But how relevant are the blueberries?

Intellectual reader, I invite you to imagine with me a malleable set of declaratives. By this I mean a set of logically related statements that can be altered for the purposes of experimentation; we can take away, add, or reposition declaratives and observe what becomes of the rest of the set. Our first observation will be the way in which each component part is related to each other. Only two sorts of logical relationships may exist between any given pair of statements, though these relationships may be described multiple ways and are best expressed as magnitudes, not booleans. In other words, it is best to discuss the extent to which a certain relationship exists rather than the fact of its existence or lack thereof.

Figure 1

We will here only discuss one of the two relationships: that of logical consequence. To describe this relationship, we may refer to declaratives as either “following from” one another or else “being contained” within each other. A concrete example is in order: suppose I held before you a black pen; if I were creative enough, I could talk about the pen forever, because there are infinite truths that may be said of this black pen of mine. But suppose, of all the possibilities, I chose to say to you, “this pen exists”. The use of the demonstrative pronoun ‘this’ brings into language all the infinite qualities that the pen possesses; hence, “this pen is black” follows from, or if your prefer, is contained within “this pen exists” because the former is a subset of all the infinite truths contained within the latter.

Figure 2

So picture the two declaratives as a venn diagram; in this instance, it is not a conventional-looking image (figure 1). But if we were to consider another example, the diagram would look more familiar: suppose instead I said to you, “this pen uses black ink, and all pens that use black ink write clearly”. Now you might reply, being the clever reader you are, with another fact that follows and is contained within the previous two; “if that is so,” you would answer in your decorous manner, “then this pen writes clearly”. Aside from our admiration for what a sensible and insightful logician this response makes you out to be, we are now struck by the complexity of a logical phenomenon. Presently we have two statements that intersect to form a third (figure 2), so “this pen writes clearly” follows from the union of “this pen uses black ink” and “all pens that use black ink write clearly”.

Kindly notice that each bubble in the diagrams above may vary in size, depending on what order of infinity it represents. Notice further that, in our second example, A and B share certain common facts, which set of declaratives we call C, but also have some differences. So how closely related are A and B? The answer is a simple measure of area, and it describes a notion that I will call ‘gravity’. To express the formula for gravity, I will refer to the area of a statement X with the symbolic convention, ∫X. So the gravity between A and B in our example is Γ = ∫C / (∫A + ∫B).

This expression solves two important problems. The first is that of defining a scope, a sector of reality that is coherent. Consider an example: you tell a friend that, on theological grounds, you believe it was immoral for him to steal blueberries from Mr. Dimmesdale, and in his contemplative manner, he says, “but ‘God works all things together for the good of those who love Him’, so my deed will ultimately come to good”. You are both right, but he has misapplied a teleological perspective to an analysis of the action itself. The fact that he brought up exists in a larger scope than the matter you are discussing. And defining a scope is no subjective matter, to express it mathematically, we must first make one more definition: a “gravitational average” is the average gravity that one statement bears on each other member of a set. With that in place, a scope is any set of declaratives that exists such that each member has an equivalent gravitational average.

The second issue that gravity solves is that of distinguishing normal functioning from dissociative functioning. Dissociative functioning is a section of a proof of actions on which an alternative declarative bears greater gravity than the primal premise. For a more in-depth discussion of this, see Is Hypnosis Self-Evident? A Concise Philosophical Inquiry, in which post I describe the concept of gravity in different terms that nonetheless mean the same thing.

It seems prudent to define one last term: the Quantum Model of Reality. If we picture reality as a black-board with an infinite area, on which each infinitesimal point represents a fact (and those combine to from larger facts), by the Quantum Model of Reality, we are able to draw lines on the board to sector it off into quantum regions contained within one another; in other words, we can draw a larger circle around a smaller one ad infinitum, where each circle represents a valid scope that is defined in terms of a gravitational average. This is why, elsewhere on this blog, we have referred to reality existing in ‘levels’. In practical application, “God works all things together for the good of those who love Him” can only be discussed in relation to other notions of equal size, and Mr. Dimmesdale’s blueberries still ought be returned.

Ref #2: What’s a Fractal

Draw your own fractals with my fractal drawing software, TWM Fractals.  With the new “Animate” feature, you can watch as your fractals iterate before your very eyes!  (It’s quite entertaining.)  Click the link to download.  (It requires java JRE.  I’m not really sure which version, but as long as it’s not super old, it should work.)

