# An Endless Knot – 2nd Movement

In a previous article, I introduced an analytical framework for thinking about art, using a song I wrote, “An Endless Knot“, as the case study. I’m working on a follow up article that formalizes all of these ideas into a mathematical model that can be applied directly to AI, but in the interim, I also wrote a second movement to the song, which I really like, that continues to develop the elements introduced in the first movement.

You’re the best thing I know of.

The best thing that I’ll ever find,

Darling, you’re the best thing I know of, Hanne –

The best thing that I’ll ever find,

Darling, you’re the best thing that I know of.

The best thing that I’ll ever find.

Our lives are intertwined,

In an endless knot in time.

Darling, you’re the best thing that I know of.

The best thing that I’ll ever find.

Our lives are intertwined,

In an endless knot in time.

Darling, you’re the best thing I know of.

# An Endless Knot (Video)

I’ve made a music video for my song, An Endless Knot.

I did a bit more work on the audio as well, but kept the rough scratch track that I recorded on my iPhone, with no editing, only mixing levels and mastering the audio. The results are fantastic, and it shows that it’s possible to cut a professional quality track with just an iPhone, if you have access to software like Logic Pro, or Pro Tools.

I shot the video by hand on my iPhone, with a single shot, and the roughness of the shot works quite well with the roughness of the audio, particularly at the end.

Even though Apple’s obviously changed quite a bit as a brand in the last few years, creating this track reminded me that there was a time when the brand was justifiably associated with the arts, and they still make great products for artists.

# How I Think About Art

My theory of art is a practical theory rooted in basic psychology and information theory. The primary tools I use to think about a piece are mathematics, to analyze the structure of the piece, and basic distinctions between exogenous signals, internal biological responses, and the psychological associations that follow. It’s probably closer to how a copywriter, psychologist, or propagandist, would think about art, than, say, a literary theorist would.

The core animating principle of my theory as a creator of art is to use literally everything as a medium to store information. So in the case of a piece of music, if the opportunity presents itself, why not write a melody that has a physically meaningful shape in some embedding, so long as it doesn’t otherwise interfere with the other information you’re trying to convey. This allows for supplemental expressions that have to be sought out, and allows for non-traditional uses of traditional elements. For example, below, I’ll discuss how a simple, one note harmonic can be used as a symbol in the most literal sense, in that it can be used to trigger associations in the listener that have nothing to do with music. This turns a single note, which in isolation can be encoded as an integer, into a trigger for a complex set of ideas and emotions, that would otherwise require a much more complex encoding to convey.

As a historical matter, the recent incorporation of visual art into music, in everything from music videos, to simply associating an image with a song, through use of an icon or an album jacket, has completely changed the context in which music is consumed. Music is now almost always associated with visual art, so we need new, practical ways, to think about these things. Further, digital audio processing, and hip-hop in particular, have created genres in which it’s common to literally copy and paste a sound into different locations in time, and even transpose the key or frequency of the sound at will, creating a literal musical collage in the time and space of the piece.

All of these norms and techniques are going to change the way people think about music, whether it’s conscious or not. It also creates rich opportunities to play with expectations and associations, that I think can be formalized in a way that is useful to both consumers and creators of art.

The case study I’ll use is a simple piece for acoustic guitar and vocals that I wrote recently. The recording is a scratch recording that I made on my iPhone, mastered just a bit to get good levels and decent equalization, and otherwise left unedited, leaving in leading and trailing audio that I’d ordinarily edit out. But rather than ask you to suspend judgement, the fact that it is just a scratch recording will play into the analysis, to demonstrate how I try to take into account all aspects of an artifact. I’ll keep the actual music theory to the bare minimum, so that non-musicians can follow along.

An Endless Knot

You’re of course free to do whatever you want, but I would suggest first listening to the song, “An Endless Knot”, and reading the lyrics below as it plays. After you’ve done so, we’ll examine the facts of the narrative at a high level, which is clearly about two lovers.

You slip your hand in mine –

The smell of ocean on your breath,

Windswept, dried, and cracked wide open.

As two soft tongues are tangled in,

An endless knot in time.

Your hair tangled up in mine –

Drips in bits, down, like wine,

Sweet from having known your skin.

Your wet clothes pressed against mine,

As two souls lost upon a sea,

Tie their knot in time.

Your skin tangled up in mine –

The smell of ocean on your lips,

Unwashed, salted, and alive,

Windswept, wild, and cracked wide open.

Oblivion within us,

Nature roars,

And celebrates us with a storm,

That paints us both in salt and water.

