A Bayesian Look at the Physico-Theological Argument (Column 506)

With God’s help

Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.

The next two columns are devoted to different perspectives on the High Holy Days. The first will deal with a mathematical “coronation” of God—naturally, in honor of Rosh Hashanah—and the second will take a different, more topical angle on processes of repentance, in anticipation of Yom Kippur. In this column I’ll discuss the existence of God, at the intersection of physics, philosophy, and probability.

Over the past two weeks, two recordings were published of a discussion I had with Prof. Elam Gross on physics and God (see Part A and Part B). In preparation for our conversation I read a paper he published in 2011 in Odyssey on science and God. The main points came up briefly in our talk, and here I wanted to expand on them a bit more. The second part of the column is a little formal, but it really requires no special background, and I think it provides important tools for probabilistic thinking about various claims and stances.

“God of the gaps”

Gross devotes a fair bit of his article to the “God of the gaps.” This is the common label for an approach that tries to prove the existence of God from gaps in scientific knowledge—i.e., if science can’t explain something, that proves there is a God. Quite a few creationists adopt this approach and rejoice at every scientific gap they think they’ve found. But among sober religious thinkers today, it’s widely accepted that this approach is both incorrect and unhelpful. The arguments for why it is unhelpful are simple (because when new scientific knowledge is discovered that fills the gap, people will conclude there is no God). But for me these are irrelevant considerations. The important question is whether these arguments are correct—not whether they are tactically helpful or harmful.

On the substantive level, one can challenge such arguments in several ways. First, the fact that we don’t understand something says nothing about its origin. There are scientific processes we don’t yet know or understand. Does that mean their source is God? Conversely, is something we do understand not from God? Second, scientific knowledge accumulates over the years, and therefore the existence of a gap only tells us we should increase research efforts to try to close it. In the past, people could “prove” the existence of God from the whole world because they understood almost nothing scientifically. Today our scientific knowledge is far broader, and so the “proofs” of old are now revealed as nonsense. There’s no reason to think the same won’t happen to contemporary “proofs” that rely on today’s gaps.

This is why I oppose—both tactically and substantively—the war that creationists wage against evolution. Not because all their claims are wrong (as the neo-Darwinists assert). Some of them are good claims. My problem is the very methodology of building faith in God on gaps in scientific knowledge. Incidentally, for the very same reason I do not accept the parallel assumption of atheist neo-Darwinists that if evolution explains everything then there is no God. They are making the same mistake as their fundamentalist creationist opponents. God is not part of the scientific explanation and is not supposed to be. Therefore, a lack of explanation does not prove God’s existence, and by the same token, the presence of an explanation does not render Him unnecessary.

Gross on “God of the gaps”: an immanent God

Gross recounts a meeting he had with several other physicists who were dealing with God and science:

In 2007, five physicists sat down for lunch during a conference on the statistics of hypotheses related to the search for the Higgs boson. We discussed a book titled “The Probability of God,” which had come out at that time and purported to calculate—using Bayes’s formula—the probability of God’s existence.

Although we all dismissed the book as utter nonsense, the conversation evolved and we found ourselves sliding into a heated discussion. After hours, we reached a kind of agreement on the question: “What is the scientist’s God?”

I don’t know the book in question, and as for Bayes’s formula as a proof of God’s existence, I’ll get to that below. Immediately thereafter he continues and describes that group’s attitude toward the “God of the gaps”:

Gaps in knowledge are the intellectual food of every scientist. Wherever a question is posed or a theoretical problem in an explanation is discovered, that is where the scientist steps in to propose a solution. Thus, for example, Marie Curie studied Becquerel rays, which turned out to be radioactive radiation. Scientists seek to close these gaps in knowledge and to offer explanations for phenomena that previously were explained only by invoking the power of God.

“God of the gaps” is therefore threatened by science. Even if St. Augustine said it in jest, he didn’t say for nothing that before God created the heavens and the earth, He created hell for those who delve too deeply into inquiry. Scientists necessarily shrink the space left for the “God of the gaps.”

