Chapter 1: Infinity Explained in Terms of the Relative Equivalence Taken by itself, the word "is", as "isness", implies the broad expanse of the infinite beyond: transcendental being. But in normal usage, "is" is an exclusivist's attempt to equate singularities. It is the intellect failing to recognize the "big picture" by focusing on details. "Is equivalent to", or any other version of "approximately equal to", overlooks differences in order to associate singular "things" by way of parallelism. It is completely opposite to the process of equality. The questions involved with equivalence are: how much difference can be overlooked and still maintain relevance; what is relevant; and how abstract are we willing to go to expand the scope of relevance? Traditionally, we make a mistake every time we equate one thing with something else. I suppose we get away with it provided we always assume that we are equating ideas or symbols and not things, because nothing in the relative is the same as anything else; everything is unique to some degree. But we can never be sure that our thought about something doesn't also evolve or change into another thought----never remaining the same. Strictly speaking, we would need to redefine our use of "is" when used in the course of normal conversation. The only thing that becomes, or is equatable to, something else is the degree to which things pertain to some quality. Presence or absence of a quality is the degree to which something is full or empty of that quality. For most things, quality is relative. But when taken to extreme (and assuming we can match nature's ability to measure degree), presence or absence of a quality become their opposite. We have reached nature's capacity, and our's as well, to distinguish differ- ences----there is nothing else left to contrast. Only here is equality opera- tive; all else is equivalence. This is the real world. Unfortunately, the unreal world of mathematics doesn't affect directly the real world so long as it is trying to equate mere numbers. The real world of numbered things are only as accurate as they are assumed to be----they are never exact. Not because nature lacks precision, but because all topics are relatable to something else and in the scheme of things all topics are brought to bear on integrating a contextual excuse for the existence of some thing, or even some process (since in science's view, everything is a process: a verb, not a noun). This is where aesthetics jumps in: it associates equivalences in the relative and allows a broad range of styles---it deals with equivalence and not with equality. These styles will be referred to as harmonic systems----there are infinitely many of these; as many as there are polynomials of integer coefficients in a single unknown. More specifically, there are infinitely many different versions of expressing a system, since approximation (or equivalence) is a trend towards equality and not a single attempt at it. Equality need only be pursued once in a while here and there, but equivalence is an ageless task. That is why this book opens with the Latin quote: Art is long, Life is short. But there are also the potential for infinitely various systems of mathematics to express aesthetics in, since all math is based on some base of digital logic: from one to infinity. But here, esoteric wisdom steps in to put a lid on variety: only ten bases are extent. There is more to this structure of mathematical linguistics covered in chapter two: humanistic mathematics. *************** Four Views of Infinity Division by zero has been difficult to define because zero and infinity were never defined properly. Existence (and measurement, for that matter) only exist with contrast. In this instance, not only can we have it both ways, but we must. If we measure something, we are measuring how much it is being displaced by something else--- and not an imaginary something else, either. For example, when we measure the mass of air in a give volume of space, we imply that we also know the mass of prana (although not its pitch) in that same space, since air displaces prana. (Don't everybody go leaping for the highest mountains all at once; the body needs to accustom itself gradually to a lack of oxygen. I caution thus for prana spells LIFE. Some may want too much of a good thing.) ? (Prana is a bandwidth of frequencies similar to and beyond the electro- magnetic. It is the energy side of looking at matter/energy; the wave of par- ticle/wave in physics. A body, or plant, without prana is a corpse waiting to decay. It has prana too, but the energy of electrons and protons, not the energy needed to organize living matter: the stuff that dreams are made of. This is specifically what prana is. Generically, prana also refers to the relative vibrating of Creation known as consciousness. Without vibration, all is but a virtual reality---a form that sleeps. Only then is space partially empty. But now, in the day phase, it is completely full of relative amounts of both energy and the intelligence necessary to manifest and organize form. (This analogy breaks down within the virtual night of creation, for even space is a quantifiable something far beyond the bandwidths of either physicality or ether(prana), yet still within the objective bandwidth of creation.) Getting back to the discussion, Emptiness only refers to a supposed lack of something. But if we had done our measuring a little differently, we would find that any volume (conceptual, or otherwise) is filled completely with many different things and no emptiness. Thus, a positive assertion about something is usually considered to be the reference, while its lack refers to what could be--but isn't. The lack is in terms of a potentiality. So long as something exists along with its lack (side by side, of course; not in the same place) then that something exists in relative form. As soon as something is taken to the limit of capacity, be it full or empty, then it falls out of the range of measurability. Capacity is something, but unlike the