The demonstration of God's existence on scientific and mathematical grounds is a topic that, after captivating thinkers like Anselm and Gödel, reappears in the recent book by Bolloré and Bonnassies. However, the book makes a completely inadequate use of science and falls into the logical error common to all arguments in support of so-called "intelligent design."

In the image: detail from *The Creation of Adam* by Michelangelo. Credits: Wikimedia Commons. License: public domain

The demonstration of God's existence on rational grounds is a subject tackled by intellectual giants, from Anselm of Canterbury to Gödel, including Thomas Aquinas, Descartes, Leibniz, and Kant. However, as is well known, these arguments are not conclusive. It is not surprising, then, that this old problem, evidently poorly posed, periodically resurfaces.

Among the latest attempts, the book by Bolloré and Bonnassies [1] seems to have gained particular success. Published in 2021, it was translated into Italian in the spring of 2024, and we continue to read glowing reviews of it in the press and digital media. The subtitle speaks of “the dawn of a new revolution.” But is there anything new to say on this subject? Are there scientific breakthroughs capable of filling the gaps left by previous attempts? While the book has sold well and attracted interest, it does not seem that there are any real novelties compared to Anselm (who died at the beginning of the 12th century). What we find instead is a rhetorical and inappropriate use of some scientific results, along with a logical error common to all arguments supporting so-called "intelligent design."

## Some Coordinates

The authors of the book argue that, based on science, it is now possible to unequivocally prove the existence of a creator deity. At the opposite end of the spectrum, people like Daniel Dennett assert without hesitation that religious faith is a form of mental illness [2]. To avoid confusion and unnecessary controversy, we do not believe that either of these two positions is necessarily correct. Religion, or more broadly spirituality, undoubtedly holds importance for some individuals, but it does not need to have a scientific basis. And it is certainly understandable that those who cherish both faith and science might find it stimulating to justify their belief on scientific grounds. However, the blending of faith and science does not always yield positive results. A remark by psychiatrist Giovanni Jervis seems to illustrate the general point well: “The longing for infinity is respectable, but if it does not turn into high poetry, it immediately becomes something lower and banal, namely sentimentality and rhetoric.” [3]

Bolloré and Bonnassies persist with rhetoric as old as it is inconclusive: because there is a long list of important believing scientists, it is entirely possible to prove the existence of a creator deity on scientific grounds. Needless to say, from a logical perspective, this argument holds no relevance. The personal beliefs of those who made scientific history have nothing to do with rationality. Science is a human activity, born for various reasons and practiced by people who lived immersed in their historical periods. Thus, as we browse through the annals of scientific history, we find that great scientists belong to all possible human categories: atheists (Laplace), religious (Maxwell), bigots (Cauchy), heretics (Newton), warmongers (von Neumann), pacifists (Richardson), conservatives (Gauss), revolutionaries (Landau), and, closer to our time, even racists and sexists (categories excellently represented by Watson, but unfortunately not only by him).

But this is not the only argument of Bolloré and Bonnassies, who develop a line of reasoning they label as “revolutionary.” They acknowledge that for nearly five centuries, scientific discoveries have suggested the possibility of explaining the Universe without the need for a creator God, but they note that recently things seem to be heading in the opposite direction. A particular role is assigned to the discovery of thermodynamics, from which they deduce that the universe is degrading, moving toward thermal death. According to them, this represents a radical change in perspective, from which the existence of a creator deity would follow. Without delving into the history of physics, it is appropriate to frame certain well-known results in their historical context: thermodynamics is not exactly a young science. It emerged in the early 19th century, and its formalization was completed by the end of that century. Similarly, the theme of thermal death, which began with a fundamental work by Ludwig Boltzmann in 1872, was widely discussed by great scientists like Kelvin already in the late 19th century. The hope that thermodynamics might be a *game changer* in the prospects of proving the existence of a creator deity on scientific grounds thus seems misplaced.

## Scientific Proof is Not Suitable for Theology

The issue is not limited to thermodynamics or other areas of scientific research mentioned by Bolloré and Bonnassies. Rather, it lies in the fact that mathematical demonstration and scientific proof, by their nature, are not suitable tools for proving the existence of the God who would have created the object of scientific inquiry itself.