If so, can a religious scientist believe in the “God of the gaps”? In my view the answer must be no. The “God of the gaps” is transcendent by nature and definition. He is disconnected from corporeal man and is almost withdrawn. This is a God who doesn’t want the accelerator to work, because the accelerator’s work would close a huge gap in knowledge, thereby necessarily reducing His presence.

I’m entirely with him in these paragraphs. I truly do not believe in a “God of the gaps” who feels threatened by science. A scientist, like anyone else, should not build his faith on gaps in scientific knowledge. Such gaps should spur vigorous research to try to close them, as I explained above.

But then he immediately continues and speaks of the only alternative available to the believing scientist:

By contrast, there is another possibility—an immanent God. This God is personal, fills only the gaps in the soul, and has mystical roots; therefore it is hard to argue with Him or to support Him scientifically. If a scientist has a God, He must, therefore, be immanent.

For him, if God does not belong to the scientific plane, what remains is a subjective, personal God that pertains only to the experiential-mystical realm.

First, a note on terminology. Gross calls such a conception “immanence,” but that is not what is usually meant by this term. In religious and philosophical thought there are two different conceptions of God: transcendence (God above and beyond the world—what Kabbalistic terminology calls the “Sovev”) and immanence (God within the world—the Kabbalistic “Memalei”). Gross doesn’t mean either of these, since both address a God who exists in objective reality and whose existence is a factual claim. He is speaking of a psychological-experiential God (who himself could be immanent or transcendent).

Talk of such a God can be read in two ways: (1) factual atheism (since factually and scientifically there is no God) with a religious feeling (which, as is known, exists among atheists as well); (2) the feeling and mysticism are non-scientific indications of God’s factual existence. As noted, option (1) is atheism in disguise, but option (2) is quite common among believers and especially among perplexed scientists whose science contradicts their beliefs (in our conversation he mentioned several such cases). The “obvious” way out for them is to move the discussion from the plane of facts and science to mysticism.

All this may be true for certain people, but Gross presumes a necessary dichotomy, and in that he errs.

The status of philosophy

Gross assumes that if one cannot bring scientific evidence for God’s existence, what remains is mysticism and subjectivity. But he ignores another, third category that is very important for our purposes: philosophy. There can be philosophical arguments for God’s existence even if one cannot reach Him by scientific means. Philosophy is not mysticism, and it does not necessarily belong to the subjective-experiential plane (see my series of columns defining what philosophy is, 155 – 160).

In my book God Plays Dice, in the third conversation of my book The Prime Existent, and in many other places, I drew a distinction between a physico-theological argument “within the laws” and one “outside the laws.” The “within the laws” argument is essentially the “God of the gaps”: gaps in scientific knowledge—cases or phenomena that the natural laws we know cannot explain—lead to the conclusion that there is a God. This is an argument within the laws, since it keeps the discussion within the scientific framework governed by natural laws. But as I suggested there, the more up-to-date physico-theological argument is an argument outside the laws. If the world indeed contains special natural laws that allow for chemistry, biology, and life, that itself shows there is a God who created them. Note that this argument does not play inside the framework of the laws but outside it. It hangs on the question of why the laws are as they are at all, and not on what happens within the framework of the laws. I won’t get into the refutations and details of the argument, as I’ve discussed it extensively elsewhere. My purpose here is merely to situate it within the discussion and explain its place in it.

This argument does not belong to the “God of the gaps” genre, since it is not built on a gap in scientific knowledge. Science does not deal with explaining the laws but with explaining phenomena by means of the laws (Karl Hempel’s deductive-nomological schema illustrates this well). Even when a new natural law is discovered, it is done by constructing a framework that allows us to explain phenomena, not on the basis of more fundamental laws that explain the new law. Moreover, even if at some point Einstein’s heart’s desire were fulfilled and a unified field theory were discovered—that is, a single comprehensive law responsible for all of physics (and perhaps through it for all natural laws in general, if you’re a reductionist)—that law itself would still require explanation.[1] That explanation would not be scientific but philosophical, since a scientific explanation is always given in terms of some system of natural laws. The explanation for the laws of nature themselves is the province of philosophy. In other words, this is a philosophical gap that cannot, in principle, be closed by science; therefore it is not a “God of the gaps” argument.

