An Introduction to Jazz Music

Introduction

In a previous article, I presented what I believe to be a brief, but nonetheless representative compilation of the classical tradition. In this article, I’ll present an analogous compilation of Jazz music, again focusing on the history. However, because Jazz is in some sense defined by its technical traditions, I will discuss some of the very basic musical concepts and aspects that distinguish Jazz as an art form from the classical tradition.

Duke Ellington

Take the “A” Train (1939)

Like any history, the starting point is not obvious, and is in some sense arbitrary. This is especially the case for Jazz, which is the product of a totally intractable set of facts. I’m not a scholar, but I know enough to say that as a practical matter, 1939 works, and so does Duke Ellington. The brand of Jazz where we’ll end up is the one that I’m most familiar with, which is Bebop Jazz, where a canon of standard songs is studied, practiced, and ultimately improvised over in all twelve keys. This is not that type of Jazz: this is a song, written by Duke Ellington, that I’m sure involves a decent amount of improvisation, but is nonetheless being performed by a large band, that probably sticks to a fairly regimented performance, save for the soloists.

At a high level, you could say that it just doesn’t sound like classical music, which is correct, but we can do better. One of the first things you should notice is that there’s a drummer, as opposed to a percussion section. This is a hallmark of modern music, and pop music wouldn’t exist at all as we know it, without the notion of a drummer, as opposed to a percussion section in an orchestra –

Jazz music took the entire percussion section, and made it one guy’s job.

You can imagine that this was probably the result of economic constraints, where you didn’t have access to timpani drums, but instead had only smaller tom drums, and perhaps a few cymbals, that you then combined into a single, standalone instrument. It’s really difficult to overstate this change in format, because the notion of a drummer ended up being a corner stone of rock music, which was ultimately abstracted into the electronic percussion that defines most modern pop music.

The next observation is that the bass is all pizzicato – i.e., he’s plucking, and not using a bow. This creates a persistent thumping, rhythmic, low-end sound. Classical composers have always made use of pizzicato, but it’s not this persistent, as you’ll note that the bass player doesn’t even seem to have a bow that’s visible on set. The expectations have therefore been reversed: pizzicato is the standard, and the bow is the exception.

Lastly, I’ll note that the orchestra is almost exclusively comprised of brass instruments. You might wonder why this is the case, and I can’t be completely sure, but by parable –

When I was a teenager, I was hanging out at Smalls, which is a Jazz club in New York City, but also a bar, and I accidentally kicked over Wynton Marsalis’ trumpet. Now, Wynton Marsalis is a very successful guy, and his trumpet is super expensive, but he didn’t care. Why? Perhaps it’s because I was a kid, and he knew that it was an accident, but also note that a trumpet is made of brass, unlike a violin, which is made of wood, and therefore probably doesn’t do as well in rough conditions, like the bars and clubs that Jazz originally developed in.

Ella Fitzgerald

Jack Strachey (Composer) and Eric Maschwitz (Lyricist), These Foolish Things (1935)

Though the high energy of early big band Jazz undoubtedly influenced later Bebop Jazz musicians, the big band format is not where things ended up. Instead, the Bebop format ended up closer to the more intimate, small band format that was typical of artists like Ella Fitzgerald and Billie Holiday, where you had a handful of performers, as opposed to a giant brass band. Ella Fitzgerald typifies the charm of this early brand of Jazz, that is almost ready-made for live performance in some Art Deco bar. For some reason, that I honestly don’t understand, other than as a product of history, Bebop artists made heavy use of songs like this, that really were originally intended for a popular audience –

But they completely transformed them into totally bizarre works of art.

The Jazz Standard

John Coltrane, Blue Train (1957)

Bebop Jazz is a unique format of music that arguably doesn’t have much precedent, outside of the obvious, that it’s rooted in the classical, twelve tone system. Though improvisation is not unique to Jazz, and plenty of classical composers improvised, and certainly when entertaining friends, and drinking; and while the origins of Jazz clearly involve a history of bars, clubs, alcohol, and even drugs, Bebop musicians are phenomenally competitive artists, that turned improvisation into a high art form, with an extremely high barrier to any meaningful participation in the genre.

The core concept is to learn a song in all twelve keys, to the point that you can spontaneously create improvised versions of the piece. Transposing a piece of music on guitar can be annoying, but transposing a piece on piano is a serious undertaking, and Bebop Jazz musicians can not only do this in all twelve key signatures, over an enormous canon of music, but they can also improvise while doing so, effectively making up brand new music in any key signature, on the spot, without warning.

This is serious stuff, and these are serious musicians, and many of them are simply astonishing geniuses. I love the actual underlying standard Jazz songs, and think they’re some of the most charming songs I’ve ever heard, irrespective of genre. But Bebop entered an entirely new realm of seriousness, where even though the music is often extremely beautiful, the musicianship is so extreme, that it’s difficult to describe as charming, since the intellectual athleticism required to pull it off borders on disturbing.

Because the focus of Bebop is on improvisation, the specific instance of the song in question is arguably paramount, since the underlying song is now in essence treated as an abstraction –

I.e., it’s a set of chord changes, and a melody, used as a guide.

Bud Powell, performing Anthropology, by Charlie Parker and Dizzie Gillespie (1945)

So when you talk about the performance of a Bebop song, it’s not just the song that you’re referencing, it’s the specific performance of the song that you’re referencing. As a result, the genre inherently places great emphasis on performance, arguably over composition, since these musicians are so brilliant, they can take anything, and turn it into a science project.

