Subtle Planes of Reality
Do numbers exist? I can count three mangoes, or four pebbles. They’re right there, I can pick them up, I can touch them with my hands. But where is this number called 3 or 4?
Well, you might say that the mangoes may rot and the pebbles may get lost, but there is something eternal, a unique value behind the multiple mangoes, called the number 3. Similarly the number 4 is the representation of the four-ness of pebbles. If the pebbles themselves are lost, the four-ness persists, across other things that are found together in groups of 4. So yes, numbers exist.
You might also convince me by saying that these numbers help us keep count and settle transactions. You may lend 3 x 100 rupee notes to someone, and they might give you 6 x 50 rupee notes back, and you understand that even though they’re giving back different physical items, they’re returning to you the same abstract value of 300.
But do you see, the first time somebody proposed the idea of 0, or of negative numbers, some people may have reacted in the same way as I did for natural numbers. “Have you ever seen -3 cows?”. “Why do you say there are 0 cows when there is nothing, why even talk about cows? Are you mad?”. “Negative numbers are not found anywhere in nature”. And so on.
And similar reasoning can help us understand their existence and utility. Negative numbers balance our accounts. We can count one direction (of taking money) as positive, and the other direction (of giving money) as negative, and balance them up to find how much total money we have left. The physical reality exists indifferent from the existence of these numbers. They’re just abstract concepts that help us, the physically real beings, in manipulating and managing our physically real things. We could do things the harder way too. You could maintain two different accounts for giving and taking instead of introducing the negative sign and using a single account. You could even not have names and sounds corresponding to numbers, and point to actual physical 3 cows. But doing that is so much harder, isn’t it?
Similarly, the first time somebody said that we could use √-1, it seemed crazy. You can’t multiply two numbers to get a negative number, so where does this √-1 come from? Why talk about it or use it? Well these imaginary numbers help us immensely with vector multiplication. Representation and calculations in signal processing and quantum mechanics would be unnecessarily complicated if we stick to real numbers, as real numbers are one dimensional, on a number-line. But complex numbers can naturally represent 2 dimensions if you see i = √-1 as perpendicular to the number line. Thus any multiplication of -1 on the number line can be seen as one 180° rotation, which can be broken down into two 90° rotations with i. And we could use combinations of real and imaginary numbers for rotating any arbitrary amount of degrees, representing it in very tidy exponential notation. So instead of relying on complicated 2D vector representation using real numbers, we could use a single complex number, and the multiplicative math all checks out much cleaner.
So, do these imaginary numbers really exist? Or are these just tools that help us? Ask your neighbouring physicist or engineer if they would like to give them up. If we loosen up or stretch the rules in this abstract space of mathematics, we stumble upon concepts that are quite handy. Who are we to claim one of these concepts is less real than the other, when ultimately they all help us navigate the real world?
Now, moving beyond just numbers, we know that summing together an infinite number of numbers in a series can sometimes converge on a single point, like 1 + ½ + ¼ … = 2. But for other series like 1 - 1 + 1 - 1 + 1 … the sum doesn’t converge. It fluctuates between 1 and 0 depending on where you stop. So, it may be a stretch, but we can say the sum is “=” ½ in some sense of “=” equality, which is looser than the one we used for convergent series. Using just this one loosening, we can derive that 1 + 2 + 3 + 4 .. “=” -1/12. A divergent series, which surely approaches infinity under conventional addition, can be said to be equal to -1/12 in some sense of the word “equal”.
“This is a cheap trick” one might say. It is surely nonsense and not “real”. But well it does make sense to loosen the definition of equality, as long as we maintain some constraints, see 3B1B’s chapter on analytic continuation for more. And ultimately, it makes it easier for us to understand real, physical phenomena (more on that later, I promise).
Modern mathematics is mostly playing around with these abstract ideas that have no real utility, until they do. There are many examples where math has preceded real physical discoveries. For example, non-Euclidean geometry might seem like nonsense at first glance, as it can have weird behaviours like sum of angles in a triangle not summing to 180°. “Surely you can’t draw or find me a triangle like that in the real world” you might say. But these non-Euclidian geometries are what preceded Einstein’s theory of relativity with space-time curving due to mass. The theory that explained measurements that Newton’s idea of a gravitational force couldn’t. It even predicted black holes and gravitational waves, which are real physical phenomena that were confirmed by experiments much later.
