Emptiness and Math
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In mathematics the notion of emptiness finds expression in the number zero, as well as in contemporary set theory. The concept of zero was discovered in India prior to the sixth century A.D. The "Arabic" number system we use today is neither Arabic nor Greek in origin. In fact, the digits 0123456789 go back to India where they were first created. The ancient Indian number system distinguished itself from other positional systems by virtue of allowing the use of zero as a legitimate number. Interestingly, the number zero did not exist in Greek mathematics, because the Greeks were essentially geometricians and had no use for the mathematical concept of a non-entity, neither did it exist in Egyptian mathematics. The Arabs, who encountered the Indian number system during their early conquests in India, found it superior to their own traditional system which used letters, and thus adapted it to develop Islamic mathematics. The Arabic word for zero is "sifr", meaning "empty." In the 12th century, the Italian mathematician Leonardo Pisano Fibonacci studied Arabian algebra and introduced the Hindu-Arabic numerals to Europe. The word "sifr" thus became "zephirum" in Latin and "zero" in English.
In the ancient Indian context, the number zero did not originally refer to nothingness or nullity. The Sanskrit word for zero is shunya, which means "puffed up, hollow, empty." The zero stands for emptiness suggestive of potentiality. The discovery of the mathematical zero concurred with the emptiness of prajna-intuition in India around 200 BC. Both signify polar opposition between being and nonbeing. Zero is that which contains all possible polarised pairs such as (+1, -1), (+2, -2), etc. It is the collection of all mutually cancelling pairs of forward and backward movements. Put it another way, zero is fundamental to all existence. Because of it, everything is possible. Zero is the additive identity, the focal point of all numbers; without it, numbers cannot be created. India alone, among the great civilisations of antiquity, was able to fathom the depth of emptiness and willing to accept its consequences in mathematics.
Following the introduction of the Hindu-Arabic numerals into Western culture, zero became a number that was used in calculations like any other number. Consequently, it lost some part of its original meaning, namely the part that suggests potentiality. Today, most mathematicians do not associate the notion of emptiness with zero, but with the empty set, which is a construct of set theory. A set is a collection of objects or numbers. For example, the set { 1, 2, 3, 5, 8 } is a set of numbers containing five elements; it is therefore said to have the "cardinality" of 5. The empty set { } is a collection that contains nothing and has the cardinality 0. The mathematician John von Neumann (1923) invented a method, known as von Neumann hierarchy, which can be employed to generate the natural numbers from the empty set as follows:
Step 0: { } (empty set)
Step 1: { { } } (set containing the empty set)
Step 2: { { }, { { } } } (set containing previous two sets)
Step 3: { { }, { { } } , { { }, { { } } } } (set containing previous three sets)
Step 4: { { }, { { } } , { { }, { { } } }, { { }, { { } } , { { }, { { } } } } } (etc.)
This sequence is obtained by iterating a functor that creates a new set from the union of the preceding two sets, thus generating sets with the cardinalities 0, 1, 2, 3, 4, ad infinitum. In less mathematical terms, the principle can be described as follows: Beginning with emptiness (step 0), we observe emptiness. Through the act of observing we create an entity containing emptiness (step 1). Now we perceive emptiness, as well as an entity. From the combination of the former two we create another entity by observation, which is different from the first entity (step 2). This process is repeated again and again. Interestingly, if we define suitable operations on the obtained sets based on union and intersection, the cardinalities of the resulting sets behave just like natural numbers being added and subtracted. The sequence is therefore isomorphic to the natural numbers - a stunningly beautiful example of something from nothing.
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I'm convinced that once one obtains a full and complete understanding of John Conway's surreal numbers, samadhi is a no-brainer...
Steve
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manifestation is merely an imbalance.
an excess or deficiency that thows off the equilibrium.
zero cant be emptiness or 10 would =1 (1 with nothing after it would still be one.)
hence the tree of life. -
What is emptiness then?