In geometry, a fractal is the infinite iteration of a recursively defined figure.  That is, it is a figure whose sides are defined recursively and iterated to infinity.  A simple example of a fractal it Koch’s Snowflake .  Koch’s Snowflake is a geometric fractal based around an equilateral triangle.

The algorithm for turning such a triangle into a fractal is as follows:  subdivide each side into four equal parts such that, in the middle of each side, a triangle protrudes that is similar to the original, only missing one side. It will look like this:

Once this is completed, the figure is said to have undergone one iteration.  Now we repeat the process for each side, include those newly formed sides:

and so on to infinity…

Once the figure has been iterated to infinity, it is considered a fractal.  This means that every part of its perimeter has the exact same structure (while some parts are smaller and others larger).  Fractals are often considered to be fraction-dimensional figures.  This is because, in any integer dimension, an infinite sum of infinitesimal parts (that is, 0over0) is an integral, which always has a finite solution.  But in the case of a fractal, we have an infinite sum of infinitesimal parts (still 0over0) that has an infinite solution.  This means that the perimeter of Koch’s flake has an infinite length (as each iteration increases the length by a factor of 4/3).  This is because the order of infinity that describes the number of sides is higher than the order of inverse infinity that describes the length of each side.  It is often thought of as a paradox that a figure, such as a fractal, can have a finite area but infinite perimeter and an infinite perimeter made only of infinitesimals.

Ref #1: What’s Recursion

It was recently brought to my attention, thanks to the much appreciated input of a commenter that not everyone reading this blog knows what a fractal is.  As I began to think about how to explain the concept, I started to realize that there may be many such topic that I frequent in my writings that readers are unfamiliar with.  Although I try to give enough background information within each post, perhaps that is not always entirely sufficient.  Therefore, I have decided to create a series of “reference posts” explaining various such things for the sake of increasing the overall accessibility of this site.

Recursion: see recursion.

On its basic level, a recursive algorithm is an algorithm that is somehow defined relative to the same system it is creating.  For example, one might define a geometric sequence recursively as follows:

A1 = 2

An = A(n-1) * 2

This definition would produce the following sequence:

2, 4, 8, 16, 32, 64 … 2^n

In this sequence, I have defined each term relative to the term before it with the exception of the first term which I have given a set starting point.  Since each term is found by doubling the previous term, the overall sequence can be said to be recursive because each part of the sequence is defined relative to another part of the same sequence, and so, more generally, the sequence is defined relative to itself.

Recursion is, however, a broader concept than this and can extend beyond the world of simple algebra.  In fact, recursion can be found all over nature.  A simple example is when two mirrors are held so that they reflect each other.  One ends up with the image of a mirror inside a mirror inside … This is because the image on the mirror is defined relative to itself.

Another simple example of recursion is proof by induction.  In a proof by induction, one proves that an algebraic expression of n is equivalent to each respective term of a sequence by plugging it in recursively.  I will use the same geometric sequence above as an example:

Prove: if   A1 = 2  &  An = A(n-1) * 2  then  An = 2^n

2, 4, … A(n-1) * 2 = An

Assume: An = 2^n

2, 4, … A(n-1) * 2 = 2^n

∵  (2^n) * 2 = 2^(n + 1)     ! plug-in 2^n for the sequence on the left and apply the algorithm

!of multiplying by two to find the next term.  See if that is equal

!to the definition one the right incremented by one.

(2^n) * (2^1) = 2^(n + 1)

2^(n + 1) =  2^(n + 1)

Q.E.D.

Another example of recursion is the algorithm used by a scientific calculator to parse a formula into a computable expression.  That is, if I enter the expression

3 *2+4*3*2 +2

into a scientific calculator, it solves it recursively.  It might, for example, have an evaluate method that takes the first number and applies to it the solution of the rest of the expression (which it finds using the same evaluate method) using the given operation.  If you are familiar with computer science, the code might look something like this (in summary):

double evaluate(String expression){

if(getFirstOperation(expression) == MULTIPLICATION)

return getFirstNumber(expression) *  evaluate(getRestOfExpression(expression))

+ evaluate(getExpressionAfterFirstPlus(expression));

else if(getFirstOperation(expression) == ADDITION)

return getFirstNumber(expression);

//the code would be written a little differently

//inorder to accommodate for other cases

// this is merely an example that would solve

//the above problem

}

Recursion can also be thought of more generally as any process that references itself.  For example, phycology might be considered recursive because it is using the mind to study itself.  Recursion occurs in levels (see “Levels of Recursion”) or iterations.  Every time a recursive algorithm is completed, the system is said to have undergone one iteration or advanced one level of recursion.

