On Humanity and Recursion

Having discussed the essentiality of rhetoric to humanity, I now wish to further generalise and universalise the claim.

Notice that existence is the foundation of perspective.  We might define a person’s perspective as “the way in which that person exists”.  In other words, a person has all sorts of attitudes that make up his perspective, but these attitudes can be understood as qualitative descriptions of his existence—he exists in a way such that he favours existence over nonexistence.

It follow then, that underlying this principle of rhetoric, which is the foundation of humanity, is the principle of recursion.  Rhetoric is the power to observe the perspective from which observation takes place—to observe one’s own existence.  Likewise, morality is the power to act in observation of the perspective from which action is taking place, and love is the power to do so on a larger scale.  It is this principle of recursion that gives rise to the concept of a moral agent.  A moral agent is an entity that posses the power to observe its own existence.  For this reason a universalised morality is one in which maxims are formed in observation of all moral agents—being a self-similar construct to a personal morality.  Morality dictates that our actions observe that which observes itself.  In this way, morality is merely the method of creating a self-observant nature.

This relates nicely to the biblical doctrine of the Trinity.  In John 14:11, Jesus tells us that He is in the Father and the Father is in Him.  In other words, God is that which contains Himself.  Hitherto, we have seen that reality is made up of self-similar layers, and that these layers define each other and themselves though causality.  Hence, the Primal Cause is that layer which defines itself through causality, and ergo, causes itself.  In metaphysical terms, we might say that God is the Deification of the principle of self-observation, and in so being, is likewise the Deification of morality, reason, and love.

The fact that a rationally sound reality is necessarily self-similar helps us understand the doctrine of Imitatione Christi (trans. in a manner that imitates Christ).  All that follows from the Primal Cause must be similar to it, and must therefore observe all those things which observe themselves, which equates to acting morally, rationally, and lovingly—in short, acting Imitatione Christi.

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.

The Mockery

Perhaps some of you who have keen memories, or who enjoy filling your average sized memories with superfluous information, will recall that I have written in an earlier post that I was intent on addressing this subject at some future point.  I’m sure you will be overwhelmed with excitement and thrill to find that said future point has arrived.

This post is merely an attempt to share with you a concept, that of The Mockery, which I think is quite important.  Therefore, it will be a little less polished than one might hope, as I don’t really have the time or energy to compile all the sources I will draw from in an organized matter, yet I think I should try to share this with you all anyway.  Thank you for your understanding, or failure to share a misunderstanding, regarding this incompetency.

I cannot take full credit for the title of this concept; it comes from a belief held by a friend of mine that he once shared with me saying, “I believe that all beliefs ought to be mocked as well as celebrated,” (or at least, he said something to that effect; unfortunately, I personally fall into the latter of the two categories I mentioned at the beginning of this post).  The two of us then continued our conversation to point out how silly that was (for if all beliefs should be mocked, than even the belief that they should be mocked should be mocked, thus you have your infinite recursion that we all love).  But in the end, one must come to the conclusion, concerning this issue, that the concept does hold some merit.

Indeed, by every model I’ve presented on this blog site, the idea must hold merit, for I’ve always acknowledged that all the theorizing we do is ultimately an insufficient model of the truth†, as “there are more things in heaven and earth than are dreamt of in philosophy.”  Thus, I am “mocking” the very beliefs that I have concluded on.  And of course, if I want to be all postmodern about it–which I have no particular inclination of being–I should really also mock my self mocking those beliefs, and mock myself doing that, and so on to infinity.  But that’s obnoxious.  I personally feel that recognition of the first level of the recursion is perfectly sufficient, as it ‘implies’ every subsequent layer (and the layers of layers, as discussed in “Orders of Infinity”).  Therefore, there is no need to say, at the end of an argument, “and all this might be wrong, but it is the best I could come up with, and that maxim might be wrong also, as well as that one, and that one …”  If you did do this, you would just drive readers as crazy as I am probably driving you right now.

