Essence: Mathematics & Connection
Introduction
Music, Art, and Mathematics. Is there a connection between them? And if so, why is there a connection in the first place? They have nothing in common. Right? Well, in this blog post, I will write about an abstract structure that appears within these three subjects.
This blog post is based on a book published in 1979 by Douglas Hofstadter titled Gödel, Escher, Bach: An Eternal Golden Braid. In this book, the author describes/defines these abstract structures as strange loops that appear within the lives and works of the mathematician Kurt Godel, the artist Maurits Cornelis Escher, and the musician Johann Sebastian Bach.
This structure, however, isn’t some regular pattern, but rather a description that can help us describe the meaning behind DNA replication, the structure of language, and even the soul of human behavior. Yes, I mean consciousness. So, before I get into describing these strange loops, let’s define it.
A strange loops is a phenomenon [that] occurs whenever, by moving upwards (or downwards) through the levels of some hierarchical system, we unexpectedly find ourselves right back where we started.
-Godel, Escher, Bach: An Eternal Golden Braid
By analogy, imagine climbing a mountain only to end up right where you started. Weird stuff hue? This is precisely what happens in the works of Godel, Escher, and Bach.
Music
Johann Sebastian Bach (1685–1750), mostly known for his famous works in the baroque era, was a German composer and musician. Many consider him the greatest composer of all time. His music, to this day, is highly celebrated and studied.
The following video is a musical piece of a Canon a 2 per Tonus, from Bach’s “Musical Offering.” Take a listen:
By looking at this musical excerpt, notice that Bach tends to change keys after every 8 measures (every 8 square boxes). He goes from C minor, D Minor, E minor, F # minor, G# minor, B flat minor, and finally back to C minor. The fascinating thing is that Bach ties the last two minors so elegantly that you don’t even notice that you’re back in the beginning. These changes in tonality seem as if you are always anticipating the next move in keys. Unknowingly, you hear the same six keys over and over again.
This is one of the strange loops Hofstadter instantly notices in music.
Art
Maurits Cornelis Escher (1898 – 1972) was an amazing painter. He was a Dutch graphic artist who made mathematically-inspired artwork. Here we will take a look at two of his art pieces.
Ascending and Descending, by M. C. Escher (Lithograph, 1960)
In Ascending and Descending Escher displays a group of monks walking forever in an endless loop. If you take a look at the top of the building one group of monks is ascending while the other is descending. The remarkable thing is that Escher was allowed to display the illusion as if the monks keep moving.
Waterfall, October 1961, lithograph
Similarly, take a look at the Waterfall, which is another painting by Escher. In this photo, the water seems to flow uphill but once it reaches the tip of the tower it begins right at the beginning. This process then continues forever in this six-step strange loop. Remarkably, Escher and Bach display these strange loops in two completely different forms: Music and Art.
Mathematics
The more difficult strange loop to notice would be in mathematics. First, let’s take a look at the peculiar loop that appears within our own English language. For example, read the following:
The statement below is false.
The statement above is correct.
Notice that if the first statement is true, then the second statement is false. But the second statement tells us that the first statement is true, but we just said the second statement is false. Then the first statement must be wrong. So, the second statement is true? Quite a confusing loop, right? Surprisingly, this loop continues forever. Try it out!
This situation is displayed within mathematics but in a much more complicated way.
First, let’s understand that in the foundations of mathematics, there is a system of rules that numbers follow. The most obvious ones tend to be called axioms. For example, we all know that 1+2=2+1. This axiom is called commutative law of addition, which might be familiar to you. You may also interpret an axiom as a rule in which we believe how numbers work.
Secondly, you must understand that mathematics largely hinges on the idea of proving its statements. What is the point of math if it doesn’t work, right? Mathematics is mostly proof-based for this reason so that we have a system of laws that are evidently true.
Now, to get to the point, going back the axioms (foundational mathematical laws), a famous mathematician in the 1930’s named Kurt Gödel (1906 – 1978) ended up proving that statements within the foundations of mathematics (near the axioms) do not have a proof. This idea ended up recurring over and over again, to the extent that we have a never-ending loop.
Implication
All these processes in music, art, and math appear to move up in a hierarchy, but then all suddenly, paradoxically, we are back right where we started.
Hofstadter has this argument that suggests that a strange loop is what creates consciousness.
How? Well, to describe it vaguely, he argues that our neurons (individual brain cells) and synapses (reactions) in our brains form the bottom level of hierarchy (C minors). The higher levels (B flat minors) would be our feelings, ideas, hopes, etc. The reason consciousness arises is that this hierarchy can twist back on itself: higher levels can influence the lower levels that determine them. This tangling of regimes gives a sense of a self. Such as how feelings and ideas have a real physical impact.
But the strangeness doesn’t stop there. Hofstadter describes how strange loops are even found in the realm of nature. For example, ant colonies.
Ants individually don’t have thoughts or intelligence, but as a combination, they form an intelligent system. Similar to how our brain behaves. A neuron, personally, is not necessarily capable; however, millions upon billions of neurons may create intelligence. So as weird as it may sound, you may think of a brain as an ant colony, according to Hofstadter.
Similarly, going back to our English language, we don’t read every single letter individually and then put them together into words. You can read a whole word at once, and it can have numerous meanings, despite a single letter having no meaning. This is the same as the subject of mathematics.
Hofstadter even describes these processes from individual strands of DNA to the fundamental particles that make our entire universe. The strange loop theory seems to prevail/show up time after time.
Conclusion
So as strange as these findings might be in first hindsight, it is, of course, a beautiful idea. But if Hofstadter’s explanation of consciousness is correct, can intelligent behavior eventually be programmed into technology? Can the strange loop theory help us obtain artificial intelligence that can be self-creative? Overall, creativity is at the soul of human endeavor.
In the words of Hofstadter,
Without doubt, strange loops involving rules that change themselves, directly or indirectly, are at the core of intelligence.
Can this be the case?