In our little piece of nothing,

God relinquishes us,

For just a few moments more.

They begin with some nominal distance between them, fully clothed, and end up on top of each other, naked. Though I don’t make any suggestions as to the genders of the two lovers, when writing the piece, I imagined myself together with a woman, subjectively, not as an exogenous observer, which explains the visceral nature of the lyrics. Ultimately, the portrait I’m trying to paint is the one you would experience in that position, not as an observer, but as a participant, complete with all the sensory and emotional information associated with that experience.

As for setting, I imagined all of this taking place on a small, cheap, wooden sailboat, at night, causing the boat to appear black in color, but the lyrics could support any setting that is at least partially outdoors, and near the ocean. The fact that it’s nighttime is partially implied by the line that rain “drips in bits down like wine”, which is intended as a poetic device, but also literally, in that the rain appears to be a dark liquid, due to the darkness of the setting.

Though I didn’t have an entirely specific person in mind, since it is a work of fiction, I did make use of elements drawn from actual memories of my life.

The First Verse

As noted above, the lyrics are deliberately loaded with sensory information, and this is immediately apparent, as the first verse conveys information about proximity, scent, texture, contact pressure, and taste. The goal is to convey as much sensory information as is possible in the lyrics, given the constraints of the structure of the piece, with the emotional response triggered by the music itself intended to color all of that sensory information, by creating an emotional context in which that sensory information exists.

This is achieved by stating the facts of the environment, but also in the word selection itself. By saying “smell” rather than “scent”, you conjure something that is a touch more vulgar, though in this case it’s obviously meant to be a positive feature. The point is, there’s a scale of emotional response elicited by a word, that is independent of the meaning of the word, and vulgar words often elicit a higher level of response than polite words, making vulgarity a powerful tool of communication.

Ultimately, providing visceral information allows the listener have to a more complete sense of moment, where the tangible and intangible fundamentals of the moment in question are presented, and hopefully, imprinted upon the mind of the listener, creating a sensory and emotional portrait of what’s happening.

You slip your hand in mine –

The smell of ocean on your breath,

Windswept, dried, and cracked wide open.

As two soft tongues are tangled in,

An endless knot in time.

In addition to presenting environmental facts, visceral information can also be used to plant inferences. For example, the sensation of kissing cracked lips is very different from the sensation of kissing smooth lips, and part of the reason I provided that information, is to trigger a sense of what it’s like to actually be there, in the moment, because you’ll immediately imagine kissing someone with chapped lips, which will cause you to be engaged in the piece, and attach the song to your own memory, and your own life. But visually, it conjures a minor injury in the woman I’m kissing, especially because I said her lips are, “cracked wide open”, suggesting that they might even be bleeding. But because we’re still kissing, it therefore implies the woman’s indifference to the risk of any related pain, and my indifference to the risk of the imposition of pain, suggesting that both of us view what’s at hand to be of greater importance than the minor inconveniences of our condition. That’s a lot of information that’s unpacked by simply saying that one of us has chapped lips. The point being, that by using visceral information, you can plant the seeds of a complex set of facts with just a few words.

Kissing a total stranger is a fundamentally different experience than kissing someone you are profoundly attracted to, or in love with. Each of these two experiences will generate very different internal responses, one possibly not so pleasant, and the other presumably wonderful, though each will consist of a complex set of exogenous signals, internal responses, and psychological associations. It is in my opinion the job of an artist to have the ability to convey accurately which of these two very different experiences is being described for consideration, using only visceral information, without having to say explicitly, for example, that, “I think this person is beautiful”. The implication from some kernel description should make the context clear, or deliberately ambiguous – the point is to develop the power to describe context using upextrapolated, raw, visceral information.

The Harmonic

For those of you that don’t play stringed instruments, a “harmonic” is a weird feature of stringed instruments, where if you simply lightly place your finger at certain points along a given string, without pressing down to actually play the note, a special third note will be played if you pluck, that is not the same as the note played by plucking the open string, or the note played by actually pressing down on the string and plucking. To be honest, I don’t know the physics behind this, but it happens at a few places along each string, and most importantly, it sounds very different from an ordinary note, and sounds more like a bell, especially on guitar. The harmonic in this song appears in two places, once in the verse, and once in the chorus. We’ll consider only the harmonic in the verse.

The first harmonic appears in the first verse, at the 0:16 mark, and really cuts through and sounds like a bell’s chime. It appears right after I say the word “breath”. Let’s just treat everyone like Pavlov’s dogs, and assume that by the end of the song, we can get people to associate that chime with whatever words are proximate in time to it. So if you go through the song, the words near the harmonic are –

(1) Breath,

(2) Tangled in,

(3) Skin,

(4) Cracked wide open.