A note on infinite regress

I said I wouldn’t get into the details of the argument and refutations of it. Still, I’ll briefly remark on a point that arose in our discussion. Arguments of this type assume that if there is a world, there is probably something that created it. The alternative is infinite regress—i.e., something created the world and something else created that, and so on ad infinitum. In the second conversation of my book The Prime Existent I explained why such an explanation is unacceptable to most philosophers. I argued there that in my view it’s not only unacceptable; it’s not an explanation at all but an evasion of explanation. The best-known parable that illustrates this is William James’s story about the Greek physicist who explained to his listeners that the world stands on the back of a giant turtle. When asked what the turtle stands on, he calmly replied: on another turtle. And so he was asked again and again until his patience ran out: Don’t you understand?! It’s turtles all the way down.

In the first part of our talk Gross mentioned this story and implicitly rejected the “turtles all the way down” thesis, but that didn’t stop him in the second part from wondering where I got the assumption that infinite regress is a problematic type of explanation. Later he even proposed the thesis of a “breathing universe,” i.e., universes that are created and perish one after another (although he conceded there is no empirical hint of this and that it doesn’t constitute a sufficient explanation)—which is just another version of “turtles all the way down.” Proposing such a picture indicates that in his view the existence of the universe indeed requires explanation (hence we generate other universes that begot it), while at the same time he ignores the problematic nature of infinite explanatory chains.

Good, we’ve now reached our topic, whose title, as mentioned, is Bayesian reflections on the physico-theological argument.

Prelude: conditional probability

Conditional probability and Bayes’s formula have come up more than once in my columns (see, for example, column 402). So here I’ll review them only briefly.

In general, the probability of any event depends on the prior information I have about it. Additional information changes the probability of the event. If we look, for example, at a fair die, the probability that it lands on 5 is 1/6 (because that’s one of six possible outcomes). But if I know the result is odd, the probability rises to 1/3 (one of three possibilities). Additional information changes the probability of every event and usually improves my ability to predict it in advance.

Let’s denote the event “the die landed on 5” by A. Its probability is P(A) = 1/6. Denote the event “the outcome is odd” by B. The conditional probability that the result is 5 given that we know it was odd is: P(A|B) = 1/3. This is the probability of A given B. It is, of course, different from the probability of A without any prior information (other than that the die is fair—which itself is prior information).

But conditional probability is tricky. There is a difference that’s very easy to miss between the opposite conditional probabilities: P(A|B) and P(B|A). For example, if we ask what is the probability that the result was odd given that we know it was 5, the answer is of course 1 (we have full information, so we can know the answer with certainty). Thus, there is no simple relationship between conditional probabilities when you reverse the order of the variables. Consider another example. Suppose I don’t know whether the die is fair. Denote the claim “the die is fair” by C. Assuming the die is fair, what is the probability of an odd result? The answer is, of course: P(B|C) = 1/2. Now ask the reverse question: given that half the results were odd, what is the probability that the die is fair? That is the conditional probability P(C|B). It is very hard to know. It depends on the number of rolls, on the nature of the die’s unfairness, and more. Consider a concrete example. Suppose we rolled a die twice and got 1 and 4. Does that tell us the die is fair? Not at all. Such a tiny number of rolls tells us almost nothing about the die’s nature.

The important lesson for our purposes is that knowing one conditional probability does not tell us much about the reverse conditional probability. These matters are very confusing, and I have given other examples in the past (see, for example, columns 144 – 145, 176, and more).

Another prelude: Bayes’s formula

The way to reverse the order of variables in conditional probability is by means of Bayes’s formula (or the formula for total probability). To understand it, we need to consider the probability that both events A and B occur, P(A∧B). This can be calculated in two different ways: (1) the probability that A occurs times the conditional probability that B occurs given A; (2) the probability that B occurs times the conditional probability that A occurs given B.[2] Thus, the results of these two calculations must be identical:

P(A∧B) = P(B|A)·P(A) = P(A|B)·P(B)

We have found a relationship between the conditional probability P(A|B) and the reverse conditional probability P(B|A). Naturally, the relationship depends on the unconditional probabilities of A and B. For example, in the case above (A = the result is 5; B = the result is odd), we know three probabilities: P(B) = 1/2, P(A) = 1/6, P(B|A) = 1. From this you immediately get the fourth: P(A|B) = 1/3. That means that if the result obtained was odd, the chance it was a 5 is 1/3.