Google has a pretty funny joke about recursion.

Theology of Nonbeing

A friend of mine was recently discussing hell with me, a delightful subject, and we came to the issue of being and nonbeing.  I realise I can often sound kind of crazy when I present one of my models on just about anything, so I thought I’d post about it in the hopes of straightening things out a bit.  If I do not achieve that end, at least the rest of the public will be able to benefit from the entertainment.  Everyone loves a good crazy-man-ramble; that’s why Shakespeare is so full of them!

I think that’s the first exclamation mark I’ve used in this entire blog!

So, to be clear, the question I will intend to address is: what does the Bible tell us that hell is?

In quantum physics, there is a theoretical phenomenon called quantum superposition.  In summary (and, if you’re a physics person reading this, forgive me for not explaining this well), it states that a physical system exists, in part, in each of its possible states or configurations at the same time, though when measured, results will only suggest one particular state.  It is my understanding, though I am much more a calculus person than a physics person, that there is not currently a satisfying model as to how this is possible, only a mathematical model for how physical systems behave under this theory, and the probability of a given measurement returning a given result.  The phenomenon is often likened to some sort of analogy, a common example being that of “Schrödinger’s Cat,” which goes as follows: a cat is placed in a box where it cannot be seen along with a lethal vial of poison that will automatically be broken after an unknown, finite duration of time, killing the cat.  While the box is closed, the cat is therefore considered to be both dead and alive (‘superpositioned’ in these two states simultaneously).  It is not until the box is open that a measurement reports a single one of these states.

In some ways, these sorts of theories seem to me like the attempts of physicists to jam together a whole lot of facts that we just can’t seem to make proper sense of, but in other ways, they seem to make perfect sense based on the empirical information we have and the accompanying math.  I suppose the theories can be described in both those ways at the same time.

I bring this up because my model of hell is very similar in nature to these sorts of theories.  There is a conflict in a lot of the most essential writings of theology and Christian literature (and even an implied conflict in the Bible) between hell being a real place that God sustains for the continued punishment of evil and hell being the conceptual place of nonexistence.  Let us call the first of these the “classical” theory and the second the “annihilationist” theory.  There seems to be particular issues with each of these models: in the classical theory, we have two main issues (1) how could a loving God be so mean as to actively torture his own creation? (2) how could any place be a hell if God himself is sustaining it? wouldn’t his very presence make it a heaven?  And in the annihilationist theory, we have two main doctrinal issues: (1) how is it justice if the unsaved just cease to exist? it seems like they got away with it. (2) what about the lake of fire and all that? doesn’t the Bible Itself completely debunk this theory?

I shall discuss each of these issues before arriving at my model.

As for the first issue of the classical theory; that is, how could a loving God be so mean as to actively torture his own creation?  Most doctrines suggest that God is not doing anything of the sort.  Those who are damned are simply allowed to do as they chose, and however painful such a choice may be, they much prefer it over having to answer to a Higher Authority˚.  Of course, this presents a problem, because it is impossible to not answer to God, as He is the very definition of Reality, so part of hell is irrationality.  There is no logical way for one to justify one’s own damnation, therefore one turns to the illogical.  This is one of the many things that Milton portrays so well in his Paradise Lost.  In the second book, when the demons are holding council, after having been damned to hell, to determine a course of action, Mammon (one of the demons) says this: “Let us not then pursue / By force impossible … our [former] state / Of splendid vassalage [under God in Heaven], but rather seek / Our own good from ourselves, and from our own / Live to ourselves, though in this vast recess [that is hell], / Free, and to none accountable, preferring / Hard [or painful] liberty before the easy yoke / Of servile Pomp” (Milton II.249-257).  Of course this is completely irrational, as there is no joy to be found in hell, and to be confined to flames is certainly not freedom, but hell is a matter of illogical rhetoric, and the demons are good at this.  Even a parting from reason is an abolition of freedom, for as the classical model of the soul suggests, a will is not free without reason.  Logic is what makes the act of willing pointed.  Without it, a will is just a random decision maker; there is no point to willing something if one does not do so for a reason.  (What’s the difference between saying “I had a point in doing it” and “I did it for a reason”?)