If you take the intersection of associations across all of these words, one fair conclusion is that each of these phrases is associated with the human body. The harmonic by its nature has a popping sound, which is part of the reason it stands out. This makes it a natural choice as a symbol for any discrete change, like a bubble popping, or a kiss ending with a “smacking” sound.

For someone that’s listening closely, every time the harmonic chimes, the set of ideas associated with the harmonic will build, aggregating to a set of bodily interactions that involve discrete changes. Ultimately, in the third verse, at the 2:34 mark, the harmonic occurs when I say, “cracked wide open”. In this case, I don’t say what it is that’s “cracked wide open”, which leaves the listener searching for an object, which probably sends them to the first verse, conjuring an image of a split lip. But, the context of the third verse suggests otherwise:

Your skin tangled up in mine,

The smell of ocean on your lips,

Unwashed, salted, and alive,

Windswept, wild, and cracked wide open.

Beginning with the first line, the obvious implication is that we’re naked, but the more interesting way to get there is to consider how the first line of each verse changes over the course of the song:

(1) You slip your hand in mine –

(2) Your hair tangled up in mine –

(3) Your skin tangled up in mine –

This gets to how people unconsciously perceive things, which is if one thing is currently in the place that some second thing used to be in, you naturally compare the two things. In this case, this will necessarily create a transition from some initial state, to some final state, which the listener will make sense of by filling in the gaps between the states, ultimately creating a timeline where two people begin with the normal trepidation of first holding hands, and then end up having sex. And again, the amount of information provided is minimal, but because of the targeted, visceral nature of the information, and the position of that information in the piece, which repeatedly occurs in the same place, the information necessarily generates complex inferences in the mind of a sensible, careful listener, ultimately constructing a real world narrative, with just a few words.

The implication is probably clear at this point, and the harmonic is now a symbol for the act of beginning to have sex. If you listen closely, you’ll hear that just after the harmonic chimes, I stomp twice, counting before the bridge begins. This entire sequence could be fairly be interpreted as penetration (the harmonic) followed by the percussion of sex (the stomping). So association takes us to the unsaid, and allows for the subtle expression of the impolite. It’s a more sophisticated version of the adult joke in a kid’s movie, though this is certainly not a song any child would understand.

The overall point is, even though it’s a piece of music, effective symbolism is still possible, which allows for a drastically wider set of signals to be communicated by piggybacking on ordinary associations, or building associations within a piece, like we did with the harmonic, until the musical symbol can ultimately be expected to produce the right associations in a sufficiently clever audience.

From the perspective of information theory, this allows for a single note to carry an incredible amount of information, which would ordinarily require a much more complex assemblage of signals to convey. The point being that, by the third verse, the listener now has a set of associations that can be exploited, allowing you to deliver a single, simple signal, and in turn trigger an extremely complex set of associations.

The “Introduction”, the Pause, and Missing Information

The Introduction

The song begins with me saying, “Norway, or some shit like that”. Ordinarily, I would never publish the gibberish that I say before and after a track. Artists do this sometimes, and Biggie Smalls comes to mind, yelling at his poor engineer to turn his mic up (in far less polite terms), but I did it because I knew that I was going to use the piece as a case study for this article, and I thought it would serve as a great example of how missing information is treated in the context of interpreting art.

First off, I’ve never heard anyone refer to the idea of missing information in art, and instead, people simply speculate what an artist intended to convey, with varying of degrees of believability and basis in reality. But what’s at work is actually missing information, and in this case, I said a thing that has no apparent connection to the rest of the piece, and therefore, no apparent meaning in the context of the piece. This leaves the listener either indifferent, because they don’t care what I think, which is fair, or curious why I would mention a seemingly random Scandinavian country in the beginning of a song that has no apparent geography, other than being in a place that is near the ocean and warm enough for two naked people to be outdoors. And you can certainly make the case that Norway, sometime in August, would fit the bill, and maybe that’s why I said it. But I’m not going to tell you why I said it. Instead, I’d like to consider what happens to a listener when information is known to be missing from a piece of art.

So to begin, imagine that I said nothing, and just started the song by singing. In that case, you would have no idea that there was perhaps some secret context that would reveal, for example, the geography, or subject, of the song. In that case, knowledge of the existence of any missing information would be simply unknown to you. In contrast, in this case, I’ve created the impression of missing information in the listener, because I’ve said something that has no obvious connection to the piece. When you consider the mechanics of that, what’s happening is that you’re taking this piece of information, that is apparently unrelated to the rest of the work, and trying to bridge the gap between the information and the rest of the work.