Sergei Prokofiev, Cello Sonata in C major, Op. 119 (1949)

Classical composers were certainly interested in Jazz, and Jazz musicians, in particular Jazz pianists, were certainly students of the classical repertoire. Gershwin was arguably a Jazz composer, and Dvořák was known to have been influenced by Jazz music. But you don’t really hear that in Dvořák’s music. In contrast, if you listen to Cello Sonata in C major, Op. 119 by Sergei Prokofiev, it’s obvious that he was heavily influenced by Jazz music. He makes use of the same longwinded, preposterously busy melodies that are exemplified in Anthropology above. And this shows up in a lot of Prokofiev’s work, but this song in particular even includes a blatant use of pizzicato in the third movement that could easily be an excerpt from a Bebop Jazz performance.

Alexander Scriabin, Piano Sonata No. 2, Mov. 1 (“Sonata Fantasy”) (1898)

I know from my personal experiences, that Bebop Jazz pianists were heavily influenced by Scriabin, but I’ve never heard anyone make the connection to Prokofiev, but to me it’s totally obvious, but perhaps that’s because I spent most of my life around cerebral Bebop artists, as opposed to the more stereotypical Jazz you find in movies, and T.V. shows. What is to me most interesting, is that Prokofiev’s Cello Sonata was written in 1949, which is arguably contemporaneous with Bebop Jazz artists making use of these same stylistic techniques in the U.S. The point being, that two very different traditions, in two totally different locations, ended up producing remarkably similar artifacts.

Frank Sinatra

Bart Howard, Fly Me to the Moon (arr. by Quincy Jones) (1964)

As Jazz developed in America, it achieved a significant degree of commercial success, and arguably defined the careers of household names like Frank Sinatra, Sammy Davis Junior, Doris Day, and others, all of whom used the genre to produce legitimate popular hits. “Fly me to the Moon” is a classic, and I’ve been listening to it for years, but it wasn’t until a few months ago that I realized that it was actually arranged by Quincy Jones. It turns out, that in addition to a spectacularly successful career as a pop music producer, Quincy was a fairly prolific composer, and had a simply unbelievable career, including time studying and composing in Paris.

Dorothy Ashby

Essence of Sapphire (1965)

I think of Dorothy Ashby’s music as coming full circle, back to the original song format of early Jazz, but rich with improvisation, and the worldliness and intellectualism of Bebop Jazz. You can clearly hear Ravel in her music, and you can also clearly hear the heavy handed influence of earlier Bebop Jazz artists, particularly in the quality of her performance, which is academic in caliber. The net result is a more approachable product, that anyone can appreciate, that is nonetheless rich with opportunities for musicians to unpack. It’s also just a simply wonderful piece of music, that I just happened to stumble upon, while I was living in Stockholm.

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An Introduction to Classical Music

Introduction

Putting together complitions of this sort is by its nature a bit self-important, but, the reality is, I’ve spent my entire life studying, playing, writing, and recording music. So, I am in fact qualified to put something like this together: I was classically trained at Manhattan School of Music as a child in piano, voice, and composition, and I spent my high school and college years working as an audio engineer in a professional recording studio, studying music informally during those years by simply hanging out with musicians constantly. This included artists from every genre imaginable, from hip-hop producers, to harmonium players. During this period, I also spent a significant amount of time with my uncle, who is a professional jazz guitarist and music professor. He had an enormous impact on not only my guitar playing, but also my ideas on harmony and music generally, introducing me to both jazz guitar playing, and jazz music theory. As a result, though my overall background is decidedly non-traditional, I grew up listening to almost nothing other than classical music, and I’ve spent a significant portion of my free time studying the genre ever since.

What I’ve put together below is a collection of music that I think will allow you to think about classical music, and music generally, by providing a context in which you can evaluate the next piece of music you listen to. The commentary is generally limited to historical information, with no real technical analysis. The point is to present a body of works that will acclimate you to the overall classical tradition, and its development over the last three centuries, and introduce you to a few of the main characters in this incredible story.

Johann Sebastian Bach

Cello Suite No. 1 in G Major (1717 to 1723)

Bach’s Cello Suites are the cliché introduction to classical music, but sometimes there’s a reason things are popular: Paris is actually beautiful, and so is Rome, and as a result, a lot of people go to these cities. Similarly, Bach’s Cello Suites are amazing, because he’s making use of a single instrument, that nonetheless fills the space, with nothing to hide or dress up –

It’s just perfect music.

Harpsichord Concerto in D minor (1738)

Though the title clearly indicates that this piece was originally written for harpsichord, it is my experience that this piece is almost always played on piano. Though the piano was already in existence at the time Bach composed the piece, as an instrument, the piano was a relatively new invention, and like many new inventions, it took time for the piano to gain in popularity. Presumably, the reason that many of Bach’s pieces that were originally written for harpsichord are now played on Piano, is the present day popularity of the piano, and the relative obscurity of the harpsichord. This transition from harpsichord to piano shows that despite having written records of the plain instructions of composers, the history of music is subject to what are in effect trends in popularity.

Wolfgang Amadeus Mozart

Symphony No. 5, Mov. 1 (1765)

Mozart was only nine years old when he wrote this piece, and you’ll note from the title that it was his fifth symphony. He wrote his first symphony at the age of eight, and so in the time span of about a year, despite being a small child, he wrote five symphonies. As a general matter, his pace of production was simply astonishing, composing what would likely be a mountain’s worth of music if it were printed and stored in one place.

Rondo in D Major (1786)

Pianist Vladimir Horowitz was a rock star in his time, and would draw simply enormous crowds wherever he played. You can see his remarkable confidence in this performance, as he casually tests the piano in front of a packed house, and simply begins playing, with no announcements, or other introductions whatsoever.