But I digress, the point is, hell is a place of reversals; what seems to be freedom is oppression, and what seems to be oppressive torture by God is really just free choice by man.

As for the second of the classical issues (how could any place be a hell if God sustains it?):  Somehow, God must create a distance between Himself and that evil, for it is an Omnipotent impossibility (Absolute Nonsense) for God to be united with evil.  Therefore, God must be entirely absent from hell.  But how can something exist  apart from God?  For “without Him, nothing was made that has been made,” and in a timeless reality, as we are lead to believe eternity exists, to be made must really be the same as to exist.  Thus it seems that nothing should exist without Him, and indeed it is commonly excepted doctrine that without God, we would cease to exist.  So how can hell exist where God is absent?  This issue seems to confirm that our model is required to contain some element of annihilationism.

The two issues with the annihilationist theory do not bring up anything of significance themselves other than that our model must include some element of the classical theory.  However one piece of evidence directly in favour of the annihilationist theory is Jesus’ repeated reference to the saved being blessed with “eternal life” (ex. John 17), the implication that the damned to not receive this.

So it seems hell must be a place where God is entirely absent, and evil suffers from the very nature of being evil (for to be evil is to suffer).  Thus we have two elements of the model that need to be jammed together like the quantum physics model discussed earlier.  And thus, hell is a place where the nonbeings exist–the failed creations, those who were given the option to be a part of the Body of Christ, to live eternally, and chose not to.  Clearly there is a sense in which, while Heaven is everlasting life, hell is well thought of as everlasting death*.  Thus it is described as an abyss (Revelation 9), or a bottomless pit (Revelation 20) where one may not rest (Isaiah 57), and evil is often associated with falling (Proverbs 24:16).  All seems to invoke a sense of infinite nothingness, and this is indeed what one should mathematically expect it to be like, for since there is such a thing as infinite somethingness, anything aside from that somethingness must be infinitely nothingness.  Another common biblical image is the likening of evil to darkness and Good to light.  Darkness is nothing in itself, but is the absence of light.  In this way, we can imagine something outside of God which, while it is nothingness because it is apart from God, still exists relative to God, and is thus neither self-sustaining nor nonexistent, just as darkness could be said to be sustained, as an independent entity (or non-entity), by the existence of light while it is still nothingness.  One thing worth noting with this analogy is that just as light and darkness, Good may exist without there existing evil but evil may not exist without there existing Good, for darkness is only defined relative to light (it is the absence of light), but light is a thing in itself†.

So there we have a good chunk of the biblical basis of the model, but let us continue to develop images for it, now secular ones, in order to better isolate it.

There’s always calculus to do this for us.  Let us imagine and infinite God that creates a finite being which of its own free will, chose to part from its maker.  Thus, the finite being can be represented by lim x->∞ 1/x.  And thus, it can be said both to exist and not exist in the same way that the limit is always zero but the function never is.  This relates nicely to the image of constant falling in a bottomless pit˚.

Lastly, let us consider an image for you folks that don’t care so much for math and technicality.  A story:  Once upon a time, there was a boy named Jimmy.  Jimmy loved to smoke, eat fries, and listen to a single rap song, which he played very loudly.  All these things were, for him, the greatest joys in life, and he did all he possibly could just to get more of them.  Then, one day, Jimmy suddenly broke down; he couldn’t stand his life any longer and he didn’t know why.  At therapy, he was introduced to classical music and forced to start eating better and stop smoking.  Suddenly, he realised that he had never known pleasure till that day, but had always been in a lot of pain.  It was as if his life had changed from black and white to colour. He no longer enjoyed smoking or eating junk, nor did he consider that rap song to be music.  Thus, there was some sort of paradox between the existence of pleasure in those activities and the absence.  It seems that they are existing pleasures, but whence one compares them to something much greater, their value completely vanishes.  So may it be with Heaven and hell.  Hell may exist, and creatures damned to it may consider it to constitute as existence (though, unlike in our story, they will find it very painful), but if they were to be graced by but a single delicate ray of Heaven, they would blaspheme their very beings and call it death.  It is clear which of the perspectives is the most correct one.  God is Reality, and since everything exists relative to Reality, that is, to the Absolute Truth, Heaven must have the final say.

Like much of my writing, this may just seem like a whole lot of work over something trivial, after all, all we really need to know about hell is that it’s a place we don’t want to end up, but this could perhaps help some people reconcile the mystery of infinite love and divine justice (two elements that Christianity is the only religion I know of to sufficiently address, and does so primarily through its unique doctrine of Redemption), but the main point of this theorising is to create functional models that may be used in other circumstances for the accomplishment of more practical ends.

