Samantha Fund

Limits of the System

Oftentimes accessing difficult domains even as a causal bystander yield benefits that go well beyond a full comprehension or command of the specific subject.

Accessing those systems enables to shoot for something higher than the system itself, and in this respect the system acts almost instrumentally.
Sometimes it’s the system itself that contains the seeds to enable it to grow well beyond itself. Into a meta-system able to actually define the rules for the system that it was generated from and just surpassed by.

Formal system – as shown by Kurt Gödel – contains truths it cannot prove.
You can also hear this limitation.
In Die Kunst der Fuge, Johann Sebastian Bach builds a system that seems complete, self-contained, perfectly ordered.
And yet, as the music unfolds, it begins to point beyond itself—toward something it cannot fully resolve.
The fugue is one such example that does not just express music.
It exposes the limits of the systems.
Transcending the limits of such systems is – as I will show in this brief article – one way we can intuitively inference and justify the emergence of conscience.

The Persistence of the Amateur

Music and math have a lot in common.
Music is instinctual; almost all human tribes and races have developed it in one form or another.
Yet there is a more profound aspect of music, once you skim past the surface and delve down into its inner workings.

As a musical practitioner, I have always enjoyed playing an instrument, although I do not sport the least inclination towards any degree of musical talent.

One reason I persist is that I read somewhere that music stimulates an area of the brain that is inactive in all other humans. Apparently, your neural cells somehow rewire, and you get to activate a portion of the brain that is only seen active in musicians.
To stimulate this part of the brain you need not be a proper musician; you just need to be attempting to be one. Once we realize that we seldom use 100% our cerebral capacities, activating an area otherwise dormant sound like a great idea.

The Art of Surviving Calculus I

I learned persistence at university.

Calculus is one of these disciplines to daunting that it teaches you to hold on tight an get ride along.

The system speaks a language you do not understand – Calculus for a high school graduate in classics is a mysterious and rather inaccessible language.
The choice becomes either give up or find another through. Memorise what you can. Borrow from those who are better equipped. Fake what you cannot yet grasp. Ask for forgiveness where relevant. Most importantly: find your way to survive.

The gatekeepers will know you are not one of the chosen. But you do not need their blessing. You only need to get familiar with their subject and move forward.

The system is never as inaccessible as it pretends to be.
Its language is not the only language.
Its rules are not the only path.
Computing is – or at least was – another one of systems for initiates.
As any exoteric domain, it projects an allure and creates attraction.
I was an early fan of the ZX81 that only had 1KB of memory with a possible 16k extension – less than the average jpg today – and yet it opened up the window into an entirely new world.
A world of rules and tools that helped you access and master multiple other domains.
And it that sense, it transcended itself.

You do not need a conservatory degree to enjoy a fugue and exploit music artistically.
You do not need to master Calculus to appreciate its logic and use it to solve scientific and technical problems.
You do not need to know every detail about roms, rams cpus and operating systems to exploit a computer and enjoy its limitless applications.

Those who persists and gets exposure to these domains eventually tap into benefits that go well beyond the domain.

The Hidden Mathematics of Music

Once we scratch the surface, music offers a lot of theory.
A lot of complex and almost mathematical theory.
So complex — majors, minors and sixths, chords, scales — that in effect, it takes a lot of effort event to get a faint exposure to them.

Math and music do share a common structure. And just as you meet logical philosophers who are great at numbers — it is basically the same skill, and I know very well one who happened to be my boss — so you do find quite a few physicists and mathematicians who happen to be quite good at playing instruments. Prominent examples of mathematicians and physicists who had some degree of musical talent include Einstein, who was a great piano and violin player, but also Planck and Heisemberg both excellent pianists. Does this musical talent of mathematicians and physicists have more to do with their social upbringing, where musical education is considered part of an elected formative curriculum, or does it point to a more inner relationship?

Whatever the link, a link is there to be found.

The Fugue as Formal System

The second twist of this argument goes again after one of my favourite authors and books: Gödel, Escher, Bach by Hofstadter.

The fugue is a musical contrapuntal composition, where two or more voices are introduced and, through an array of different transformations and alterations, come to constitute an ensemble that is not perceived by the listener as separate voices but as texture and melody. In The Art of Fugue, Bach encompassed his most complex and subtle exercise, exploring musical technique that would consign him to eternity. He both introduced a hint of self-reference in the main theme—Contrapunctus XIV characterized by the B‑A‑C‑H theme, B‑flat, A, C, B‑natural—and, by reversing and altering the themes to be played as new voices, and in Contrapunctus X, the ever‑ascending scale, a musical equivalent to infinity, by altering the tone of the voices so as to create an ever‑ascending scale. By using the circle of fifths, the music sounds like it is ever ascending without ever coming down.
What makes this passage extraordinary is that the ascent is not merely a technical trick. It creates a psychological effect: the listener feels the music escaping its own gravitational field, climbing toward something that can never be reached—only approached.

Gödel, Incompleteness, and the Limits of Knowledge

This resonates very well with Gödel’s theorem and its use of self-reference to ascend from within a formal system to outside it. The central take of Gödel’s theorem is that there is at least one statement within a closed formal system that cannot be proved within the system, thus invalidating the completeness of the system. And the proof is achieved from within the rules and mechanisms of the formal system itself. Thus Gödel uses this approach as a form of epistemology, and potentially extends it to negate positivism and the ability for mankind to come to absolute truths.
If positivism claimed that all meaningful knowledge can be grounded in empirical verification, Gödel showed that even within the most rigorous formal system there are truths that escape the system’s own grasp. The fugue, in its infinite ascent, performs the same revelation.

Self reference and infinite iterations into the emergence of Consciousness

The fact that all this reasoning is achieved by self‑reference and extending a proof to infinity—in Gödel, in Bach—is a very interesting mechanism. This mechanism of self‑reference and infinite ascension—covered in another post in this blog—is, in this author’s view, strictly linked to the emergence of consciousness in both humans and machines.
Bach’s final fugue remains unfinished. Perhaps it could not be finished. And perhaps that is the point: the search for absolute truth, like the fugue, is an infinite ascent—one we continue because we cannot do otherwise.