Pythagoras’s Theorem and what it means to ‘know’ Dhamma

Pythagorean theorem. 13th-century Arabic manuscript page showing a proof of the Pythagorean theorem by Persian polymath and mathematician Nasir al-Din Tusi (1201-1275).

What does it mean to be knowledgeable in Dhamma? What is the kind of knowledge that brings freedom from suffering? We must learn how to make the distinction between the ordinary form of knowledge, and knowledge with wisdom, which of course are two entirely different things. We can often unconsciously confuse them when it comes to Dhamma, and think we ‘know’ when in fact we know nothing at all.

By way of an analogy that points towards the nature of the knowledge of Dhamma, it happens to come in handy to have studied mathematics, for there are some useful comparisons to make between the nature of mathematical knowledge and the nature of wisdom. (I hereby offer my sincere apologies in advance to anyone who hates mathematics; and I hope they may be able to understand the explanation that follows in spite of it…)

Let us take as an example the above image, which a diagram of Pythagoras’s Theorem (actually, a proof of it): something that pretty much everyone learns when they are in school, and probably the one mathematical theorem of which most people retain at least a vague memory (even if it’s only of how much they despised it).

It is quite easy to understand what the theorem says. It says that in any triangle which has an angle of 90 degrees, the square of the length of the side of the triangle opposite the 90 degree angle, is equal to the sum of the squares of the two other sides. More concisely: c2 = a2 +b2.

When you learn this off, you’re able to do some calculations using it, given in the exercise book in school, and the calculations you do, assuming you do them correctly, will give you the answer that you find at the back of the textbook.

Now you might think you ‘know’ Pythagoras’s theorem. You even might think you understand it. But this ‘knowledge’ is not really knowledge at all; it’s a piece of information that has been handed to you by somebody else.

The Nature of Knowledge

The truths of mathematics, usually presented in the form of statements or theorems, all share two important features:

  1. They can all be derived.  This means that anyone, given the right tools, some training and someone to point him in the right direction, is capable of finding out the law of, let’s say Pythagoras’s theorem, for himself. That means that you don’t have to (and shouldn’t) depend on mere trust in what you are told; with a little effort and calculation, you can see how Pythagoras came to his conclusion, tracing yourself the path that he took to reach it.
  2. Any mathematical statement, any theorem like Pythagoras’s theorem, also has to have a proof. The entire why behind this statement, this equation, is open, clear, laid bare for anyone to see.

A mathematical proof is something that is immensely strong. You can’t, for example, prove Pythagoras’s theorem by paying somebody to draw 90-degree angle triangles all his life, and showing that the theorem holds true for each of them. What if it is a coincidence? What if there is some triangle out there where the theorem does not hold true? To prove something mathematically means that you prove its truth in all cases, in a way that leaves absolutely no doubt whatsoever for the one who has understood it. Try arguing with someone who has understood the proof of Pythagoras’s theorem; try to make him doubt it, or question its truth; if that person has really understood the proof, your attempt will be fruitless. To prove something mathematically means to clearly show, leaving no room for doubt, that this cannot be otherwise.

The same is true of the teachings of the Buddha. They are not meant to be accepted at face value and learned by heart and repeated. They are as clear, transparent and open as a mathematical statement, meant to be discovered for oneself, found out for oneself, known as something you have seen, not as something you have heard or read. With the help of some guiding points and a start on the right path, anyone can trace the path that is taken to reach the vital knowledge, somewhat like deriving a mathematical statement.

Further, as with someone who has studied a mathematical proof, or even better, found one for himself; for the one who has really understood the Dhamma, who has seen it himself, proved it to himself, there can also be no room left for doubt in it. He is possessed of a kind of knowledge which is based on understanding of the nature of things, nothing less and nothing extra, no need for interpretation, no ‘philosophy.’ He knows because he has seen that things are this way and cannot be otherwise.

