On Inexactness

Philosophy, and maybe even human thought as such, has long been a quest for exactness; for figuring out the exact nature of reality and the forces that shape it; for figuring out the exact workings of language; for figuring out the exact meaning of happiness, justice, and the good life. Us humans, we gauge, compute and quantify: “you can’t manage what you can’t measure”.[1] Give us a world and we’ll hold a yardstick to it. Show us a universe and we’ll show you our metrics. Nothing escapes the ruler, the meter, the stopwatch, the word. But how exact do we want these systems of representation – our measurements and language – to be? Are we beings of exactness? Not in an absolute sense, for sure; nor all the time, it would seem; and, in some regards, maybe not at all. In order to substantiate these claims, I’ll take you through some thoughts on statistics, language, and ethico-political philosophy.

Wittgenstein looking at a blackboard saying 1+1=3

En route to an answer, we must first endure another question. This – slightly annoying yet ultimately unavoidable – ur-philosophical query concerns, of course, what we understand ‘exact’ to mean. The Merriam-Webster dictionary has two entries for its adjective form:

         1: exhibiting or marked by strict, particular, and complete accordance with fact or a standard

         2: marked by thorough consideration or minute measurement of small factual details.[2]

Both entries talk about a certain adherence or compliance to factuality. The first entry, however, adds ‘accordance with a standard’ of some sort. Let’s hone in on this addition. It appears to introduce some kind of relativity (to a certain praxis) into the seemingly more objective requirement of factual accordance. So what are these standards precisely? How do they come about, and how do they function? We will start off by looking at what, at first sight, seems to be a very unphilosophical subject, the practitioners of which ostensibly obsessed with exactness: statistics[3].

Statistical significance and standards for exactness

Every student of statistics has, at one point or another, come across a thing called ‘p-value’. Some of these unfortunate souls will, like me, still be haunted by the specter of this central component to statistical analysis, waking up after decimal-filled fever dreams, bathing in the coldest of night sweats. So what are p-values?

A p-value is a tool to test statistical significance. More specifically, it tests the null-hypothesis, (H0) which is the hypothesis that (usually) contradicts the specific theory one wants to prove. In other words, it holds that there is no effect or relation between the groups or phenomena under scrutiny. Until evidence proves otherwise, the null-hypothesis is assumed to hold true. As such, it functions as a default baseline position with regard to your alternative hypothesis (H1), which generally does hypothesize this effect or relation. Even though it does not validate it, being able to reject the null-hypothesis is a crucial step in proving the specific theory or effect you formulate in your alternative hypothesis. Rejecting the null is basically indirect evidence of an experimental hypothesis being true. The p-value indicates the probability that, under the assumption that the null-hypothesis is true, you have obtained a test statistic that is at least as extreme as the one that is calculated from the sample data. In other words: it shows you how rare your results would be in a world in which the null-hypothesis is true. The lower the p-value, the less likely it is that this is the case, and the more solid your case for rejecting the null.

Now for the interesting part: this p-value threshold for rejecting the null-hypothesis is rather arbitrary. Researchers can never definitively rule out the null-hypothesis, and due to this fact have simply agreed upon a specific probability-threshold. Within the social and life sciences, the 0.05 significance level has become the accepted convention. As such, we have basically agreed upon the fact that it is ok for statistical research to have a 5% or lower chance to incorrectly reject the null-hypothesis. Why set the mark at 5%, you may ask? There are no satisfying answers here, I’m afraid. With absolute certainty a distant pipe dream in most scientific disciplines, what we’re left with is convention, based on requirements of reproducibility and demands for scientific ‘output’, among other things. As far as this particular standard is concerned: it was first developed by Ronald Fisher in the 1920s, who insisted that prudent leniency was sometimes required in interpreting p-values. He saw the p-value as providing an index that establishes whether further investigation of a phenomenon was justified or not. Instead of heeding his advice, the 0.05 threshold is used by researchers as a strict cutoff point, the pivot point holding up a rigorously dichotomous significant/not-significant seesaw – always skewed, never balanced.

That being said, while 0.05 can be seen as a general standard, the threshold for p-value based statistical significance does in fact vary between scientific disciplines, with .01 being more common in natural sciences and chemistry. In particle physics the threshold is even lower, with the general threshold in reporting ‘evidence of a particle’ located at p<0.003, and the standard to report a ‘discovery’ at p<0.0000003. Recently, a vast number of studies in the social and life sciences have proven difficult or entirely impossible to replicate or reproduce. Seeing as reproducibility is one of the core scientific principles, it didn’t take long for a word that defines our era to surface once more; another crisis was born. This ‘replicability crisis’, ‘or crisis of reproducibility’ is the reason why a group of 72 prominent researchers have recently pleaded for changing the p-value threshold from p<0.05 to p<0.005.

