Over 2000 years ago, when mathematicians were struggling to determine the area of a parabolic segment (merely a few decades after being able to determine the area of a circle), Archimedes, perhaps the original math geek, solved the problem. It might be tempting to explain how Archimedes solved the problem via it being a stroke of luck, raw genius, or pure intuition, and other such generic terms. But in his letter to his mathematician-friend Eratosthenes, Archimedes describes his thinking process and how he arrived at the area of a parabolic segment, step-by-step (Strogratz, 2020). Essentially, Archimedes thought of a parabolic segment not as a shape (which it indeed is), but as an object to be weighed. At the time mathematicians already knew how to calculate the area of triangles, so Archimedes weighed this parabolic “object” against one triangular “object” on a lever. Then he kept adding triangles until the lever balanced – ultimately expressing the weight of a parabolic segment in terms of multiple triangular weights; thereby expressing the area of a parabolic segment in terms of the area of a triangle.
It is worth going over a process such as this because it gives us a better sense of how the theorems that we have built much of our world based on (our buildings, our devices, our classrooms) are usually conceived. They typically do not appear out of thin air. What we usually attribute as intuition or genius can generally be explained more concretely, as the research outlined in “Advancing mathematics by guiding human intuition with AI” (Nature, 2023) suggests. This study leveraged artificial intelligence, namely machine learning, to thereby illustrate the patterns and relationships, once hidden in vast swathes of data, within the fostering of mathematical discoveries. Zooming into specifics, one such pattern is the ability to stretch insights across different fields, like Archimedes did when thinking about area not as area, but instead as weight to be balanced. The same applies outside of mathematics, too. Consider Steve Jobs, who stretched what he learnt in a calligraphy class in college into the realm of computing, which is a big reason why we now have a range of fonts to choose from on our computers (Isaacson, 2011) . Ultimately, it is generally possible to logically explain where insights, initially perceived as genius or intuition or magic, come from.
If so, how do we then explain a mathematician whose inspirations do not have any semblance of logical origin? For one of the most prodigious mathematicians in history, whose mind was surely connected to the empirical world of numbers and logic, claimed that it was an entity with no apparent basis in science – God – who appeared in his dreams and communicated those theorems to him. How does one explain this enigma, the enigma of the mathematician Srinivasa Ramanujan?
Ramanujan was a peculiar case; as Kanigel (1991) writes in his biography of the mathematician:
“The South, as educated North Indians were wont to see it, was backwards and superstitious, scarcely brushed by the enlightened rationality of Bombay and Calcutta. And yet, somehow, out of such a place, from a poor family, came a mathematician so alive with genius that the English had practically hand-delivered him to Cambridge, there to share his gifts with the scholars of Trinity College and learn whatever they could teach him”.
Srinivasa Ramanujan was an Indian mathematician who had almost no formal training in pure mathematics, developed his mathematical research for the most of his life in isolation, and output theorems at among the fastest rates for any mathematician who has lived ever. As a child, when asked to explain his mathematical gifts, he would credit his hometown deity, Namagiri – “Namagiri would write the equations on his tongue. Namagiri would bestow mathematical insights in his dreams”, and when asked how he had come up with his theorems at Cambridge, for he “routinely telescoped a dozen steps into two”, and most mathematicians could not comprehend his working, Ramanujan gave the same reply (Kanigel, 1991).
This is puzzling – how can such a logical mind draw such critical inspiration from the seemingly irrational world of faith, with no semblance of a “logical explanation” for how these insights came about?
The founder of analytical psychology, Carl Jung, put forth a theory of the collective unconscious which can help us explain why such a Ramanujan might have been able to source inspiration from a source like religion. Jung suggested that the unconscious mind is much more than a store of repressed thoughts – it is a wellspring of creativity, and in particular, our subconscious mind can communicate ideas to our consciousness via dreams. Jung writes the following: “Dreams are impartial, spontaneous products of the unconscious psyche, outside the control of the will. They are pure nature; they show us the unvarnished, natural truth, and are therefore fitted, as nothing else is, to give us back an attitude that accords with our basic human nature…” (Jung, 1964). Jung’s theory prompts us to grapple with the idea that the God Ramanujan saw in his dreams was in fact the personification of his subconscious genius; a genius even more intense and creative than what Ramanujan’s conscious mind had access to; a genius that emerged from a deep cognitive reservoir. Like Archimedes, perhaps Ramanujan was stretching insights across fields, and thus there indeed was a novel but logical flow to his ideas that allowed him to reach conclusions and produce new theorems. However, this process for Ramanujan occurred in his subconscious mind, and his dreams, while seemingly irrational and mystical, were indeed his subconscious mind communicating complex mathematical ideas in the symbolic language of the unconscious.
Alas, one assumption that permeates this interpretation based on Jung’s work is that religion and logic are inherently at odds, and Ramanujan’s religious inspiration must therefore be explained logically, in this case via psychology. Furthermore, this interpretation tends to interpret religious or spiritual experiences as manifestations of the inner psyche rather than as experiences that might have an independent, external truth. Is it possible for us to consider that the sense of divine Ramanujan experienced was, to some extent, genuine and not entirely a manifestation of a subconscious unknown to him?