Ludwig van Beethoven

Piano Sonata No. 14 (“The Moonlight Sonata”) (1801)

I watched a separate video of pianist Daniel Barenboim discussing this piece, and he seemed unsure of Beethoven’s motivation for writing it. I’m not Daniel Barenboim, but it seems likely to me that this was a romantic offering from Beethoven to a woman named Giulietta Guicciardi, given the rather glowing terms he used to describe her role in his life:

“… a sweet, enchanting girl, who loves me and whom I love. After two years, I am again enjoying some moments of bliss, and it is the first time that – I feel that marriage could make me happy, but unfortunately she is not of my station – and now – I certainly could not marry now.”

Moreover, the piece is explicitly dedicated to her:

Beethoven_Piano_Sonata_14_-_title_page_1802

So, it sounds to me like a bittersweet song, offering affection, together with a lamentation over a relationship that never got to run its full course.

Johannes Brahms

Six Pieces for Piano, Op. 118 – 2 (1893)

Though I hate playing the, “who’s your favorite composer?” game, if I had to pick, it would be Brahms. The balance between ferocity and delicateness in his music is mentally exhausting, forcing even the most competitive people to concede that he is without question the superior composer, because you’re confronted with someone who is so overbearingly intense, and then suddenly, and inexplicably gentile. His strategy seems to be one of chaos, where you’re forced to simply trust the music, because you simply cannot anticipate what is going to happen.

Violin Sonata No. 1 (1879)

My understanding of the history of this piece* is that Brahms composed it around the time that Felix Schumann died. Felix was the child of his close friends, and fellow composers, Robert and Clara Schumann. Though I’m not aware of any formal dedication of the piece, Brahms was aware of Felix’s poor health, and Clara was aware of this piece of music, as is made clear in their correspondence. Brahms is known to have had a deep affection for Clara, possibly romantic in nature, so it is not inconceivable that he wrote this to alleviate Clara’s suffering, having to watch her son die.

My personal affection for this piece of music cannot be exaggerated, in particular, for the second movement –

I discovered it under thankfully happier conditions, right around the time that I met a woman that I fell madly in love with, and ended up writing an arrangement of the second movement for her.

*The following link takes you directly to Violin Sonata No. 1.

Maurice Ravel

String Quartet No. 1, Mov. 1 (1903)

This piece in my opinion marks the beginning of modern music –

Whether or not you know anything about classical music, you can tell that it just doesn’t sound the same as the other pieces above. Music theorists can undoubtedly tell you a lot more than I can about the evolution of harmony, and there’s no doubt that these kinds of intellectual developments matter. But there’s also a practical side to the development of the arts, and in this case, Ravel was composing in a cosmopolitan society that had moved away from an economy dominated by the Catholic Church. As a result, he managed to find an economic and political freedom to compose that Mozart and Bach never knew, simply because they were subject to the whims of either the Church, or royalty.

So while I don’t intend to dismiss the importance of intellectual traditions, there is also the practical, economic and political context in which people live, and because artists are people, these factors can also completely change the things they create, simply because artists, like everyone else, can be either constrained, or liberated by the contexts in which they live.

Claude Debussy

Suite Bergamasque, Mov. 3 (“Clair de Lune”) (1905)

As I mentioned above, I left formal classical training during my teenage years, and started working at a recording studio. Though I didn’t deliberately leave classical music behind me, the new sounds I was exposed to occupied a significant portion of my listening time, and as a result, as a practical matter, I spent much less time listening to classical music. However, I was also experimenting with home recording at the time, through a ridiculous set up that I had concocted using a computer that my dad had recently bought for me, which featured some horrible preamp from 1960, and a guitar cabinet that I had rebuilt into monitors. While poking around the audio files on my computer, I found a midi file entitled, “Clair de Lune”. I had no idea what this was, but I clicked on it out of curiosity, since I knew it was a music file, and though the sound libraries were positively awful, it didn’t matter –

The music was just astonishing, I’d never heard anything like it.

Sergei Rachmaninoff

Rhapsody on a Theme of Paganini (1934)

As a general matter, Rachmaninoff’s music is sweeping, cinematic, and full of life, with thunderous, impossible-to-play piano parts, and soaring orchestration, that all comes together to make you feel as though your life has been subsumed into some immaculately produced, Art Deco era movie. Though this piece was not, to my knowledge, written for any particular film, Variation No. 18* is a piece made for film, and is in fact featured in a film, “Somewhere in Time“.

*The following link takes you directly to Variation No. 18.

 

On Distributions of Complexities

Last night, I came to the rather obvious, but in my opinion fascinating conclusion, that there are more binary N \times N matrices than there are graphs with N vertices. Stated differently, not all binary matrices correspond to graphs. This is because of the way graphs are generally represented as matrices, where entry (i,j) is 1 only if vertex i is adjacent to vertex j. In an ordinary, undirected graph, a vertex is never adjacent to itself, and because the edges are undirected, if vertex i is adjacent to vertex j, then vertex j is adjacent to vertex i. As a result, if a given matrix represents a graph, then it must be the case that entry (i,i) = 0, since a vertex is never adjacent to itself, and entry (i,j) = (j,i), since the edges are undirected. Therefore, the set of N \times N matrices that correspond to graphs is smaller than the set of all N \times N matrices.

In fact, you can easily show that the portion of N \times N matrices that correspond to graphs goes to zero as N increases. There are a maximum of N choose 2 edges in a graph of order N. Therefore, the number of labelled graphs on N vertices is 2^{N \choose 2} = 2^{\frac{N(N-1)}{2}} <  A = 2^{\frac{N^2}{2}}. In contrast, the number of N \times N matrices is B = 2^{N^2}. The ratio \frac{A}{B} = 2^{- \frac{N^2}{2}}, which approaches zero as N approaches infinity. Since you can trivially construct a binary string from a binary matrix by appending row i + 1 to the end of row i, it follows that the portion of binary strings of length N that correspond to graphs approaches zero as N approaches infinity.