relative presence or absence of a thing, it is unmea- surable. Measurement needs contrast; and capacity has no relative contrast--- it either exists, or it doesn't. When it doesn't exist, the transcendental value of reality is being considered. When it does exist, both fullness and emptiness of a quality are taken to their extremes and fuse into one. Absolute presence and absence are indistinguishable, one from the other. Only in relative form do they take on practical meaning. Just because capacity is immeasurable, doesn't make it infinite or zip---it is very much a finite existence. Yet, in practical terms, that is precisely what we do: we name the impossible. So the terms: infinity and zero, imply the immeasurable capacity of an existential system to be both full and empty at the same time, although we don't measure them that way. We usually say something (or some conceptual quality) is either full or empty, present or missing, but not both---again, for practical reasons. We take the immeasurable and treat it as if it were measurable. We measure or acknowledge that something is either present or missing, but beyond our capacity to measure distinctly, when it is really probably still there outside of our measuring range. So we make a semantic distinction between infinity and zero when no such distinction exists. We are really taking something relative and naming it as if it were absolute in capacity. This is O.K., so long as we recog- nize what we are doing. Up until now, I don't think we have. Our blunder is not a big one; but in mathematics it has created confusion whenever we attempted to divide by zero, or in rarer circumstances, by infinity. *************** The Definition: How to Divide by Either Infinity or Zero [1] Reflexive Property of Equality: A number is equal to itself. a = a [2] Substitution Property: If values are equal, one value may be substituted for the other. If: a = b and b = c, then: a = c [3] Distributive Property: A math operation between an expression and a set of expressions; the individual expressions within the parenthetical set are separated from each other by the same math operation; the primary math operation outside the set may be distributed individually among the elements within the set. If a, b, c, and d represent expressions, then: a * (b + c) = a * b + a * c While: a * (b + c * d) not= a * b + a * c * a * d [4] Addition Property of Equality: Addition is distributive over equality. a + (b = c) >>> a + b = a + c (b = c) + a >>> b + a = c + a [5] Subtraction Property of Equality: Subtraction is distributive over equality. a - (b = c) >>> a - b = a - c; a + -1 * (b = c) >> From [4] above. (b = c) - a >>> a - b = a - c; -a + (b = c) >> From [4] above. [6] Multiplication Property of Equality: Multiplication is distributive over equality. a * (b = c) >>> a * b = a * c (b = c) * a >>> b * a = c * a [7] Division Property of Equality: Division is distributive over equality. a / (b = c) >>> a / b = a / c; (*) a * (b = c)^-1 >>> a * (b^-1 = c^-1) [3] >>> a * b^-1 = a * c^-1 [3] (b = c) / a >>> b / a = c / a; (b = c) * a^-1 >>> b * a^-1 = c * a^-1 [3] Both use [3]. Exponentiation is distributive over equality if it is considered to be the same as multiplication. But it has to be used twice at the (*) starred line, because of exponentiation's conventional precedence over multiplication. Pg. 27, Selected Answers Division by Zero The world of magnitude is a circle, not a flat line segment. All systems of enumeration are relative and have a finite capacity for fullness and emptiness. Wherever a system partakes of an upper or a lower limit, there it touches absolute magnitude----the indefinable. But magnitude is a ring, so there are two versions of relative absolute: infinity and zero. Both are versions of the same thing seen from two different perspectives. Imagine sitting on a point marked on a circle; this is its reference. The circle is marked with a fixed number of magnitude divisions; this is its capacity for relativity. No markings would constitute a transcendental n absolute frame of reference. The divisions run positively and in increasing positive integer value in one direction; negative with increasing negative integer value in the opposite direction. Although at any one time it is either one version of a ring or its compliment, potentially it is a double ring linked at every marking including the reference. First, choose a positive view: the reference will be named zero. Any motion away from the reference will be by adding positive integers to zero. Eventually, the capacity of the system will be reached: infinity, which had been the starting point. Second, take the reverse condition, negative: the reference will be infinity. Any motion away from the starting point will be making an ever widening hole in infinity by subtracting an ever increasing something from the upper limit. Eventually, zero will be reached (originally infinity). Notice there are two ways of changing a limit into its opposite: by counting from one extreme to the other or by multiplying the reference by a -1 when standing on the reference and merely inverting direction. If the circle has only ten markings on it, one of them the reference, then: positive >>>> +1 >>> +0, +1, +2, +3, +4, +5, +6, +7, +8, +9 direction -9, -8, -7, -6, -5, -4, -3, -2, -1, -0 <<< -1 <<< negative direction Notice that the single digit values are paired in each column with a compliment. In computer architecture, this is called "one's compliment". For the purposes of subtraction, a negative number is inverted into its one's compliment, then a special form of wrap-around carry-over addition is performed reflecting the structure of a ring. Inversion is the equivalent of multiplying by a -1. Working with a three digit capacity and single digit examples, the left two digit positions filled in with either 0's or 9's act as sign digits; 0 means positive and 9 means negative. Usually, the left-most digit is reserved for signing by keeping it off-limits to addition or subtraction, but allowing it to invert into