To learn more about how religion and logic can come to terms with each other, we turn to peers of Ramanujan – brilliant scientists who did not feel that logic and religion had fundamental conflict, perhaps at all (though of course, these people did not go to the extent that is equating their inspiration to the appearance of God). We first turn to a mammoth in physics, Albert Einstein, and then we turn to Werner Heisenberg, who was awarded the 1932 Nobel Prize in Physics “for the creation of quantum mechanics” (Nobel Prize, n.d.).
Very quickly we realize that our priorly drawn line between logic and science versus religion and God is a false one – in a 1930 interview, Einstein declared: “I am of the opinion that all the finer speculations in the realm of science spring from a deep religious feeling, and that without such feeling they would not be fruitful.” – Einstein himself has definitely replied to us that it is indeed possible, and probably rational to heed religious inspiration in the pursuit of scientific discovery. Surely then, our initial question is wrong! Furthermore during a dinner in 1927, Einstein similarly said, when asked whether he was indeed deeply religious and was being heard around at the time: “Yes, you can call it that. Try and penetrate with our limited means the secrets of nature and you will find that, behind all the discernible concatenations, there remains something subtle, intangible and inexplicable. Veneration for this force beyond anything we can comprehend is my religion. To that extent, I am, in point of fact, religious.” (Jammer, 1999); from this it is obvious that while it is extremely unorthodox for a mathematician to credit his theorems to divine intervention, it is not so rare for a logical person to also have religious sentiments. What we can learn from Einstein is that religion and God have various kinds of interpretations, and they vary deeply from person to person. Depending on the interpretation itself, a belief in “God” can be considered increasingly logical. Is it very in line with the logic we apply in science to believe in a personal God who interferes in our personal affairs? Einstein did not believe so. However, he did believe in a “subtle, intangible and inexplicable” force and categorized that as God (Jammer, 1999). Thus, in Ramanujan’s world that we cannot entirely comprehend, it is indeed possible that Ramanujan’s feelings of divine intervention were genuine, depending on how he personally defined the divine.
This weaves in well in the context of Ramanujan’s religion, Hinduism. Radhakrishnan (1953), a renowned philosopher and former President of India emphasizes that the personal Gods evident in Hindu temple statues or mythological stories are fundamentally a representation of a greater force. He writes how all Gods are one, how God is everywhere, and that the representation of God in a human form is simply meant to allow us humans to provide a characterization that humans can comprehend.
Having discussed how the line between religion and logic becomes blurred depending on one’s interpretation of religion via referencing Einstein, we now turn to Werner Heisenberg, another scientist with nuanced views on the interplay between science and the divine. As he famously said: “The first gulp from the natural sciences will make you an atheist, but at the bottom of the glass God is waiting for you” (Otremba, 1979) (interestingly, this is an apt description of Einstein, who was deeply religious when young, became atheistic in his middle years and returned to have a nuanced view towards God afterwards).
It was, in fact, a theory Heisenberg himself formulated that led him to this view – namely, the uncertainty principle. This principle essentially states that at the quantum level, i.e. when we go really, really, small, it is impossible to simultaneously know both the exact position of a particle and the exact velocity of the same particle.
Before this principle, scientists believed that they could measure everything about a particle precisely. However, the uncertainty principle introduced a fundamental limit to this knowledge, suggesting that there are inherent uncertainties in the way we can know the world. This was a radical shift from the deterministic view of classical physics that everything could be predicted with certainty if enough information was available. Ultimately, this posited a radical revelation: there are limits to objective knowledge. Furthermore, not only were there limits to humans’ ability to gain objective knowledge, but also no longer was the line between objective and subjective clear.
To understand why, here is a brief and simplified explanation of why one cannot know both the position and velocity exactly at the same time: it is because the act of a person measuring a particle’s position directly impacts that particle’s velocity. This blurs the lines between what we see and what actually is, i.e. subjective experience versus objective observation. So if we cannot know everything about the smallest parts of our universe with absolute certainty, then how can we make definitive statements about the bigger things (after all, the bigger things are made up of the smaller things!).
This led Heisenberg to believe that the universe was filled with fundamental mysteries and uncertainties, prompting him to wonder whether like the quantum world, the divine could exist in a way beyond our full understanding. An even simpler analogy is this: we know that a fourth dimension exists, and there is even a video online in which a fourth dimensional world was built on a Minecraft server (Mashpoe, 2022). This video gives us a glimpse into how it would feel like to live in a four dimensional world. Extrapolating this further, the video also makes it apparent that a five dimensional world is possible (as supported by mathematical proof) – but it is simply impossible for us to ever have a video that lets us visualize the fifth dimension. In essence: there are things we know exist that are beyond our understanding. If so, is it not reasonable to assume that there may be things in forms we cannot comprehend that are beyond our understanding?
