Mathematics of miracles: when zero meets infinity

According to NASA, there are hundreds of millions of planets located within their stars’ habitable zones in our galaxy — yet life is only known to exist on Earth. The fact that there is no significant evidence of extraterrestrial life suggests that life on other planets either does not exist or is so underdeveloped that we are unable to detect it. Of course, there remains the possibility that an advanced civilization exists somewhere in a distant galaxy, but we simply haven’t encountered them yet — perhaps because it would take an unimaginably looooooong time to travel to our cosmic neighbors for a “sleepover.”

So, the scenario of another intelligent civilization existing somewhere in a different galaxy seems quite possible. And if at least two intelligent civilizations exist in the universe, separated by vast distances, they may have arisen independently through some random sequence of physical, chemical, and biological interactions that eventually led to the formation of organic molecules — and finally, to intelligent life.

There are many scientific estimates regarding the probability of a civilization like ours emerging in the universe, and these estimates vary dramatically — from 10⁻²⁴ to 0.61. Most of them assume that life arose randomly, following the laws of nature. However, despite decades of research and experimentation, scientists have not been able to synthesize a reproducible form of life — even when deliberately attempting to do so. We might only be able to better estimate the probability of life’s random emergence once we ourselves can create reproducible life in a laboratory, since that would allow us to understand the precise steps required for life to appear. And once we know those steps, we could estimate the odds of such an event occurring amid cosmic chaos. For now, let’s assume that this probability is very low — somewhat greater than 10⁻²⁴, but still extremely small.

It becomes even more fascinating when we consider the probability that the Big Bang could produce a universe like ours — complete with planets in habitable zones. This probability ranges from an unimaginably small number (I can’t even type it out; see image below) to about 10⁻²⁴.

This value is known as Penrose’s number — one of the smallest meaningful probabilities ever proposed in physics. If we multiply these two probabilities, the result becomes even smaller, implying that the chances of the Big Bang producing a universe like ours, complete with intelligent life, are almost impossible.

We might compare this to a blind person wandering across a basketball court, tossing the ball randomly, hoping by sheer chance to make a basket. What are the odds that such a person would score? Now imagine instead a sighted, skilled basketball player deliberately aiming for the hoop — how different are their chances of success?

Many would say that the random emergence of the universe and life resembles the blind person’s desperate throws: the probability of success is vanishingly small. In contrast, a universe and life brought into being by a divine Creator resemble the skilled player’s shot — purposeful, precise, and virtually certain to score.

So now we face a dilemma: which is true? Did the universe and life appear randomly through the laws of nature, or were they deliberately created by God? For a random event to succeed, it might require countless failed attempts; for God, it might take only one. But according to Judaism, that’s not exactly the case.

According to the Zohar, one of the most mystical and sacred books in Judaism, God created this world through many attempts: “Before the Holy One, blessed be He, created this world, He built worlds and destroyed them… and He was pleased with the work of His hands when He created this one” (Zohar I:30a-b). A similar idea appears in the Midrash and Talmud. Moreover, the Talmud gives the exact number of previous generations: 974 (Chagigah 12a-b).

If we look deeper into why it took God so many attempts to create the world, we find this explanation: “At first God wished to create the world with judgment, but saw that it could not endure, so He combined mercy” (Bereishit Rabbah 12:15). It’s hard to grasp the meaning of this without translating it into scientific language.

According to Kabbalah, judgment corresponds to the laws of nature — with their limits and restrictions — while mercy corresponds to the miraculous, which transcends those laws. In essence, Judaism teaches a simple truth: the world could not endure through the laws of nature alone; it required the addition of divine mercy — something beyond natural order. And this message aligns beautifully with modern science, which tells us that the probability of our existence is almost zero — yet here we are. Isn’t that a miracle?