It is inexactness’ turn in the limelight, and by God it’ll bathe in it.

Any measuring instrument is subject to varying degrees of measurement uncertainty and instrument error. Statistical tests are no exception. Standards for exactness are tricky things, marred as they are by both arbitrariness and necessity. In statistical analysis, even though we submit to rather arbitrary probability levels of measurement error, exactness in its absolute form is the ideal. This Splijtstof issue, however, is not here to confirm or conform to orthodox ways of looking at things, but to upend such status quo’s. We could give a rat’s buttocks about exactness; it has received plenty of applause. It is inexactness’ turn in the limelight, and by God it’ll bathe in it. We’ll take a bold lil’ peek into the abyss of inexactness. In other words, let’s move on to the departments of life in which exactness is a less strict requirement. Time to turn to language. Time to holler at Ludwig Wittgenstein! 

Language and (in)exactness

What does it mean for language to represent reality in an exact manner? That, it turns out, varies tremendously between various contexts. Whereas words, phrases, expressions, and distinctions can become more salient, this does not necessarily lead them to be well-delineated. This is not to say that there are not crisp, well-delineated concepts out there. In fact, Wittgenstein says, we can alter the rigidity of concepts’ boundaries:

[We can] use the word ‘number’ for a rigidly bounded concept; but I can also use it so that the extension of the concept is not closed by a boundary. And this is how we do use the word ‘game’. For how is the concept of a game bounded? What still counts as a game, and what no longer does? Can you say where the boundaries are? No. You can draw some, for there aren’t any drawn yet. (But this never bothered you before when you used the word ‘game’.)[4]

While we can make concepts as exact as we want, this is often not what they are for. We can make it so that ‘a pace’ means a distance of 75 cm, but the use of, and for, concepts such as ‘a pace’ or ‘stand roughly there’[5] is precisely their non-exactness, their fuzziness. Meaning does not presuppose rigid boundaries. Often we use expressions that do not have a fixed meaning, and this does not impair their use.[6]

Nevertheless, Wittgenstein notes, inexactness remains suspicious in the eyes of many, and exactness is praised – even though there is no single measure or ideal of exactness to be found. In this, he is indubitably on point. Why measure at the level of the millimeter when a nanometer – or a femtometer for that matter – is more exact? Despite this, we use centimeters (or two decimals of a meter) instead of micrometers when specifying the height of our bodies; kilometers instead of millimeters per hour to gauge the speed of our automobiles[7]; weeks instead of hours to indicate the age of an unborn human baby (before gradually shifting to years a few years after birth)[8]. The (in)exactness of our metrics is undeniably related to their practical use. The world does not seem to care for specific levels of exactness – we do. The apparent ‘crystalline purity’ of logic, for example, was not something we discovered, Wittgenstein argues, but was presupposed as a requirement

Not only does meaning not presuppose rigid, exact boundaries, such boundaries would in fact cripple us. Requirements of crystalline purity leave us on slippery ice, Wittgenstein warns us, “where there is no friction, and so, in a certain sense, the conditions are ideal; but also, just because of that, we are unable to walk. We want to walk: so we need friction. Back to the rough ground!”[9] Language, in order to keep up with the world and our ever-changing place within it, needs jagged edges, fuzzy, messy, grainy and vague. This is exemplified by the role of simile and metaphor in our attempts to understand new phenomena. Take writings about physics (and science as a whole), for example: by way of experiment, try to talk meaningfully about a black hole without taking recourse to metaphor.[10]

The modes of use of words (our linguistic practices) show family resemblances instead of essences, Wittgenstein tells us. Family resemblances, unlike essences, have blurry edges. Their boundaries are fuzzy, permeable, and apt to change over time. Like actual families, conceptual family resemblances are dynamic. They can lose, spawn, or adopt new members. The fact that meaning is not fixed forevermore, however, does not mean that anything goes: there are rules. These rules – when it comes to games and language alike – stand “like a signpost”[11] in the sense that while they ‘point at something’, they nonetheless potentially allow for doubt and multiple interpretations. At the same time, there has to be a certain agreement or commonality at play, since “a person goes by a signpost only in so far as there is an established usage, a custom.”[12] Like signposts, rules are in order if they fulfill their purpose under normal circumstances.[13] In other words, it is not the case that fuzzy concepts or concepts based on family resemblances – Wittgenstein talks of “blurred edges”[14] – are unregulated. Like games, they are simply not bounded by rules in every aspect.