This is probably the same mechanism that allows a listener to take a limited set of facts, and construct a realistic narrative, but in this case, there’s simply not enough information to do that conclusively. In contrast, above, when the lyrics suggested an initial state of holding hands, and an end state of having sex, the listener can construct a realistic, detailed narrative using that limited information. But in this case, there’s just not enough information to work with, especially because of the expletive, which doesn’t fit the rest of the song. As a result, the introduction seems to be a totally unrelated piece of information. It turns out that the statement isn’t totally unrelated, but I’m not going to give you any more information than that, and that’s just how it is.

The Pause

At the 1:33 mark, I pause for a little longer than seems natural, and significantly longer than I did in the first verse. It turns out that I was scrolling through the lyrics on my laptop, because I just wrote the song, and don’t know the lyrics well enough to recite them from memory, and if you listen closely, you can hear my clicking away at the down button on my laptop to find my way to the next verse. I’m telling you that this was an error, but if you take the view that all opportunities for expression, and all opportunities for encoding information, should be exploited, if it’s meaningful to do so, then in the absence of knowing that this is in fact the result of an error, you would naturally search for a meaning to the pause, and perhaps even the clicking. That is, does the pause, which is anomalous to the second verse, mean something? Stated differently, should I associate something with the occurrence of the pause?

This is similar to the symbolic harmonic and stomping, except pausing is already an accepted tool of the musician, but what arguably isn’t an accepted tool, is a bizarely prolonged pause. Nor is an explicitly symbolic pause.

Without fighting over degrees of acceptability, the point is that the introduction of the unexpected creates the impression of meaning, since the logical inference is that if there’s a low probability aspect to some work of art, then it was deliberate. In this case, I’m deliberately giving you a semi-finished product to force the issue, and present a framework for thinking about art that, is, I suppose, rooted in classical expectations, but permissive of deviations, and, attaches significance to those deviations. But to have a probability, you need an expectation in the first place, which means you need a body of work to ascertain what is normal, and what is unusual. I think, the framework should be, that you can pause for as long as you like, but a listener has the expectation that you will pause for some normal amount of time, and if you go beyond that, which is fine, then there will be significance attached to that decision.

This would allow for, as a general matter, artists to have essentially unlimited freedom, but still have a traditional context in which choices made will be evaluated. Expressed differently, I’m expecting Bach, but I’ll be able to interpret Zappa, because I’ll evaluate Zappa using the groundwork and the context of the Western, classical tradition. I’m not saying that this is the only way to think about art. I’m instead saying that it’s an extremely practical way to think about an incredibly wide variety of art, and, also, to create an extremely wide variety of art. In short, this view permits modern art, but uses an objective framework in which it can be evaluated, thereby organizing the disaster that is unbridled human expression.

The Philosophy of the Narative

When considering the facts of a narrative, we’re invariably going to invoke economic, existential, and natural philosophy, even if we don’t do it by name. This particular song is a celebration of minimum economics, and maximum intervention of nature, where massive, primordial joy is experienced using the absolute minimum economics necessary to generate it – two naked people in a shitty boat, and a storm.

It is a celebration of companionship, and the human body, and wild, uninhibited love. Uninhibited both in terms of having no fears, apprehensions, or hangups about the human body – in fact suggesting quite the opposite, that rain is only improved in taste by contact with the skin of the partner – and in terms of having no apprehension in succumbing to, and being physically painted by, the wild thrusts of nature, both internally, in terms of emotional state, and exogenously, by being physically sprayed with salt water. This is not always wise, since both nature, and people, can be dangerous, especially in shitty boats pelted by a storm, but it is at times necessary, in my opinion, to live a full life, and I suppose the only hope is to get lucky, in all respects.

The theme of the knot is rooted in my ideas on time, specifically, that time simply doesn’t work the way it seems to at our level of existence, and that the future can interact with the past, and vice versa, in a manner that is physically meaningful, and still allows for objectivity. This is not something that’s worth discussing in any detail in this note, but the import for the narrative of this song, is that two lovers are drawn to a moment in time, and have been, through a set of interactions that connected them, despite being initially separated by some distance in space. This is a theme that’s obviously been explored before, I just happen to have a more formal explanation for how these things could be physically possible, than your typical artist. Ultimately, nature celebrates the outcome, by tossing salt water at them, out of sheer necessity, because this is really the only sensible way for nature to safely communicate with two naked people having sex in the middle of the night in a shitty boat.

