence. It is incorrect then to imagine that the rules of evidence we might apply to determine if a room contained a chair, or if the world contained a unicorn, can with equal measure be applied to G-d, the ground of being itself. One argues for the existence of G-d in this sense by pointing out that, being needs a ground, that if anything exists, there exists a ground thereof.

There are various ways in which the notion ‘ground of being’ might be understood. The obvious way is as a ’cause’ and there is a long philosophical/theological tradition according to which G-d is the first cause. The difficulty with a first cause argument for G-d’s existence is that if nothing can exist without a cause then nor can a first cause exist. A more sophisticated form of the same style of argument (due to Leibniz) starts with the distinction with the distinction between contingency and a necessity, where a contingent thing depends upon prior causes, and a necessary thing is independent or self-caused. G-d in terms of this distinction is the necessary thing that lies at the root of the causal chain of the events comprising the history of the universe. Bertrand Russell’s rebuttal -expressed during a classic 1948 radio debate with Jesuit priest and philosopher Frederick Copleston- is that if anything can exist without a cause then it may as well be the world as G-d. Copleston had no compelling counter to this objection, but then Russell had no compelling counter to Copleston’s point that a causal chain needs an origin. By Russell’s reckoning, if we know the cause of every member of a series then there is no need for any overarching cause, but the story is told of Alice who possesses a certain book but borrowed it from Bob who borrowed it from Dylan who borrowed it from Jack… This process could go on indefinitely into the future, but no one imagines it could go on indefinitely into the past.

Their discussion puts us in mind of Kant’s antinomies of pure reason, and his suggestion that neither the idea of a causal chain that goes on forever in the direction of the past, nor one that is grounded in a first cause really makes sense because, Kant would say, both notions are attempting to go beyond the bounds of possible experience. So while it make sense to argue that the book in the above analogy must have had a first borrower, it doesn’t make sense to argue in an analogous way that the universe had a first cause. The difference is that book-borrowing is an empirical process whereas universe-creation isn’t. If Kant is right to see a paradox here, and if going beyond the bounds of experience results in a paradox, then Kant’s own analysis, and philosophical inquiry generally, are themselves paradoxical. I think Kant is right to see a contradiction, but that nonetheless a potentially endless chain of contingencies, and a necessary first cause can and must exist side by side. This is to say, a solution to the seeming paradox must be sought that allows for two distinct notions of infinity to exist side by side: the uncompletable infinity of plurality which underlies contingency, and the completed infinity of singularity that underlies necessity. Without a marriage between these seemingly dual notions of infinity, both theism and atheism, or any position on the question of G-d’s existence -or indeed any philosophical question- is bound to result in a seeming paradox. Both sides of classical philosophical debates about the existence of G-d, freewill and determinism, the mind-body problem etc. have valid arguments, but the arguments aren’t sound because they all involve the false assumption that these two notions of infinity are incompatible. This points us I believe toward the answer to our original question: it is a condition of rational inquiry that G-d in the sense of a infinite and singular ground of being exists.

Many reasons can be adduced for saying so, but most powerful of these emerges from work of the great 18th century Swiss mathematician, Euler. Euler produced a result in mathematics called appropriately by John Derbyshire “The Golden Key”. Euler argued that the sum


it is equivalent to the product

\frac{2\ 3\ 5\ 7\text{...}}{2\ 4\ 6\text{...}}

I interrupt the flow of numbers here to tell an apocryphal story about Euler. It concerns the French philosopher Denis Diderot was visiting Russia on Catherine the Great’s invitation. Euler who was employed at the Academy of Saint Petersburg was asked as a brilliant man and a committed Christian and apologist for Christianity to confront the philosopher, whose arguments for atheism were influencing members of the Empress’ court:

Diderot was informed that a learned mathematician had produced a proof of the existence of G-d: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced this non-sequitur: “Sir


hence G-d exists – reply!” Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress.