While Einstein nudged us to consider that religion and logic are non dichotomous, where depending on interpretation the two may very well be in synchrony, Heisenberg extends on this and makes us consider the prospect that it is most logical to believe in some interpretation of God. It is then sensible why a stupendously logical mind like Ramanujan’s may have been deeply religious as well – to the point where his mathematical ideas were sourced from religion.
In fact, referencing Einstein’s quote earlier – “the finer speculations in the realm of science spring from a deep religious feeling” – it is worth further investigating whether it may in fact be not just merely logical, but also rather optimal to source scientific breakthroughs from what may, on the surface, seem like unscientific areas such as religion.
“Psychedelics, Meditation, and Self-Consciousness” (Milliere et al., 2018) aids this investigation. This study discusses in detail about how meditation and psychedelics, depending on length of practice or dosage, similarly disrupt narrative self-consciousness, either through a loss of autobiographical memory and self-related beliefs or a reduction of self-referential thought. In essence, they allow people to step outside their personal narratives. This nudges changes in brain activity and allows people to access altered states of consciousness. Drawing parallels with our interpretation of Jung’s work earlier on, psychedelics and meditation can make it easier for people to access the wellspring of creativity hidden in their unconscious, when they step outside their conscious, personal narratives.
Venturing beyond the study’s parallels with Jung, however, we also recognize that the reduction of self-referential thoughts has links with consistent religious practice. Namely, religious practices can reduce self-referential thought. This strongly supported by a study in which the brain regions of religious folk attributed to self-referencing were compared with those of non religious folk, and the latter displayed much less brain activity in said region (Inoue, Takahashi, Burr, & Kawada, 2011).
The implications of this are significant – if scientific and mathematical insights can emerge from states of consciousness that are associated with religious experiences, then the boundary between the scientific and spiritual becomes even more porous. This porousness allows for a cross-pollination of ideas that can enrich both fields. Cognitive phenomena, be it arising from religious devotion, deep meditation, or psychedelic use, can facilitate a kind of cognitive flexibility that is conducive to paradigm drifting insights. In this case it makes even more sense why Ramanujan was so uniquely creative in being able to generate theorems seemingly out of thin air – he abided by religious practices: as an infant he would not eat except at the temple; as a teenager he would fall asleep on the temple steps in the middle of the day, “his notebook with his pages of mathematical scrawl tucked beneath his arm, the stone slabs of the floor around him blanketed with equations inscribed in chalk”; in line with Hindu tradition, every morning, he shaved off his front portion of hair, bathed in the river paying attention to cleaning his eyes, ears, and nose in particular, and prayed afterwards (Kanigel, 1991). Ramanujan’s upbringing was rooted in religious practice and spiritual connection.
At this juncture we have traveled from questioning how someone so logical could be inspired by something as illogical as religion, to positing that being inspired by religion might be among the most optimal strategies for creative problem solving. This begs a culminating question – if us humans draw artificial even between things as fundamental as logic and religion, are we drawing artificial lines in sub-realms as well? Certainly. Right versus left in politics, nature versus nurture in psychology, science being objective versus art being subjective, emotional versus rational decision-making – these are all examples of artificial lines that have been drawn, lines that exist only in our binary comprehension of things that are not truly reflective of how things probably are. Then, how much creativity are we withholding from ourselves when attempting to solve many of the problems that face us (be it climate change, human response to artificial intelligence, and so on) by not recognizing the porousness of different realms?
Ramanujan was a rare gem who broke these boundaries. To many, mathematics remains akin to a standard textbook: systematic, uniform, and devoid of personal essence. It is a collection of established truths and methods to be learned and applied devoid of personality. But Ramanujan’s relationship with mathematics was more than about solving problems – “An equation for me has no meaning,” he said, “unless it expresses a thought of God” (Kanigel, 1991). To Ramanujan, mathematics was a creative process akin to a poet composing verses, where theorems and formulas were like lines of poem that only he could have written. His story challenges the artificial boundaries between logic and religion, suggesting new realms of creative potential in mathematics and elsewhere.
Srinivasa Ramanujan returned home to India in 1919 due to his deteriorating health from the harsh English climate and the scarcity of familiar comforts in a foreign country. He died a year later in Kumbakonam, India. He has since been celebrated as one of the greatest mathematicians of all time.
There continues to be a nagging question among mathematicians and biographers since his death: what might have been had Ramanujan been discovered a few years earlier (he was in Cambridge sharing and developing his insights for a mere five years!), or lived a few years longer?
We will never really know. But if we are able to break out of our own pre-established lines, perhaps we may be able to get a glimpse of the answer to that question ourselves.
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References
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Strogratz, S. (2019). Infinite Powers. Atlantic Books.
Inoue, Y., Takahashi, T., Burr, D., & Kawada, R. (2011). Neural consequences of religious belief on self-referential processing. Social Neuroscience, 6(5-6), 559-570. https://doi.org/10.1080/17470919.2011.569772
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Millière, R., Carhart-Harris, R. L., Roseman, L., Trautwein, F.-M., & Berkovich-Ohana, A. (2018). Psychedelics, Meditation, and Self-Consciousness. Frontiers in Psychology, 9, 1475. https://doi.org/10.3389/fpsyg.2018.01475
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