003. infinity: measuring singularity.

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The natural state of affairs, of observation affecting change, and change affecting observation (as described in 002), is what science has come to term "Observer Affect." While Einstein placed limitations on change and or observation, Quantum Mechanics (QM) has since broken these limitations, and in return placed new and different limitations on observation and or change. It is important, I think, to take a closer look at the limitations which observers of the quantum run into. 

Initially QM determined that all interactions between all entities occur in a quantized manner. That is to say, that on the smallest level, things exist in little packets. For example, an atom most always give off energy in the form of some number of photons. An atom may never radiate half a photon. The important thing to note is that QM is basically saying that the universe is like a huge lego construct. If you keep looking for smaller and smaller scales, eventually you will hit the smallest possible scale, which can no longer be divided. QM really goes one step further to say that one one lego sculpture would like to communicate with another lego sculpture they must exchange at least one lego piece in order to communicate. 

While these two premises are not quite so complicated, especially not here where I have oversimplified (perhaps beyond what is tolerable) , they give rise to another aspect of reality which is actually the first thing people now think of when they hear the words "Quantum Mechanics." This property of existence is called the "Uncertainty Principle." It seems that when one tries to break the world down into finite shapes and finite interactions, some element of uncertainty enters into the equation. It becomes impossible to obtain all the information that there is to know about a specific object. Or rather, the information obtained becomes increasingly inaccurate the more information you attempt to gather. Again this is a gross oversimplification of a very complex principle but this isn't a discussion of QM, it is a discussion of the inner workings of the world. 

Now it would be fairly easy for me to explain away the uncertainty principle based on what I discussed in 002 (about the way change and observation are interwoven to create the reality we know), but instead I would like to suggest that it may shed some light on another aspect of the world. I would like to focus on the oversimplification I have stated above. The information obtained becomes increasingly inaccurate the more information you attempt to gather. I think that statement is a more specific form of the more generic understanding 'less is more.' This phrase has far-reaching applicability. Efficiency can be defined by this phrase. It is certainly true of human resources and software development, and it finds expression in black holes as well. Most important though is that 'less is more' is an essential expression of a paradox of reality. 

The universe is an interrelated organic whole. It is an interwoven palace of simplicity. It is a singular design that is infinite. When we examine any one portion of the universe, no matter how small, we are looking at the universe in its entirety. When we try and understand how one microcosm of the universe interacts with another, or in reality, interacts with itself in another scale or perspective, we are bound to see results that become increasingly useless as we cause feedback. Just as two microphone+speakers will cause feedback as they try to amplify eachother's signal, so to, when we try and separate the universe into two (or more) specific entities, the identical information contained in both and related to itself causes mathematical or scientific noise. This is the measurement paradox. 

The measurement paradox is the wall between ourselves and HaShem. Measurement is very similar to aspects of the sephirah of Binah, understanding. It is understanding which hides knowledge from us. When you say that A relates to B in the manner defined as C, what you actually say is this: Everything (E) does not relate directly to A or to B. Furthermore you are implying that A relates to B only in C, and not in any other way. In actuality, everything relates to everything in every way. Definitions and formulas are the enemy of wisdom. When you define or prove an axiom, you have painted yourself into a corner, a fixed flawed view of the world. 

You might say that this line of argument serves no purpose because it in itself falls prey to flaws that it points out. There are in fact two specific points to this expression: 1) the first is a simple lesson: To attain wisdom you must open your mind to all possibilities 2) There may be a structure of which to speak: 

In my confusion about infinity in the eyes of modern mathematics I came across a perception that I believe is crucial to any understanding. But first a little mathematical background: In set theory, infinity is a property of a set. If a set contains a subset which can be put in a one-to-one correspondence with itself, it is infinite. What this really means is that: lets say I take the set of numbers greater than zero, and of this set I take a subset, all numbers divisible by two (and greater than zero). Now, if I can give you a number greater than zero and divisible by two for any number you can give me that is greater than zero, by the definition of infinity in set theory, the set of all numbers greater than zero would be infinite. And we know that this is true, for any number greater than zero that you give me, I can multiply that number by two and give it back to you as a number that fits in the specified subset. Sure enough, the set of all numbers greater than zero is infinite.

This idea sounds very nice in that it is a very specific definition of infinity. However, when one plays with this definition at length, resulting in things like Cantor's infinite levels of infinities, one loses all sense of the meaning of the word. There is one aspect of numbers that this perspective ignores and that is the idea of containment, which is fundamental to the whole idea of a set to begin with. What happens with Cantor is that he tries to talk about the size of an infinite set, something which from the onset we know to be meaningless. Somehow along the way, a meaning is attributed to the 'size' of infinity, and this is where mistakes and circular logic come in. Here is the heart of a paradox. Whenever you feel the logic of a problem as if it were a skater in a half-pipe, you know you are trying to comprehend a paradox. Let us examine this paradox: The set of all integers clearly contains the even integers, so it must be larger. The set of all integers can be put into a one-to-one correspondence with the set of even integers which seems to say they are the same size. How can two things of equal size fit within one another? There are only two answers: 1) you have mispercieved something or 2) you are witnessing a paradox. The answer is that it's meaningless to try and give a definition to the 'size' of infinity. However, infinity itself can be defined. 

This is where I believe the set theory definition of infinity is deficient, infinity bears a connotation of containment. Even though it would seem that we cannot compare the size of the set of integers to the set of even integers, we can clearly state that one dwells within the other. In the same way, even though we cannot break all of reality down into little bite-size formulas that can explain all of existence, we can speak of levels of existence. Certain levels contain other levels which contain still other levels. The paradox is a container that allows two levels to coexist, like the set of all integers is made up of the set of negative integers and the set of positive integers. 

So, even though it may seem that this expression points out the futility of understanding, it also suggests that there are entire levels we may comprehend through understanding. However, a total perception is still limited to those attained by self-negation. Of course, an ideal total perception is impossible, as the Torah promises: Lo yirani haAdam va'hai. ('Man may not percieve me because he lives.)