Contrary to legend, Diderot was mathematically literate, and actually wrote on this subject, but to most people mathematics -at least mathematics of this sort- really is gibberish. The non-mathematical takeaway from Euler’s argument is that the process of subtracting the reciprocals of prime numbers and all of their multiples from the series 1+1/2+1/3 +1/4… is the process of contracting the series to 1. But unlike subtracting 8 from 9 to make 1 this is no ordinary subtraction. The number 1, after these subtractions have been performed, is undivided. Because every integer is either prime or the product of primes -the primes have been called ‘the atoms of arithmetic’- it follows that the primes are distributed throughout the number line in a manner comparable to that in which a fixed quantity of dye is watered-down and thereby is able to spread out. When there is no water, a state of infinite prime-density exists, and at this point the very distinction between the physical and the mathematical, between the world and the mind that surveys it, vanishes. The principle governing the distribution of the prime numbers in the number line is therefore the principle governing the co-existence of the plurality and singular, and in the same way that there can be no denying either composites or primes, there can be no denying either the plural or singular aspects of this principle. Without both plurality and singularity, there can be no numerical distinctions, and the apparatus of logic and philosophy in particular, and the phenomenon of consciousness in general, are conditional on the existence of these distinctions.

Philosopher Alvin Plantinga has developed a neat form of this general ‘ontological’ style of argument which goes as follows:

(1) G-d = the being that exists in all possible worlds;
(2) It is possible that G-d exists, i.e. a being satisfying this definition exists in at least one possible world;
(3) But if G-d exists in one possible world then he exists from (1) in all possible worlds;
(4) And if G-d exists in all possible worlds he exists in the actual world.

This, I was once told by an atheistic logic professor, is the closest thing to a proof of G-d’s existence, we will ever see. The counter-argument is simply that since it is possible that G-d doesn’t exist, then we can using the same line of argument draw the contrary conclusion. But is the possibility that G-d doesn’t exist a legitimate possibility? If we identify G-d with a state of infinite prime-density, the possibility of the non-existence of G-d is equivalent to the possibility of arithmetic without prime numbers. And if prime-density is to energy-density as the numerical is to is the physical, then the possibility of the non-existence of G-d is equivalent to the possibility of a material world without energy. So as a conscious material being I am committed to the reality of an underlying state of infinite prime and energy density.

Consider the act of cutting a pie into smaller slices.

It will readily be agreed that all the pieces of a pie add up to the whole pie, to be represented by the number 1. It is clearly not true however that 1/2 the pie + 1/3, of the pie + 1/4 of the pie… add up to 1. These fractions add up to a sum greater than 1 because they overlap, because they involve repeated quantities of pie. 1/2 the pie contains 1/3 and 1/4, and every smaller fraction of the pie, and so there is no sense in using this form of addition if one wants to combine the fractions of the pie to arrive at the correct total. The appropriate method of summing the parts of the pie involves adding its unique divisions. In the example above


A unit can be subject to infinite divisions, but if we follow this rule we find that no matter how often or by what means we divide the pie the sum will always converge to 1

\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)\text{...}\to 1

When we add all the slices of a physical pie in the manner described we arrive at 1, but it is clearly not a unified 1, it is a 1 shot through with gaps, and a tortuously complex affair. A unified 1 is what remains if all prime divisions are removed. This is the thought we need to distinguish a physical pie and a metaphysical pie: it must possible to operate on the slices of a metaphysical pie in such a way as to re-form a single unified whole. Any mathematics that is inadequate to this task is not the mathematics written in Galileo’s the book of nature. Otherwise put, the slices of the metaphysical pie are unlike the slices of a physical pie in that they don’t exist in a spatio-temporal/arithmetic framework, rather they are the pieces of this framework. They are not like pieces of a pie, but like the pieces of the pieces of a pie. They have the property that they are proto-arithmetic.  That’s the simple way to look at something a little more technical. We can turn Euler’s Golden Key by using mathematics surrounding the concept of an area unit. If a circle is supposed to have an area of 1, then a light source E located at the center will posses the same strength from center to circumference for E/1 = E. This is the same thing as saying that there is no difference between center and circumference. If there is no such difference, then either the circle has no area and no radius, or infinite area and an infinite radius; E either has either infinite strength or no strength. But in reality the strength of E always lies between these extremes. By the \pi r^2 formula, we know that a circle of area 1 has a radius of  \frac{1}{\sqrt{\pi }} or 0.56419, but there is no clear way to extract sufficient variation from \pi to permit a potentially infinite number of energy levels less than infinity but greater than 0. \frac{1}{\sqrt{\pi }}\approx e^{-\gamma }=\frac{1}{\sqrt{e^{2 \gamma }}}=0.561459. Since gamma is the limit