These are my personal views, which are a hodgepodge of informal and formal theories of nature, and common sense. And though we can take things to an academic level, I’m not sure that’s practical with a more universal theory of art, because you need to have a practical level of exposition for a large number of topics, rather than an academic level of exposition for a single topic, unless you’re planning to write a tome about every Björk video, which is not practical. This doesn’t make academic analysis unimportant, it’s just a different way of creating and thinking about art, which is probably closer to what goes on at an ad agency, than a fine arts college, which in my opinion allows for the creation of more holistic pieces that draw on a wider breadth of genres, mediums, and philosophies.

Thinking about art in this way allows you to use a simple recording device, an acoustic guitar, a human voice, some stomping, and to ultimately create a piece of art that goes way beyond a typical folk song. All of this forces us to be more universal artists, natural philosophers, perhaps even mathematicians, to really understand the full complexity of modern art.

The Bridge

Let’s just accept my use of harmonics and stomping as a clever, but ultimately crude symbolic sequence intended to conjure the mechanics of sex. What follows this is a bridge that is deliberately busier, and more aggressive, in terms of the guitar work, than the rest of the song, and intended to convey nature’s celebration through the music itself, which is a strange exercise.

Composers that I absolutely love claim to describe physical objects, like people, or water, all the time, and while the results are often beautiful, I can’t help but call bull shit, because what they’re typically doing is simply calling something a “tone poem”, or a “portrait”, without really explaining what the mapping is from their song to the person or thing they’re supposedly describing using music. In the case of my piece above, the bridge is definitely in the bull shit category of describing physical phenomena with music, but it has some physically meaningful elements, in that I increased the rate at which the base note changes, in an attempt to evoke the sound of heavy rain rapidly falling during a storm.

In fairness to musicians, physically describing things is generally not their job – they write music, which is inspired by particular people or things, and generally not subject to the constraint of actually encoding some physically meaningful aspect of the source of inspiration. In fact, part of the magic of music is that it is ludicrously abstract, but nonetheless generates an incredible response in people, often driving them to tears. When you think about that, it’s a weird fact, because it’s a sequence of tones, without any obvious physical meaning, but if they happen to hit you at the right moment, they could leave you emotionally devastated.

Returning to the lyrics, the act of sex is equated with oblivion, through the annihilation of opposites, due to the combination of physical, sexual opposites, of man and woman. The intent is to convey the occurrence of a psychological annihilation where the two lovers become part of a larger canvas that includes all of nature itself, painted by nature, which includes their internal, subjective experiences, as well as their external appearance, which nature alters by calling up a storm, and tossing salt water at them, celebrating them, and making explicit, nature’s hand in their decision to join each other on the night in question.

This piece is a shameless celebration of the notion of creation, and of an interventionist theory of nature, that on occasion celebrates us, and more generally, interferes in our lives. This is obviously true in some sense, the only question is whether or not nature engages in conscious choice, rather than happenstance. This song takes the view that nature is as conscious as we are, in fact, more so, and vastly, and incomparably, more powerful.

# Imitating and Expanding Datasets

Attached is an updated version of an algorithm I introduced in a previous post that can imitate a dataset, thereby expanding the domain of a dataset. This version is incomparably more efficient than the previous version, since it uses my new categorization and prediction algorithms. In the example I’ve attached, which imitates the shape of a 3D vase, the algorithm generates approximately 140 new points per second, and copies the entire shape in about 11 seconds, running on a cheap laptop.

Below are images showing the original shape, the point information, and the replicated shape generated by the algorithm.

11-4-19_NOTES

# Superstition, Information, and Probability

Coincidence has always annoyed me, since it is not causation, and therefore, lacks a normal physical explanation. Nonetheless, it has some of the superficial elements of causation, making it particularly infuriating if it occurs often, for those that are generally scientifically minded. From what I’ve read, Bach would use random outcomes as the seeds of his compositions, and I often do something similar, which I finally took the time to think about. And upon reflection, I think there’s actually good reason rooted in information to use random exogenous signals as “inspiration” for creative works.

Causation

If I throw a ball at a window, and it breaks, there’s a narrative that can be constructed, and expressed in mathematics, that explains the transfer of momentum from my body to the ball, from the ball to the window, and ultimately, the shattering of the glass. In short, causation as a concept is really coextensive with physics, at least in my mind, which means we can construct a mathematical model that has a one-to-one correspondence to some set of measurements made on the system. In less technical terms, I can tell you how the system behaved using mathematics, and therefore, I can predict its behavior going forward.

