Our complex relationship with God is reflected in the wide variety of names by which God is known. Two of the most common appear in nearly every prayer we utter: Adonai (“Our Master”) and Melech (“Our Ruler”). As our master and ruler, God issues commandments and, more broadly, rules that govern almost every aspect of our lives — rules collectively known as Halacha, the laws of Jewish behavior. Torah study is devoted to understanding and interpreting these laws in order to uncover God’s intentions.
God is also called Hamakom, meaning “the Place,” which is often understood to represent the world or the universe. This aspect of God is associated with a different set of laws, those which were perfectly designed to govern the uncountable, mostly hidden processes that make life possible. The laws of nature reflect those of God’s characteristics which instill a sense of awe and wonder, such as order, beauty, and harmony.
Then there is the most mysterious of all God’s names — the one we are neither allowed to say nor even to write. This ineffable name leads us into the abstract realm of Kabbalistic thought and mysticism. The difficulty we encounter when trying to understand Kabbalistic ideas lies in the fact that they are not connected to the things we know – that is, to the natural world. Instead, they point toward deeper, transcendent truths that are disjointed from ordinary experience.
Mathematics is also known by different names corresponding to overlapping subfields, each with its own distinct character, yet all governed by the same underlying principles — the rules of logic and the axioms of set theory. The traditional broad categories into which the field is divided are “Applied Mathematics” and “Pure Mathematics.” I would like to propose that the relationship between mathematics and Judaism can be similarly understood through both the applied and the abstract lens — one that reflects “Hamakom” (the immanent, encompassing God) and one that reflects God’s ineffable name – God’s more mystical, transcendent nature.
The “applied” perspective asserts that in order to approach an understanding of God, we should begin at the beginning. More specifically, in the beginning God created the universe, and if we seek to understand His creation, it is imperative we first learn mathematics. This idea is encapsulated in Galileo Galilei’s famous quote: “Mathematics is the language in which God has written the universe.” Many scientists throughout history have expressed similar beliefs. For example, Euclid said, “The laws of nature are but the mathematical thoughts of God,” and Kepler wrote, “The chief aim of all investigations of the external world should be to understand the rational order which has been imposed on it by God, and which He revealed to us in the language of mathematics.” Paul Dirac was even more direct: “God is a mathematician.”
These views reflect the belief that mathematics serves as a means of understanding the rational, ordered nature of the universe, and by extension, the divine order that governs it.
That mathematics is the key to understanding not just God in general, but the Jewish conception of God, in particular, has long been emphasized by great Torah scholars. Rabbi Baruch Schick of Shklov, in his translation of Euclid’s treatise on geometry from Latin to Hebrew, quotes his teacher, the Vilna Gaon, as saying, “If one is ignorant of geometry, one is a hundredfold more ignorant of the wisdom of the Torah, for the two are inseparable.” Maimonides, in his Guide to the Perplexed, wrote, “It is certainly necessary for whoever wishes to achieve human perfection to train [them]self first in the art of logic, then in the mathematical sciences according to the proper order, then in the natural sciences, and after that, in the divine sciences.” Similarly, Abraham Ibn Ezra argued that only after learning logic, mathematics, and science can one “ascend to the great level of knowing the secret of the soul, the secret of the supernal angels, and the concept of the worlds to come in the Torah, the Prophets, and the sages of the Talmud.”
These great scientists and rabbis are telling us that to gain an understanding of the Jewish God, we must first understand God’s greatest creation — the universe. Moreover, this understanding can only be achieved when we are able to speak God’s language — mathematics.
The first reason mathematics is essential for understanding the natural world is that, as Galileo emphasized, it is the language in which the laws of nature are written. Newton’s theory of gravitation tells us that any two masses exert a mutual force of attraction. While this qualitative statement provides us with an basic level of understanding of this fundamental force of nature, it doesn’t explain why planets orbit the sun in elliptical paths, how the moon’s gravitational field causes the rise and fall of the tides, or how to calculate the launch speed and angle of a projectile required to reach a specific target. However, by understanding that the force of gravity between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them, we can derive precise answers to all of these questions, and many more.
Beyond its practical applications in the natural world, mathematics provides a framework for understanding aspects of Judaism such as miracles, prophecy, and God’s existence — some of the most difficult concepts for skeptics to accept. Examples of “prophecy” abound in the mathematical aspects of theoretical physics. A year after Einstein published his seminal paper on general relativity, he received a letter from Karl Schwarzschild, who at the time was serving in the German army on the Eastern Front. In the letter, Schwarzschild was able to produce an exact solution to Einstein’s field equation, something Einstein had tried and failed to accomplish. Einstein agreed that Schwarzschild’s formula was mathematically correct, but was disturbed by the fact that it contained singularities — that is, terms that became infinite when massive stars died, collapsing under their own weight, creating regions of space where time ceases to exist and the laws of physics break down. At the time, Einstein and other leading physicists argued that while these singularities had to appear in the formula for the gravitational field for mathematical reasons, they had no physical significance. Schwarzschild did not object — he was merely following the hidden path revealed to him by the illuminating light of mathematics.