\gamma =\lim_{x\to \infty } \left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)

a measure that looks better able than \pi to perform the function we require is e^{2 \gamma }. Gamma is the limit of a potentially infinite number of values, so instead of

\pi \sqrt{\frac{1}{\pi }}^2=1

we may write

\lim_{x\to \infty } \left(e^{2 \gamma } \sqrt{\frac{1}{e^{2 \left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)}}}\right){}^2=1

and redefine \[Pi] whilst effectively preserving its traditional value. From here it a hop, skip and a jump to the generalization

\lim_{x\to \infty } \left(e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)} \left(\left(\frac{1}{\exp \left((s+1) \left(\sum _{n=1}^x \frac{1}{n^s}-\int_1^x \frac{1}{n^s} \, dn\right)\right)}\right){}^{\frac{1}{s+1}}\right){}^{s+1}\right)=1

And to a distinction between a spiral that unfolds forever (s = 1), and a spiral that unfolds for a finite time and the begins to intersect itself:

We can say the following: if s = 1, then there is an approximately symmetrical relationship between energy and space, and the new formula will yield predictions which are similar to those yielded by 1/r^2. If however s != 1, the balance is strongly tipped toward energy (extreme example the singularity of concentrated light at the root of the universe), or conversely toward space (extreme example the interior of black holes), and the new formula makes entirely different predictions than 1/r^2. When s !=1, the region of space described by the new law curves back on itself. In these light or space dense environments, curvature -as a function of density- is far greater. But these non-classical, short-ranged, extremely curved environments are aspects of classical, long-ranged, and moderately curved environments of the solar system and every day life. To turn Euler’s key consider that

e^{2 \gamma } \sqrt{\frac{1}{e^{2 \gamma }}}^2-e^{2 \gamma } \sqrt{\frac{1}{e^{2 \left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)}}}{}^2\approx \frac{1}{x}

and that 

Then we create two kinds of matrices, one such that the values on the center diagonal are the terms of the harmonic series, and therefore a potentially infinite matrix:

U=e^{2 \gamma } \sqrt{\frac{1}{e^{2 \gamma }}}^2

0=-\left(e^{2 \gamma } \sqrt{\frac{1}{\exp \left(2 \left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)\right)}}\right){}^2+\epsilon +e^{2 \gamma } \sqrt{\frac{1}{e^{2 \gamma }}}^2+1

\left( \begin{array}{cccccc} 0 & -U & -U & -U & -U & \ldots \\ U & 0 & -U & -U & -U & \ldots \\ U & U & 0 & -U & -U & \ldots \\ U & U & U & 0 & -U & \ldots \\ U & U & U & U & 0 & \ldots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{array} \right)

u=e^{2 \gamma } \left(\left(\frac{1}{e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)}}\right)^{\frac{1}{s+1}}\right)^2

0=-e^{2 \gamma } \left(\left(\frac{1}{\exp \left((s+1) \left(\sum _{n=1}^{p_x} \frac{1}{n^s}-\int_1^{p_x} \frac{1}{n^s} \, dn\right)\right)}\right){}^{\frac{1}{s+1}}\right){}^2+e^{2 \gamma } \left(\left(\frac{1}{e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)}}\right)^{\frac{1}{s+1}}\right)^2+\frac{1}{s}+\epsilon

\hbar = the point at which the gap between e^{2 \gamma } \left(\left(\frac{1}{e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)}}\right)^{\frac{1}{s+1}}\right)^2 and e^{2 \gamma } \left(\left(\frac{1}{\exp \left((s+1) \left(\sum _{n=1}^{p_x} \frac{1}{n^s}-\int_1^{p_x} \frac{1}{n^s} \, dn\right)\right)}\right){}^{\frac{1}{s+1}}\right){}^2+\frac{1}{s} ceases to decrease.

