On Measurement
The work-in-progress arose from the Alter-Eco seminars and studios that were convened virtually during the Spring and Summer of 2020.
Thesis:
Measurement cannot be disentangled from intent: an “effective” measurement is not an effective measurement of an objective matter, but rather, an effective objectification of one’s intent to make a matter objective. In the text that follows, we conceptualize the mathematical notion of a “spectrum of operators” in order to interrogate the entanglement of measurement and intent.
What is a spectrum of operators?
To begin approaching the idea of a “spectrum of operators,” let us consider something deceptively simple: a square.
We are all familiar with the spectrum of rotation operators under which a square remains “invariant”—that is, those rotations after which the square appears unchanged. This spectrum includes the 0° rotation, the 90° rotation, the 180° rotation, the 270° rotation, the 360° rotation, the 450° rotation, and so on. To say that a square is invariant under all these rotations is to say that one cannot distinguish between a square rotated 0°, 90°, 180°, 270°, 360°, 450°, etc.—so long as the rotation is a multiple of 90°, the square appears the same.
Within this spectrum, the 0° rotation is what we might call the “fundamental” or “ground” rotation operator. And yet, the full rotational timbre of a square—its characteristic signature, its expressive presence in the space of transformation—is not defined solely by this ground rotation. Rather, it is defined by the entire spectrum of rotation operators under which the square is invariant. After all, the “fundamental” 0° rotation is also the ground rotation of a circle, a triangle, and a borromean knot, each of which possesses a distinct rotational timbre.
What distinguishes these shapes from one another is not just the set of rotation operators they include, but also the ones they exclude. For example, the spectrum of a circle includes every conceivable rotation: 1°, 2°, 3°, 90°, 137.5°, 999°, and so on—no rotation alters its appearance. A square, on the other hand, is not invariant under 1°, 2°, 3°, 5°, 7°, or any rotation that is not a multiple of 90°. So while the circle and the square share many rotational invariances, including all those multiples of 90°, their rotational timbres remain distinct because the square excludes an infinite set of transformations that the circle includes.
Now consider a third shape: the equilateral triangle. Its spectrum of rotational invariance includes 120°, 240°, 360°, and every rotation congruent to those modulo 360°. Again, we encounter a different timbre, one that distinguishes the triangle from both the square and the circle—not through what it includes alone, but through what it forbids.
Let us now complicate the situation by introducing time intervals and sense impressions. Once more, we regard the same square:
When you and I observe the square, we are likely to assume that it is undergoing a 0° rotation between each of our sense impressions of it. That is, we take its fundamental, or unchanging, appearance to be stable across time. However, it is entirely possible—at least logically—that the square has undergone a 180° rotation, then a 90° rotation, then a 450° rotation, then a 900° rotation, and finally a 360° rotation in the time intervals between our observations.
You might scoff and say, “Sure, that’s possible, but it’s totally friggin’ improbable.” You might even insist, “I can prove to you that there’s no friggin’ way the square has rotated 90° between any two observations.” And to that I respond: “Prove it.”
Your proof might proceed by marking the corners of the square like so:
Then you might argue: if this square were rotating by 90° in the intervals between our impressions, we would witness permutations like this:
Yet the letters—A top left, B bottom left, C bottom right, D top right—remain fixed. Therefore, you conclude, the square can only be rotating by 360° or by factors thereof. There is no way, you claim, that it has rotated 90°.
How do I respond? I say this: by marking the corners as you have, you have altered the shape. You have not proven something about the unmarked square—you have proven something about a new shape: the square with alphabetically ordered corners. The square you began with had a spectrum of invariance that included any and every rotation that is a multiple of 90°. But by marking it, you have constrained its spectrum. You have made a new shape, one that is invariant only under 360° rotations or their multiples. You have changed its rotational timbre.
What you have done, in essence, is not to prove anything about the original square’s behavior. Rather, you have objectified your intent—you have made something measurable by restricting what it can be. The “square with alphabetically ordered corners” becomes a proxy for the square, but it is not the square itself. It is an effective objectification of your intent to make the square’s rotation measurable and determinate.
We can thus regard this act—marking the corners—as a paradigm of taking a measure. To measure, in this sense, is to make a matter’s spectrum of operators more exclusive. Measurement, therefore, is an act of narrowing: it is the deliberate exclusion of possibilities in order to stabilize a preferred version of reality.
If this is so, then to measure something is also to ask:
• What possibilities am I excluding?
• How precisely am I excluding them?
• Why am I excluding them?
At the same time, measurement is not only an act of restriction—it is also creative, ontogenic. To take the measure of a matter is to make something new out of it, just as the alphabetically marked square was made out of the initial, unmarked square. In this sense, measurement is a generative intervention.
Thus, one must supplement the above questions with:
• What am I making out of the matter?
• How precisely am I making it?
• Why am I making it?
To measure, then, is not to reveal an objective truth already present in a thing, but to enact an intent that gives form to a set of possibilities. The spectrum of operators—a concept from mathematics and physics—becomes a way of illuminating the field of transformations we admit, as well as those we banish, in the name of clarity. And perhaps, more often than we admit, in the name of control.