Einstein had a similar reaction to the work of the Belgian physicist Georges Lemaitre, whose theory showed that not only do singularities have meaning, but that the universe actually began as a singularity. Einstein’s famous response to Lemaitre was: “I have read your article. Your calculations are correct, but your physics is abominable.” It turns out, however, that the predictions made by mathematics were discovered to be correct: Schwarzschild’s singularities predicted the existence of black holes, and those of Lemaitre became the basis of the now-widely-accepted Big Bang Theory.
It is striking that a mathematician, manipulating equations in a small dog-eared notebook — in a dimly lit office containing no telescope, no test tubes, no computer, no experimental equipment of any kind, the window shades drawn, cut off from the outside world, late in a quiet night, notices that two terms unexpectedly cancel, a complicated expression simplifies considerably, a mysterious formula emerges containing a singular term that seems to make no physical sense — and yet, a sequence of hypothetical syllogisms forces it to appear, a term that predicts the existence of regions of the universe that are thousands of light years from earth, that are infinitely dense, a region where the laws of physics break down, or a term that predicts a massive explosion 14 billion years ago, which produced the universe we inhabit.
Now let us turn to “Pure Mathematics”, otherwise known as “Abstract Mathematics.” Unlike applied mathematicians, pure mathematicians are not concerned with real-world applications; they focus on structures that exist only in the realm of the imagination. A simple example is the geometric concept of a sphere. We can define a perfectly round sphere rather easily, and we can picture it in our minds. But no one has ever seen or touched a sphere. Objects like billiard balls or soap bubbles may resemble spheres, but they are finite, whereas a mathematical sphere is infinite — it consists of an infinite number of points. In this sense, spheres are abstract objects; they exist in thought, but not in the physical world. In fact, nothing in our physical world is truly infinite. Whether we count grains of sand on a beach, stars in the sky, or atoms in the universe, all are finite. Infinity, however, is central to abstract mathematics. Concepts like limits in calculus, set theory, topology, geometry, and algebra, are all based on the idea of infinity.
The reason many of us struggle with the concept of God is that God is inherently abstract, while idols are tangible and (literally) concrete. God’s abstract nature is encapsulated in God’s infinitude, a core tenet of Jewish belief. The first words we say upon waking, the Modeh Ani prayer, begin with “I give thanks before You, King living and eternal.” Before reciting the Shema, we affirm that God’s kingdom is “forever and ever.” In the Kedusha, we proclaim that “God will reign forever,” and in Adon Olam, we are told that God has no beginning and no end. The mathematical concept of infinity, when translated into Hebrew, is “Ein Sof” — also the Kabbalistic name for God.
The modern theory of infinity was developed in the late 19th century by Georg Cantor. One of his most groundbreaking discoveries was that some infinities are greater than others. In fact, there are infinitely many sizes of infinity. Cantor, rather interestingly, denoted the smallest infinity with the symbol ℵ0, and larger infinites by ℵ1, ℵ2, etc. Cantor also introduced the concept of “absolute infinity”, which is larger than any other infinity. Cantor described Absolute Infinity as being beyond all mathematical determination, and something that could only be comprehended by the mind of God. He believed that his work in mathematics was bringing humanity closer to understanding the divine.
Mathematics is inherently abstract, but Cantor’s theory was so abstract that for a long time, it was rejected by the world’s leading mathematicians, philosophers and theologians. Poincaré said it was a “grave disease.” Kronecker called Cantor a “scientific charlatan” and a “corrupter of youth.” Wittgenstein said Cantor’s theory was “laughable” and “wrong.” There were even objections from the clergy: Cardinal Franzelin, with whom Cantor had an extensive correspondence, told Cantor that his theory may be suggesting Pantheism, which would be unacceptable to the church.
Cantor was eventually vindicated. Bertrand Russell recognized him as one of the greatest intellects of the 19th century, and David Hilbert, in a clear reference to the Garden of Eden, declared, “No one shall expel us from the paradise that Cantor has created.”
The infinity embodied in God’s nature, and the infinity that underlies the foundation of mathematics, are both abstract concepts, and it is understandable that we, at least initially, resist, often quite strongly, their acceptance.
In conclusion, mathematics — both in its applied and pure forms — offers a profound way of understanding the universe and, by extension, the divine. In Jewish thought, mathematics is not just a tool for scientific inquiry; it is a means of approaching the infinite wisdom of God. Whether through the precise application of mathematical laws to the natural world or the contemplation of abstract mathematical structures, mathematics serves as a bridge between the finite and the infinite, between the physical world and the divine.
(This post is part of Sinai and Synapses’ project Scientists in Synagogues, a grass-roots program to offer Jews opportunities to explore the most interesting and pressing questions surrounding Judaism and science. Dr. Jacob Sturm, a professor of mathematics and computer science at Rutgers University, is a member of Congregation Agudath Achim in Bradley Beach, NJ.)
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