We must first remember that there is no mathematical demonstration or scientific proof that does not begin with some hypothesis assumed to be true. The validity of a scientific conclusion depends on two factors: the correctness of the reasoning and the plausibility of the assumptions. That’s why a fundamental part of scientific work concerns the justification of assumptions.

In mathematical demonstration, definitions and rules of inference are taken for granted. For example, we can conclude with certainty that a number is odd (B) if we know it is not even (A), because we know that all numbers are either even or odd (A or B). But why do we know this? Because we define what it means for a number to be even. For the same reasons, we know that all numbers are either even or odd (assuming we agree on what numbers are). Further questions about what justifies these assumptions are answered by millennia of mathematics and everything successfully built upon it.

Experimental scientific reasoning is more complicated because the truth of the premises always carries a degree of uncertainty, which any scientific proof transmits to its conclusion. Therefore, one of the many things taken for granted in scientific proof is the concept of practical certainty, through which an apparently contradictory move is made: assuming the veracity of the data collected while knowing that they may not be entirely “true.” This makes any experimental reasoning probabilistic.

With these premises, we can ask what form a mathematical demonstration and a scientific proof of the existence of a creator deity would take and what they would assume. Spoiler: they would assume the existence of something very much like the deity (creator).

## Mathematical Demonstrations

Regarding the first, the effort is spared by the greatest of modern logicians, Kurt Gödel. In his mathematical reworking of Anselm's ontological proof, we find a series of very strong assumptions, including one on the necessity of the existence of certain properties that, during the demonstration, lead to the conclusion that properties of the “divine type” necessarily exist. As with all conclusions obtained through mathematical demonstration, Gödel's is persuasive insofar as the truth of its premises is persuasive. In this regard, we leave the word to another logician, certainly less famous than Gödel but among the most brilliant we have had in Italy, Roberto Magari [2]: “In essence […] Gödel correctly deduces his thesis from certain axioms (although we must first agree on what ‘God’ might mean), but there are no more reasons to believe the axioms than there are to directly accept the thesis itself.”

## Scientific Proofs

What would a scientific proof of the existence of a creator deity look like? Let’s sit once again on the shoulders of giants. One of the first examples of a statistical test of a scientific hypothesis, now one of the central tools in the experimental methodology toolbox, was conducted by John Arbuthnot in the early 18th century. By examining London’s baptism records from 1629 to 1710, he noticed that in all 82 years, more boys than girls were registered (and thus likely born). The data appeared to conflict with the hypothesis that boys and girls were equally likely to be born, as if gender were the result of repeatedly flipping a balanced coin. If we assume these hypotheses are correct, a simple calculation shows us that the probability of observing more boys than girls consecutively for 82 years is very, very, very low—1 in 2^{82}. Therefore, Arbuthnot correctly concludes that it is reasonable to assume an imbalance at birth that makes it (slightly, as we now know) more likely to have a boy. So far, so good. But Arbuthnot goes further and asks why this imbalance exists. He finds the answer in what he already believes: divine providence. By introducing more males than females into the community, divine providence allows (among other things) the latter to observe the sacrament of marriage despite the significant losses of the former, who often do not return from war.

## The Rejection of Chance

Arbuthnot's reflection, published in the *Philosophical Transactions of the Royal Society*—one of the first modern scientific journals—contains a logical error that, four hundred years later, we find essentially unchanged in the type of creationist argument known as “intelligent design,” and which can be found throughout Bolloré and Bonnassies' book. The pattern is as follows. We start from experimental observations, which we will call DATA. A hypothesis is formulated that captures the idea that there is no creator deity, i.e., that the DATA we observe are due to CHANCE. Then, P(DATA | CHANCE) is calculated, i.e., the probability of observing the DATA if the CHANCE hypothesis is true. Finally, we assume that this probability is very small, meaning it is extremely unlikely that our observations are due to pure chance. So far, so good, just like in Arbuthnot's case.