I was trying to think of a way to puzzle it out and decided to look at a simple solid object, such as a cup. How is a cup empty? We usually say that a cup is empty if it does not contain any liquid or solid. This is the ordinary meaning of emptiness. But, is the cup really empty? A cup empty of liquids or solids is still full of air. To be precise, we must therefore state what the cup is empty of. Can a cup be empty of all substance? A cup in a vacuum does not contain any air, but it still contains space, light, radiation, as well as its own substance. Hence, from a physical point of view, the cup is always full of something. If I understand it correctly, in the Buddhist point of view, the cup is always empty, and understanding of emptiness is different from the physical meaning. The cup being empty means that it is devoid of inherent existence.
What is meant with non-inherent existence then? Is this to say that the cup does not ultimately exist? I don't think so, I think that the cup exists, but like everything in this world, its existence depends on other phenomena. There is nothing in a cup that is inherent to that specific cup or to cups in general. Properties such as being hollow, spherical, cylindrical, or leak-proof are not intrinsic to cups. Other objects which are not cups have similar properties, as for example vases and glasses. The cup's properties and components are neither cups themselves nor do they imply cupness on their own. The material is not the cup. The shape is not the cup. The function is not the cup. Only all these aspects together make up the cup. Hence, we can say that for an object to be a cup we require a collection of specific conditions to exist. It depends on the combination of function, use, shape, base material, and the cup's other aspects. Only if all these conditions exist simultaneously does the mind impute cupness to the object. If one condition ceases to exist, for instance, if the cup's shape is altered by breaking it, the cup forfeits some or all of its cupness, because the object's function, its shape, as well as the imputation of cupness through perception is disrupted. The cup's existence thus depends on external circumstances. Its physical essence remains elusive.
Of course, I went through much Plato in college, and this goes against what I learned. Which draws me into a tangent about Magick, what "feels right" and what "I've learned." When I am trying to assume a god-form, I simply have no feeling when I attempt to do so with the Egyptian gods, but if I go to the East, or in cases even Roman or Greek, I find it infinately easier. But back to Plato, he holds that there is an ideal essence of everything, e.g. cups, tables, houses, humans, and so on.
I'm struggling on this topic. Help.
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i really dont know if i can help at all. but i think that you are right on about the cup being a cup because of it's different attributes combined that we decide to call a cup. i believe there can be emptiness of a certain thing, but it's a subjective emptiness. i dunno if i am making sense.
as for God-forms, i dont know much about it but since one god always has it's corresponding god in another culture, i would say use the gods u r comfortable with. just translate the egyptian god into it's corresponding god that you are more comfortable with and use that, i guess. essence of a thing is it's "cupness" i guess one could say. kind of just a convention needed because of how we percieve things. i mean essentially everything is just energy, and there is no reason why one couldnt transform one thing into another. i know i'm just babbling here, but i'm tryin.
probably someone else can come up with a better answer. -
I took Jim's advice and have spent much time meditating on the subject. First I want to give Plato some credit by assuming that the essence of cups ultimately exists in the realm of mind. After all, it is the mind that perceives properties of an object and imputes cupness onto one object and tableness onto another. It is the mind that thinks "cup" and "table". Does it follow that the mind is responsible for the existence of these objects? - Apparently, the mind does not perceive cups and tables if there is no visual and tactile sensation. And, there cannot be visual and tactile sensation if there is no physical object. The perception thus depends on the presence of sensations, which in turn relies on the presence of the physical object. This is to say that the cup's essence is not in the mind. It is neither to be found in the physical object. Obviously, its essence is neither physical nor mental. It cannot be found in the world, not in the mind, and certainly not in any heavenly realm, as Plato imagined. We must conclude that the objects of perception have therefore no inherent existence.
If this is the case for a simple object, such as a cup, then it must also apply to compound things, such as cars, houses, machines, etc. A car, for example, needs a motor, wheels, axles, gears, and many other things to work. Perhaps we should consider the difference between man-made objects, such as cups, and natural phenomena, such as earth, plants, animals, and human beings. One may argue that lack of inherent existence of objects does not imply the same for natural phenomena and beings. In case of a human being, there is a body, a mind, a character, a history of actions, habits, behaviour, and other things we can draw upon to describe a person. I then divided these characteristics further into more fundamental properties. For example, I analyzed the mind and saw that there are sensations, cognition, feelings, ideas. And, I analyzed the brain and found that there are neurons, axons, synapses, and neurotransmitters. However, none of these constituents describe the essence of the person, the mind, or the brain. Again, the essence remains elusive, and I'm right back where I started.