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Segments in a straight line have non-zero lengths but all of them have zero area i.e. they are "empty" when viewed as embedded in 2-dimensional plane. Similarly, all 2-dimensional shapes viewed as embedded in 3 dimensions appear as zero-volume entities, although in their own plane of 2 dimensions they do have intrinsic substance (area).
So my first question is: is this a valid analogy for the buddhist concept of "emptiness"?

Naturally, the analogy can be continued to infinity: at each step, objects are "substantial" within their own plane but "empty" as regards the higher planes. This urges us to think that in infinite dimensions we have finally arrived at perfect substantiality, as no higher plane can exist to destroy it. But we can move further on, to spaces with uncountably infinite dimensions, where "orders of infinity" can indeed provide new dimensions to our enquiry, to a point where our human thinking begins to be at a loss as to what is going on...
So my second question is: has this extension to infinite dimensions caught the attention of buddhist philosophers?

exp8j
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  • Your question is posed in a mathematical way. See Volume or Determinant for a mathematical understanding. I'm new to Buddhism, so I can only guess "emptiness" should be translated to "absence". – Gyro Gearloose Mar 05 '24 at 20:22

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I think this takes the analogy in the wrong direction. If we look at a circle drawn on paper, we actually have a line (a substantial form) that separates inner and outer spaces that are themselves empty. This is the nature of mind: every thought and mental object is a sketched-out, semi-arbitrary delimiter we use to sort the world into manageable clumps. But we don't seek to contract that circle into nothingness; it's more that we expand the circle until we no longer see the form of it, so that all we have is the emptiness.

I've always liked the metaphor they use in the daodejing (chapter 11). We make a cup out of clay, but the clay isn't the important part of the cup. The empty space the clay defines is the important part. Form defines empty space, empty space makes form useful. The trick (for a Buddhist or a Daoist) is to recognize those forms as mere forms, so that we can recover that space in its unconstrained form.

Ted Wrigley
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  • As for recovering space in its unconstrained form, I also sometimes think that as my body is a closed form dividing the whole of space into an interior and an exterior, I choose to name "me" the interior of that form, but why not make an inversion to suddenly become the infinite exterior? – exp8j Mar 04 '24 at 19:16
  • Or, on a more modest level, to read a book by concentrating on the empty space between the letters. Now such playful emptiness is good for providing a relaxed & amusing perspective for working with forms and putting them in order, in beautiful collections of any kind... But another perspective also appears: to see emptiness as the field of potentiality for the infinite varieties of forms... Evidently, I do have a hard time grasping the value of emptiness per se! – exp8j Mar 04 '24 at 19:16
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Emptiness in Buddhism is a complex topic with different ways of viewing it. As a Buddhist, what is important is how these perceptions of emptiness affect our thinkings, decisions and behaviours. Buddhism is known for its pragmatism and practical approaches. I believe the geometric example of emptiness given in the OP is valid. However, I struggled to see its practical applications. Below, are some practical applications of emptiness.

Emptiness as a meditative dwelling. If we have a chance to go to a quiet park or a place where there is little human activities, we get a chance to experiment and experience this. Or if we happened to live in a place where there is winter, go to a large, open space after a heavy blizzard. Empty our thoughts and listen intently to the surroundings. We will notice the quietness in the air. Slowly, as we willingly stay with this quietness, we’ll notice our mind becoming very still. And the strange thing is the mind actually enjoys being in that state of stillness. No inner chattering, discourse and arguments about the past or future. Just being present and enjoying it. Honestly, this is one reason why I (and I believe many people) enjoy going to the parks and being surrounded by greeneries. I strongly believe if we can maintain this state of mind throughout the day, it will do wonders for our mental wellbeing.

