Joe Pagano

What the Buddha Knew About Mathematics But Never Told You



Posted: Thursday, April 28, 2011

by Joe Pagano
Math by Joe

"Form is emptiness, emptiness is form." ---Heart Sutra

At the heart of Buddhism is the concept that stillness reigns supreme, that by virtue of stilling the mind and emptying it of its contents, one can become enlightened. What is mind-startling about this premise is the truth that rings loudly from its core. What we shall see here through a mathematical example is how the Buddha was indeed right: form does arise from emptiness and emptiness is actually form. It is all in the perception.

The famous mathematician, John von Neumann, created a method, known as von Neumann hierarchy, of generating the set of Natural numbers {1,2,3,...} from...essentially nothing. From this example, we see how form arises out of emptiness, and thus reinforces the precept of the Heart Sutra quoted above. To carry out this example, all we need is some basic knowledge of set theory.

A set is a group of objects. Thus the set A = {3, 5, 7, 9} comprises the odd numbers from 3 through 9 inclusive. We can also talk about the set {}, which is called the empty set. The empty set is the set which contains no elements; this set is sheer "emptiness." Now sets can contain sets as elements. For example, take the set Aabove and now add the element {12} which is the set which contains the element 12. Form the set B = {3, 5, 7, 9, {12}}. The set B contains five elements, one of which is itself a set.

When we talk about the union of two sets, we simply merge the elements of each set into one, not repeating any common elements. Thus if C = {1, 2, 3} and D = {3, 4, 5} then C union D, or CUD, in which the symbol "U" means union, is the set E = {1, 2, 3, 4, 5}. Now the example that follows is going to show how we create something from nothing, or form from emptiness. We have the following steps:

Step 0: {} Empty Set

Step 1: { {} } Set containing the empty set

Step 2: { {}, { {} } } Set containing the previous two sets

Step 3: { {}, { {} }, { {}, { {} } } } Set containing the previous three sets

Iterating in this way, we create a sequence in which the next element is the union of the two previous sets. Now at Step 1, we have a set with one element; at Step 2, we have a set with two elements, and so on. In this way, we can generate the set of Natural numbers {1, 2, 3,...}

Thus from emptiness, namely the empty set, which contains nothing, we have created form, that is the Natural numbers, which contain an infinity of numbers. The only difference is how we look at the two sets: our perspective. In mathematics, there is a very special word for what we have just done between the sequence generated by the empty set and the Natural numbers. We say the two sets are isomorphic, which basically means that the elements of each set are the same except in the way we perceive them. This is much the same as the isomorphism that would exist between the numbers 1,2,3... in English and the same numbers called by another name in a foreign language such as French or German, or even Macedonian, if you will.

Thus emptiness is form, depending on our perspective of course; and form, the Natural numbers, is emptiness, again depending on our perspective. Indeed the Buddha did know something about mathematics that he never told us; but now even we know some of his ancient secrets. Now maybe if we apply them to everyday living, everybody will be a lot better off
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