As a result, graphs, as a class of mathematical objects, are fewer in number than binary strings for any given order.

This might seem like it’s just some high school combinatorics, but it points to something profound, if you consider this fact in the context of complexity. Specifically, you can show that most strings of a given length are random in the sense that their Kolmogorov complexity is approximately equal to their length. Stated mathematically, if x is a string, and x is Kolmogorov-random, then K(x) \approx ||x||. This implies that compressible strings are actually rare, despite the fact that most of what we see in the world is compressible. That is, the everyday objects you see have significant symmetries, which necessarily implies that they are not Kolmogorov-random, using any sensible representation of the object. For example, the laptop I’m using to write this article is symmetrical both horizontally and vertically, down to reasonable levels of measurement. As a result, I can reconstruct a reasonable image of the entire laptop given only a fraction of an image of the laptop.

This is also true of physics itself at our scale of observation, in that the outputs of Newton’s equations are never Kolmogorov-random, since they are by definition computable. That is, because Newtonian physics is written in the language of computable functions, it is necessarily the case that the behavior of Newtonian objects is compressible. For example, if you track the path of a baseball thrown across home plate, you will produce a curve that is Newtonian, and therefore, of a very low complexity, when constructed using any sensible scale of observation. It is, therefore, at least anecdotally fair to conclude that the distribution of complexities of everyday objects differs significantly from the distribution of complexities of binary strings. Specifically, most binary strings are Kolmogorov-random, whereas most everyday objects are not.

This suggests a purely mathematical question about the distribution of complexities over sets of mathematical objects. Specifically, in this case, what is the distribution of the complexities over the set of graphs of a given order?

Intuitively, it seems like graphs should have a distribution of complexities that is similar to the distribution of complexities of binary strings. Let’s imagine a binary string s = a_1, a_2, \ldots, a_N. If each a_i = 0, then the string will have a low complexity. Similarly, if each a_j = 1, then the string will have a low complexity. In fact, the complexity of the compliment of any given string can differ by at most a constant from the original string, since we can write a simple program that can take the compliment of any given string, the complexity of which will not depend upon the string in question. Therefore, the average complexity over all strings with a given number of bits set to one is symmetrical as a function of the number of bits that are set to one.

Now let’s consider graphs using analogous reasoning, and consider a graph as a set of N \choose 2 edges, each of which is either included or not. If we begin with an empty graph, intuitively, this graph should have a low complexity, since it contains no edges. This is supported by the associated graph matrix, which will consist of nothing but zeros. Similarly, if we consider a complete graph, which will include all possible edges, this graph should again have a low complexity, since it has a simple rule of construction: connect every pair of vertices. This is again supported by the associated graph matrix, which will consist of nothing but ones (except along the diagonal). Again, the complexity of a graph and its compliment can differ by at most a constant, since we can write a simple program that calculates the complement of any graph, that will not depend upon the graph in question. Therefore, the average complexity for a graph with a given number of edges will again be symmetrical as a function of the number of edges. Note that we are ignoring yet another detail, which is that we will need a separate program that takes a matrix, and actually displays a graph. But again, you can write a rudimentary program that does exactly this, the complexity of which will not depend upon the graph in question.

As a result, we have established that the distribution of complexities over the set of binary strings necessarily has some structure in common with the distribution of complexities over the set of graphs. At the same time, the portion of strings that correspond to graph matrices approaches zero as the length of the string approaches infinity. This suggests that the set of binary strings contains a subset that is, in terms of the distribution of its complexities, similar to itself.

Copyright Policy Reminder

Just a reminder, though I’ve shared the code for my algorithms, they remain subject to my Copyright Policy.

Also, note that I’ve now officially registered all of my algorithms with the U.S. Copyright Office, and if you’re not familiar with U.S. Copyright law, it’s actually fairly protective of copyright holders.

So, while I’m happy to encourage academic use of my software, I will be extremely disappointed if I find out it’s being used for any purpose that violates my super generous Copyright Policy.

A Mathematical Theory of Creativity

I was listening to In Utero by Nirvana last night, and I was reminded by how great they were as a band. That album in particular has a really unique, and unusual sound, and is harmonically quite strange. I was also reminded of the fact that Cobain is certainly not the best guitarist in the world. Nonetheless, his songs are legitimately interesting. You could say that this is the result of working well with what you have, and in some sense, this must be true, because this is exactly what he’s doing. It turns out that we can think more rigorously about this distinction between creative brain power, and technical competence, using computer theory.

For example, if I give you a set of notes that are all in the same key, it’s trivial to construct a melody that fits them, and machines can certainly do this. If, however, I give you a set of notes that are not in the same key, then constructing a listenable melody using those notes is non-trivial, and I’m not sure that a machine can do this. This is exactly what Nirvana’s music does, which is to take a set of disparate key signatures, and connect them with listenable melodies that your average listener probably wouldn’t second guess at all. That is, their music generally sounds normal, but harmonically, it’s actually quite interesting. The same is certainly true of Chris Cornell‘s music as well, in that he also makes use of what is effectively multi-tonal music, in a manner that isn’t obvious in the absence of conscious reflection.

Creativity in the Arts

Taking it to the next level, we can think of the rules of art as constraints:

You’ve got a set of what are effectively rules of harmony, a set of expectations in your audience, which impose some more rules, and a space you populate with notes. So any final product can be viewed as a collection of notes arranged in that space, which either satisfy the rules, or don’t. What artists like Cobain, Erik Satie, and others do is to create simple, but non-obvious solutions to the problems of art. In their case, the real work is not necessarily in the performance of the piece, but is instead in the contemplation beforehand that produced the piece.