Emptiness as an attribute of objects. If we are obsessed with wealth, material possessions, physical appearance and reputation, seeing the lack of self in these things (even in ourselves) helps to weaken this strong attachment. Who is going to inherit our wealth, possessions or remember our youthful and good looks after we are gone? Does it matter? Being around people who are constantly obsessing over money, power and appearances and also people who are nonchalant about these things; do you ever wonder why there are such huge differences both in attitude and behaviour over wealth, reputation and gain? When trying to understand the psychological make-up of these two groups of people (whom I know intimately) and I realized that fundamentally they have very different valuations of things. If we see possessions as intrinsically devoid of real value (i.e. emptiness) other than for sustaining our existence; would we become hysterical if we lose these possessions? Yes, I may envy the former group but the most enjoyable times were spent with the latter group.

Emptiness as a type of awareness-release. Personally, I don’t have concrete experience with this. But I believe this is a central theme in Mahayana tradition and must, therefore, been a source or an inspiration for liberation for many practitioners over the ages.

So, if there are practical applications in a geometrical definition of emptiness, it would be more relevant. With Metta.

Desmon
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  • I like "Emptiness as an attribute of objects". And of mind too. But this assertion is valid only for someone living in the fullness of a plane that contains & extends, enriches above emptiness that which he refers to. In this way we can explain "paradoxical" statements like "emptiness is form, form is emptiness". Emptiness in an embedded plane is indeed form in a container-plane. Form in an embedded plane is indeed emptiness in a container-plane. Just like 2D figures are part of 3D, but flat, although still existing (fully in their own plane, "emptily" in the container-planes). – exp8j Mar 06 '24 at 09:53
  • Now a second way to go is to forget about embedded planes and see only one "space" that includes everything, but has the ability to become denser or thinner here and there, resulting in the appearance of "forms in a void". But ultimately form and void are of the same "etheric" substance. Perhaps this view is compatible with the first, but I can't say right now. – exp8j Mar 06 '24 at 09:54
  • enriches above emptiness that which he refers to. I think it can be either way. Sometimes, seeing emptiness enriches the mind. Other times, a mind at peace, fulfilled and gladdened, happens to notice the lack of substance in external phenomenon. – Desmon Mar 07 '24 at 05:27
  • Perhaps the fulfilment and gladness you mention is a kind of 'entrance gate' to the richer container. Inhabiting a richer container permits the restriction of attention to any of its 'empty' sub-worlds, for amusement, rest or additional creativity. – exp8j Mar 07 '24 at 14:29
  • And perhaps the 'sound of one hand' of the famous koan could be linked to that of a hand tapping its reflection in a mirror, where there is one hand in two different planes. – exp8j Mar 08 '24 at 15:59
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Your analogy does not demonstrate emptiness. Emptiness is the emptiness of phenomena. All conditioned and unconditioned phenomena are empty of Self. Do you imagine line or area or volume or infinite dimensions as Self ? No , you don’t. First you need to understand the phenomena which is under question. The language of line , area or volume was learned by you. You were not born with it. It is a mental phenomena. If you stick to mental phenomena then I should remind you that mental phenomena are also empty of Self. Mathematics is not me , mine or myself because understanding and cravings for mathematical knowledge is impermanent. You may have to call line is an area if you insist mathematics is the Dhamma.

SacrificialEquation
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  • The geometric example was meant only as a quantitative analogy to a qualitative reality. Mineral seems embedded in plant, plant in animal, animal in human. It seems that at each level, something crucial is added that makes the previous one seem rather "flat" or "empty". And we have reached only number 4... Isn't that wonderful? Perhaps that sequence is already a part of the Infinite Self! Sure there is thirst for the absolute, there is loneliness, loss and sadness at the human level. What are these feelings if not sure signs of their own overcoming? – exp8j Mar 05 '24 at 19:23
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Evidently, I do have a hard time grasping the value of emptiness per se!

The doctrine of emptiness seems to me a generalization of the parable of the Blind Men and the Elephant.

Its "value" is several:

  • To keep the men from disputing ("I am right!" -- "No, I am right!")
  • To keep each man from being distressed by their own view, whatever it is ("I am right, and I don't like it!")

There's probably more that I'm not grasping but even this is a useful beginning, isn't it?

ChrisW
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  • Yes, emptiness and fullness, humility and self-assertion, seriousness & playfulness in a kind of "golden balance". – exp8j Mar 06 '24 at 10:05