New ways of speaking can lead to new experiences. Having said that, new experiences aren’t necessarily what we’re after during surgery.

Football, for example, has rules, but these do not dictate every possible move: there are no rules limiting how fast or how high you can shoot a ball. And while the rules forbid (field)players to use their hands when touching the ball, they do not oblige players to only use their feet. The fact that rules do not form a ‘closed’ system of action allows for creative play. In fact, the dimension for creative action that this opens up is almost universally admired. One would have trouble finding a football fan that fails to admire Ronaldinho Gaucho’s repertoire of elasticos, pedaladas, and no-look passes, or the less refined but equally brilliant Cuauhtemiña (or ‘Blanco trick’) and nalguiña (‘little hunchback flick’) first executed by Cuauhtémoc Blanco. In those moments, even though everyone on the pitch is involved in playing a game, the element of play was more saliently present in the actions of these creative minds. They direct our attention to the difference between playing by a game’s rules, and playing with a game’s rules. They might play on grass, but have Wittgensteinian rough ground underneath their feet all the same.

Similar to game regulations, the open-endedness of the meaning of certain concepts guarantees the possibility of a creative use of language. Just as there are games that allow for more creativity than others, there are language games that are more strictly regulated than others. The language used in a hospital’s operating theatre is more strictly regulated, more closed, than the language games played in art galleries or poetry slams. There is power in the innovation that ‘regulative openness’ facilitates. New ways of speaking can lead to new experiences.[15] Having said that, new experiences aren’t necessarily what we’re after during surgery. In other words, our desire for creativity and novelty – which presuppose a certain degree of inexactness – are determined by our practices and goals in specific contexts. These practices, ‘the way we do things’, evolve and change, and the levels of exactness that underlie them can change with them – their thresholds constantly being renegotiated. This is true for our scientific ‘language’, as the p-value standards discussed in the previous section show, but just as true for our everyday language, albeit oftentimes a bit less explicit.

Inexactness has proven a tough cookie, robustly (omni)present as it appears. We’ve seen that there are domains in which it is imposed on us (as statistical tests show), and that there are domains that permit it. But there are also domains that demand it, as shown by language – especially by the creative types of language games. In the next section, we will continue to excavate the site of this demand for inexactness. Let’s resume our staring contest, and see whether inexactness gazes back dead-eyed from the abyss, or ogles us with a sprightly smirk.

Regulative Ideals and inexactness

In ethics as well as political philosophy inexactness can take on a certain normative significance. This is arguably most clearly exemplified by what was originally a Kantian concept: the regulative ideal. In her study of this concept, The Role of the Unrealisable, Dorothy Emmet defines Regulative Ideals (note the capitalization) as:

concepts not realisable in particular instances but which have a role in setting standards for practical reason, including thinking when seen as an intellectual practice. They are not fully instantiable, but they set a direction for a practice and prevent us from settling for surrogates. Purported realisations of Regulative Ideals are likely to be surrogates, and their effect can be to diminish the practice.[16]

Regulative Ideals, Emmet holds, have a role in our practical reason even though they are unrealizable, by setting standards for teleological orientation. She distinguishes between truth and good lowercased and Truth and Good capitalized. The first two can be instantiated, the latter two, she holds, cannot, and as such qualify as Regulative Ideals[17] (Regulative Ideals clearly don’t participate in your lower-case peasantries). Emmet likens them to other concepts that are also marked by non-instantiability – concepts without designated empirical instances in individuals or phenomena – such as constructions (for example: the average married man in Britain, who has 2.5 children), ideal types, and idealizations or abstractions.[18] What sets Regulative Ideals apart from these, however, is that they can be seen as (or at least come close to being) ends in themselves, instead of mere means or tools in service of other goals.

Regulative Ideals clearly don’t participate in your lower-case peasantries.