Coincidence

But coincidence is something different. If I had to define coincidence, I would say it has two components: (a) a low probability, and (b) contextual relevance. For example, when I was on holiday during law school, I was driving with some friends through the interior of Sicily, listening to house music on the radio. Then suddenly, the station changed on its own (presumably because we had left the physical boundaries of the original radio station) and Le Tombeau de Couperin, by Ravel started playing. The event of a radio changing stations on its own is, I would say, low probability, and this piece is relevant to me, because it’s one of my absolute favorite pieces of music.

We can use these criteria to construct other examples. Imagine walking out of a store having just purchased a bright orange hat, when suddenly, someone throws an orange at you. Both events are low probability in the ordinary course, and the latter event of getting hit by an orange is relevant, because it intersects in property with the item you just purchased. And you would be completely certain the event was deliberate, even if it seemed superficially impossible for that to be the case.

Last night, in New York City, the wind was blowing extremely hard, and knocked over at least one item in the shower, and I remember immediately attaching significance to the label of the item. This is a strange thing to do, and certainly superstitious, but upon reflection earlier today, I realized it’s actually quite sensible as a mechanic. When a low probability event occurs – e.g., something loud, unusual, or bright – the brain probably treats the objects in question as likely to contain information, which is perfectly rational, since you might have to quickly size up a threat, or an opportunity in the face of a low probability event.

So I think of superstition as a sort of perversion of this perfectly reasonable instinct to treat a low probability event as a signal worthy of attention. But, I think creativity can turn this round again in a reasonable way, by using low probability signals as a seed for a creative train of thought. And this is because if something is low probability, then it should contain a lot of information, if our brains encode experiences efficiently. This could explain novelty seeking behavior as well, as the brain’s desire to consume novel information, which can in turn be used as the seed for new deterministic inferences.

Poets, musicians, artists, and even some mathematicians, are notorious novelty seekers, often degenerates because of it, which suggests that there’s probably something physically meaningful to their conduct. People generally write this behavior off as some kind of aberration typical of a creative type, but dismissing this observation turns what might actually have a scientific explanation into mere coincidence. I think creative people might have wells of information derived from random, low probability experiences, that then get restructured into novel artifacts. In short, my hypothesis is, that creative types probably have heightened senses, meaning they’ll be even more drawn to novelty, pleasure, drama, etc, precisely because they’re going to pull way more information out of those experiences than a normal person. This will in turn build up a well of information that can later be drawn upon to generate creative artifacts.

Writers do this more literally, by having unusual experiences, and sometimes simply writing about them. But painters, musicians, and even more abstractly, mathematicians, I think do the same thing, except the information becomes more and more abstract as you progress from the literal to the mathematical.

I think this is probably how it works, because deterministic thinking will never produce an original idea, since it is by definition, mechanical inference from a set of known assumptions. The truly hard part is coming up with a new and correct assumption in the first place, like F = MA. Solving for acceleration once this is known is just not as impressive, or as uncommon as stating the equation in the first instance.

As a result, creative people have to be doing something else. I think some creative people can take things even further, but I certainly use random, low probability events, as the source of inspiration for ideas, and Bach did the same, and I’m sure plenty of other people do something similar.

So whether or not the story of Newton’s Apple is true, it certainly makes a lot more sense in light of the work of Shannon.

# A Simple Model of Infinitesimals

I did some work while I was in Stockholm on distributions over countable sets, but stopped in the interest of finishing up what seemed to be, at the time, more important work in physics and A.I. But yesterday, I realized that you can construct a simple model of probability on countable sets using the work that I had abandoned.

Uniform Distributions on Countable Sets

The primary offender for probabilities on countable sets is the uniform distribution. This is because it is impossible to select a real number probability for a uniform distribution on a countable set, since there is no real number r that satisfies,

$\aleph_0 r = 1.$

So instead, my proposal is to drop the requirement that the sum over the probabilities equals 1, and instead to require only that the product of (x) the number of trials and (y) the pseudo-probability of the signal equals (z) the frequency of the signal in question.

So, if each element of a countable set of signals appears exactly once in a countably infinite number of observations, then our pseudo-probability p must satisfy:

$\aleph_0 p = 1.$

Obviously, $p$ cannot be a real number, but this of course doesn’t prevent us from assuming that such a number exists.

There’s an additional wrinkle, which is that for countable sets, there are an infinite number of “uniform distributions”. For example, each element of a countable set could appear exactly twice over a countable number of observations, or three times, etc.

