Tag: Buddhism

Zero — Invented or Discovered?

It seems almost a bizarre question. Who thinks about whether zero was invented or discovered? And why is it important?

Answering this question, however, can tell you a lot about yourself and how you see the world.

Let’s break it down.

“Invented” implies that humans created the zero and that without us, the zero and its properties would cease to exist.

“Discovered” means that although the symbol is a human creation, what it represents would exist independently of any human ability to label it.

So do you think of the zero as a purely mathematical function, and by extension think of all math as a human construct like, say, cheese or self-driving cars? Or is math, and the zero, a symbolic language that describes the world, the content of which exists completely independently of our descriptions?

The zero is now a ubiquitous component of our understanding.

The concept is so basic it is routinely mastered by the pre-kindergarten set. Consider the equation 3-3=0. Nothing complicated about that. It is second nature to us that we can represent “nothing” with a symbol. It makes perfect sense now, in 2017, and it's so common that we forget that zero was a relatively late addition to the number scale.

Here's a fact that's amazing to most people: the zero is actually younger than mathematics. Pythagoras’s famous conclusion — that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides — was achieved without a zero. As was Euclid’s entire Elements.

How could this be? It seems surreal, given the importance the zero now has to mathematics, computing, language, and life. How could someone figure out the complex geometry of triangles, yet not realize that nothing was also a number?

Tobias Dantzig, in Number: The Language of Science, offers this as a possible explanation: “The concrete mind of the ancient Greeks could not conceive the void as a number, let alone endow the void with a symbol.” This gives us a good direction for finding the answer to the original question because it hints that you must first understand the concept of the void before you can name it. You need to see that nothingness still takes up space.

It was thought, and sometimes still is, that the number zero was invented in the pursuit of ancient commerce. Something was needed as a placeholder; otherwise, 65 would be indistinguishable from 605 or 6050. The zero represents “no units” of the particular place that it holds. So for that last number, we have six thousands, no hundreds, five tens, and no singles.

A happy accident of no great original insight, zero then made its way around the world. In addition to being convenient for keeping track of how many bags of grain you were owed, or how many soldiers were in your army, it turned our number scale into an extremely efficient decimal system. More so than any numbering system that preceded it (and there were many), the zero transformed the power of our other numerals, propelling mathematics into fantastic equations that can explain our world and fuel incredible scientific and technological advances.

But there is, if you look closely, a missing link in this story.

What changed in humanity that made us comfortable with confronting the void and giving it a symbol? And is it reasonable to imagine creating the number without understanding what it represented? Given its properties, can we really think that it started as a placeholder? Or did it contain within it, right from the beginning, the notion of defining the void, of giving it space?

In Finding Zero, Amir Aczel offers some insight. Basically, he claims that the people who discovered the zero must have had an appreciation of the emptiness that it represented. They were labeling a concept with which they were already familiar.

He rediscovered the oldest known zero, on a stone tablet dating from 683 CE in what is now Cambodia.

On his quest to find this zero, Aczel realized that it was far more natural for the zero to first appear in the Far East, rather than in Western or Arab cultures, due to the philosophical and religious understandings prevalent in the region.

Western society was, and still is in many ways, a binary culture. Good and evil. Mind and body. You’re either with us or against us. A patriot or a terrorist. Many of us naturally try to fit our world into these binary understandings. If something is “A,” then it cannot be “not A.” The very definition of “A” is that it is not “not A.” Something cannot be both.

Aczel writes that this duality is not at all reflected in much Eastern thought. He describes the catuskoti, found in early Buddhist logic, that presents four possibilities, instead of two, for any state: that something is, is not, is both, or is neither.

At first, a typical Western mind might rebel against this kind of logic. My father is either bald or not bald. He cannot be both and he cannot be neither, so what is the use of these two other almost nonsensical options?

A closer examination of our language, though, reveals that the expression of the non-binary is understood, and therefore perhaps more relevant than we think. Take, for example, “you’re either with us or against us.” Is it possible to say “I’m both with you and against you”? Yes. It could mean that you are for the principles but against the tactics. Or that you are supportive in contrast to your values. And to say “I’m neither with you nor against you” could mean that you aren’t supportive of the tactic in question, but won’t do anything to stop it. Or that you just don’t care.

Feelings, in particular, are a realm where the binary is often insufficient. Watching my children, I know that it's possible to be both happy and sad, a traditional binary, at the same time. And the zero itself defies binary categorization. It is something and nothing simultaneously.

Aczel reflects on a conversation he had with a Buddhist monk. “Everything is not everything — there is always something that lies outside of what you may think covers all creation. It could be a thought, or a kind of void, or a divine aspect. Nothing contains everything inside it.”

He goes on to conclude that “Here was the intellectual source of the number zero. It came from Buddhist meditation. Only this deep introspection could equate absolute nothingness with a number that had not existed until the emergence of this idea.”