Trying to put doubt in someone who has this kind of knowledge in him, is as useless as trying to persuade a competent mathematician that Pythagoras’s theorem is not true: he will simply laugh at you, maybe show you the proof if he is feeling generous, and not even bother to argue back.

The purpose of trust

For someone who is working on the study and practice of the Dhamma, the aim is to work to attain this kind of knowledge. Such a study does, to be sure, require a certain level of basic trust in the teacher, the teaching and the practice from the beginning, otherwise it would be impossible exert the required amount of effort, to have the discipline and determination that is needed to reach the goal. The same goes for mathematics; figuring out the proof of, say, Einstein’s General Theory of Relativity could be the work of a lifetime, and someone who whose mind from the beginning is full of nothing but doubt in the discipline of mathematics, who does not have a basic trust in its validity, will never succeed in this endeavour.

In both cases, however, this basic trust is a necessary starting point, but only that; and if we don’t go beyond it, we will never progress. Most people pass through their so-called ‘study’ of mathematics without bothering to try to understand the proof of the things they are told, and without trying to find them for themselves. Why bother? It’s so much effort to do it, when you can just take the statement happily in your hand and remember it: c2 = a2 +b2; now you can get all the right answers at the back of the book without doing anything more. Even if they are forced to study the proof of such a statement, most people prefer to memorise it line by line, rather than making the effort of understanding it for themselves. It seems such a tremendous effort to ask why all the time. It’s much easier to just accept things as they are told to you and do the exercises and get the right answers at the back of the book.

In both mathematics and dhamma, though, this kind of attitude will get you nowhere. You might be able to talk for hours about the teachings of the Buddha, you might be able to get all the right answers in the back of the textbook, but when the real test comes, you are lost. If you think to reach the end of suffering by learning by heart the Eightfold Path, the Four Right Exertions, the Seven Factors of Enlightenment, the Five Faculties or the Four Bases of Power – think again.

How to “derive” the truths of the Dhamma?

How to see them for oneself? How to arrive at the clear and unshakeable knowledge of it that does not allow any room for doubt; as one who has understood the proof of Pythagoras’s theorem?

It is necessary to get out of the habit of taking things at face value and assuming we know the answers. For example, take the question: is it possible for one who is a Sotapana to kill?

We all might think we know the answer, but where does this knowledge come from? Do we understand why a Sotapana will choose death of himself rather than killing? Being able to give the ‘right answer,’ the one you find at the back of the textbook, is worthless in this matter.

Understanding of Dhamma should also not be understood as referring to the mere ability to analyse a theoretical subject. There are plenty of university professors who write technical papers on Dhamma, and yet do not even keep the five precepts. But if one genuinely comes to understand why a Sotapana will never kill, then that person will never again dare to kill any living creature. Their understanding is not theoretical; it applies to their own being; it affects and changes their mind for good.

It is said (in the Angutarra Nikaya 5.26) that there are five occasions when one may see the Dhamma and attain freedom from suffering: when one is

  • being taught by a teacher;
  • teaching others;
  • Repeating the teaching one has heard;
  • Contemplating the teaching;
  • Observing a meditation object that is properly grasped, attended, and seen with wisdom.

In each case, it is not just the fact of listening, teaching, repeating, contemplating or meditating that gives rise to concentration and freedom from defilements. It is said that ‘Exactly as the teaching is heard (or taught, repeated, contemplated, observed and properly understood in meditation); in just that way he experiences and sees the teaching and its meaning.’’

In other words, the one who pays proper attention and remains heedful, ardent and resolute, experiences and sees exactly that fact, that reality which is pointed to in the teaching he hears (or teaches, repeats, contemplates or observes as an object of meditation).

Where to find this reality, and the eyes to see it? If we try to search for it with our eyes, our ears or in words or memory, we will never find it. Therefore, first and foremost we need to find how to look within ourselves to see the site where the Dhamma is applied, in order to thoroughly understand it, no matter by what means. The mind must learn to look into itself. Without this, we will never be able to go beyond the level of textbooks and theory, which are worthless for one whose aim is freedom from suffering.

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