The Human Symphony

How Calculus Suggests Mortality is a Trick of Perception

Key Takeaways

1. The notion that there is something illusory about our experience of space and time runs from Plato’s Allegory of the Cave through Newton’s invention of calculus to Einstein’s field equations.

2. Just as the first and last notes of a symphony still exist, even when they are not being played, our youth and old age exist, no matter where we are in our lives.

3. The interoperability between temporal and spatial dimensions suggests that our mortality may be a trick of perception rather than a fundamental limit.

Plato, Newton, & Einstein

In his Allegory of the Cave, the Greek philosopher Plato compared our perception of the universe to shadows flickering on a cave wall. A shadow is a 2-dimensional projection cast by a 3-dimensional object. While Plato’s concern was metaphysical, his use of shadows as an allegorical device uncannily foreshadows the modern concept of dimensionality.

Two thousand years later, Sir Isaac Newton was a key figure in a long tradition of Christian Neoplatonism, which was especially strong at Cambridge University during his time. To this day, scholars debate how much of an impact his Platonism had on Newton’s independent co-discovery of calculus.

In calculus, variables like space and time are combined into rates that describe real-world experiences, demonstrating a deep connection between these dimensions. Though perceived differently, space and time are connected, like interlocking Lego bricks. Distance over time (speed) might be measured with a simple rate, like miles per hour. The rate of speed over time (acceleration) is a compound rate, which could be measured in miles per hour per second.

Calculus is the mathematics of rates, and rates of rates. Whether measuring rates of change or accumulation, it’s indispensable to modern physics. Without it, Einstein could never have formalized his field equations, in which tensor calculus fuses space and time to describe the dynamic geometry of spacetime.

Einstein’s theory of relativity, which describes this profound interplay, was his most famous contribution to modern science. His work was only possible because of calculus, arguably the most significant of Newton’s many earth-shattering contributions to the Scientific Revolution. And Newton, in turn, was heavily influenced by the post-Renaissance resurgence of Plato.

At the Symphony

In 1824, Beethoven premiered his 9th Symphony at the now-demolished Kärntnertor Theater in Vienna. Auguste Mandlick painted a similar performance at the nearby Musikverein, which still stands today. His work serves as the Title Card for this essay.

To explore the relationship between time and space as perceived experience, imagine sitting in a concert hall listening to Beethoven’s 9th. You’re stuck in the time signature with the rest of the audience, listening to each note in sequence. With no ability to rewind or fast forward time, audience members can hear only the current note being played in each moment. In the middle of the symphony, its very first and last notes are inaudible. But those notes haven’t ceased to exist; they’re merely hidden around a corner in time.

Reading Beethoven’s sheet music frees us from the confines of the time signature. With enough space on the page, your eyes could take in the first and last notes of the symphony at the same time, along with every note in between, rather than having to experience those notes in sequential order.

Sheet music works by swapping out the temporal dimension—which we measure with a stopwatch—for a spatial one—which we measure with a ruler. One second might be represented by one eighth of an inch on the page. Once you see that the symphony can be expressed through both a temporal and a spatial medium, the idea of a deep correspondence between these two dimensions becomes obvious. And the implications for humankind are astounding.

The Music of the Human Story

The experience of hearing a symphony versus reading its sheet music offers a powerful metaphor for a philosophical idea: that our linear experience of time might be a trick of perception. Physics describes a fundamental relationship between spatial dimensions and time. Our experience of that relationship is key to this perspective.

We’re all equally powerless to fast-forward or rewind time when we listen to a symphony. But a glance at the sheet music confirms that the first and last notes of the symphony do exist in some sense.

Similarly, we cannot fast-forward or rewind our lives. We’re stuck in the time signature along with everyone else. But though they may be hidden around a corner of time, this perspective suggests our youth and our old age still exist…somewhere, just like the first and last notes of a symphony.

When you begin to think in terms of this correspondence between distance and time, you become conscious of your ancestors stretching out behind you into the mists of the distant past. And of your descendants fanning out before you, proceeding into the uncertain future. Like the first and last notes of a symphony, these people, too, have an existence hidden from us by the way we perceive time.

The grand trajectory of the human story, therefore, has a shape to it, sculpted in the medium of time. Dimensionality and calculus allow us to glimpse that shape, in our mind's eye, by imagining the sheet music to the human symphony.

Conclusion

We experience our lives as a fleeting melody, mourning the notes that have passed and fearing the silence to come. But calculus winks at us. It hints that reality is a multi-dimensional manifold, which our limited brains must interpret as a story with a beginning, middle, and end. The combined efforts of Plato, Newton, and Einstein afford us a peek at the sheet music of the human symphony. Next week’s essay will take a closer look at the shape of that symphony.

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Further Materials

With his culminating masterpieces now complete, Beethoven longed for opportunity to present them to the public. But Rossini had so captivated Austria in 1823, and Viennese audiences were now so enamored of Italian melody, that no local impresario dared risk a fortune on two compositions so difficult as the Missa solemnis and the Choral Symphony. A Berlin producer offered to present them; Beethoven was about to agree, when a combination of music lovers, led by the Lichnowsky family, alarmed at the thought of Vienna's outstanding composer being forced to go to a rival capital for the premiere of his latest and most prestigious works, agreed to underwrite their production at the Kärntnerthor Theater. After hard bargaining on all sides the concert was given on May 7, 1824, before a crowded house, and with a stoic program: an overture ("The Consecration of the House"), four parts of the Missa solemnis, and the Ninth Symphony with a stentorian German chorus to crown it all. The singers, unable to reach the high notes prescribed, omitted them. The Mass was received solemnly, the symphony with enthusiastic acclaim. Beethoven, who had been standing on the platform with his back to the audience, did not hear the applause, and had to be turned around to see it.
Will & Ariel Durant, The Age of Napoleon, 1975, page 584