A physicist, mathematician and Zen master walk into a bar

An inquiry into the surprising connection between Albert Einstein, Georg Cantor and Dogen Zenji.

It never ceases to amaze me how many non-trivial similarities exist between ancient spiritual writings and modern discoveries in physics and mathematics. This gives credence to the age old adage that the truth is reshaped and reformulated for every new generation. I want to examine how a few paradoxical passages from Dogen’s famous Zen text Genjo Koan can be seen through the lens of two triumphs of modern thought: Einstein’s Theories of Special and General Relativity and Georg Cantor’s work in Set Theory.

Space and Time

If I am running and you are stationary, time is passing more slowly for me than for you. If you are standing on the top of a mountain and I’m on the surface of the Earth, time is passing more slowly for me than you. These are both unintuitive consequences of Einstein’s theories of relativity. Time is slowed in the presence of a gravitational field, and slows down proportional to your velocity. There is no universal clock against which all events are measured, nor is there a container of space in which all events occur. Space and time are relative and interconnected, and Einstein merged them into a singular field known as spacetime. Spacetime can curve, stretch, warp and twist. Your position in this continuum is fundamentally unique and you are a node in this universe-sized network. Einstein’s insights were seen as groundbreaking, yet below we see Dogen writing about the exact same thing, hundreds of years prior.

“For time to fly away there would have to be a separation [between it and things]. Because you imagaine that time only passes, you do not learn the truth of being-time. In a word, every being in the entire world is a separate time in one continuum.

Time does not pass uniformly everywhere, it depends on your velocity, how close you are to a gravitational field, and your reference frame. So everyone has a separate time, in the continuum of spacetime.

“Firewood becomes ash, and it does not become firewood again. Yet, do not suppose that the ash is after and the firewood before”

This perplexing quotation by Dogen is an intuitive notion for Einstein: “The distinction between past, present and future is only a stubbornly persistent illusion.” What they are both getting at is the fact that distinctions between past and future are not fundamental to the structure of reality. The radical difference between firewood and ash is only striking because of our specific human embodiment. Our view of reality is deeply blurred, in that most of the microscopic details of a state, be it firewood or ash, are ignored by our being. Existence would be utterly overwhelming if we were conscious of all the data of experience.

We can only see a tiny fraction of the electromagnetic spectrum.

If every detail of the state of firewood and the state of ash were available to us, they would not appear so radically different.

If questions pertaining to the unreality of time interest you, I would highly recommend reading The Order of Time by Carlo Rovelli.

Infinity

Before diving into the issue of infinity, some groundwork must be laid. Let us consider the following set: A={1,2,3,4,5}. The set A has five elements in it, the numbers 1,2,3,4,5. A proper subset of A is a set that contains only combinations of elements of A, but is not identical to A. So some examples of subsets of A are: {1,2}, {1,2,3,4}, {1,3,5} and so on. It should be clear then that for a set with a finite number of elements, a subset cannot be the same size as the original set. This rule does not apply for infinite sets.

Georg Cantor began working on transfinite arithmetic at the end of the 1800s. His remarkable conclusion was that there are different sizes of infinity. Namely, there are more Real Numbers than Natural Numbers. Recall that Natural Numbers are the counting numbers: N={1,2,3,4,5,6,7,…..}, and Real Numbers are all those numbers, as well as all decimal expansions and fractions. Using his famous Diagonal Argument, he showed that there are more Real Numbers. A consequence of his work is also that a proper subset of an infinite set can be the same size as the original set. In Set Theoretic language, they can have the same cardinality.

Are there more Natural Numbers or Even Numbers?

N={1,2,3,4,5,6,7,…} & E={2,4,6,8,10,….}

Two things appear obvious at first: E is a proper subset of N, and it looks like E has half as many elements as N. In fact, N and E are the same size. You can match every element in N with an element in E. In Set Theoretic language, N and E can be put into one-to-one correspondence. The moral of this brief foray into Set Theory is this: there are things in nature from which you can take away from, yet lose nothing in the process. Here is another passage from Dogen’s Genjo Koan, in which he explains a similar idea:

“Enlightenment is like the moon reflected on the water. The moon does not get wet, nor is the water broken. Although its light is wide and great, the moon is reflected even in a puddle an inch wide. The whole moon and the entire sky are reflected in dewdrops on the grass, or even in one drop of water. Enlightenment does not divide you, just as the moon does not break the water…Each reflection, however long or short its duration, manifests the vastness of the dewdrop and realizes the limitlessness of the moonlight in the sky”

The point of this passage is to demonstrate that the nature of enlightenment is inexhaustible and cannot be diminished once accessed. It can be divided, yet each division retains the depth of its original source. The process of being reflected in a puddle does not detract from the moon’s vastness, nor is the reflection any less vast. This is similar to how taking out the Even numbers from the Natural numbers does not detract from the size of the Naturals. This is certainly a counterintuitive notion, yet Cantor provides rigorous proof for its validity. Comparing these exotic concepts helps to shed light on the transcendent truth that they both point towards. Whatever truth may be, it transcends language. Bare reality cannot be pigeonholed into rational thought. Yet, from within the confines of words, Einstein, Cantor and Dogen are urging us to go beyond.

Works Cited

Dōgen, and Bokusan Nishiari. Dogen’s Genjo Koan: Three Commentaries. Counterpoint, 2011.

Rovelli, Carlo, et al. The Order of Time. Riverhead Books, 2018.