I do anticipate mathematics catching up to this concept in the future; it’s a wonder we don’t already have numbers that are simultaneously entirely real and entirely imaginary.

__________________________

˚ Conceptually, there is a difference between God creating and sustaining a place of torture away from Himself to which all are given the option to go, and Him actively torturing them.

* Thus Milton also writes “Then who can create thee lamenting learn / When who can uncreate thee thou shalt know” (Milton V.894-95), and thus revelations talks of damnation as the “second death.”

† An interesting aside is how this relates to eastern philosophy.  It upholds the idea of nonbeing, or whatever they prefer to call it, but refutes the idea of Yin and Yang, because while nonbeing is dependent on being, being is not dependent on nonbeing, and thus the one is evil and the other Good.  There is, therefore, an essential difference between the two that results from logical necessity.

˚ It is interesting to note, however, in this case we are literally plugging in infinity for x, and therefore we arrive at an absolute infinitesimal (that is one for which, like a recursively defined infinitesimal, rate of change is irrelevant), and in calculus, we like to say that infinitesimals are both nonzero and zero quantities.

Please note: on further examination, it was found that there had actually been twenty-five other exclamation marks on this blog preceding the ones above, so I was a little off; I didn’t realise I was such an exciting writer!

On Probability

That’s such a stuffy tittle.  You might call it “The Possibility of Probability,” or “What Happens With Chance,” if you like.

I thought I might be of some use to my readers if I were to write a brief article on this subject.  Please note that this is primarily a philosophy paper and not a mathematics one; though, of course, the two subjects are, as all subjects, inseparable and hard to distinguish from one another.

It is, and rightly so, the most commonly accepted model across all subjects that all probability is metaphorical.  From basic physics, we reach the self-evident conclusion that all physical systems have a predetermined out come from the moment they are set up.  Therefore, when this principle is applied on a macro-level, we reach the modern understanding of Chaotic theory; that is, that the entire universe is one giant physical system, composed of the interactions of countless smaller systems, that has had a predetermined course of action from the moment it was set up.  Thus, according to our understanding of physics, there was a one-hundred percent chance, since the dawn of time, that you would be sitting in the exact spot that you are currently, at this exact time, reading this exact sentence, and thinking the exact thoughts that you are thinking right now.

Of course probability is still a very useful concept in many cases.  One excellent example is genetics.  We still use the assumption that inheritance is “random,” and therefore, that the mathematical principles of probability can be applied to determine the “likely-hood” of one trait being passed on versus another.  This method of evaluation is very practical because the chaos involved in the system is so developed that it can be assumed to be random–it functions much like a small angle approximation.  However, according to our larger model, there is no such thing as randomness in the literal sense.

It is imperative that we understand the universality of this principle, even as we venture into metaphysics.  In the absolutist’s scope, probability is still only metaphorical when it comes to metaphysics, and one can use the physical metaphor of probability as an analogue to the metaphysical one.  As I have indicated in my post on fractal reality, there is a fine line between the metaphorical and the literal, and perhaps even no line…metaphorically.  But in the case of probability, there definitely is a line, as the actual relationship between the metaphor and the reality it represents can be entirely understood by the human mind (that is, if anything really can).

In metaphysics as in physics, the whole principle of metaphorical probability is designed entirely for the sake of convenience, and is not actual believed when it comes down to what is really happening.  Thus, in physics, we can estimate the probability that a cannon firing a tennis ball will hit its target, while in reality, we know that if we had every specific detail about the set up of the system to an infinite degree of accuracy, we could know for sure whether it would.  Likewise, in metaphysics, we can say that there is a freedom of the will such that at any given time it has a certain chance of making one decision over another, but in reality, we know that if we knew every single detail about the soul (which is fractal), we could say for certain which decision it would make.  That’s not to say that the soul isn’t free, but that its freedom is not bound to time.  All of time is a metaphor when it comes to metaphysics.  The whole story of a soul choosing between death and life is thus an embodiment of the soul in the medium of time, all though, a complete knowledge of the soul outside of such a scope would allow for a knowledge of the soul’s every decision “before” it was made.  It is as if, in both metaphysics and physics, time is merely a way of looking at a complex system part by part.