its opposite upon multiplication by a -1. Using the above two strings of digits as a table of compliments: Example 1: 111 >> carry-over digits; since the +8 = 008 = 008 left-most carry-over 1 has -4 = 004 * -1 = +995 nothing to add to, it is --- --- carried "around" to the +4 003 right-most digit and added +001 << here; (this wrap-around can --- occur as many as three times 004 until the dust settles); --- the result is interpreted as +4 >> being a positive four. Example 2: *** >> nothing to carry-over; +3 = 003 = 003 -7 = 007 * -1 = 992 --- --- -4 995 * -1 = -004 = -4 >> Voila! (french: behold) Example 3: *** >> nothing to carry-over; +1 = 001 = 001 -1 = 001 * -1 = 998 --- --- 0 999 * -1 = -000 = -0 (Of course in computers everything is done in base two, but one's compliment will work with any base so long as the math system has an even number of different possible signs.) Mathematicians, in the early days of computing, unfortunately freaked-out over the occurrence of negative zero. They decided to conform to the standards of fear and ignorance that is strictly enforced to this day by performing what is called "two's compliment" arithmetic within computer operations---thus, saving face. Computers never have this little problem now, but they fail to teach us anything about the universe either. Calculators never have this problem, they do things differently. So, in a one-digit universe: 0 = lower limit; 9 = upper limit; -0 = upper limit; -9 = lower limit 0^1 = lower limit; 9^1 = upper limit; 0^-1 = upper limit; 9^-1 = lower limit Although multiplying any number inverts it into its one's compliment, raising a number to a negative power doesn't automatically invert it. It has to be a limit in order to invert: 2^-1 not= 7^1, but 2^-1 = 1/2 Also, 2^-1 = 002^998 >>> this is useful for doing logarithms, but it still has to be interpreted correctly at the end of operations. Raising a limit to a negative power doesn't have an analogue in computerese, but is illustrated subjectively by an example: x * y = 1 Solving for y yields: y = 1/x Using x as the input value from 1 to infinity yields a range of y from 1 to zero. At least this helps to define division by infinity. But what about division by zero? Looking at the graph of this function begs the question: what do we have against dividing by zero, especially since the graph points out the trend to where the answer is headed; namely, infinity. Of course we could have substituted a negative infinity in place of either of the two zero's to postu- late a situation and attempt a solution in a different manner to see what happens: Using: 9 = infinity; 0 = zero; And: -0 = infinity; -9 = zero; y = 1/x 0 = 1/9 9 * 0 =? 1 [%] 9 * 0 = 0 9 * -9 = ? (8 >> carry-over wrapped-around 9 * 9 * -1 = 9 *9 --- 1 +8 >> to here and added --- [*] 9 << gives, * -1 >> but when the multiplication ----- is completed the same result 0 >> occurs; no matter what we divide by, zero or infinity, the constant gets squashed to zero when the function is put back into the form of multiplication: infinity = (1: old constant) / zero Or: zero = (1: old constant) / infinity zero * infinity = (zero: new constant) >> line [%]; What in effect has happened when we divide by a limit is the function flattens into an asymptotic function We already know: zero * zero = zero And: infinity * infinity = infinity >> line [*], but these examples don't help with the problem of defining division by zero. Of course, we are allowed to take a step back and see how it looks: Using: & = infinity; 0 = zero; y = 1/x Yields: 0 = 1/&, as x approaches infinity; & * 0 = 0 But wait; we weren't allowed to do the inverse: y = 1/x Yields: & = 1/0, as x approaches zero; 0 * & = 0 Should there be any difference between: 0 * & = & * 0 = 0? Is there any problem with solving for a function alternately, first for x and then for y, and then making a composite graph of the two solution systems? We do that all the time by setting the function equal to zero and alternately solving for x and y: x * y = 0 y = 0/x Yields: 0 = 0/x, no matter what x is; And: 0 = 0/y, no matter what y is. This points to only one conclusion: That infinity and zero are limits to magnitude and are compliments to each other. This thinking isn't new. The "I Ching", the Chinese "Book of Changes" states: whenever anything goes to far in the extreme, it changes into its opposite. This forms the basis for a closed, looped vision of the universe (the dragon biting its tail is a traditional Oriental image). The capacity of a closed system to embody relatively absolute limits (parameters of operations) is suggested by an idea stated as the topic of Maharishi Mahesh Yogi's 18th course? on The Science of Creative Intelligence: "...two fullnesses: fullness of fullness and fullness of emptiness". *************** ?"...two fullnesses: fullness of fullness, fullness of emptiness...", Maharishi Mahesh Yogi, excerpt from The Science of Creative Intelligence course Lesson ?18 Zero is just another form that infinity takes in the relative. There are four ways to categorize the inclusion of elements within a set: all, nothing, some of all/nothing, and neither all/nothing. The transcendental form of infinity is beyond any representation possible, thus it is the "neither" type. It also cannot relate to the relative in relative terms, so it has no relative function. The capacitive -ant form of infinity is not concerned with how much of itself is filled with either all or nothing, but how much capacity it has for embodying all or nothing relative to other "capacities". This is where infinity starts looking like the relative, but is not quite fixed as a relation just yet. We are still in the realm of broad abstractions to form the basis for relative operations. An example of relative capacity would be the number of digits on a calculator readout, or the number of slots in a crate for holding apples that could be filled or empty. *********** Historical Precedence The I Ching.One's Compliment Arithmetic.Definition of Results Obtained By Dividing With Infinity Or Zero.