Elam Gross and colleagues on conditional probability and Bayes

We saw above that the five physicists in that story mocked the attempt to prove God’s existence using Bayes’s formula and conditional probability. In another passage, Gross goes into more detail:

The hypothesis of God’s existence should be tested by the scientist by answering the question: given the data we have (the Earth, the Sun, the Moon, stars, a galaxy, cosmic background radiation, and so forth), what is the probability of God’s existence?

We must formulate our questions in a way that they can be answered, and in both cases—physics and theology—the questions are: assuming there is a Higgs boson, how well do the collision data fit its existence? Or: assuming there is a God, how well does the amazing universe that envelops us fit the fact of His existence?

In mathematical terms we can also ask: what is the probability of the universe’s existence assuming God exists? That, of course, is a trivial question whose answer is 100%. That is not the question that interests us. The question that interests us is the reverse: what is the probability of God’s existence given the universe as we understand it?

Up to here, he is pointing out a very important nuance that’s easy to miss. The question “given that I see a world as it is, what is the chance that God exists?” is quite different from the reverse: “given that God exists, what is the chance the world would be as it is?” These are two different questions, since they are conditional probabilities with the variables reversed. However, he errs on the number. He claims the answer to the second question is 1 (trivial), but that’s not correct (at least if we have no further knowledge about God). God could have decided to create a different world or not to create a world at all; in that case God would exist but not this world—or any world. Still, the distinction itself is, of course, correct. So what do we do with this?

Now he explains the connection to the physico-theological argument:

The classic “return to religion” argument says: Look how beautiful the butterfly is; look how complex the human being is—could such things have arisen by chance? Of course not! And behold a proof of God’s existence. In other words, given such a wondrous universe, one must posit a God who created it.

This argument relies, of course, on a common statistical mistake—that the probability that, given the butterfly, God exists equals the probability that, given God, the result is a wondrous butterfly. The latter probability is 100%, but the former is certainly not.

This is truly bizarre. First, no one claims that the physico-theological argument yields probability 1. There is a very small chance that the world arose by chance, and therefore there is no necessity that God exists. But beyond that, the truth is exactly the opposite: the physico-theological argument asks the first question—given the butterfly, what is the chance that God exists? And as we saw, the reverse question (given that God exists, what is the chance the butterfly would be as it is?) is ill-defined (it’s not a statistical question at all; the world that would come to be is the world God decides upon—there are no random processes there). The physico-theological argument, in its standard formulation, does not tie these two questions together. It merely says that the chance that something complex arose by accident is negligible, and therefore it is reasonable that it did not arise by accident. What’s wrong with that argument? What exactly is it “reversing”?

He continues:

Bayes’s theorem provides the link between the two probabilities. The probability that the butterfly is beautiful given God must be multiplied by the probability that God exists at all, since if there is no God there’s no point in any probabilistic discussion. The probability that the universe’s existence points to God’s existence equals the probability that, given God, the universe exists—but only on condition that God actually exists.

The solution he proposes is that the “religious persuader” failed to take into account the prior probability that God exists. As we saw above in Bayes’s formula, reversing the direction of conditional probabilities depends on the unconditional probabilities of the two events. That’s certainly true, but it has nothing to do with the “religious persuader’s” argument.

I’ll now try to employ the principle of charity (see column 440) to propose a plausible reading of what Gross and his colleagues might have meant—i.e., the connection of the argument to conditional probability and Bayes’s theorem. In the course of the analysis I’ll clarify why, even so, they are not correct.

First formulation

Let me introduce two propositions: A = God exists. B = a complex world exists. Now let’s ask what probabilities, conditional or not, we can assume for these events. The unconditional probability that God exists, P(A), is of course unknown to us. The probability that a complex world exists is P(B) = 1. As for the conditional probabilities, that’s trickier. We seek P(A|B), the conditional probability that God exists given that a complex world exists. We saw above that, contrary to Gross’s claim, the reverse conditional probability P(B|A) is unknown and not really well-defined. So what do we do?

