The Mathematical Proof of Emptiness
Emptiness is the teaching that the things we observe have no absolute, independent existence. Emptiness is a cornerstone of Mahayana Buddhist thought, yet this tradition is not the only that points to it. The mathematical and scientific traditions of the West are now becoming aware of it.
Math is counting. The simplest mathematical system is the counting numbers, also known as the natural numbers: 1, 2, 3, 4, 5, and so on. Math also refers to the manipulations used with these numbers, which are nothing more than symbols, to arrive quickly at results that could also be obtained by counting items directly. For example, say that we want to know the number of squares on a chessboard. We could simply count all of the squares. Alternatively, we could count the number of squares in each row, then observe that each of the rows are the same, then add the number of squares in each row to itself for every row. Alternatively, we could perform the symbolic operation 8 × 8 = 64. Each of these is an equivalent way of counting the squares. Math is shorthand for and equivalent to simply counting objects.
Counting is a very ancient way of thinking; there is evidence of basic forms of counting even in prehistoric times. It is certainly a tribute to the human mind that such a perfectly logical and convenient system seems to have arisen naturally even at the earliest stages of our history. The general acceptance of observations such as these was a source of great surprise when mathematics was proven to lack inherent logic. In 1931, young Austrian logician Kurt Gödel published his First and Second Incompleteness Theorems. These theorems dealt a deathblow to the idea of the inherent logic and perfection of mathematics.
If we can describe the basis of mathematics – the counting numbers – by a system of simple rules, then we have a logical definition of the counting numbers and of mathematics that is more rigorous than simply saying that the numbers are 1, 2, 3, and so on. Gödel essentially proves to us that there are basic truths of arithmetic, the shorthand for counting, that can never be proven, no matter what system of rules we use to define it. An example of a basic truth is 1 + 1 = 2. Notice that the word “truth” is commonly used, along with the phrase “unprovable but true.” These are observations of the fact that, although Gödel has proven that we cannot define counting such that all of the statements such as 1 + 1 = 2 can be proven, we simply “know” that they are true. How couldn’t we? Who cannot see that if we have one object, and then another, that now we have two? Hence, such statements are called “arithmetic truths.” But Gödel proves that they are not logical truths. We must therefore conclude that counting, or arithmetic, and all of the higher mathematics based upon it appear as obviously true to our minds, but in reality they are not…