Similarly, Paul Dirac sought a relativistic wave equation for the electron. The equation he came up with produced solutions for both positive and negative energy states, which was problematic, as how can something have negative energy? Instead of rejecting the math, he proposed that it could be possible that there may exist particles with opposite charge but same mass as electrons (positrons). Nobody had ever seen any such anti-particle before, but lo and behold, ~4 years later Carl Anderson discovered it accidentally during his experiments with cosmic rays.
Let’s take a small detour, and talk about nothing for a moment. I promise this is related to the divergent series before. So what exists inside nothing? Let me be more precise and ask, what exists in a vacuum in deep space? Far, far away from any galaxy, where there is no trace of thermal energy (0 Kelvin). One would assume it is nothing. One would also assume it is static. In fact, such a physical location is the best way to solidify the abstract concepts of “static” and “nothing”.
But let’s think deeper for a moment. Based on the science we already know from high school we say a magnet can attract other metallic objects from a distance, and it happens without a medium. Same for the forces between electrical charges. If we have two charges in empty space at absolute zero, there is nothing between them, and yet they are pushed together or pulled apart. Similarly, we say that light travels as an electro-magnetic wave in nothing. There was a big search for a material medium or “ether” for light, like we have water or air for waves and sound, but ultimately all observations pointed to the fact that light is indeed waving in nothing.
Well, it may be physically empty, but we do represent the underlying medium as a Field, which is a mathematical abstraction. The idea is that every point in space, denoted by some coordinate system, has some value (be it scalar, vector or tensor). With this idea, we no longer have to deal with each particle influencing each other particle, and can simply use the underlying Field for book-keeping the total aggregate forces acting at any point in space. Even if this “Field” is not a physical medium made up of matter, it is as real as the numbers we use to measure the physical forces.
But does this field really exist in deep space, with no charged particles, where the value at every point is surely 0? Well yes, just like I can point to my empty stable and say I have x horses, where x = 0. A trickier question to answer intuitively is whether the values of this field remain static in absence of any charged particles. And also, are particles born out of this field, or is the field just a made-up abstraction to represent the behaviour of real particles. The answer to these depends on whom you ask, or how you look at it. A great example is the Casimir Effect proposed in 1948, as explained below.
If you take two neutrally charged sheets of metal, in empty space, at 0 Kelvin, and place them very very close (microns apart), they are attracted to each other. This is counter-intuitive if you ask me, as I would have thought in absence of any external force or thermal energy, matter should just be wherever it is. Static and unmoving. But reality says no, and we can even calculate the exact force of attraction using two ways:
The first, and easier way (in my humble opinion) is to look at it macroscopically, which involves an assumption that there are fluctuations in the EM field even in a perfect vacuum, known as the Vacuum Energy. This implies that reality, even when empty of everything, is not static but vibrating. It is vibrating with all possible frequencies. So when we bring two sheets of conductive metal really close together, we constrain the EM waves present in the space between these two sheets. These are 3 dimensional standing waves, but they can be understood analogous to the 1 dimensional standing waves that happen when a guitar string vibrates.
In the guitar’s case, the string is constrained at two points. The main pitch of what we hear comes from the vibration of the whole string. But what we hear is not just a single wavelength, which if you listen using a sine-wave generator sounds very bare. Instead, the physical string is free to move every which way between those two points, so we get smaller and smaller waves layered on top of the main one. These higher pitched vibrations happen at integer ratios of the string, also known as harmonics (so the wavelength is 1/2, 1/3, 1/4 … and so on) and gives the guitar its fuller sound.
Similarly, in empty space we have all possible frequencies of EM waves, but constrained by two sheets we have only the limited harmonics in the region enclosed by them. This creates a force that pushes the sheets closer together. If we try to calculate this difference, even with a toy 1D model, at one point we have to sum up the energy of all the harmonic standing waves. Energy is proportional to frequency, so we add up integer multiples of the base frequency (so 1x, 2x, 3x …), which is a divergent sum. There is a boundless infinity of waves pushing the sheets from the outside, and a constrained infinity pushing them from inside. But we know there are no infinite energies in physical reality, so we know that these two infinities should “cancel out” to give us a finite, inward pushing force on these sheets. And if we just plug in 1 + 2 + 3 + 4 … = -1/12 as we derived before, the math “works out”, giving us negative energy in the interior region, pulling them closer. Sorry for the lack of rigour in my explanation, but I hope you got a rough idea.
Experimental validation for this effect was done much later (in 1997) to measure the exact force, and it gave the same value as predicted by these calculations, within a few percent of error. So, does this mean that 1 + 2 + 3 … is equal to -1/12 in a real sense? And that empty space is really vibrating with all possible frequencies?