But this type of brain power is certainly not unique to art, since we can imagine mathematical constraints that together pose a difficult problem, that nonetheless has a low complexity solution. That is, the solution to a computationally hard problem could itself have a low Kolmogorov complexity. There are real problems like this, such as Diophantine equations, which as a class are non-computable. We can easily state the answer to a Diophantine equation, since it’s just a set of integers. We can also quickly confirm that it is in fact the correct solution, since that requires only basic arithmetic. As a result, the solution to any Diophantine equation will have a trivial Kolmogorov complexity, and confirming its accuracy is also computationally trivial. Nonetheless, because Diophantine equations are non-computable as a class, any given problem could require infinite time to solve, or, a non-Turing Equivalent machine.

So, we know these types of problems exist in mathematics, where the complexity of the solution to a problem is low, but the computational power required to find that solution is extremely high, and possibly infinite (at least on a UTM). My opinion is that great artists are capable of exactly this type of thinking, where a superficially impossible aesthetic problem is solved by a particular piece. I think Prokofiev takes this approach to the extreme, producing works of art that are not only non-obvious, but also technically complex, in that you really have to be a technically competent musician to play one of his pieces. As a general matter, I would say that Prokofiev’s music solves complex aesthetic problems in non-obvious ways, using complex solutions. In contrast, I would say that Cobain and Satie solve complex aesthetic problems, using simple solutions.

Creativity in Mathematics

Nature itself imposes constraints that distinguish between true and false claims about mathematics. This is an abstract way of thinking, but it is necessarily the case. Just consider a simple example: “all even numbers are divisible by two”, versus, “all odd numbers are divisible by two”. If you have a collection of objects that is even in number, you will find that it can in fact be divided into two equally sized collections of objects. In contrast, if you have a collection of objects that is odd in number, you will find that it cannot be divided into two equally sized collections of objects. This seems trivial, but it’s actually quite profound: it shows that theorems of numbers operate as primordial rules of our reality, that are arguably even more fundamental than the laws of physics, since there is absolutely no possibility that they will ever be refined.

As a result, we can think of mathematicians as solving problems imposed by the most fundamental constraints of reality itself, in turn generating claims about mathematics that are demonstrably true in our external world. The most interesting theorems of mathematics are the ones that seem to follow only from these primordial rules, where you’re forced to wonder how it is that the mathematician in question ever conjured such a result. In contrast, the least interesting theorems of mathematics are those that can be derived mechanistically from known theorems. For example, if I tell you that A is false, you know mechanistically that (not A) is true, but that’s certainly not interesting.

Stated differently, if you begin with a known result, and proceed mechanistically by evaluating all possible consequences of that result, whatever inferences you arrive at, are, frankly, boring, because it doesn’t require any creativity to do that – a Turing Machine can do that, and much faster than a human being can. In contrast, if someone identifies a new and correct theorem of mathematics, that arguably doesn’t follow from any known result, you’ll be surprised and amazed, precisely because it’s not clear how people do these things. In fact, even if you yourself do this all the time, it’s still surprising when someone else does it.

A few examples of theorems that I think are of this type are Kuratowski’s Theorem, Ramsey Theory generally, and of course, the Graph Minor Theorem. Taking Kuratowski’s Theorem as an example, imagine a machine trying to solve this problem, even after it’s already been stated. A truly creative mathematician will begin by asking interesting questions in the first place, thereby stating interesting problems, and it’s not clear that even this initial step is computable. For example, in the case of Kuratowski’s Theorem, why would there necessarily be a finite set of forbidden sub graphs? You’d have to first state this question, in order to prove that it is the case. But even if we skip this step, it’s just not realistic to assume that a Turing Machine could winnow down the set of forbidden sub-graphs to a finite set, which is exactly what Kuratowski’s theorem achieves. In particular, determining whether two graphs are isomorphic is a hard problem, and that’s just a small part of what’s necessary to solve a problem like Kuratowski’s theorem, as a practical matter.

A Mathematical Theory of Creativity

All of this allows us to distill precisely what makes great mathematicians, and great artists, so impressive, in mathematical terms:

They begin with an enormous, or possibly infinite, number of possibilities, and somehow present you with a correct one. This suggests that true creativity is the power to solve computationally hard problems, and possibly, non-computable problems.

I say this all the time, but it’s pretty obvious, to me at least, that some human beings are simply not equivalent to a Turing Machine, but are instead, categorically superior. (Note that a reasonably intelligent human being is always at least as powerful as a Turing Machine). I think a good place to look for this kind of brain power is in musicians and architects, since both are forced to solve complex technical constraints in a manner that is more than just functional, since they must also satisfy aesthetic criteria. Obviously, mathematicians and scientists would be a great place to look for this type of brain power as well, but regrettably, I get the sense that this bloodline has been massacred.

Which makes you wonder –

Why is it that so many of our finest musicians seem to end up dying long before their time?

Perhaps I’m not the only person aware of the connections between creativity in the arts and in mathematics, and maybe others don’t like the economic and military risks posed by too many creative people, giving them an incentive to ensure that we don’t live long enough to have too many children.

Using Fiberoptic Cable for Lighting

Fiberoptic cable is widely used in telecommunications, but it can also be used to transmit light for purposes of generating luminosity in a space. And it dawned on me earlier today, that you can use a centralized lighting source for a space, and then transmit that light using fiberoptic cable to different locations in the space. So for example, if you have a few rooms, rather than pump electricity through wiring, you pump light from a central source into the rooms using fiberoptic cable. You can still have bulbs for aesthetics, but the luminosity is generated elsewhere, in a centralized place. To simulate current lighting, you could have a switch in a room that causes the centralized source to increase in luminosity, and cause the related fiberoptic cable to open up, allowing light into the space in question.