Note that we do know another conditional probability that is also relevant. We know that without God, the chance that a complex world exists (i.e., that it arises randomly on its own) is very small. This is the assumption of the physico-theological argument. Thus: P(B|¬A) = x, where x is very small.

But for our discussion what matters is the reverse probability: P(¬A|B). That is, the chance that there is no God given the existence of our complex world. Why is this useful here? Because the complementary probability follows from it very simply: P(A|B) = 1 − P(¬A|B). If so, let’s try to compute the first probability using Bayes’s theorem. From above we have:

P(B|¬A)·P(¬A) = P(¬A|B)·P(B)

Substitute our data to get:

P(¬A|B) = x·P(¬A)

Remember that every probability (conditional or not) lies between 0 and 1. That means the product on the right-hand side is very small. Hence, the conditional probability that God does not exist given that our complex world exists is very small. Therefore, the conditional probability that God does exist under these circumstances is very large (almost 1):

P(A|B) = 1 − x·P(¬A)

Q.E.D.

So there is a connection here to conditional probability and Bayes’s theorem, and even to reversing variables in conditional probabilities. But ultimately the physico-theological argument does work, and the unconditional probabilities don’t play a major role in this analysis. Therefore I’ll suggest another formulation in which all those components do play a part.

Second formulation

From another angle, we can use the formula for total probability. This is just a further development of Bayes’s theorem. The probability of B (the world’s existence) can be written as the sum of two cases: assuming ¬A and assuming A (by the Law of the Excluded Middle, there is no third option). Thus I write it as a sum of two possibilities:

P(B) = P(B|¬A)·P(¬A) + P(B|A)·P(A)

If we substitute this into Bayes’s theorem, we obtain the formula for total probability (in the case of only two possibilities):

P(¬A|B) = P(B|¬A)·P(¬A) / { P(B|¬A)·P(¬A) + P(B|A)·P(A) }

Let’s try to estimate this probability. It obviously depends on the unconditional probability of God’s existence or non-existence (clearly P(A) + P(¬A) = 1). If we assume that the unconditional probability of God’s existence is very small (as Gross claimed in Part B of our conversation), you’ll immediately see that on the right-hand side there’s a contest in the denominator between two small factors, P(A) and x, and a small factor in the numerator, x. The result depends on the ratio between these two; and if the probability of God’s existence is much smaller than the probability of the world’s random arising (as Gross claimed), then the conditional probability of His existence can also be small—that is, the conditional probability of His non-existence is large. If the probability of His existence is large (as I claimed), the conditional probability naturally comes out large as well.

Here there is a very important step in the discussion. I told Gross that his assumption—that the probability of God’s existence is much smaller than the probability of the world’s random arising—is a begged question. He assumes there’s no chance that God exists, so it’s no wonder he concludes that the proof is poor and the conditional probability of His existence is small. Of course I too beg the question when I say that the probability of His existence is large. What can we do to obtain a meaningful argument that does not beg the question?[3] I suggested to him a simple model that assumes both probabilities are 1/2. Essentially, we assume we have no a priori knowledge regarding God, and we ask whether the total-probability formula can move us toward one of the two outcomes (that He exists or does not). This seems a very reasonable model to test the quality and implications of the argument.

So now we need to substitute into the last equation: P(A) = P(¬A) = 1/2. This factor cancels in numerator and denominator. We get:

P(¬A|B) = P(B|¬A) / { P(B|¬A) + P(B|A) } = 1 / { 1 + P(B|A) / P(B|¬A) }

Recall now that P(B|¬A) is tiny, and that we have no information about the value of P(B|A). Clearly, the probability that the world exists is higher when God exists (whatever His nature) than when He does not. Therefore, the fraction on the far right is a very large number, and the result for the conditional probability that God does not exist, given the existence of our world, is likely tiny. That is the physico-theological argument. Q.E.D. Incidentally, in Gross’s view P(B|A) = 1, in which case this proof would be perfect. As noted, that assumption lacks basis, but there is some logic to assuming it when we search for an alternative to random emergence: the alternative is that there is an agent who wanted precisely this world and therefore created it as it is. If that’s the assumption, then indeed the probability is not merely much larger than the probability of emergence without God—it’s actually 1.