Using the natural numbers as our underlying set of signals, we could observe the uniform distributions,

$1,2,3,4,5, \ldots,$

or,

$1,1,2,2,3,3,4,4,5,5, \ldots.$

In both cases, each signal appears an equal number of times over a countable number of observations.

So as a general matter, we can instead simply require that,

$\aleph_0 p = K$ (Equation 1),

where $K$ is the number of times each element appears over a countable number of observations.

So our pseudo-probability is a number, that is not a real number, that when multiplied by the number of observations, which is assumed to be $\aleph_0$, produces the frequency of the signal in question.

This definition implies some clean, intuitive arithmetic over these numbers. For example, if signals $i$ and $j$ have probabilities $p_1$ and $p_2$,  respectively, then their combined frequency in the distribution is given by,

$\aleph_0 p_1 + \aleph_0 p_2 = \aleph_0 (p_1 + p_2) = K_1 + K_2$.

There is, however, a special case where each signal appears a countable number of times. That is, it is possible to have an infinite sequence of observations in which each signal appears a countable number of times.

For example, there are an infinite number of primes, and every prime number can be exponentiated a countably infinite number of times, producing a unique, countably infinite set associated with each prime number.

So let’s assume the primes are our underlying set, and our observations are generated by reading the set of integers in increasing order. If we spot a prime, or a power of a prime, we list the prime number itself as an observation. Otherwise, we write nothing. So if we see 2, we write 2; if we see 9, we write a 3; if we see 16, we write 2; if we see 10, we write nothing.

Reading the first 6 numbers, we would write,

$1,2,3,2,5$.

Though we read 6 numbers, we made only 5 observations, since 6 is not prime or a power of a prime.

Reading the number line from left to right in this manner will cause us to list each prime number an infinite of number of times, and because there are an infinite number of primes, we will generate a set of observations that contains a countably infinite number of signals, each of which appears a countably infinite number of times.

Returning to Equation (1) above, we would have in this case,

$\aleph_0 p = \aleph_0.$

Any real number would satisfy this equation, but not only is it not intuitive to use a real number in this case, it also implies that there’s more than one solution to the equation, which is pushing the boundaries a bit far, even for a pseudo-probability concept. As a result, my proposition is to use the number,

$\Omega = \log(\aleph_0).$

Note that $\Omega \neq \aleph_0$, since $2^{\aleph_0} \neq \aleph_0$, at least under normal assumptions. Also, it’s easy to show that $\Omega \aleph_0 = \aleph_0$, and therefore, we have a number that is not a real number, but nonetheless satisfies the equation above. This would imply that whenever you have a countably infinite number of observations, and a signal that occurs a countably infinite number of times within that set of observations, the pseudo-probability of that signal is $\Omega$.

Developing an Intuition for Probabilities on Countable Sets

Intuitively, this makes sense, since if you know that something occurs an infinite number of times in an infinite sequence of observations, the number that describes your expectations for each observation you make in that case should not be a real number probability, since, as shown above, this implies that each real number is just as good as any other real number.

At the same time, the number shouldn’t be an infinitesimal, since the portion of observations in which the signal occurs does not have the same relationship to the total number of observations that an infinitesimal does. That is, if a signal occurs once in a countably infinite sequence, intuitively, if that signal has a well-defined probability, then it should be an infinitesimal, since our expectation is that the event is extraordinarily unlikely to be the current observation, but not truly impossible, since it is actually certain to eventually occur. In contrast, if a signal occurs a countably infinite number of times within a countably infinite sequence of observations, then intuitively, we would not expect the pseudo-probability of that event to be an infinitesimal, since the event could happen often.

For example, if we consider the probability of observing an even number when reading a permutation of the natural numbers, it would be unreasonable to say that the probability is infinitesimal, since there are by definition an infinite number of even numbers in the set that we will eventually observe. At the same time, it’s not quite right to say that the probability is $\frac{1}{2}$ either, because it is possible to observe an arbitrarily long sequence that contains no even numbers at all.

And unlike with a finite set of observations, there is no limiting argument that justifies our intuition that the probability of observing an even number increases with the number of observations. It’s simply not true, since for any given sequence of odd numbers, you can construct a permutation on the natural numbers that contains an even longer sequence of odd numbers. As a result, there are also what I would call philosophical reasons to believe that this probability should not be a real number, or an infinitesimal.

And $\Omega$ is not an infinitesimal, nor is it a real number. It is instead an interesting number that has some of the same algebraic properties of the infinite cardinals, but some other unique properties that are not shared by the infinite cardinals. As a result, I think it’s a good candidate, and on balance, this model of probabilities on countable sets is algebraically clean, simple, and intuitive.