The advantages seem significant, because there’s no resistance to light moving through fiberoptic cable, and therefore, no ambient heat. This should save money and electricity. The same is true of the generation of luminescence itself, which is now centralized, meaning you don’t have piping hot bulbs hanging in the space itself, and instead have a concentrated heat source that you can manage, by either placing it outdoors (like a centralized A.C. unit), or otherwise ventilating it. You could be super efficient, and pipe the heat generated back into the space during the winters. You could also supplement the luminosity generated by the unit with natural sunlight during the day, which will change its color, but because you have total freedom with the delivery of the light into the space, you can probably correct for that. Also, natural sunlight looks nice, and it would probably be astonishing to see natural light literally piped into a space.

I thought of this initially as an artistic device, since it would allow for light to be placed anywhere in a space, since fiberoptic cable can be made quite thin. This means that even though you can use bulbs, you don’t need to, and can instead get really creative, using, e.g., treated papers, plastics, metals, etc., since the light is already generated, and now all you’re doing is treating it, and placing it somewhere in a space. This would provide interior designers with real freedom to use light like any other component in a space, placing it at will in any location, with essentially any texture, shape, or color.

On the Implausibility that the Human Mind is Turing Equivalent

Regrettably, I have much better things to do than write this article, like try and get funding for my A.I. venture, but upon reflection this morning while exercising, I realized that the idea of the human mind being Turing Equivalent is comically implausible. In fact, I was instantly offended once I understood why, because people really take these incredibly stupid ideas seriously, and this is often how people justify the awful manner in which they treat each other.

Here’s the mathematical argument:

Can a program produce a copy of itself, if it does not have access to its own code?

This is distinct from a program that produces a copy of itself, because it contains print instructions that end up reproducing itself. What I am instead asking is whether there is a program that can reproduce itself, or its mathematical equivalent, when the original code for the program does not contain any descriptions of itself?

So for example, a program that generates all possible programs up to some certain length would not qualify as a solution, since in order for this program to terminate, it would have to know its own length, which is forbidden by the problem.

Now assume that the program somehow did reproduce itself. How would it test this? It can’t: since it can’t read its own code, there is no way for it to know for sure, and it can only test some finite number of inputs, and develop some measure of belief that it has in fact succeeded.

Therefore, it’s not possible to write such a program, since the program can test only a finite number of inputs.

Now consider a program that can tell you how long its own code is, without access to its own code. This is also impossible, because by definition, the best you can do is to write a program that simulates the underlying program for some finite number of inputs. But there could be multiple programs that produce the same outputs for all inputs. Therefore, even if you produce a correct mathematical equivalent of your underlying program, you can’t be sure it’s the same length as the underlying program.

It must be the case that any sufficiently intelligent person is at least as powerful as a Turing Machine, since I can teach any such person a Turing Complete language, and then that person can perform any algorithm written in that language, given enough time. In summary, it must be the case that sufficiently intelligent, literate people are at least as powerful as a Turing Machine.

Now assume that the mind of Alan Turing is Turing Equivalent. That is, Alan Turing was a Turing Machine. Alan Turing produced a paper that describes the Turing Machine. This is exactly the scenario described above, where a program (i.e., Alan Turing) wrote another program that is mathematically equivalent to itself (i.e., his description of the Turing Machine) without having access to his own code (i.e., Alan Turing had no idea how his brain actually operates).

The epistemology here is a bit subtle, in that the proof above tells us only that there is no way for a program to know that it has produced another program that is equivalent to itself, if it doesn’t have access to its own code. However, in the case of a Turing Equivalent program, you do have information about the program, even if you don’t know its implementation, since it is by definition Turing Equivalent. As a result, it is in this case at least possible for a UTM to test whether or not a given program is Turing Equivalent, since this can be established by deduction, which can be mechanized.

But Alan Turing didn’t know what Turing Equivalent was until he described the Turing Machine. So we’re back where we started, which is that Alan Turing had no idea, and no way to test, whether or not what he produced was in fact equivalent to himself. And neither does anyone else, absent experimentation.

Again, human beings are necessarily just as powerful as Turing Machines, for the reasons described above, but there’s plenty of experimental evidence suggesting that Turing Machines will never produce the types of artifacts that some human beings produce constantly, in particular, novel mathematics. This doesn’t imply the non-existence of other types of machines beyond a UTM that might be able to do these things, but the point is, human beings can do all of the things that a UTM can do, but it seems quite clear that a UTM cannot do many of the things that some people can do.

This suggests the obvious: some people are more powerful than UTMs.

The contrary view assumes that Alan Turing just happened to produce a copy of his own code, even though this is not a testable hypothesis. The contrary view assumes that Alan Turing just happened to accidentally reproduce all of that which describes his mind, without having access to that information, and despite any means by which this hypothesis can be tested.

This suggests the obvious as well: some people find it morally convenient to assume that we’re machines, and would rather not consider the issue any further.

Using A.I. to Generate Hypotheses on Compressed Representations of Physical Systems

As noted, I’ve now turned my attention to thermodynamics, but rather than simply describe the physics alone, I’ve decided instead to supplement my work with software that actually implements the models of physics I’m describing. This software is in turn rooted in my work in A.I., which together, allow for sophisticated scientific computing to be executed on cheap consumer devices. In this note, I’ll discuss an observation that I made in my original paper on physics about the nature of apparently random behavior.