Either way, this analysis truly integrates conditional probabilities and the importance of the difference between them, and its result depends on the unconditional probabilities. It therefore seems plausible that the Five Musketeers had this claim in mind. Yet if one does not beg the question regarding the unconditional probability of God’s existence—as they likely did—the conclusion of the calculation is that the Bayesian formulation of the physico-theological argument safely leads us to the conclusion that God does indeed exist. The Bayesian proof they mocked is excellent. Such an argument will, of course, fail to prove God’s existence to someone who assumes in advance there is no chance He exists. But for one who has not formed a firm stance and remains open to both possibilities, this argument is an excellent proof. That is all one can reasonably expect from a logical argument.

A methodological conclusion

We saw above that Gross and his colleagues mocked attempts to prove God’s existence in a Bayesian manner and claimed that such a proof ignores the difference between the two types of conditional probabilities involved in the discussion. Surprisingly, it turns out they were wrong. For someone who does not come with a fixed, firm stance, this argument does indeed lead him to the conclusion that God exists. Needless to say, every argument rests on premises, and one who does not accept them will not be compelled to adopt the conclusion.[4] That cannot be considered a flaw in a logical argument. A probabilistic argument is supposed to increase the likelihood that the conclusion is true for someone who is open to that possibility, and that is exactly what happens here. If this isn’t an excellent argument, I don’t know what is. There is no ignoring of conditional probabilities here, and Bayes’s theorem is used entirely correctly.

In the previous column I dealt a bit with the disputes between Beit Hillel and Beit Shammai, and with the heavenly voice’s ruling that the halakha follows Beit Hillel. I mentioned R. Yosef Karo’s explanation in his Rules of the Talmud that the halakha was set like Beit Hillel because they were closer to the truth, even though Beit Shammai were sharper and more incisive. The reason is that Beit Hillel’s methodology was to present Beit Shammai’s words before their own, to weigh the opposing position, and only then to form their stance. Such a methodology can improve one’s chances of reaching the truth even if he is less gifted than his counterpart. Those who entrench themselves in a group that hears only one voice and does not seriously weigh counter-arguments are prone to reach false conclusions, even if they are brilliant people.

Gross’s description of the physicists’ meeting aroused precisely these feelings in me. I suspect it was a group whose members all echoed the same way of thinking and were attuned to the same type of arguments, while dismissing out of hand anyone who argued otherwise. In such a situation, what engineers call “positive feedback” (a destructive phenomenon) is created; and therefore, even if they are brilliant people and excellent scientists, they are liable to arrive at mistaken conclusions. If you label other views as “arguments of religious propagandists” and reject them outright because of that label without seriously examining the considerations (as happened at least once in our conversation), then even a group of brilliant people can reach the absurd conclusions we saw here. Note that in this case it was a simple mistake, and I assume that without the bias they wouldn’t have made it. Let that teach you how important it is to hear opposing opinions and weigh them seriously.

A word to readers of Haaretz, readers of Makor Rishon and Besheva, and all the victims of Google’s algorithms who are fed day and night with information continually tailored to their views and inclinations and who are always given positive feedback. It may be pleasant, but it is truly destructive (I discussed this in column 335 and also in columns 450 – 451). You have been warned!

[1] In the books mentioned I explained that the alternative is that such a law would be logically necessary—i.e., it would be a product of mathematics and logic alone, with no need for observations. If that were to happen, science would become a branch of mathematics and the distinction between empirical science and mathematics would vanish. As far as I know, there is no sane scientist today who seriously believes this. Therefore I ignore this possibility here.

[2] If the two events are independent (i.e., the conditional probability of an event equals its unconditional probability), then both calculations just give the product P(A)·P(B).

[3] In column 461 I discussed the relationship between Bayesian entailment and material implication, and between statistics and logic regarding causality and begging the question.

[4] For example, here we assumed that the probability of the world’s random emergence is very small. That is the basic premise of the physico-theological argument, and there are quite a few atheist challenges to it (which I discussed extensively in the sources cited). But here I am not dealing with the argument as such, only with its Bayesian-probabilistic structure, which is the focus of Gross and his colleagues’ attack. As we have seen, this attack surely fails.