As a part of the correspondence with my friend on the subject of this concept, I wrote him a letter.  Among the things I tried to accomplish in that letter was a differentiation between Mockery and Comedy.  Here is an excerpt (which I reserve the right to have modified from its original form) from that letter in which I took a stab at that task:

It is painstakingly obvious from the way we are made, that we ought to laugh at the silly, eclectic combination of mortal bodies and immortal souls, “everlasting splendors,” (or, for now, whatever your philosophical equivalent is).  A few pieces of music come to mind: Beethoven’s ninth symphony and Brahms’ first symphony (both of which, if you haven’t heard, you ought to drop everything and listen to right now).  I was literally thinking these exact words when listening to Beethoven’s ninth most recently: “The second movement is both magnificently powerful, and boldly comical.  This is part of what makes the piece so brilliant.  If he [Beethoven] weren’t so comical with his style while also being serious, we would almost feel tempted to worship the sounds like pagans, and this would detract from the beauty because it is unnatural and wrong˚, but by also taking his music lightly as he does, Beethoven allows us to get swept away in aesthetic rapture, and then laugh at ourselves for being so moved by something so finite and mortal.”  This is one of the things that really excite me about postmodernism.  My music and thinking is sometimes almost hard to distinguish from the postmodernist.  It is, truly, in complete opposition; however, my hope and prayer is that it is what ultimately becomes of postmodernism.  More on this later.

I’d like to take this idea of comedy a bit farther.  I also mentioned Brahms’ first Symphony.  His comedy is a little more deeply rooted into reality.  There is nothing particularly funny about his music, but oh how beautiful the last movement is.  If you’ve heard it, perhaps you’ll understand what I cannot really say about it in words (this is one of the things about music; if I could explain everything music has to offer in words, it would hardly be worth writing in music).   Here is my best attempt: the music concludes with a brilliant, warm, rich finale theme.  The symphony began in a dark, minor key and builds the whole way to this conclusive ending.  It’s like death and resurrection.  This is a portrait of the comedy of life.  All the horrid things of now mean nothing in the end beyond the extent to which they pointed us to the brilliant One who does mean something, and life eternal is free and beautiful; we will one day find that we truly had nothing to fear the whole time.  This is another sense of comedy (if you can understand it from my vague and abstract description, or perhaps you’ve heard the piece and can see the connection) that I believe in.  It is the same kind that exists’ in Dante’s Divine Comedy.  The other type of comedy (the one described in the previous paragraph, I’ll call it mockery) is also present in this one.  The mockery of Brahms’ first symphony is that it ends, and we are still here with all our fears and pains, and even during the symphony, we never fully left, the symphony never entirely met our longing for Heaven (because it is not heaven, it is mortal).  Parallel mockeries exist in Dante as well.

That distinction being made, I went on to discuss its ramifications:

I think we should laugh at both comedy and mockery.  We laugh with joy at comedy and with something more like scorn at mockery.

Thus we have our two types of philosophical humor, if you will.  Now, to begin unraveling the mess of philosophy to which end, as I mentioned earlier, I hope postmodernism is bound.  Yes, it is the opposite.  Where postmodernism looks to deny an absolute reality in place of our self-mocking individual ones, it denies, or “mocks,” our individual views in place of the One absolute.

It is a rejection of the temporal rather than of the eternal.  Dante and other thinkers (and artists, and the rest) of the past don’t seem to have seen this.  Dante is so very caught up in mortal things.  He thinks that everything can be measured.  Every sin or noble doing is like a withdrawal or deposit in a check book, and the final balance determines exactly to which “level of Hell, Purgatory, or Heaven” you belong.  This is not the case.  As C. S. Lewis writes in the intro to his wonderful fantasy novel, The Great Divorce, there is no in-between Heaven and Hell, no going to Heaven with “a scrap of Hell in our pocket.”  No, all the universe is ultimately binary (the opposite of the postmodernism) and yet infinite binary (a compromise).  There is right, there is good, there is beauty, and there is wrong, there is evil, there is ugliness.  That’s it.  All these things ultimately boil down further into eternal states of being and non-being.  0 and 1.  But this is getting a bit ahead of myself*.