Sequences of States

In, A Computational Model of Time-Dilation [1], I presented a model of physics rooted in information theory that implies the correct equations of physics, while nonetheless making use of objective time. The net result is a simple, and ultimately correct model of physics that is, even for scientific purposes, indistinguishable from relativity. It has the added benefit of being entirely quantized, which makes it ideal for scientific computing. In the final section of [1], I introduced some tangential ideas that I didn’t want to leave out, but didn’t fully unpack, since they were not necessary to the main result of the paper, which is a mechanical model of time-dilation without space-time. Specifically, I noted that what seems to be random behavior, could nonetheless be generated by deterministic rules, politely suggesting that perhaps the notion of physical randomness is not really well-described (see Section 6.4 of [1]).

The intuition is as follows: just imagine a simple Newtonian projectile that we’ve observed N times along its path from launch till landing. Because of Newton, we can write down an equation that will produce a curve that has what is, for all practical purposes, a one-to-one correspondence to the measurements of its positions as a function of time, provided that we know its mass and initial velocity. Therefore, we can define a function that takes the initial conditions of the projectile as its inputs, and generates each point along the path as its output. Expressed symbolically, the function P(t) = F(m,v_0,t) would generate the position of the projectile at time t, given its mass, initial velocity, and the time in question. This is how you normally do physics, which is to take a set of initial conditions, and solve for some future state of the system, applying a rule to those initial conditions.

Now imagine instead, that we simply created a dictionary of the observations of the projectile over time, where the first entry of the dictionary contains the first observed position of the projectile, the second entry contains the second observed position, and so on. Note that if we take this approach, then we don’t need a rule of physics, because we can instead simply describe the path of the projectile using the indices of the dictionary in order. That is, the path of the projectile can be described by the sequence of integers (1, 2, 3, \ldots, N), since these integers correspond to the entries of our dictionary, that in turn contain the positions of the projectile over the course of its path. Expressed symbolically, P(i) = dictionary(i), which expressed in plain English, means that the i-th position of the projectile is in entry i of our dictionary.

If we’re modeling a simple Newtonian projectile, then this is probably a pointless exercise, because we are by definition simply repeating our observations. That is, this approach simply takes a set of observations over time, and indexes them in that same order. But, if we’re modeling a complex system that either doesn’t have a closed-form equation, or is too complex to allow for one to be discovered, then this approach could be useful.

Specifically, we can use the software that I introduced in a previous article, to compress the observed states of a system into some small number of what are really macro-states. And then, going forward, we can describe the current state of the system using some small number of macro-states that we can index in a dictionary. This will allow us to say, as a practical matter, that, “the system is in some category of states that looks like this”. Moreover, it could allow for patterns to be discovered in the behavior of the system that might be obfuscated at the micro-state level. That is, a system might have some enormous number of micro-states, and so any sequence indexing those states might appear chaotic, whereas sequencing the macro-states might yield periodic, or otherwise regular behavior, that will be easier to identify, simply because we’re now considering a sequence over a much smaller set of integers than would be required to index the micro-states of the system.

So in summary, by compressing the micro-states of a system into some manageable number of macro-states, we can take an intractable set of observations, and reduce them to a tractable set of representations. This in turn, allows us to test hypotheses over a manageable set of macro-state representations. Even if this is still too complex for a human being to make sense of, it might nonetheless be possible for machine learning techniques to be applied to observed sequences of macro-states, which could allow for the discovery of predictable macroscopic behavior, in what is an otherwise, superficially chaotic system.

Expressed in simple terms, we use A.I. to first compress initial observations of a system into some tractable set of macroscopic representations of the system. We then make a subsequent set of observations, and again use A.I. to search for useful hypotheses regarding the macroscopic behavior of the system in those observations, that can then be tested experimentally. We can automate the hypothesis testing by using the model of error I introduced in a previous article, which will allow a machine to dismiss a hypothesis as simply incorrect, allowing the machine to focus on refining other hypotheses that are merely imprecise.

Thomas Jefferson on the Utility of Mysticism

At this point, I’m sure regular readers of this blog realize that I’m a pretty unusual fellow, given that I routinely publish industrial quality A.I. software, and have casually rewritten essentially all of known physics, while suggesting the possibility that pop stars are deliberately dropping references to my life in their works. I’m obviously also an extremely cynical person, with years of experience as a Wall Street attorney, which has made me a brutal realist, that is regrettably familiar with the shortcomings of this life, and the pointless inadequacies of our species. So, common sense suggests the truth, which is that I do these things on purpose, to create an even more ridiculous persona, that is layered atop what is already a decidedly ridiculous human being. And this cuts to the utility of mythology, which is that it can be used as a tool to garner attention, and take an already remarkable narrative, and elevate it to symbolic heights. I know this works, because I spend a lot of time thinking about how human beings think.

I have the liberty of living in the United States of America, which I owe to men like Thomas Jefferson, who risked their lives, their fortunes, and their reputations, so that I could conduct this research, that I know will one day completely change the course of human history. And I make a point of being unashamed about my confidence, because to do so is in my opinion to be plainly honest about what I believe, and unapologetic in my honesty. To do so is decidedly American – to dispense with the nonsense and insecurities of others, and to instead have the right say what you think, at all times, and to have that right secured by the power of our Government.

Censorship by the state is rooted in dishonesty, and delusion, by insecure men, for whom the truth brings only pangs of inferiority, and not the joyful echo of what is natural, and plainly the case. For me, the truth is, at a minimum, always useful, since it allows you to identify what is unfortunate, and resolve to change it. Other cultures have decided that it is instead easier to pretend, and to ride a donkey, while pretending you are atop a majestical horse. This way of thinking is not American, and we have to resist the temptation to allow our insecurities to unwind the legal protections against censorship embodied in our Constitution, that have led to a sea change in human knowledge, competence, and happiness. The practical reality, is that the slow push to censorship is the design of lesser cultures attempting to stifle American intellectualism, because they have no hope of competing with us on the merits – it is simply a joke of a contest.