To take this a little further, consider a four-dimensional cube.  We can only express such an object in the form of a hypercube, which is a three-dimensional object that changes shape over time, thus expressing each of the different four-dimensional angles form which the real object can be viewed.  But in four dimensions, all those angles are present without any need of morphing.  Likewise, a fractal-dimensional physical or metaphysical system must be expressed by morphing a “three-dimensional,” or what I will call, “normal-dimensional” (to avoid a bias towards physics), one over time.  (and I said this article wasn’t about math)

But when we talk about the limits of either of these systems, we must set aside our normal-dimensional perceptions and likewise our metaphorical probability.  There is no “chance” that the end or beginning of time did or will look one way or another physically or metaphysically.  It just was, will be, and if we are to be most literal, is.  Thus, when we talk about the qualities of the Omnipotent, there is no chance that they are one thing or another, they just are what they are.  This understanding excludes the possibility of an arbitrary Omnipotent “happening” to cause an intelligent humanity.  In the literal sense, nothing about metaphysics is random, and because when we discus the Omnipotent’s first action of causing, we are referring to a limit, even the metaphor of chance is senseless.  Therefore, if we are to say that humanity is intelligent, then we must also say that the Omnipotent is intelligent.  For an understanding of logic cannot “happen” to arise from nowhere.  It must, at the limit of causality, have been present in the origin of reality.

As for the objection that computers are “more intelligent” than the human beings that have created them (this objection was posed by a commenter).  My answer is, no they are not.  The intelligence that this, and all my arguments on the Omnipotent, are referring to is the one upon which the scope of the argument depends.  That is, the argument depends on the fact that human logic is capable, to some extent, of finding and understanding metaphysical truth.  That we can build a computer that emulates some of the mathematical algorithms of the human mind does not mean that we have created something “intelligent” in this sense of the word.  It only means that we have found a way of putting that human intelligence into general terms, much like writing it down.  If I throw a baseball through a window, I have broken the window, not the ball.  In the same way, it takes a human to know that a particular circuit pattern will perform a particular task, but the circuit pattern doesn’t know the first thing about the matter, it is only a tool which is being used by someone who does.  Therefore, computers, which have no understanding of human logic, are not intelligent at all.

Orders of Infinity

You had to know it was coming…another calculus post!

If you have absolutely no interest in calculus, then I don’t recommend reading this, it will probably just be frustrating.  I don’t really expect anyone will follow this, but I did my best in wording it; it is a difficult concept.  I might rather call it an anti-consept because it is not an established idea, but an idea of ideas of ideas of …

Here it is:

Allow me to begin with a definition: an “arithmetic dimension,” n, is an algorithm of numeric manipulation that is defined according to a series of summations or subtractions of the input from or to itself repeated to the n-th level of recursion.

Therefore, the first arithmetic dimension is addition/subtraction, the second is multiplication/division, and the third is exponentiation/root.  These are the only commonly used and defined dimensions, but there are in fact infinite arithmetic dimensions.  Consider it this way: addition is pre-defined, multiplication is the addition of the multiplicand to itself repeated the number of times indicated by the multiplier, exponentiation is the multiplication of the base by itself repeated the number of times indicated by the power.  Therefore, it is clear that the fourth arithmetic dimension is the raising of the input to the power of to itself repeated the number of times indicated by the “secondary input.”  Thus it is recursively defined: the n-th arithmetic dimension is the application of the (n-1)-th dimension using the input as both the input and the secondary input, repeated the number of times indicated by the secondary input.

I bother presenting this definition before we begin because I am not aware of its existence elsewhere.  Therefore, I will invent a notation for it: let the n-th arithmetic dimensional operation applied to an input a and a secondary input b be expressed as

a n: b

and read as “a dim n b.”

I am interested, at present, in only the application of the addition-based side of each of the arithmetic dimensions, so this notation and reading will assume a positive based definition and only acquire a negative one if 0≤b<1 as is inherently true from the nature of arithmetic.  The concept I wish to use this for at present (though I’ve already found it has many applications beyond this concept) is that of the orders of infinity.

It is often said that there are infinite subsets of infinity.  This statement, while true, only looks at the negative-based arithmetic dimensions.  That is, we can subtract, divide, or root infinity by any finite number and get an output of infinity.  What is looked at less often is the opposite, the positive-based arithmetic dimensional operations applied infinity.  Infinity can be added to, divided by, or raised to the power of any finite number, and once again, the end result is infinity.