Which is to say, certain properties of the zero likely were understood conceptually before the symbol came about — nothingness was a thing that could be represented. This idea fits with how we treat the zero today; it may represent nothing, but that nothing still has properties. And investigating those properties demonstrates that there is power in the void — it has something to teach us about how our universe operates.

Further contemplation might illuminate that the zero has something to teach us about existence as well. If we accept zero, the symbol, as being discovered as part of our realization about the existence of nothingness, then trying to understand the zero can teach us a lot about moving beyond the binary of alive/not alive to explore other ways of conceptualizing what it means to be.

What Is Meditation?

meditation

I've been putting this off for years. Now is the perfect time.

Meditation offers a path toward increased happiness, creativity and mindfulness. Steve Jobs was a lifelong practitioner, reading deeply on the subject and taking meditation retreats. Each year he'd re-read his copy of Autobiography of a Yogi.

I've always put off exploring meditation until later. I convinced myself I was too busy. That I'd explore it when things were less hectic. But I was wrong for two reasons. First, logically, if meditation is beneficial enough to practice at some point, it is probably valuable enough to practice now. That is if I think it will add value to my life, choosing to start becomes a matter of opportunity costs. What would I do with the time I'm spending meditating versus the benefits of meditation. The only way to really evaluate that is to do it and see the results. Second, and somewhat less logically, is that I've met some amazing – and incredibly busy people – in the past year and one thing that seems to keep coming up is meditation. None of these people “makes time for meditation.” Rather they all see meditation as an enabler of everything else they do.

As I reflect on my life, my priorities, my friendships, and my place in the world, there is no better time than now to start exploring meditation.

Dan Harris has the best practical advice on how to meditate but I wanted to explore it a bit more. In Buddhism for Beginners, Thubten Chodron does a good job looking at meditation.

Let's start with what meditation is not.

Nowadays meditation is sometimes confused with other activities. Meditation is not simply relaxing the body and mind. Nor is it imagining being a successful person with wonderful possessions, good relationships, appreciation from others, and fame. This is merely daydreaming about objects of attachment. Meditation is not sitting in the full vajra position, with an arrow-straight back and a holy expression on our face.

[…]

Meditation is a mental activity. Even if the body is in perfect position, if our mind is running wild thinking about objects of attachment or anger, we're not meditating. Meditation is also not a concentrated state, such as we may have when painting, reading, or doing any activity that interests us. Nor is it simply being aware of what we are doing at any particular moment.

Why do we meditate? What are the benefits?

The Tibetan word for meditation is gom. This has the same verbal root as “to habituate” or “to familiarize.” Meditation means habituating ourselves to constructive, realistic, and beneficial emotions and attitudes. It builds up good habits of the mind. Meditation is used to transform our thoughts and views so that they are more compassionate and correspond to reality.

[…]

By building up good habits of the mind in meditation, our behavior in daily life gradually changes. Our anger decreases, we are better able to make decisions, and we become less dissatisfied and restless.

What is meditation? How do we learn to meditate?

Some people think they can invent their own way to meditate and don't need to learn from a skilled teacher. This is very unwise. If we wish to meditate, we must first receive instruction from a qualified teacher.

[…]

First, we listen to teachings and deepen our understanding by thinking about them. Then, through meditation we integrate what we have learned with our mind. For example, we hear teachings on how to develop impartial love for all beings. Next, we check up and investigate whether that is possible. We come to understand each step in the practice. Then, we build up this good habit of the mind by integrating it with our being and training ourselves in the various steps leading to the experience of impartial love. That is meditation.

Meditation is of two general types: stabilizing and analytical. The former is designed to develop concentration and the latter to develop understanding and insight. Within these two broad categories, the Buddha taught a wide variety of meditation techniques and the lineages of these are extant today. An example of stabilizing mediation is focusing our mind on our breath and observing all the sensations that occur as we breathe. This calms our mind and frees it from its usual chatter, enabling us to be more peaceful in our daily life and not to worry so much. The visualized image of the Buddha may also be used as the object upon which we stabilize our mind and develop concentration. While some non-Buddhist traditions suggest looking at a flower or candle to develop concentration, this is generally not recommended by Buddhist traditions because meditation is an activity of our mental consciousness, not our sense consciousness.

Other meditations help us to control anger, attachment, and jealously by developing positive and realistic attitudes towards other people. These are instances of analytical or “checking” meditation. Other examples are reflecting on our precious human life, impermanence, and the emptiness of inherent existence. Here we practice thinking in constructive ways in order to gain proper understanding and eventually go beyond conceptual thought.

Buddhism for Beginners is a great place to start and offers “a manual for living a more peaceful, mindful, and satisfying Life.”

(Image source)