[A Novice’s Reflections]
“The doctrine that number is the essence of all things, passing through the prism of a thousand philosophical tracts, remains the central insight of Western science, the indispensable key of coordination.” (8)
- David Berlinski Infinite Ascent (2005)
“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 characters in which it is written.
“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 it, one is wondering in a dark labyrinth.”
- Galileo Galilei (1564-1642)
Numbers, equations, computations, shapes and their like may not be what generally comes to mind when you think of spirituality, yet there is a long tradition that sees these two arenas of human endeavor as being connected in one way or another. The early Greek mystic and mathematician Pythagoras (c. 580 – c. 500 BCE), once said:
“Number is the essence of all things.”
Pythagoras is believed to have formulated the theorem named after him that describes the relation that exists between the sides of right triangles (i.e., that the sum of the squares of the two sides equals the square of the hypotenuse). Yet it is also true that Pythagoras was a mystic as well as a mathematician. He founded a religious community in Italy that lived by an ascetic rule; its members engaging in mathematical work and meditating upon mathematical ideas as well as pursuing more ‘traditional’ spiritual goals. Fifteen centuries later, the mediaeval Christian mystic and mathematician Nicholas of Cusa (1401-1464 CE) ruminated that:
“God is a circle whose center is nowhere and whose circumference is everywhere.”
Nicholas wrote about ‘knowing God’ through ‘learned ignorance;’ a knowing beyond attained knowledge. He worked on the problem of squaring the circle and also wrote about the human-divine relationship in symbols and metaphorical language drawn from mathematics. Thus in both Pagan and Christian mysticism mathematics and spirituality have been linked.
Beginning about the time of Galileo Galilei (1564-1642), however, mathematics and spirituality seem to have gone down somewhat more divergent paths. I know of no mathematician-mystics in the centuries since the Renaissance; perhaps that is just my ignorance [if you know of someone, let me know]—but nowhere in all of my spiritual, religious and theological studies have I happened upon someone who found in meditations upon modern mathematics or its implications a pathway to the Transcendent. On the other hand, there seems to me a certain ‘mystical’ inclination amongst scientists; most of whom would not call it that, and many of whom would eschew the word as having nothing to do with their work. Yet – as I’ve suggested in other blogs – there are scientists who seem to have had essentially mystical experiences in their engagements with Nature via experiment, research and theoretical work.
Perhaps the reason why mathematics and spirituality have ended up on divergent paths in the last 400 years is that mathematics & science have so shaken and then undermined the ancient worldview in which all of the great traditions of human spirituality were originally framed. Nicholas Copernicus, Tycho Brahe and Galileo Galilei are among the names that define the “modern worldview” at its inception; a way of understanding the world that is rooted in rational explanation and empirical evidence. Those who came after them continued the dismantling of ancient cosmologies, laying out in the process the map of how the cosmos really is, beyond our human preconceptions and desires. The tools of this revolution were science and mathematics. The fruit of the revolution has been progressively deeper insight into the nature of Earth & Cosmos and ultimately the realization that we ourselves are a manifestation of the physical universe. The revelations of science began by shaking the foundations of the ancient worldview. They have resulted in the emergence of an entirely new worldview; one in which the traditional spiritualities of our species have so far been hard pressed to find themselves at home.
Indeed, there has been a long quarrel between the advocates of the new worldview – the scientific one – and the old worldview – the ‘religious’ one (for lack of a better term). Those who have owned the revelations of science have generally embraced mathematics as the primary language of Nature. Those who have held onto the older cosmologies – perhaps out of a sense of security, perhaps because of the spiritual benefit to themselves – have not paid as much attention to mathematics as perhaps mystics living before the modern period had done (e.g., Pythagoras and Nicholas of Cusa). As mathematics is wedded to science in the modern worldview it has come to seem to certain spiritual people as irrelevant at best, or at worst an ‘enemy’ of spiritual wellbeing and flourishment.
What I want to suggest is that mathematics and spirituality can be brought together on the same path; not by some safe and secure ‘return’ to outmoded conceptions of the world that would require amputating all of the breadth and depth of scientific revelation, but by gleaning the basic and essential truths from the great spiritual traditions of our species on this planet, and then replanting them in the rich soils made available by the revelations of modern science & mathematics. The truths of human spirituality are primarily subjective and inter-subjective in their purview; in the most general terms they refer to the journey of the ‘soul’ toward ‘perfection,’ ‘holiness,’ ‘wholeness,’ ‘fulfillment,’ ‘salvation,’ – and they refer as well to the various practices that contribute to making this journey successful. Human spiritualities from around the world all reference disciplines that are said to effect change in the human seeker; aiding that individual in the quest for self-realization. The communities that have practiced the great spiritualities provide a network of support for the seeker, as well as a context in which the individual mystic can experience the symbols, stories and rituals that will facilitate their transformation.
