Tangible Math

Physicality of Symbol Use

One implication of the theoretical framework we (Ricardo Nemirovsky and Michael Smith) use is that concepts normally considered to be abstract must have perceptual and motor components.  In other words, there must be some sense in which even mathematical ideas are physically perceived and interacted with.  This suggests for instance that the symbols that mathematicians use are not representations of some intangible abstractions; instead, the abstractions in question are constituted in a given context by the physical interactions the mathematicians have with the symbols they’ve chosen to employ at that moment.  That is, there’s some kind of physicality in the use of mathematical symbols, even after they’re drawn.

We can shed light on what this perspective offers us as researchers by examining some of the details of how some mathematicians engage in the physicality of their symbol usage.  This allows us to notice similar instances when they appear in others whom we observe working with symbols, thereby helping us to better understand what those symbols mean for those whom we observe.

For this particular case study, we’ll share an instance of a topologist describing how the idea of “spin” differs between the classical mechanics of a rotating ball and the quantum mechanics of an electron.  In the following images, the transcript excerpt is shown at top and the part of the transcript that corresponds with the image is shown below the image.  (The image occurs at the end of the transcript fragment.)  The orange arrows indicate the mathematician’s hand movements from the previous image to the current one.

The topologist draws something meant to depict a sphere with an equatorial line and then draws an arrow out the top of the sphere.  The arrow is meant to indicate the rotation of the sphere by the right-hand rule: if you point your right hand’s thumb away from your palm and rotate your right hand so that your thumb points in the same direction as the arrow, then your fingers will curl in the direction of the sphere’s rotation.  The length of this arrow (or “rotation vector”) indicates the speed of rotation.  Notice how once the topologist has drawn the vector, he holds the chalk in the same direction as the arrow is intended to point and then twists his hand in the direction of the rotation, as though the chalk is meant to be the rotation vector and his hand the sphere.

At the same time he says “vector”, he does a lifting movement upward as though to indicate the direction the vector would be pushing in if it were a force vector.  He then returns to using the chalk as the rotation vector but moves upward it as though again to emphasize its arrow-like nature.  He then extends his middle finger and swings his hand upward as though to position the middle finger as the vector as he prepares to refer to the axis of rotation.  He draws a little curved arrow around the base of the vector to indicate the direction of rotation, and once more mimics the use of the chalk as the vector.

What we see here are at least three different ways that this mathematician has elected to embody the idea of the rotation vector: once by using the chalk as the vector, once with an upward swing of the open hand, and once with the middle finger.  You can even see here how he transitions from emphasizing the ball’s rotation that the vector indicates to highlighting the direction of the vector when he starts referring to it as “the axis about which it spins”.  All of this is explicitly and repeatedly tied to the marking on the blackboard as though to emphasize that all these physical movements are to be understood as tied to his drawing.  The fact that he felt the need to do this in order to explain his diagram suggests that his movements here are part of his intended meaning of his diagram.

Notice, too, that this tells us quite a bit more than simply that movements and body positions are somehow tied to meaning here.  We can perceive to some fair degree how they relate to meaning.  The twisting movement is tied to the axis, but the axis has a dynamic directionality to it; it would mean something different if the axis were drawn as something non-directional like a line segment, and it would also mean something different if this topologist had insisted on a static grip on the axis instead of this movement upward.  This is not to say that one must understand rotation vectors in terms of movement in the direction of the arrow’s pointing, but it does indicate that this is part of the meaning for this mathematician in this situation, and thus this understanding of rotation vectors is one that at least some mathematicians employ at least some of the time.  By recognizing this, we gain insight into a range of possibilities for how at least classical rotation can be understood.