Pure mathematics is commonly viewed as the pinnacle of rigorous formal reasoning. It relates abstract concepts to each other, with no necessary connection to any world, whether physical or metaphysical. But it was not always this way. The roots of mathematics reach deep into the foundations of the cosmos. It was only through a modern process of radical abstraction and refinement that mathematics came to be viewed as independent of any connection to the world.
Geometry originated as an empirical science of earth measurement (which is the etymological meaning of the word geo-metry). It was a practical tool for surveying plots of land. When Euclid formalised geometry as a mathematical system, it was understood as an axiomatic science of physical space. The Pythagorean theorem was not merely a result of pure mathematics. It was viewed as a law describing spatial properties of the real world. And this view lasted for over two thousand years. Galileo, writing about natural philosophy, put it this way in Assayer:
“Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.”
It was not until the modern mathematical discovery of non-Euclidean geometry that the meaning of geometry was abstracted from physical space to include mathematical spaces of various kinds. Similarly, modern mathematics also abstracted the meaning of number from natural numbers and rational numbers. These number systems were originally physical sciences of counting. They arose from the practical need to count objects (such as animals, goods, and products) and periods of time (such as days, months, and years). In modern mathematics, however, the notion of number was abstracted to include other types of numbers, such as complex numbers and transfinite numbers which have counter-intuitive properties.
Free from its tether to the physical world, pure mathematics exploded into a marvellous and infinite realm of abstraction. Surprisingly, some of this new mathematics turned out to be remarkably effective in the physical sciences. This suggests that, despite its disconnection from its empirical roots, there remained a profound and mysterious link between pure mathematics and the deep order of the world.
When I began university studies, my ambition was to become a physicist so that I could understand the deep order of things, the nature of reality. So, when I took my first course in quantum mechanics, I was not content merely to solve the Schrödinger equation. I wanted to understand what quantum mechanics was saying about physical reality. After courses in the philosophy of physics, however, I realised that physics could never ultimately satisfy my deep yearning to understand the nature of reality. For example, I realised that there were dozens of different interpretations of quantum mechanics, but no possible way to empirically determine which one was correct. Physics could never provide a definitive answer to the philosophical question of what kind of world quantum mechanics was describing.
Meanwhile, I was also reading more widely in philosophy, both Western and Eastern, philosophers such as Kant, Nagarjuna, Plato, Shankara, Proclus, Eriugena, and Nicholas of Cusa. As a result of these studies, together with contemplative investigations, it became clear to me that the ultimate nature of reality could not be contained in any philosophical system. Just as there are many philosophical interpretations of quantum mechanics, but no way to empirically determine which one is correct, similarly, there are many philosophical descriptions of reality, but no way to know with certainty which one is true. A pure conceptual system can be internally consistent, but there is no way to know for certain whether it is making true statements about the world. This parallels the situation from mathematics, which is why Einstein wrote in his 1921 essay “Geometry and Experience,”
“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
My yearning for certain knowledge of reality could not be fulfilled by any conceptual system at all. Conceptual systems are, in the end, relationships between pure concepts that have no certainty external to themselves. Although there is freedom to choose definitions and assumptions, and truths can be known with certainty within the limited context defined by those choices, any claim that those concepts describe some reality external to themselves can never be certain. Reality, in other words, cannot be known with certainty through concepts. In short, it is ineffable.
This insight, I came to understand, was in harmony with the testimony of many spiritual philosophers. For example, the Upanishads declare, “He comes to the thought of those who know him beyond thought, not to those who imagine he can be attained by thought.” Similarly, Lao Tzu writes, “The Tao that can be told is not the eternal Tao.” On the other hand, the statement ‘reality is non-conceptual’ is itself a conception about reality. So, reality must transcend even the duality between conceptuality and non-conceptuality. As it says in the Heart Sutra, “Form is emptiness, emptiness is not different from form, neither is form different from emptiness, indeed emptiness is form.” Just as Shiva’s dance is not different from Shiva, and waves are not different from water, conceptual form is not ultimately separate from reality.
So, although concepts and thoughts may seem to hide ultimate reality, they are also manifestations of it. Concepts that are confused serve to veil, while concepts that are clear serve to reveal. This is why the rigorously clear concepts of mathematics have been viewed as having spiritual significance. Like a perfect crystal, they refract the light of reality with orderly purity. This was my motivation for embarking on graduate studies in mathematics. It was not a search for some true conceptual understanding, but a practice of harmonising the mind with the most refined and transparent forms of manifestation.
The roots of mathematics extend not only to the earth and its measurement, but also to the principles governing the heavens. The mathematical term logic and the religious word logos share the same etymological root. Both point to a principle fundamental to the creation and order of the world. In the Gospel According to John, for example, it is said that the logos, translated as the Word, is a generative power in creation:
“In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by it; and without it was not any thing made that was made
The remarkable effectiveness of mathematics in the physical sciences suggests that there is, indeed, a deep connection between the manifested world and this logos. Similarly, in the Pythagorean tradition, the ordering principle of this cosmos is the logos, or, more specifically, number. Thus, it is said in the Pythagorean tradition that number is the principle, source, and root of all things. This order is exemplified in the harmonies of musical vibrations, which correspond exactly with the quantitative ratios of numbers, as well as in the ‘music of the spheres’, i.e., a musical harmony of the heavens. In the Pythagorean tradition, the fundamental principles of number are at the root of manifestation: Number in time is music. Number in space is geometry. Number in space and time is astronomy. Legend has it that above the door of Plato’s academy was the inscription “Let no one ignorant of geometry enter.”
The study of mathematics, according to Plato, is a prerequisite to a life of philosophy, whose ultimate purpose is to bring to birth, in the soul, a transcendent vision of the Form of the Good. Because mathematics is a domain of pure intelligible truths that are not contingent upon time, place, or sensory experience, the study of mathematics helps to develop the power of abstract thought and to turn the soul away from the transient world of the senses and toward the transcendent world of eternal forms (Republic 527b). So, for Plato, the study of mathematics is an essential spiritual practice.
This view of the sacred significance of mathematics can be found centuries later in the writings of Nicholas of Cusa. Cusa says in his De Docta Ignorantia that mathematics plays a special role in the power of the mind:
“Since there is no other approach to a knowledge of things divine than that of symbols, we cannot do better than use mathematical signs on account of their indestructible certitude.”
In modern times, the 20th century American mystic and philosopher Franklin Merrell-Wolff also recognised the spiritual significance of mathematics. Echoing Plato, Merrell-Wolff writes in Pathways Through To Space,
“The greatest achievement of western genius has been in the development of the abstract thought which has its crown in higher mathematics. The freeing of thought from dependence upon the sensible image is an accomplishment of the very greatest difficulty. Until thought has won this power, it cannot penetrate into the Realm of Imageless Consciousness.”
Thus, the remarkable effectiveness of mathematics in the physical sciences is a sign of an even more profound role in mirroring order at the deepest levels of reality, beyond the physical, where it reflects the fundamental order of creation in all its diverse possibilities.