Something has gone wrong, however, with our spiritual quest. It has become divorced from the objective world. Spirituality is seen to be something exclusively – and perhaps even militantly – ‘personal;’ a ‘private’ sanctuary. As such, it belongs to a realm that science is thought not to touch. When we are ‘being spiritual’ (an unfortunate misnomer that evinces the alienation to which I am alluding) we are in a ‘safe’ little world all our own; shared inter-subjectively – if at all – with others of a like mind. We go into our little worlds and there find succor; but then we have to go back out and into the wider world and work and face a reality that is all too often in contradiction with what we have nurtured in our ‘safe little spiritual haven.’ I fear that this is a more common dilemma for ‘spiritual’ people than is usually guessed.
It has not always been like this. I’m always amazed that ancient Christian theological adventurers – the great ‘fathers’ (and sometimes ‘mothers’) of the Church – sought to frame the truths of Christ-centered experience in the best cosmology and philosophy of their time. They knew the Greek philosophers and they knew of Ptolemy’s model of the universe, which was the most advanced view of the cosmos at that time. They accepted the best scientific understanding of the world available, and they argued that the revelation of God in Christ conformed to that understanding! They argued, in contradistinction, that the pagan religions of the ancient world were less true because they were out of touch with the best science and philosophy of the day.
Somewhere along the way, however, Christian cosmology got frozen into dogma, and when Galileo Galilei saw those moons orbiting Jupiter and when he saw that the Earth’s Moon had irregular shapes on its surface and was thus not a ‘perfect sphere’, something cracked; the mirror of medieval cosmology was shown to not reflect the world as it actually was—as ‘God’ had actually ‘made’ it. With this, the modernist revolution began! For four hundred years, science & mathematics have been revealing the world – the Earth & Cosmos – in all of its objective glory, undermining the pillars of older cosmologies and raising new ones in their place, progressively exploring the universe in all of its wondrous particulars. Those scientists from Isaac Newton to Dmitri Mendeleev and their many successors, from Einstein and Neils Bohr to Stephen Hawking, revealed the wonders of the physical world. From Charles Lyell and Charles Darwin down to the present, geo- and bio- scientists have been revealing the wonders of the living world.
Unfortunately, spirituality has remained largely unaffected by this ongoing and awe-inspiring revolution. While spiritual teachers and mentors have made progress in the understanding of the human quest for transformation and self-realization, and while many people have benefitted in numerous ways from the practice of traditional spiritual disciplines, all too often they have had to ‘privatize’ their spiritual efforts as a result of framing their understanding in a cosmology that is essentially a ‘fiction’ at this point in time. There is definite subjective benefit from the practice of traditional spiritualities, but this benefit is often ill-fitted to life in the world because its logic is still tied to the ancient cosmologies that spawned them. Spiritual people all too often find themselves living in a subjective reality that is out of touch with the objective world around them. A community of ‘like-minded’ seekers can somewhat alleviate the stress that this disjunct creates, but this only raises the ante to another level; instead of ‘me’ against the world, it becomes ‘us’ against the world. One major consequence of this disjunct is fundamentalism in all of its dysfunctional forms; the ‘urge to evangelize’ others – to make ‘them’ like ‘us’ – that drives so much religious fervor in the world today.
The only way this disjunct between the subjective and objective dimensions of the spiritual life is going to be dissolved is by our wholehearted embrace of the revelations of science & mathematics. Only by acknowledging what science & mathematics have revealed about ourselves and the world can we hope for a spirituality that is holistic and ‘realistic.’ The subjective spiritual truths that we find most meaningful can only be deepened by opening them to the light of scientific understanding. Like the Church Fathers & Mothers, we are going to have to try and plant the truths of our spiritual traditions in the rich soils of a modern understanding of the cosmos. What would it mean to embrace mathematics as something revelatory? What might it contribute to our self-understanding? Our understanding of ourselves and others? Our understanding of the world around us?
What does mathematics have to do with spirtuality?
I think that any genuine spirituality begins in an engagement with life as it is. A spirituality is a praxis – i.e., a set of disciplines and rituals – that clarifies one’s self-image, edifies and guides the practitioner along a path ultimately tending toward their fullest humanity. The right practice of a genuine spirituality reveals the practitioner’s deepest human potential; it ignites the embers of our best possible selfhood. If a spirituality is to be more than a satisfying, personal illusion or a privileged private escape from a reality one doesn’t want_ or have the courage_ to face, it must link the practitioner to the world-as-it-is; not construct some merely comforting version of the world such as we are all too often wont to create – a ‘world’ in our own image, after the manner of our personal predispositions and our ‘wants’ and ’needs.’