Of course, the calculus literate know that while all this is true–that is, while any finite operation applied to infinity outputs infinity–the qualities of the infinity outputted by these different operations vary.  That is, while infinity squared still equals infinity to the first power, the ratio between infinity squared and infinity to the first power is equal to infinity, but the ratio of infinity to the first power and infinity to the first power is equal to 1.  Therefore, when we apply any positive operation of an arithmetic dimension higher than 1 to infinity, we get a higher order of infinity that can be appreciated via other arithmetic operations.  (Notice this statement excludes the first arithmetic dimension because infinity + x, where x is finite, is still the same order of infinity.)

This is a pretty big deal considering the following:  In a very important sense, all the orders of infinity are not equal to each other–they are in fact infinitely different, but that, I will admit, is irrelevant.  The real issue in considering them equal is that it would disrupt a pattern when we start using infinite secondary inputs.  That is, ∞ -1: ∞ (also written as ∞ – ∞) is equal, not to infinity, but to zero, but ∞ 1: ∞ would be said to equal ∞.

But now consider something like ∞ ∞: ∞.  You shouldn’t be reading this sentence yet–you should still be considering.

…Ok you can go on reading now.

That above mentioned quantity is the highest arithmetic dimensional highest order of infinity.  However, the raising of such a concept introduces a second set of dimensions: We have thus far defined an arithmetic dimension relative to the use of arithmetic operations, but we might also now consider an “arithmetic dimension” its own operation which can be used, in a similar respect, to define an arithmetic dimension dimension.  I know, the terminology is silly, but it’s the most natural wording that arises.

A second “arithmetic dimension n dimensional” operation with an input of a and a secondary input of 2 could be written out long hand as follows:

(a n: a) n: a

and a secondary input of 3:

((a n: a) n: a) n: a

After defining this, we could give it some sort of notation (perhaps a c: n: b), and then define the arithmetic dimension dimension dimension.  We could keep going about this to infinity, plugging the algorithm into itself, with each additional dimension requiring an additional input (though we might just default to assigning this input a value of ∞).  And then, if we really had so much time on our hands, we could begin constructing a series of that series, calling the arithmetic dimension the first in the series, the arithmetic dimension dimension the second in the series, and so on to infinity.  Then we could begin to construct a series of that series, and a series of the series of that series, and so on.  In short, there is no limit to the fractal of orders of infinity.

All that probably seemed pointless, but its not; my point is this: if one travels to a high enough order on any of these dimensions, the first order of infinity in that dimension is considered equal to zero (the trivial case is to compare ∞^1 to ∞^∞).  That concept taken and applied to the infinite sets of infinite series of recursion is a powerful thing.  It is even recursive in itself, because we could use this model of infinity to evaluate the infinite system of dimensions that we have used to arrive at the model.

Spooky.  I know.

A Singular Application of Levels of Recursion

A friend of mine recently showed me the following question which I believe can be found online somewhere (besides here):

If an answer to this question is chosen at random, what is the percent chance that it will be the correct answer?

A. 25%

B. 15%

C. 50%

D. 25%

There is actually nothing wrong with this question.  If one looks at it at the trivial case level, it actually doesn’t have an answer, and therefore, an answer must be assigned arbitrarily in order to see the rest of the system work its way out, thus any answer given is ultimately arbitrary.  The question is, in this sense, like asking “What is the correct answer to this question?” which is really just nonsense.  However, ignoring that, lets suppose we assigned our trivial case the answer “B. 15%.”

This selection, while creating an arbitrary answer on this level, the trivial case, causes a relative correct answer of A or D on the next, lets call it the second, level of recursion.  There being two correct answers on the second level of recursion makes C the right answer on the third level, and thus on the fourth level we are back to A or D.

This is not a paradox, it is just a matter of an indeterminate level of recursion, which I find, as you probably could have deduced from the title of this website, quite fascinating.

Of course, the absolute answer to this question is that is does not have an intelligible answer anymore than does the aforementioned question, “What is the correct answer to this question?”  However, if we assume the trivial case for no reason (i. e. we chose it trivially˚), then I think the most convincing answer would be the “infiniteth” level of recursion, which, because the system has no limit, no end behavior, would be best put into the words “none of the above.”

In a later post, I might well invent a less trivial application or come across it by necessity; I just found this one interesting.

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˚ O dear.  I’m really not that funny am I.