\left( \begin{array}{cccccc} 0 & -u & -u & -u & \ldots & -u \\ u & 0 & -u & -u & \ldots & -u \\ u & u & 0 & -u & \ldots & -u \\ u & u & u & 0 & \ldots & -u \\ \vdots & \vdots & \vdots & \vdots & \ddots & -u \\ u & u & u & u & u & \hbar \\ \end{array} \right)

It is illustrated below that the second kind of matrix describes quantities that are classically finite:

This is a rearrangement of the essence of Euler’s argument. The difference is that matrices of the second kind explicitly involve a finite number of unique terms of the harmonic series. What happens if per impossible we eliminate all of the finite matrices, which are aspects of the infinite matrix? Same thing that happens if per impossible we eliminate all of the reciprocals of the primes and their powers which are aspects of the harmonic series – we reduce the infinite matrix to a state we can associate to 1/0 or physically to an infinite concentration of energy. These finite matrices describe units but they are not classical units -units that can be subject to an indefinite amount of scaling- but proto-units. They describe, not the parts of the pie, but the parts of the parts of the pie. This distinction between potentially infinite and strictly finite matrices is essential to an understanding of why evolution is a flawed theory of origins because the earliest event in the universe cannot, in the light of it, have been simple. It must have possessed the same order of complexity as the present-day universe, meaning that it can only have come into existence at once, and by an act of creation. Richard Dawkins in The Blind Watchmaker focuses on the eye, often cited by detractors as something that cannot have come about as the result of a stepwise process, and is at pains to argue the contrary case. The classical universe is like a giant eye that cannot have come about as a stepwise process, and so the premise on which all of Dawkins’ arguments in The Blind Watchmaker rest -that complexity is reducible to simplicity- is false and so all of these arguments are unsound at their root. This reduction gets stuck at the moment of creation, which is the moment when the prime and energy density of the classical universe is at a maximum. There is a gap between G-d, whose kingdom if you will lies in the realm of infinite light, and the darkening kingdom inhabited by man. Dawkins doesn’t appreciate what is perhaps the most important truth behind the engineering of the physical universe: the classical/quantum divide is the divide between the arithmetic and the proto-arithmetic, and therefore the quantum domain doesn’t as he imagines exist on the time line line of the universe studied by biologists and scientists that study large scale objects – it lies outside and beneath this time line. The classical universe can’t be scaled down to quantum size, and nor can the quantum universe be scaled up to classical size, and to suppose otherwise is to make a category mistake.

The infinitely prime and energy-dense state to which we are led if seek to trace either the origin of mathematical or the physical world satisfies the traditional definitions of G-d that at first glance make this definition so tricky – immateriality, omnipotence, omniscience, omni-presence, immanence and transcendence etc.. Every day objects can not possess these properties, but energy can: to energy belongs all-power, all-knowledge, energy is everywhere, and is both immanent within the world but transcends any one locality. And energy is nothing if not the ground of being. Moreover, it does not follow that because energy is to be identified with knowledge and power and the good things in life that it is also to be identified with ignorance and weakness and bad things – the passage from a state of pure energy to a state that includes evil is due to matter, which is inherently spatio-temporal in nature, and exists only on a basis of the absence, of the privation of energy. Contrary to Anthony Flew in the Presumption of Atheism, Plantinga’s argument, with supplementation from 18th century mathematics, and the isomorphisms there are between number theory and physics, places the burden of proof in all debates about G-d’s existence squarely on the shoulders of the negative side.


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Dawkins, R (1986) The Blind Watchmaker

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Euler, L (1737), Variae observations circa series infinitas (Various observations concerning infinite series)

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Plantinga, A (1974) The Nature of Necessity

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