The problem arises in the next step, that is, in choosing an alternative hypothesis to CHANCE to explain the DATA. From a logical perspective, the answer is simple: the negation of CHANCE. But from a practical-scientific point of view, it is not at all clear what this means precisely. Let's return to Arbuthnot. The CHANCE hypothesis is translated as equal probability of M and F. Its negation involves the infinite choice of all pairs of non-negative real numbers different from ½ that sum to 1. Since probability takes values between 0 and 1, the possible choices are as many as the points on a line. In this infinity, we will literally find everything, including a creator deity. This is illustrated by a series of fundamental probabilistic results that address the question of whether disorder can be a source of regularity [5-8]. The idea is that in a binary sequence that satisfies a certain precise definition of "chance," for every N, all possible sequences of length N of 0s and 1s exist. Therefore, if we were to transcribe the *Divine Comedy*, *War and Peace*, and *The Odyssey* into binary code, at some point, the *Divine Comedy* would appear followed by *War and Peace*, then *The Odyssey* followed by the *Divine Comedy*, alternating verses from the *Divine Comedy* and *The Odyssey*, and so on through the astronomical number of combinations one can imagine. The binary sequence associated with a random number between 0 and 1 almost certainly contains more material than Borges' Library of Babel, the history of our Universe from the Big Bang to the present day, and everything that will happen until the thermal death of the universe.

Arbuthnot's error, found in creationist arguments up to the one in Bolloré and Bonnassies' book, is the following. Rejecting a hypothesis like "life is due to chance" means opening up an infinity of alternative hypotheses, not all of which make scientific sense. Quite the opposite. It is worth remembering that in scientific reasoning, the plausibility of hypotheses descends from the intricate network of facts sometimes referred to as "theory." When this is insufficient to outline a plausible explanation of the observations, the appropriate scientific attitude is to continue scientific research or suspend judgment. It is worth noting, in closing with Arbuthnot, that to this day, there is no consensus on what explains the established data on the sex ratio at birth. And perhaps there may not be an explanation.

## Improbability: Use with Caution

Arguments that leverage the improbability of observations in light of the hypothesis tested with experimental data are foundational to the construction of scientific knowledge. However, they must be applied with extreme methodological caution. Their inappropriate use, a term covering everything from "carelessness" to "fraudulent behavior," is at the heart of the heated methodological debate on statistical significance [9,10]. A clear heuristic rule emerges. Arguments based on the improbability of the hypothesis one wants to refute become less reliable the further one strays from assertions belonging to quantitative areas of science. This is why, in many legal systems, the presumption of innocence is rightly taken for granted. The burden of proof, that is, falls on the prosecution, not the defense. Unfortunately, there are many examples where this norm of civilization has been violated, leading to convictions based not on reasonable probability of guilt but on the slim probability of innocence [7,8].

#### References

#### [1] M-Y Bolloré e O. Bonnassies. *Dio. La scienza, le prove.* (Ed. Sonda, 2024)

[2] D.C. Dennett. *Rompere l’incantesimo* (Raffaello Cortina, 2007)

[3] G. Jervis, intervista su *Reset*, settembre-ottobre 2007, pag.44

[4] Roberto Magari. *Logica e teofilia in Kurt Göde*l. In: “La prova matematica dell’esistenza di Dio”. 8Bollati Boringhieri Editore, 20069

[5] P. Diaconis and B. Skyrms. *Ten Great Ideas About Chance* (Princeton University Press, 2018)

[6] C.S. Calude and G. Longo. *The deluge of spurious correlations in big data*. Foundations of science 22, 595 (2017)

[7] A. Vulpiani. *Caso, probabilità e complessità* (Ediesse, 2014)

[8] H. Hosni. *Probabilità: Come smettere di preoccuparsi e imparare ad amare l’incertezza* (Carocci, 2018)

[9] V. Amrhein, S. Greenland, and B. McShane, “Retire statistical significance,” *Nature*, vol. 567, pp. 305–307, 2019

[10] D. J. Hand, “Trustworthiness of statistical inference,” *J. R. Stat. Soc. Ser. A Stat. Soc*., vol. 185, no. 1, pp. 329–347, 2022