Levels of Recursion

Haskell Curry’s paradox, titled “Curry’s paradox,” is often stated in formal logic as follows:

let A = (A–>B)          ! A is a boolean assigned the value “if A then B,” in words this means that “A is true if A’s being true means that B is true”

A –> A

A –> (A–>B)             ! Substitution

A && A –> B

A –> B

A                                      ! Substitution

B                                       ! modus ponens

The logical error lie in the ignorance of levels of recursion.  In reality, there is no such thing as letting A1 = (A1–>B), because that is an in equality.  A1 does not equal A1–>B, it equals A1.  The algebraic analog of this principle would be something like “x=x+1” for which there is no real solution, and seeing as there is no such thing as imaginary logic, at least not yet (though there should be), the expression is utter nonsense in logic.  The first statement can remain in the syntax that it is currently in; however, it should be realized that what that statement implies is “Let A1 = (A2–>B).”  Thus the recession in logic works just the same as it does in algebraic recursive sequences (i.e. we never define “a sub n” in terms of “a sub n” but rather in terms of “a sub n plus or minus some integer value”).

Thus the proof is disproven as follows:

let A1 = (A2–>B)

A1 –> A1

A1 –> (A2–>B)

A1 && A2 –> B              ! This simplifies no further.

This understanding is essential to the functionality of logic and is very relevant allover the place.

The Power Rule

I just thought I’d put this up because most people don’t know it.  The reason I know most people don’t know it is because I’ve never been taught it or read it anywhere, and yet it’s so simple and essential, unlike most everything else I come up with. ;~)

Here it is, a proof for the power rule in deferential calculus (people take too much for granted):

What is a limit really?

In many fields, especially ones related to the physical sciences, small angle approximations are often used. For example, when x is very small, the sine of x is said to equal x and to equal the tangent of x. These conjectures are then used as theorems in a proof.  In my opinion, this is bad calculus. Such approximations generally serve their purpose well, but are merely a way to make a very long proof look very short.  In calculations, it’s fine to be a little bit off, but not in proofs.  We often come upon a difficult matter to prove and a professor takes the shortcut. While these shortcuts do work, they bring us to the right answer, they are not the real reason why that answer is right. In reality, the sine of x is never equal to x or the tangent of x unless x is absolutely zero.  Consider the following proof:

notes:

1. Because the recursion can approach infinity at any rate (even an infinite rate), and in application, x is assumed to be a finite constant just greater than one, this indeterminate quantity (0^inf) evaluates to infinity.  Thus, as we should expect, the anti-proof does not apply to an infinitely small x value or a zero x value.

2. At this point, one may evaluate the expression on the left by flipping and multiplying by the bottom, thus getting sin(0) = 0, which is true, but there is nothing stopping us from moving the cos(x) to the right side of the equation before the self-referencing recursion and thus ending with it on the other side, so the move of multiplying both sides by 1/0 is legal in this case, this way of doing it just makes it more clear.  Of course, if one did start with the cosine on the other side, the proof might also evaluate to 0 = inf.  Admittedly, this proof is somewhat ambiguous over all for these reasons, but the end result (ignoring all the paradoxes) is that a limit is a limit, and any finitely small angle will not satisfy this equation just as much as zero does not equal one, or infinity does not equal zero.  If you plugin an infinitely small value for x, all the problems immediately dissolve and the limit is proven.

3. because, as was said in the previous note, the infinite quantity can approach infinity at any rate (an indeterminate rate), here it could be said to be approaching it at the same rate that x is “approaching” zero (even though x is a finite constant), and thus the limit holds (if we want it too…ha, ha, ha).  Also note, it doesn’t present a problem for us that the power of the cosine had to approach infinity at a rate that would make the power of the cosine evaluate to an infinite quantity, because by continuing to manipulate that original rate of approach, we can also manipulate the rate at which the power of the cosine approaches infinity (which is what we need in this last step).

My Cartesian Point?

My point ultimately with all this silliness that no one actually knows (though we do, I think, know more about infinity and zero and the imagination of numbers– ok, i need to get better jokes–than we say we do) is that if you zoom in far enough (as we did here using the infinite recursion), the tangent of x, the sin of x, and x are just as far away from each other when x doesn’t equal zero as zero is from infinity.  Thus, while the approximations are useful when dealing with actual quantities, they are just bad math when used in proofs (even though everyone does it… it’s pretty much like jumping off a cliff of infinite height), because in proofs, we depend on expressions being absolutely equal, and finitely small number is infinitely larger than an infinitely small number.