Bertrand Russell famously made the argument for the non-existence of God through lack of evidence on his 90th birthday. At his party in London a woman asked him that being so close to death and an atheist what he would do if after death he found God to exist. She asked: “What will you say?” Russell gleamed at the question and, pointing his finger upwards, answered: “Why, I should say, ‘God you gave us insufficient evidence[1].’” This argument has the premises and conclusion:

  • If there were a God then we would have ample evidence of God’s existence.
  • We do not have such evidence.
  • Therefore, there is no God.

 It pragmatically objects to deism on the grounds that only evidence should regulate belief. This essay intends to question this common objection used by atheists because such premise 2 of this argument appears to ignore some of the very obvious evidence in the knowledge that humans have acquired over the past four or five thousand years; that is the role of irrationality and improbability in our reality.

Atheists argue that belief in God is irrational because there is no evidence to support it. The evidentialist argument is formulated thus:

(1) Belief in God is rational only if there is sufficient evidence for the existence of God.

(2) There is not sufficient evidence for the existence of God.

(3) Therefore, belief in God is irrational[2].

Note that the conclusion is not that God does not exist, only that if God did exist it would be unreasonable to believe in God because there is a lack of evidence to support God’s existence. Something should only be believed if it is self-evident, such as the laws of logic and arithmetic. It is argued that belief in God is not self-evident, or evident to the senses, or able to be confirmed. Therefore, as belief in God is not rational, in that it is not supported by evidence or argument, then belief in God must be irrational.

Judgments of rationality and irrationality are sometimes hard to make. Philosophers living over two and half thousand years ago reasoned that all things were made of water and used the evidence available to them to rationally speculate what comprised the main elements of the universe. To us now these ideas seem irrational and absurd when judged against contemporary knowledge and evidence. However, some of the most significant discoveries that are relevant to us today were made by ancient philosophers in mathematics and geometry, with irrational numbers being one of the main.

Irrational numbers were first realized by the Pythagorean School. It is said that Pythagoras (569-500BCE), who devoted his life to the study of mathematics, wrote on the entrance to his school “All is Number”, as Pythagoras believed the whole universe was ruled by numbers. When writing geometric proofs to find the hypotenuse of a square the Pythagorean School discovered irrational numbers. On an isosceles right triangle with a measure of one for each of the legs the hypotenuse will equal the square root of two. However this number could not be expressed as a measured length, as the answer to the square root of two can only be expressed in decimal form with an unending numerical value: 1.41421356237309504880168872420969807856967187537694807317667973799… This discovery disturbed the Pythagoreans so greatly that they refused to recognize the existence of such irrational numbers.  However, the square root of two became known as “Pythagoras’ Constant”[3]. As yet, no one has been able to find the end of this numerical value with the current record for its calculation standing at two trillion digits held by Alexander Yee in 2012[4].

Some irrational numbers are also known as transcendental numbers because they cannot be expressed as the root of any algebraic equation with rational coefficients[5]. It is common knowledge that algebraic numbers are countable while transcendental numbers are uncountable[6]. Joseph Liouville first proved the existence of transcendental numbers in 1844[7]. Transcendental numbers are uncountably infinite[8]. No rational number is transcendental and all transcendental numbers are irrational. Therefore, we assume the existence of irrational numbers whose values are uncountably infinite. While we cannot find the exact value we accept its existence because of obvious proof.

Irrational numbers are essential to humanity as they serve as a major part of industrial mathematics, in the construction of buildings and machines, chemical equations and practical physics[9]. These numbers assist human society’s understanding of our natural world and the universe that contains it. Irrational numbers such as pi or phi (the Golden Ratio) exist as an ideal that can be used in the real world. They have been essential to the development of civilization from the 3rd millenia BCE onward. Without them we would not have been able to land on the Moon. Our use of them is evidence of an intelligent life form existing on this planet.

It appears that there is a very low probability for intelligent life forms in the universe to have evolved[10]. One of the main arguments against it is that there should be many stellar systems within our own galaxy that have intelligent forms of life on them and that, considering evolutionary time frames, we should have been quite obviously visited or contacted by them up until now. In fact, it could be argued that humans are the result of an improbable set of chances. This improbability of intelligent life seems to have given rise to the concept of God, a first cause, an interventionist who created the world out of goodness and who is involved with humans in a cycle of perpetual creativity, rebellion, punishment and redemption[11]. It is this ideal cycle of creation, dissent and redemption that has preserved what some consider an irrational faith amongst those that believe in God and has allowed them to overcome adversity.

But is it irrational to believe in God when it is not irrational to believe in irrational numbers? While we can prove irrational numbers we cannot evaluate them. We assume, or have faith, that they are infinite because, as much as we have tried with the technological capacity that we have, we still have been unable to establish a finishing point. This assumption, or faith, allows us to utilize irrational numbers through approximating them to assist in our own creative output.

Humans have built and designed using the Golden Ratio such things as the Great Pyramid of Giza, Chartres Cathedral, the sculptures of Phidias, the compositions of Michelangelo, da Vinci and Rembrandt and the architecture of Le Corbusier[12], as it is used in nature to build such things as seashells, flowers and pine cones.  The irrational number e was used by Euler as the base of natural logarithms[13] and is used today as an important application in civil engineering, calculus, differential equations, probability and number theory, yet mathematicians today still cannot prove the nature of its number[14]. Again, the irrational number pi has been relevant to humans for four thousand years and its versatility is infinite being found in geometry, trigonometry, probability, statistics, complex numbers, calculus and physics[15]. These are the numbers that allow us to explore the universe and yet, after thousands of years of accumulated knowledge, we still do not understand them because of their infinite state.

The argument that Russell makes is that belief in God is irrational because of a lack of evidence, yet the existence of intelligent life in the universe is so improbable that it could be said to be the evidence of God. Like irrational numbers, there is not sufficient evidence for God but it could be argued that there is enough evidence in the improbability of the evolution of human civilization to assume God’s existence. Like irrational numbers, God is an ideal concept that profoundly assists many humans in dealing with their reality and overcoming adversity. Therefore, as it is rational for humans to accept the existence of irrational numbers, it is also rational for humans to accept the existence of God. This analogous counter argument can be set out so:

  • To believe in God is irrational because there is not sufficient evidence for the existence of God.
  • There is not sufficient evidence for the existence of irrational numbers because they have an infinite number of digits. Yet we theorize they exist as they assist us as an ideal which can be used in the real world.
  • In the same way, belief in an infinite God assists us as an ideal which can be used in the real world.
  • Therefore, God exists as irrational numbers exist.



[3]  Conway, John H.Guy, Richard K. (1996), The Book of Numbers, Copernicus, p. 25



[6] Martin Aigner and G¨unter M. Ziegler, Proofs from The Book, fifth ed., Springer-Verlag, Berlin, 2014, Including illustrations by Karl H. Hofmann. MR 3288091

[7]  Aubrey J. Kempner (October 1916). “On Transcendental Numbers”.Transactions of the American Mathematical Society (American Mathematical Society) 17 (4): 476–482. doi:10.2307/1988833. JSTOR 1988833.

[8] Weisstein, Eric W. “Uncountably Infinite.” From MathWorld–A Wolfram Web Resource.









T. Blackwell,