Science is the most powerful tool our species has developed for exploring and revealing the Cosmos. Science, when practiced well, opens our eyes to the way the universe around us really is, breaking up the scabs of self-delusion and false belief that muddy our vision. As science shows us what the universe is really like, it puts us in-context as a manifestation of Nature; members of one species among millions that stand at the end of an at least 3.8 billion year process of evolution. Science is thus a primary ground for spirituality. It reveals the objective side of the reality within which our subjective experience and inter-subjective life needs to be framed. To embrace science would be a boon to our spiritual progress. We need to identify the essential tenets and practices of our spirituality and separate them out from the metaphysical and cosmological dross of the pre-modern era, and then take the courageous step of replanting these ‘essentials’ in the soils made available by modern science.
Mathematics takes this revelation of ourselves and the world one step further; by going beyond the limits of our senses to describe the world as it is in itself, without the limiting bias of our species-based modes of experience. Mathematics is a powerful tool. I think that learning mathematics and attempting to appreciate the world around us through its revelations inspires a profound humility. I say this not as a mathematician, nor even as someone who has always been good at math; I’m not arguing for one of my own proclivities or personal biases—I’ve come to mathematics late in life, and am trying to grapple with it as something to which I have never really been attracted. Growing up, I was a fairly typical ‘creative’ personality. I was intuitive, impressionistic, spontaneous, imaginative to a fault, into all kinds of daydreams and nightmares, always wont to bring to life worlds in which I could play out one imagined or dreamed scenario or another. I didn’t pay math much mind! As I’ve turned toward science over the last decade, however – re-grounding myself, spiritually and poetically, in a naturalistic worldview – I’ve come to see the integral role played by mathematics in our present knowledge and understanding of Earth & Cosmos. I’ve had to begin to learn a new language – “the language of Nature” – in order to continue along the path.
I accept that mathematics may be ‘the language of Nature,’ not as a way of demeaning other ‘languages’ (such as poetry and philosophy) but as a way of moving ‘beyond’ them. While scientific inquiry often begins with a question or a problem, and proceeds through various stages in which there is observation and experiment, data collection and analysis, there seems to be a level of understanding at which numbers, equations and calculation take over the role of physical description and bring research to a new plateau of clarity. My impression of the workings of science is that, while verbal language plays a definite and valuable role; vivid as well as critical descriptions of phenomena and the results of experiments and experiences are integral to the seeking of knowledge about Nature—there is a sense in which mathematics takes us ‘further’ than verbal renderings of phenomena can do.
As a writer, I appreciate and use and defend literary language, so I don’t at all mean to imply that verbal language is a ‘lesser manifestation of the human spirit’ than mathematical language (nor do I think that the majority of scientists think that it is). The power of verbal language to describe and explain the world is frankly astonishing, as a deep read of the best poets, essayists, naturalists, dramatists and philosophers will attest. Verbal language reaches deep into the subjective and inter-subjective realms, and its effect is often revelatory in wisening ways. In approaching the more objective dimensions of reality, verbal expression can be honed and made critical enough to communicate much of what we know and understand about Nature. Yet mathematics has its own astonishing powers of description, revelation and narration. Mathematics seems to describe and reveal the world in a different way than does verbal language.
I have sought to understand what numbers and equations mean, and not just how to ‘manipulate’ or ‘solve’ them. As a novice practitioner I have gleaned an impression of mathematics as a revelatory language; like verbal expression—it is another kind of access to the physical reality of the world in which we find ourselves. How does it do this? I think disciplines like algebra and calculus, as well as trigonometry and geometry, describe reality because they initially have all had their origin in human attempts to describe the world in order to function in it more effectively. Navigation and commerce, astronomy and architecture are four of the pragmatic arenas in which mathematics historically developed. Yet, from out of these practical arenas came higher mathematics. Practical matters led on to theoretical ones; and then – out of the theoretical explorations of mathematicians came deeper insights into reality than the practical concerns of navigation and such had intended.
Mathematics seems to me to be sourced in number and spatial relation. Then, the patterns found in numbering things and describing their spatial arrangements is abstracted, until general principles are found. Thus, the most abstract mathematics has ties to experience and day-to-day problem solving of a variety of types. I think this has deep and profound implications for any genuine human spiritual praxis. The objects in the world around us can be enumerated. Many things can be counted; and then there is a sense in which ‘real’ things can be ‘translated’ into numbers. An equation can describe the shape of an object, the slope of a road winding down a hill, the changes in variable factors, such as the weather or the current flowing through a wire. Can we really contemplate the world; the reality in which we live, and not consider such things? I did, for a long time; and now I think that I was to a certain extent impoverished by not taking these kinds of things into consideration in my spiritual reflections and study.