Well, there is a second way to understand and calculate this force, by looking at it microscopically. The metal has so many molecules, all influenced by London Dispersion forces which act even at 0 Kelvin. These forces mean that electrons are never static, so various atoms in one sheet can sometimes instantaneously align, leading to a momentary polarization, which induces the same polarization across space on the other sheet of the metal. The two sheets then act like very weak magnets, enough to attract each other. This gives us the same numerical values with a different theoretical understanding. But even in this case we have to make some unintuitive assumptions, like electrons not being static at 0 thermal energy.
Both of these theories are weird, but ultimately these are just mathematical models of our underlying physical reality. The sheets are attracted to each other with some force irrespective of the abstractions we use to understand it. Reality is weird, so our abstractions have to match the weirdness. The microscopic abstraction at the electron level leads to clunkier calculations, while the macroscopic abstraction at the EM field level requires us to renormalize divergent sums, and using the -1/12 result. Both of these require us to believe in vibrations even at absolute zero. Temperature or heat is understood as a macroscopic measurement of kinetic energy of microscopic particles. But here we either have to believe that the electrons can never be made completely still, or see it as fluctuations in the underlying electric field. Beyond the edge of my current understanding are more modern theories like Quantum Electrodynamics that unify electromagnetism, special relativity and quantum mechanics under a single Field.
Let’s come back to Dirac’s equations predicting the existence of anti-particles. Did the positron pop into existence because of his search for a beautiful equation that combines Einstein’s relativity and Schrödinger’s wave functions? Of course not. The positron was always there, just like gravity was bending light, and light was shining, way before we came up with the mathematical models for them. This abstract plane of mathematics cannot directly affect our material plane. But I want to make a bold claim here, that the mathematical plane is also “real” in a sense, in the same exact sense of the word “real” as various other planes that philosophers and poets have sung about since millenia.
Many people, when first asked to explain what mathematics or physics is, give an explanation that is something like a “Map of reality”, where ultimately the Map is Not The Territory. For example, the place called New Delhi is where I am living, but pointing it on a map of India is not that same place. The location on the Map is a location on a piece of paper. It is just a concept or abstraction of the real territory of New Delhi, but it helps us understand and navigate real territory.
But there is something missing in this idea. A map maker has to survey the territory before creating the map. In a sense, he is bound by his experiences of the territory. On the other hand, Math is tangibly more powerful. It can shine light upon unknown territories, and even shepherd us beyond the horizon like it did with Dirac. It is not a mere map of reality that we have already explored, but another plane of reality, with its own rules and structure, its own independent reality. Exploring this plane, searching for a beautiful equation, Dirac peeked outside the boundaries of what we knew about the territory of our physical reality. From the point of view of a human who has not developed the “faculties” to interact with this mathematical plane directly, a slab of glass that helps you talk to your friend face-to-face from across the planet is not so different from magic.
In various ancient eastern philosophies we hear about the existence of other planes or Lokas of reality. Schools like Vajrayana Buddhism and Trika Shaivism are explicit in stating that the other Lokas are not at a separate physical location, as is often imagined by people who point at the sky. Instead, they are metaphorical pointers, or abstractions of complicated mechanisms, and connected to the physical reality or Bhu Loka through our thoughts and actions.
Like the plane of mathematics cannot directly cause things on our material plane, the abstract deities of light (Devas and Devis) from these other Lokas can’t hurl stones at us. Our physical reality is still shaped through the physical actions of physical beings. For example, through people who are exploring these realms, trying to connect with these deities, be it through Mantra meditation, or esoteric rituals, or drawing abstract connections between different sounds, symbols, images and forces of nature. These beings of abstract realms are similar to how Mathematical Platonism posits that numerical entities like Pi are eternal, waiting for humans to tap into their mysteries by exploring the abstract, structured landscape of mathematics. How every imperfect circle found in nature is the entry point to explore the perfect circle on this other plane of platonic ideals.
Traditions that practice Tantra emphasize the reality of these other realms or planes. These esoteric and secretive traditions understand deities to be eternally present, who can reveal themselves to humans as Mantras (sounds), their corresponding Yantras (patterns), or visual human and animal like forms, holding different objects, performing various Mudras (hand-poses). Just like numbers are more than just a tool, so are these sounds and symbols. They’re eternally present in these other planes for us to access, understand and help us discover new territories on this physical plane.
Some practices involve chanting a particular deity’s mantra, while concentrating on their corresponding yantra, visualizing the deity in their anthropomorphic form, or placing different sounds corresponding to these deities at different parts of your own body. This is believed to transform the practitioner’s reality from the ground up, metaphorically speaking. That is, transforming the subtlest aspects of their consciousness, the ingrained mental patterns and so on, until this transformation boils up, and fructifies, in the physical reality of their neurons and cells.
