In this post, I’ll describe Andrew Pickering’s notion of “disciplinary agency” from his 1995 book The Mangle of Practice: Time, Agency, and Science. I’ll use the same mathematical example he uses: Hamilton’s discovery of quaternions. My hope is that Pickering’s notion gives some insight into the once-not-exactly-common-but-not-all-that-rare idea that “the code is telling us where it wants to go.” In the modern era, perhaps it could be used to think about prompting LLMs to write code or do math?
(This is a resurrected post from 2004 in honor of Hamilton Day.)
Let’s define agency as “the capacity to do things”. People have agency. I use my agency to write this essay. You have the ability — the agency — to read it or to turn aside.
Generally, we think of agency as being something that humans have. In his book, Pickering proposes that it’s useful to think of nonhumans as having agency. For example, in an experiment a scientist creates a machine, turns it on, and watches it. While creating it, the scientist is exhibiting agency. But when watching the machine, the scientist is passive while the machine does whatever it is that it does. We can say the machine exhibits material agency. It’s no longer manipulated by the human; instead, it’s in control while the human sits back.
Another type of agency is disciplinary agency, in which a human gives up control to a routine way of reacting to patterns in the world. We can say that that routine has agency. It has the ability to do things in the world that the human doesn’t expect, that the human can only observe and then react to.
This pattern of humans exercising agency, then sitting back and watching while something else exercises agency, Pickering rather fancifully calls the dance of agency. He claims it’s a common pattern in intellectual practice.
Pickering uses Hamilton’s quaternions as a case study. In the 1800’s, mathematicians had established a correspondence between algebraic equations involving imaginary numbers, like x+iy, and 2D geometry. Such an equation corresponded to a vector from the origin to the point (x,y). (x+iy)+(u+iv) corresponded to the vector (x+u, y+v). Multiplication of algebraic equations corresponded to a two-part rule: multiply the lengths of the two vectors, and add their angles.
That’s fine for two dimensions. What a mathematician named Hamilton wanted to do was figure out how to link algebra and geometry in three dimensions. How can we talk about a point (x, y, z) algebraically?
Hamilton first modeled 3D after 2D. If a point (x,y) corresponds to x+iy, perhaps (x, y, z) corresponds to x+iy+jz. The z axis is perpendicular to the x axis the way the y axis is, so he said that j is an imaginary number like i. That is, jj=-1 as well as ii=-1.
That’s fine, but does it work? One way to tell is to repeat what you already know how to do in the new context. That is, you apply — rotely — a discipline you already know. Hamilton knew the rules for algebra, so he started manipulating equations and seeing if the results corresponded to any sensible extrapolation (from 2-space to 3-space) of multiplying two vectors.
A first thing he did was consider the square of a point in 3-space. According to the normal rules of algebra, (x+iy+jz)² is:
x² - y² - z² + 2ixy + 2jxz + 2ijyz (1)
^^^^^
This is an example of disciplinary agency. Having decided on his representation, Hamilton had no choice about how to proceed: the rules of algebra controlled, and they produced the result that they produced. Everything is straightforward, except there is a bit of a question: what does ij mean? He knew that ii=-1 and he’d assumed that jj=-1, but what was ij? He wasn’t forced to answer that question yet, so he could move on.
Now Hamilton turned to geometry. He needed to extrapolate the rules for 2D into 3D, keeping fixed the idea that multiplication means multiplying lengths and adding angles. A problem arises in adding angles. When you square, there are multiple points that satisfy the addition constraint. Which to choose? Hamilton chose the one that lay on the plane connecting the original point and the x-axis.
Hamilton has here taken back agency from the discipline of algebra. But once he makes his decision, he surrenders again to the discipline of geometry. You can now square a point and, moreover, translate the result back into algebraic notation:
x² - y² - z² + 2ixy + 2jxz (2)
Since this formula and formula (1) represent the same point (just gotten at by two different disciplines of multiplication), and (1) has the extra term 2ijyz, we know that:
2ijyz=0
Having given discipline free reign to get this result, Hamilton entertained two possibilities for why 2ijyz is zero. The first was that ij=0. The other was bolder, abandoning commutativity so that ij=-ji. (If multiply out (x+iy+jz)² under that assumption, you’ll see that the you get the same result as the geometric multiplication; that is, without the 2ijyz term.)
He assumed the first possibility and kept multiplying. This time he multiplied two different points rather than squaring the same point. He ran into another problem that required him to accept that ij=0 couldn’t work, that it had to be that ij=-ji.
But if ij isn’t zero, it still has to equal something. Hamilton made the leap of supposing ij was equal to a new imaginary number, k.
Once he’d worked out all the combinations of multiplications of i, j, and k, he’d succeeded: he had a description that allowed algebraic and geometrical multiplication to come to the same answer… except that the new imaginary k meant that he was working in four dimensions rather than the three he’d intended. He’d had one goal, but his assumptions and the results of disciplinary agency had pushed him away from it toward what he came to call quaternions. (The way goals mutate in practice is a theme of Pickering’s.)
So we see a pattern:
- Use your human agency to make some extension of what you know.
- Yield to disciplinary agency to see where that takes you.
- When you run into resistance — something isn’t working — start thinking. Make a new extension.
- Yield to disciplinary agency again, and see where it takes you now.
- Repeat until you’re happy.
That’s the dance of agency.
Note the similarity of the mathematician’s agency to Peirce’s description of the making and use of hypotheses. He generally thought of mathematics as quite close to an experimental science. As Cornelis de Waal puts it: Cornelis de Waal, Peirce: A Guide for the Perplexed, 2013, pp. 42-43. (Highlights mine.)
In sum, mathematical reasoning is experimental rather than reflective, and active rather than contemplative. Within mathematics we gain knowledge, Peirce argues, “not by a simple mental stare, or strain of mental vision,” but by “manipulating on paper, or in the fancy, formulae or other diagrams” (CP4.86). Such experience alone, Peirce argues, brings out “the reason hidden within us,” and as such “this experience only differs from what usually carries that name in that it brings out the reason hidden within and not the reason of Nature, as do the chemist’s or physicist’s experiments” (id.; also Section 9.5). Put differently, when experimenting on man-made diagrams, we just as much hit upon hard facts as when conducting experiments in nature.
We push at reality; reality pushes back.