The intuition that “number is the essence of all things” is something that I resisted for most of my life, I now think, because of the humility it would take to accept it as a spiritual proposition. To allow that number is what characterizes reality at its most basic levels is to undo our very understandable bias toward our own sense experience; our own existential access to reality. Our senses and our minds have evolved to access the world around us in a certain way – within a certain range of parameters – with a particular degree of accuracy. Our eyes, our ears, our other senses, our minds, our metabolism and our nervous system; all of these characteristics of our being have come to be what they are as a result of natural selection over millions and millions of years; Nature molding us and making us what we are. We are the kind of animal that we are because of our particular evolutionary trajectory, and while we have incredible access to the nature of Nature (as evidenced by the success of science) via our senses, there is something beyond what we can access via the body’s portals on the world; and that is what number being ‘the essence of reality’ alludes to.
We are incredibly good at accessing Nature via our senses and our conceptual, verbal-oriented minds. Yet, beyond the picture of the world we draw out of experience and that we express in verbal language, there is a dimension of reality that strikes us as non-intuitive and that puts our sense-based experience into a broader perspective; there is a wider lens—one that begins in_ and is defined by_ number. Mathematics, as part of the praxis of science, seems to reach out beyond the limits of verbal description and existential experience to grapple with the way the universe is without the familiar and therefore ‘comforting’ categories we usually use to engage with our environments. As such, a mathematical exploration of the world can complement a poetic, intuitive and experiential engagement with our environment, and actually distill it, clarify it, and focus it; so that we can see farther than we had before, and reach deeper into the nature of reality than we can with our senses. Mathematics extends the reach of experience and conceptual thinking. It acts as a ‘telescope’ or a ‘microscope,’ bringing reality into close focus.
As such, mathematics represents the ‘beyond’ of the world; the reality beyond our senses—and as such can function in spirituality in much the same way as the transcendent metaphysics of traditional spiritualities once did. To read about mathematics and try to comprehend reality as it is revealed by mathematics – the ‘real’ that goes beyond our senses and our familiarity with our surroundings – is to come to awareness of a ‘transcendent realm.’ By ‘transcendent’ I mean ‘beyond,’ not ‘above’ in the ‘supernatural’ sense. There is nothing unnatural or supernatural about what mathematics reveals. Think of it as describing the world as it is without the categories we have evolved to use to engage with our physical surroundings. In more traditional ‘religious’ language, think of it as describing the universe as how the ‘divine originator’ – by whatever name you know it – ‘made’ it. All planets and animals have interfaces with the reality that color it for them in terms that allow them to survive, thrive and flourish. We are no different from the other animals in this regard.
Seen in these terms, I suppose, mathematics might be thought of as helping us see into ‘the mind of God.’ It can open a ‘window on creation.’ To appreciate the universe as it ‘is’ beyond our senses and beyond the reach of even verbal language can facilitate a true humility; an attitude in which you see and understand yourself in proper relationship to ‘the divine,’ to others and the world. St Bernard of Clairvaux once said that true humility is “to think neither more nor less of oneself than one ought.” This is a goal that can be furthered only by understanding ourselves, others and the world, and this is what science and mathematics really promotes. If we understand the revelations of science & mathematics, we will have a greater chance at true humility than if we stay in our little subjective spiritual worlds and believe what we want about the world.
Contemplating ‘the beyond of the world,’ may well lead to experiences of wonder and even awe. I’ve only been studying mathematics and trying to understand it for four years or so, yet I have had several moments when I had to sit back and wonder at the world after hearing some mathematical truth, conundrum or anecdote explained. Simple questions still lead me into awe. What are ‘negative’ numbers? To what does the square root of 2 refer? Why are there numbers such that a2 + b2 = c2? The so-called Pythagorean triples – such as 3, 4, 5, and 8, 15, 17 – satisfy this equation. Why? What does it say about reality? Why are there Pythagorean triples for this equation but not for a4 + b4 = c4 or equations with even higher exponents?
Wonder – like humility – is one of the most common themes in human spirituality. A spirituality opens your ‘spirit’ to the cosmos in ways that are life-engendering, and this leads to both wonder and humility; a lifting up and a willing surrender of one’s biases in favor of a more real vision of the way things are. Meditation on the truths of mathematics can complement meditation on other truths and lead to enlightenment on a number of levels. While much more could be said on this topic, I hope I have said enough here to elicit interest and perhaps even inspire you to look into the mathematical side of reality.