The plane of these deities is structured with psychic and symbolical patterns, it is understood using myths and archetypes, and explored using ritual and devotion. Analogous to the plane of Mathematics and Physics that is structured with formal patterns and understood using logic and reasoning. To an outsider, this might seem like magical thinking, but that is similar to how a time traveller from the 16th century will see magic and sorcery in inventions like GPS, made possible only through highly abstract Math and Physics.
A great example to tie this all together is Ramanujan, the famous mathematician from the early 20th century, who pushed the boundaries for summation of divergent series, and had the idea that 1 + 2 + 3 + … while divergent and unbounded, could also be seen as equal to -1/12 in some sense. He was well regarded by his peers, like GH Hardy, for his exceptional intuition. He filled many notebooks with formulas and results, often arriving at them directly, while the rigorous proofs for the results were derived by him or others later on.
Whenever he was asked how he arrived at the results without step-by-step reasoning, or whenever he was awarded or credited for his exceptional work, he would attribute all of it to Devi Namagiri and her consort Narasimha, who he said communicated with him through dreams. Rather than dismissing it as mere humility, I feel we should respect his wishes, and credit Devi Namagiri for his work. Because Ramanujan’s Devi is at least as real as the mathematics that was enlightened by her, through him.
In many eastern traditions, physical reality is understood as being pervaded by subtler planes of existence. Planes of logic and reasoning, of symbols, of potential, going subtler and subtler till you reach the pure Self (Atman), the Vast (Brahman) or Emptiness (Shunyata). The etymology and boundaries of these planes might depend upon the tradition, but the overall structure is very similar.
Ramanujan got his insights through what we can understand as the subtle plane where his Devi resides. He even mentions seeing flowing blood (linking it to Narasimha), and other symbolic references in his dreams. He had this unique ability to receive insights in a language compatible with symbols of both the planes of Devas and formal Math. And of course, he was well trained in logical reasoning, to establish and solidify some of these insights as formal proofs (while also leaving many unproven in his Lost Notebook).
In the present day we have even managed to connect some of his ideas to phenomena in our physical reality. We saw one example above. To claim that just the mathematical structure is real but the spiritual structures are not, is in my opinion a grave mistake. Same as claiming natural numbers are real while imaginary numbers are nonsense.
You might object that the idea of planes might make philosophical sense, but practically speaking Science and Math have given us tangible results. The modern scientific era and the industrial revolution have given the world undeniable material progress. While symbolic thinking and dogmatic beliefs had led to countless wars, famines and diseases before.
But maybe we should take our heads out of the narrow hole of recent Western European history, and look at other civilizations in the world, across wider timespans. So many parallels exist between the language, beliefs and practices of Europe, Iran and India if we go back far enough. There has been a continuous exchange of ideas across the globe, right from the time of Bronze Age civilizations, if not before. So I urge you to pick a system that navigates these subtler realities, one that is well established, has a mostly unbroken lineage of teachers and teachings, and start exploring this other plane of symbols and rituals for yourself. Practice it, same as we practice Mathematics or Science.
For example, you could pick Vajrayana Buddhism, spend a few years understanding their philosophical lens to view reality, but more importantly doing the prescribed practice(s). If it gives you tangible results and experiences, invest more time and energy. If not, move on to some other system. One can stick to the Scientific Method and use their own subjective experience as evidence, discarding what seems like dogma in these traditions. But as a first step, one has to drop the dogma that outright dismisses ideas subtler than physical reality.
A word of caution here - the result of these practices is naturally going to be what the tradition claims as its goal. Some traditions aim towards material success and stop there, some aim towards enlightenment or freedom from suffering, through the renunciation of material world. While others are all encompassing, claiming material success, fulfillment of worldly desire, inner peace are all valid stepping stones towards ultimate freedom. So spend some time exploring the landscape before choosing your destination and charting your path.
You may dismiss Tai Chi as just glorified breath work and stretching exercises, but there are so many results published since 2020 that show that Tai Chi is significantly better at treating hypertension, anxiety and depression, better at increasing lean mass, when done in controlled trials against “non-mindful” forms of simple stretching or breathing. So why do we dismiss the reality of Qi or Prana, which are analogous to the Math behind the Engineering of Tai Chi or Yoga? You might not find physical correlates of Qi or Prana or Chakras if you dissect a body, but belief in these ideas, exploring them in their own plane of existence, has led to practices that give tangibly better results than what 20th century understanding of physiology could. Thus, I claim that Prana or Chakras are as real as Calculus or Number Theory. Just like Mathematics, these are not mere conceptual maps of our physical reality, but more powerful, as they too can lead our understanding beyond the horizons.