In its brief history, the United States has completely rewritten the human landscape, inventing basically everything that we take for granted today, including recorded audio, electricity, electric light, atomic power, the internet, space travel, and scores of other incremental achievements in understanding that bridged the gaps between these enormous leaps. The United States is also a cultural powerhouse, having invented entirely new genres of art, such as Jazz, that have broken the shackles of the classical tradition, while nonetheless pioneering arguably even greater technical competency, as Jazz pianists routinely play at the same level as a classical concert hall pianist, except they’re improvising the entire time, which is just astonishing.

No other civilization has accomplished this much in such a short amount of time, and you could make the case that this is an absolute – i.e., that no civilization has ever accomplished this much, period.

The reason for this outsized performance gap is obviously due in part to luck, since we started out with some truly extraordinary people, like Jefferson, that were proper geniuses, but also statesmen, and warriors: this is not a typical combination of skill sets, and these men obviously gave the United States a lot of runway to work with. But our success is also rooted in their philosophy, which valued personal liberty, and freedom of speech in particular. They understood how information works, and that censorship could be used as a tool to completely ruin a society, and so they rightly made it quite plain that this is a country that will not tolerate censorship, period.

In a letter to Bishop James Madison, sent two-hundred and twenty years ago, on January 31, 1800, Thomas Jefferson discussed the writings of Johann Adam Weishaupt, who is generally known as the founder of the Illuminati. Ordinarily, if you talk about the Illuminati, people, perhaps fairly, assume that you’re insane, or that at a minimum, you subscribe to unrealistic ideas about how the world works. But the truth is, these organizations were, and still are, very real, because they make sense in an environment where you’re subject to censorship.

This is plainly the conclusion that Jefferson reaches about Weishaupt, saying that had Weishaupt lived in America, rather than Germany, then he would not have resorted to mysticism and the occult, because he would instead have been free to say what he truly believes. The implication is clear, in that Jefferson knows that Weishaupt is not a mystic, but is instead using mysticism as a means of encryption, using a private language that only his followers will understand, because he is operating in an environment where he is not free to say what he truly thinks.

The same is probably true of the alchemical writings of people like Newton, who were obviously not mystics. It is instead far more likely that Newton had ideas that he knew to be of economic and military value, that he did not want to share with the world, and so perhaps instead, he used a language that was known only to people with whom he felt comfortable sharing that knowledge. There are obviously plenty of strange people in the world who believe in ridiculous things, but this does not imply that mysticism is always the product of an occult belief. Rather, in the case of someone like Newton, it’s probably indicative of encryption.

There is nothing you can do about people using secret languages to communicate, if they choose to do so. But, what we can control for is whether or not we respect an individual’s rights under the Constitution to free speech. If we don’t, we will create experts in secret communication, and we will send humanity right back into the dark ages, where our most valuable ideas remain idle in the crypts, rather than being allowed to blossom in an open market place.

Technology has created the opportunity for unprecedented connectivity, allowing ideas to cross enormous distances in a single click. This is, oddly enough, also a subject of Jefferson’s letter to Bishop Madison, in which Jefferson describes some difficulty in sending packages, concluding that the best means is, “some captain of a vessel going round to Richmond”. In contrast, we live in an incredible time, where information can be exchanged, essentially instantaneously, with no cost of transit. This is an astonishing achievement, again, brought to you by the United States.

But, technology also allows foreign governments, and malicious domestic groups, to censor individuals expressing opinions, or sharing information, that might be politically or commercially threatening, which is precisely what I do every day, all day. And I rely upon the protections provided by the Constitution, that entitle me to conduct research within the bounds of the law, to share the fruits of that work, and to claim property in that work through our intellectual property laws. Technology can be used to assist in this process, which it certainly has, in that I can quickly and easily produce professional quality documents, test software, and distribute both. But, the openness and accessibility facilitated by technology also creates a means by which it can be used to harm and suppress people.

We need to come together as a society and address these risks, because they are dangerous to not only the market place of ideas, but also to our financial markets, our personal safety, and the very electoral machinery that allows our democracy to function. We are instead being distracted by people that clearly do not have the United States’ best interests in mind, but would rather pretend, and continue to behave as if these risks don’t exist, so that they can slowly erode the intellectual dialogue that in large part powers this country’s economy, and allows our citizens to vote on an informed, and reasonable basis.

A Unified Model of Physics

I’m now turning my attention to describing thermodynamics in the language of information theory, making use of a mix between academic work, and commercial software, that would allow for fairly sophisticated scientific computing to be conducted on rather cheap machines.

This work reminded me that I have produced what is arguably a novel, and unified model of physics that is consistent with the enormous body of experiments testing relativity, but nonetheless makes use of objective time. As a result, it’s a much simpler model of physics than relativity, and because it’s rooted in information theory, and entirely quantized, it is built for scientific computing. Because it is, to my knowledge, consistent with all of the experiments testing relativity, the model I’ve presented is either correct, or as a practical matter, a more sensible alternative to relativity, since it produces results that are, even for scientific purposes, indistinguishable from relativity.

Here is the main paper on time-dilation, and here are the follow up papers that address gravity, charge, magnetism, wave interference, and color charge. Together, these papers present a single model of physics that allows for essentially all of the behavior described by relativity and quantum mechanics, in a unified conceptual framework that naturally lends itself to scientific computing.