That’s the problem with the artists (and the rest) of the past.  Not that they believed too much in absolute truth, but that they believed too little, or (for the sake of clarifying) rather that they believed they knew that absolute truth, and that it was there, present to their conciseness in its entirety.  It isn’t.  Dante is like a Pagan.  It is the business of the ancient Greeks and Romans to worship that which is human (or at least much too human-like), and even sometimes that which dies.

We must worship only things eternal.

As for my reference to Dante, this letter does not, by any means, convey my entire rhetorical view of him; I just presented a simpler one for convenience.  To “expand your model” of my view of Dante a little, I share the following:  When I wrote that “Dante is like a pagan,” I really meant, more specifically, “Dante, as I portrayed him in this letter, is like a pagan,” though my broader view of Dante is that he, on whatever level of his consciousness it may have been, understood what I have shared with you in this post, and thus, wrote the way he did in order to establish the “first level” of the recursive sequence.  That is, we cannot worship God if we have absolutely no concept of him, we instead need the largest concept of him that we can fathom.  However, as it turns out, and as Dante seemed to realize, just as “he who wishes to gain his life shall lose it,” and “the first shall become last,” so to does the simpler, more concert concept of God often constitute the larger one.  Indeed, the concept God presents us with in the bible, his very word made flesh, is often the most concrete one of all, and it is, beyond the shadow of a doubt, the largest, and most accurate.

[You don’t have to read this part if you don’t want to] And now, to write about what I just wrote in the previous paragraph, as that seems a relevant task in a post about The Mockery and all its recursiveness, notice, I wrote to you that what was in the letter was not a full model of the truth of the matter as I understand it.  Indeed, even what I’ve written in that paragraph is not complete.  Consequentially, many of you reading this may have felt like what I wrote in my letter was a lie, for it almost seems contradictory to my belief as I shared it with you.  This presents a good question: what is the difference between bearing false witness and presenting and incomplete model of the truth as you know it?  Since the question is relevant to the mockery, I thought I’d raise it here, and then those of you with the aforementioned, outstanding memories will recall it when I later write a post in which I answer it as best and honestly as I can with my most genuine attempt.

Finally, to comment on this post as a whole†: this is the strangest and worst-crafted posts of all those I’ve yet written.  It seems to make a mockery of this entire blog.  I apologize for however unpleasant that may be.

______________________________

† Not because its wrong, but because it’s not the full truth.  But, when we consider fractal reality, it can be seen as the full truth, within the bounds of the scope in which it exists; that it, its true to the degree of detail that one possesses the ability of seeing among the fractal of detail that makes up reality (like a computer programer creating ‘objects’).  Now here’s the twist: apply everything I’ve written in this footnote so far to itself.  Ultimately, once one does that infinitely, we see that we must abandon the whole issue of the mockery while we are “working on,” or formulating our models, and only turn to it in the process of “appreciating” those models, for it makes no difference to the process of thinking whether something is an object or an infinite string of binary.

˚ To mock my own writing–isn’t if funny that here I have my own thoughts in exact quotes after the remark I made earlier in the post about my memory?   On a more serious note: this particular thought that the aesthetic would be lessened by an immorality in the act of its enjoyment goes along with “The Art of Thought.”  That is, it is pleasurable, in some specific sense, for the mind to ponder truth, and likewise, for the soul to experience “righteous art.”  Thus art is an act of philosophy, for a major part, if not the only part, of aesthetic is morality.  Hence, in the context of the letter, it is pleasurable to listen to music that is good for the soul in much the same way that it is pleasurable to eat healthy foods, exercise, or tend to a wound, all of which actions are good for the body.

* If your interested in my writing on this matter (which I did not include in this post), see “Fractal Reality,” under “Metaphysics.”

† And the effect the contents of the post, and thus effect the comment about the post, once more effecting its contents …

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 and a secondary input 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 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.

__________________________________

˚ 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.