Regarding Matters of Sustainable Development
Let us now “apply” the lessons above by turning our attention to the UN Sustainable Development Goals.
Each of these goals can be regarded as an attempt to take the measure of seventeen distinct matters: (i) the matter of poverty, (ii) the matter of hunger, (iii) the matter of health and well-being, (iv) the matter of education, (v) the matter of gender equality, (vi) the matter of clean water and sanitation, […] and, last but not least, (xvii) the matter of international partnership in support of the SDGs.
When an organization such as SDG Impact takes the measure of these various matters, we must return to the guiding questions: What possibilities are being excluded from these matters? And equally, what is being made of these matters through that exclusion? In the case of SDG Impact, the answer is explicit. Their mission, as stated on their website, is: “Empowering investors with clarity, insights and tools to achieve the Sustainable Development Goals (SDGs).” That is, SDG Impact seeks to make these seventeen matters into matters of capital investment.
Which leads us to ask: What is excluded from sustainable development when it is framed as a matter of capital investment?
One thing comes to mind immediately: to the extent that sustainable development becomes a matter of capital investment, it becomes a matter for those who have capital to invest—and not a matter for those who do not. The impacts of actions taken by those without capital will likely be excluded from the scope of the measures SDG Impact takes, unless those actions can somehow be connected back to investment logic. In spectral terms: any operator on sustainable development that is not a factor of capital investment will be excluded from the spectrum of operators SDG Impact admits as relevant to sustainable development.
Once again, we must insist: there is no “hands-off” approach to measurement. To take the measure of a matter is to get a handle on it, to orient it, to frame it according to intent. An organization like SDG Impact is not revealing an objective fact when it measures the impact of capital investment on sustainable development. Rather, it is objectifying its intent—its will to make capital investment appear to have objective impact.
And while this may seem innocuous, even beneficial, it raises a crucial concern: in striving to make capital investment appear impactful, the organization must exclude that which obscures or resists its impact. But what if that which obscures the impact of capital investment turns out to be more impactful with respect to sustainable development than the investment itself?
If we do not wish sustainable development to become the exclusive purview of those with capital to invest, we must then ask: How else might we measure it?
Imagine another organization, perhaps called SDGs for Communities, whose mission is: “Empowering grassroots community organizers with clarity, insights and tools to achieve the SDGs.” Here, sustainable development would be rendered as a matter of grassroots community organizing. Capital investment, unless tethered to such organizing, would be excluded from the spectrum of operators deemed relevant. That is: any operator on sustainable development that is not a factor of grassroots organizing would fall outside the scope of this measurement.
And we might ask—shouldn’t it? Might it not be more meaningful to make matters of sustainable development into matters of collective, place-based action rather than abstract flows of capital? Might it not be better to exclude those forms of capital investment that are disconnected from community empowerment?
Substrates, Transformations, Information(s)
So again, we return to the core claim: an “effective” measurement is never an effective measurement of an “objective” matter, but rather an effective objectification of an intent to make that matter objective. This, of course, is not a novel insight. It resonates throughout the work of Michel Foucault—from Madness and Civilization to The History of Sexuality—wherein the production of knowledge is always entangled with the exercise of power.
What may be novel in our articulation is the framing of this entanglement through the notion that the substrate of measurement is not a determinate discretum (a set of fixed elements), nor a determinate continuum (a smooth, measurable field), but rather an indeterminate spectrum—a field of latent potential, a domain of possible transformations.
More precisely:
Indeterminate spectra—also known as noise—constitute the substrates of measurement.
Measurements are the discrete or continuous trans-formations that shape, partition, or extend these substrates.
Determinate discreta (i.e., digital data) and determinate continua (i.e., analog data) are the in-formation(s) produced through these trans-formations.
In short, a measurement is not a neutral act of observation—it is a creative, selective act that trans-forms an indeterminate substrate into something measurable. It makes in-formation out of potential.
Our central proposition, then, is this: intent is to be discovered not by processing the results of measurements, but by interrogating the transformations that produce those results. That is, to understand intent, one must attend not to the data itself, but to the ways in which the substrate was rendered measurable—to the decisions and exclusions that made a spectrum more or less exclusive.
This distinction is critical in a society saturated by information technologies—technologies that process and optimize in-formation, while ignoring the deeper question of how that information was formed in the first place. Ours is a culture that prides itself on “data-driven decision-making,” yet often fails to pause and ask: what intent drove the decision to gather this data in this way?
And so, many of the crises we face today are described as “unintended consequences” of well-meaning decisions. But that framing misses the point. These crises are not unintended—they are the result of a widespread failure to interrogate intent itself. A failure to ask: What are we making by measuring this way? And what are we unmaking?
To address this failure, we must shift from information technologies to trans-formation technologies—technologies that prioritize the shaping, composing, and design of transformations, rather than the extraction, storage, and circulation of information.
Such technologies would invite us to become designers, makers, world-builders—to attend with care to what, when, where, why, and how a trans-formation renders a spectrum more or less exclusive. They would teach us that every measure is a making, every exclusion a choice, and every transformation an enactment of intent.