All of this suggests a different notion of expectation when reading an infinite string, versus reading a finite string. If you’re reading an infinite string, its structure is already determined, but unknown to you. In some sense, this might seem overly philosophical, but it changes the mathematics. When observing a source governed by ordinary probabilities, the assumption is that, eventually, the distribution of the observations will converge to some stable probability distribution. Perhaps you know what this distribution is beforehand, and perhaps you don’t. But in either case, you’re operating assumption is that within finite time, some distribution will become apparent and roughly stable.

In contrast, when reading an infinite string, the structure of the string is already pre-determined. Moreover, the set of all infinite strings includes strings that have no stable distribution at all, which, therefore, cannot be described by ordinary probabilities. Instead, we can only count the number of instances of each signal within the string, as we have above, and form a different notion of expectation, that is nonetheless useful, but distinct from the notion of expectation used in ordinary probability. Also note that unless we know the distribution of signals within an infinite string ex ante, it would be impossible to have any expectations at all about any observations, other than knowing the set of possible signals (assuming we have that information ex ante).

We can also take the view that infinitesimals are no different than ordinary real numbers. In short, just like the actual number $\pi$ is the result of a countable number of calculations that can be only approximated, and not truly realized on a Turing Machine, similarly, an infinitesimal is a quantity that which, if iteratively added to itself a countable number of times, would produce a real number. The difference being that computable real numbers converge to their ideals, whereas anything short of a countable number of operations on an infinitesimal produces another infinitesimal.

# Vectorized Prediction Algorithm

Following up on my previous post, attached is a fully vectorized version of my original prediction function, which is extremely efficient, and basically instantaneous. Also attached is a command line script that demonstrates how to use it.

CMNDLINE_10-25-19

NM_pred_spot_cluster

# Vectorized Clustering Algorithm

Attached is a completely vectorized, highly efficient version of my original clustering algorithm. Its runtime is drastically shorter, and comparable to my real-time algorithms, though it is based upon my original technique of iterating through different levels of discernment until we find the level that generates the greatest change in the entropy of the categorization. Also attached is a command line script that demonstrates how to apply it to a dataset.

optimize_categories_fast_N

generate_categories_fast_N

test_matrix_cat_accuracy

10-24-CMNDLINE

# Tracking Moving Objects in 3-Space

Attached is a polynomial-time algorithm that can identify, and then track objects in three-space over time given 3-D point information from a sensor, or other input.

Because the algorithm tracks at the point level, rather than at the object level (i.e., it builds its own objects from point data), it can be used to track objects that have irregular, and even diffuse shapes, like smoke, or steam.

Running on a \$200 dollar laptop, it can track 15 Newtonian projectiles at a rate of about 3 frames per second, with an accuracy of about 98% to 100%.

Here’s a plot of one of the sets of projectiles I applied it to:

All of the other functions you need can be found in my full library here.

Projectile_CMNDLINE

# Autonomous Noise Filtering

I’ve updated Prometheus to include autonomous noise filtering.

This means that you can give it data that has dimensions that you’re not certain contribute to the classification, and might instead be noise. This allows Prometheus to take datasets that might currently produce very low accuracy classifications, and autonomously eliminate dimensions until it produces accurate classifications.

It can handle significant amounts of noise:

I’ve given it datasets where 50% of the dimensions were noise, and it was able to uncover the actual dataset within a few minutes.

In short, you can give it garbage, and it will turn into gold, on its own.

Since I’ve proven that it’s basically mathematically impossible to beat nearest neighbor using real-world Euclidean data, and I’ve come up with a vectorized implementation of nearest neighbor, this version of Prometheus uses only nearest neighbor-based methods.

As a result, the speed is insane.

If you don’t use filtering, classifications occur basically instantaneously on a cheap laptop. If you do have some noise, it still takes only a few minutes for a dataset of a few hundred vectors to be processed, even on a cheap laptop.

Attached should be all the code you need to run it, together with a command line script that demonstrates how to use it. But, it’s probably easier to simply download the my full library as a zip file from my researchgate blog.

Enjoy, and if you’re interested in a commercial version, please let me know.

calculate_std_dev

CMND_LINE

express_normalize_dataset

find_NN

find_NN_dataset

generate_categories_N

iterative_filter

iterative_row_filter

matrix2array

no_model_prediction

optimize_categories_N

PrometheusAI_DEMO

spec_log

test_data_magnitude

vector_entropy