The decision is ultimately up to the individual. An engineer can either use complex numbers and make his life easy, or stay a purist and go about the same calculation the hard way. Similarly, I can either explore these subtler planes, or dismiss anything that is not physically real or immediately intuitive.
Let’s come back, once again, to the concept of the vibration of “nothingness” we were discussing before in the context of Casimir’s experiment. You can debate that Fields are not a real thing but just empty space, and what is really real is matter. Or you can view the Field as the emptiness that is the essence of reality, where all sub-atomic particles of matter are just energy values of this field, quantized in particle-like chunks, existing in an abstract complex plane as a cloud of probability, entangling with each other to produce this tangible-ness. As Alan Watts puts it, it is a cosmic dance, with no nouns, only verbs. In my humble opinion, both of these ways of looking at things are equally “real”.
In Trika Shaivism reality is conceived as the non-dual Shiva, the underlying conscious void beneath reality. This void is static, but it is also dynamic. It is vibrating with Shakti (translated as energy or power). This vibration of nothingness is known as Spanda. If I blur my eyes just enough, I can see how this view of reality eerily rhymes with the fluctuations of EM fields in empty space.
Reality is vast and incomprehensible. It can be seen using different lenses, that is, through different abstract structures. Just as the force in Casimir Effect could be calculated with different models, one where we posit that nothing likes to vibrate, and in another we see the same nothingness as a static container, for matter that is never at rest. I am not saying that because reality is weird, we should accept any weird theory. Quite the opposite. We should accept only self-consistent theories that explain the weird phenomena of our reality.
Some traditions like Nyaya and Advaita Vedanta stuck to logic, and extensively debated scholars from other schools. While other traditions may focus on looser, symbolic reasoning, with greater emphasis on ritualistic practice or devotion. But ultimately the reality is vast, and human brains are limited, so there are necessarily going to be multiple self-consistent view points (like blind men describing an elephant).
This idea goes back quite far. Here’s a verse from Isha Upanishad (Yajur Veda Ch. 40 Verse 5), which was originally orally transmitted, with strict intonation and meter to avoid corruption, and is at least 2500+ years old:
तदन्तरस्य सर्वस्य तदु सर्वस्यास्य बाह्यतः ॥ ५ ॥
That moves (tad ejati) and That moves not (tan naijati), That is far (tad dūre) and the same is near (tadvantike). That is within all this (tad antarasya sarvasya) and That also is outside all this (tadu sarvasyāsya bāhyataḥ).
The verse may seem contradictory, but later philosophers like Shankara (c. 800 CE) wrote detailed commentaries on cryptic verses from Vedantic texts to establish well-structured theories like Advaita (Non-duality). If you are intrigued by the parallels these ideas from ancient spiritual traditions have with descriptions of reality of modern physics, you are not alone, and this is not a mere coincidence. Many of the scientists that established the foundation of modern physics, like Bohr, Schrödinger, Oppenheimer and Einstein were avid readers of Vedantic texts like the Upanishads (where the above verse is from) and the Bhagavad Gita. It was Bohr in fact, who suggested the idea of zero-point energy to Hendrik Casimir of the Casimir effect.
Similar to how Tai Chi is born from the understanding of Qi, we should pay attention to the philosophies of Vedanta, Trika or Buddhism that rhyme so well across different planes of reality. Many of these traditions see reality as a fluid, interdependent non-duality. They rhyme well with our best understanding of our material universe, even if they approach it through the opposite end, by flipping around a core dogma of modern thinking right on its head. The dogma that objective reality of matter is real, but subjective reality of first-person experience is just an epi-phenomena – something to be explained away – as mere excitations in neurons.
These traditions instead claim that the subjective first-person awareness, that is indisputably real to every living being, is the ever-present dynamic static at the base of all reality. This subject is not separate from the multiplicity of shapes, forms and experiences of objects observed and illumined by it. A practitioner of these traditions is empowered to experience this Non-Duality in their own first-person awareness, by simply listening, logically reasoning, and meditating upon these teachings. Because really, You are That.
क्षमस्वापराधं महागुप्तभावं
मया लोकमध्ये प्रकाशिकृतं यत् ।
तव ध्यानपूतेन चापल्यभावात्
स्वरूपं त्वदीयं न विन्दन्ति देवाः ॥७॥