Mathematics, movement, music and Leonardo

I’ve always been intrigued by the sensation of movement in music.  And it is fair to say that it was my first calculus class that led me to graduate study in mathematics because, for the first time, I saw movement in mathematics.  My fascination with each of these was nudged again by an interview with jazz pianist Vijay Iyer that I heard on NPR’s All Things Considered.

Once a graduate student of math and physics at Yale, Iyer’s attention eventually turned to music and the study of music perception and cognition.  It was improvisation on the piano (unlike the lack of improvisation in his training on the violin) that brought Iyer to consider that it is the relationship between the body’s movement and an abstract idea that together make music.

One thing that has kind of obsessed me at the piano for maybe the last decade or so is just listening to what my hands would do with the instrument. So the keys’ action speak… It all lays right under the hands… It comes from the physical logic of the hand at the keyboard. To me, they’re gestures. You know, they’re physical gestures. I hear the sweep of a hand in those shapes…- it’s this physical thing that’s also something that resorts to a logic that’s very nonphysical. Letting these numbers unfold through physical action is, for me, part of the process of making music.

Iyer brings it all round to some very primal things.

Particularly, what music is for us as people is it’s a sound of other people. For example, the way we perceive rhythm is by imagining ourselves moving. So there is a really primal connection between music and the body.

…it’s about the rhythms that are inherent to the body. Breathing is one of those. Heartbeat is another. Speech is another. And those are all timescales that correspond to musical activity.

It’s probably easier for us to appreciate music as an action, than say mathematics as an action.  But they each rely on movement.

Out of the early twentieth century debate between the formalists and the intuitionists sprang this claim from L.E. J. Brouwer that mathematics has its origin in the movement of time.  Expressed as the First Act of Intuitionism it calls for:

Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognizing that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time.  This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory.  If the twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities.  And it is this common substratum, this empty form, which is the basic intuition of mathematics.

Brouwer’s extreme intuitionism challenged the legitimacy of far too many mathematical constructions, to be realistically accepted by the mathematics community.  But no doubt there is a strong link between our conceptual worlds and our experiences of space and time.

The manifold expressions of mathematical movement are beautifully described in some of Fritjof Capra’s  The Science of Leonardo.

The book contains wonderful descriptions of the paths taken by Leonardo’s thoughts, and how they line up with modern developments in mathematics as well as the growing insights into cognition itself.  But perhaps these passages are most relevant at the moment:

In contrast to Euclid’s geometry of rigid static figures, Leonardo’s conception of geometric relationships is inherently dynamic.  This is evident even from his definitions of the basic geometric elements.  “The line is made with the movement of the point,” he declares, “The surface is made by the transverse movement of the line;…the body is made by the movement of the extension of the surface.”

Leonardo also drew analogies between a segment of a line and a duration of time:  “The line is similar to a length of time, and as the points are the beginning and end of the line, so the instants are the endpoints of any given extension of time.”  Two centuries later this analogy became the foundation of the concept of time as a coordinate in Descartes’ analytic geometry and in Newton’s calculus.

As mathematician Matilde Macagno points out, on the one hand Leonardo uses geometry to study trajectories and various kinds of complex motions in natural phenomena; on the other hand, he uses motion as a tool to demonstrate geometrical theorems.  He called his approach “geometry which is demonstrated with motion.”

It’s as if he could place himself completely within what he saw instead of outside of it.

One of the things I found most interesting was that, in the absence of mathematics that could describe dynamic systems, Leonardo made scientific drawings (shown in the book).  These are beautiful drawings, rich with observation.  Capra explains that these were not realistic representations of a single view.

Rather they are syntheses of repeated observations, crafted in the form of theoretical models.

I particularly like this description of Leonardo’s mathematics:

Since Leonardo’s science was a science of qualities, or organic forms and their movements and transformations, the mathematical “necessity” he saw in nature was not one expressed in quantities and numerical relationships, but one of geometric shapes continually transforming themselves according to rigorous laws and principles.  “Mathematical” for Leonardo referred above all to the logic, rigor, and coherence according to which nature has shaped, and is continually reshaping, her organic forms.

And this one of his science:

Leonardo’s studies of the living forms of nature began with their outward appearance and then turned to methodical investigations of their intrinsic nature.  Life’s patterns of organization, its organic structures, and its fundamental processes of metabolism and growth are the unifying conceptual threads that interlink his knowledge of macrocosm and microcosm.

Capra also makes the argument that da Vinci’s concept of the soul bears a strong resemblance to contemporary ideas that identify the mind, or cognition with the process of life.  What Leonardo’s soul shares with current ideas about cognition is their two-fold nature.   Today, cognition (like Leonardo’s soul) is both the process of perception and the process that animates the movement and organization of the body.  Leonardo often wrote about it in terms of action.  And in this light, Capra refers to his notes on the flight of birds.

Over many hours of intense observations of birds of flight in the hills surrounding Florence, Leonardo became thoroughly familiar with their instinctive capacity to maneuver in the wind, keeping their equilibrium by responding to changing air currents with subtle movements of their wings and tails.  In his notes he explained that this capacity was a sign of the bird’s intelligence – a reflection of the action of its soul.