Emmet distinguishes between two types of rationality that both qualify as forms of practical reason, yet differ in the type of teleology that is at play. Instrumental teleology, on the one hand, consists in opting for the most efficient means – using the information available to us – to reach an envisaged goal. Even if we do not attain the goal, we have acted rationally as long the most efficient means was opted for. The second type of teleology, on the other hand, displays more difficulty in separating means and ends, and is marked by a “purpose informing a whole course of action, and rationality is shown not only in relation to the end, but in the way in which the activity is carried out”.[19] What is done has to be done well instead of simply realized, and the thing to be achieved takes shape ever more clearly once progression is made – Emmet lists doing philosophy as an example. It is this second type of rationality that is linked to Regulative Ideals. These are open-ended practices. The ideal aspect does not provide a sufficiently specific final objective, but provides a point of orientation nonetheless, while the regulative aspect guides the approach.[20] Regulative Ideals are unattainable yet approachable. There is no pinpointing here, only gesturing.

The idea of the pure Good Will (or ‘Holy Will’) fulfills precisely such a role in Kant’s philosophy, Emmet holds. In addition, she considers Absolute Beauty, the Good, Truth, Justice, Peace, and the Common Good as Regulative Ideals. All of them have this in common: they are things that have a hard time being expressed in/on terms beyond themselves – and are near-impossible to define conclusively.[21] When it comes to politics, the exemplary concept is the General Will (or as it is more commonly referred to nowadays: ‘The Will of the People’). I will restrict myself here to discussing this particular Ideal. Viewed as a Regulative Ideal, the General Will does not function as a concrete program, but instead sets a standard for a form of political activity. The General Will cannot be fully realized or fully exemplified. It is an intentional object, existing in intellectu. The concept itself is marked by ambiguity: it can both be understood as the will for something general (the general or public interest), and as the will of something general (the general population or ‘the people’).[22]

Regulative Ideals warrant praise, but also suspicion. (Absolutist) claims of the General Will being fully realized can lead to disastrous results. These claims, and the resulting distortion of the General Will as a Regulative Ideal, can be observed in single party ‘democracies’, certain populist movements, or even majority voting. Once we think it is actualized, Emmet states, we open the door, at best, to complacency, and at its worst, to the spiritual pride which the Greeks designated as hybris. This strikes me as rather optimistic. The worst-case scenario, it seems to me, would be full-blown totalitarianism. On a theoretical level, it is not logically impossible for the entire citizenry being well-informed, fully disinterested, clear-sighted as to the common good, and in agreement. In practice, however, this situation is nigh impossible.[23] As Emmet rightly notes: 

politics is a sphere where those with different interests and viewpoints contend for power. The process cannot realistically be seen as expressing a single will. Still less could this be a single rational will; decisions are continually made with inescapably limited information, and even this is subject to different interpretations.[24]

As a result, differences of political opinion are practically unavoidable, and what counts as the common good or public interest is subject to continuous discussion. Who can accurately say what the public interest is, distinguishing it successfully from competing and usually more specific particular interests?

Even though claims of its full realization have to be critically discarded, we do in fact have criteria by which we can gauge approximation of the Ideal. As a very basic example, the higher the percentage of participation in elections is, the more ‘general’ the vote is, and the better it represents the voice of the people. There is more to the Ideal in question than voting in elections, however. Whereas public decisions are likely to be based on majority vote, and even though these decisions may indeed qualify as expressing the General Will, this cannot be concluded without qualification. The General Will goes beyond majority vote or public opinion in the sense that it requires people honestly considering the public interest – the interest of the community as a whole – with disinterested and informed concern for the common good. These requirements urge us to not only look at the procedures by which we can register public opinion, but also at the processes by which this opinion is formed. All of these factors qualify as criteria to measure approximation of the Ideal in question. At the same time, all of them, just as any practical standard, are subject to scrutiny and change.

The General Will should, I believe, be viewed in the same light that Slavoj Žižek conceptualizes the Hegelian notion of totality. This totality is not an organic Whole. Instead, it is a critical notion. Locating a phenomenon in its totality, in this sense, far from revealing some kind of hidden harmony, includes all of its distortions as its integral parts – its antagonisms, inconsistencies, and ‘symptoms’. According to Žižek, Hegelian totality is fundamentally marked by such distortions:

The underlying premise is that the Whole is never truly whole: every notion of the Whole leaves something out, and the dialectical effort is precisely the effort to include this excess, to account for it. Symptoms are never just secondary failures or distortions of a basically sound System—they are indicators that there is something “rotten” (antagonistic, inconsistent) at the very heart of the System.[25]

It might be contingent what these distortions are, but not that they are. In the case of the General Will as a Regulative Ideal, the task becomes to identify its distortions, while at the same time critically evaluating them. To what extent are they necessary integral parts, or contingent add-ons that perhaps be amended, or at least replaced by less harmful ones? What groups and opinions are excluded? What opposing views exist, and how are they incorporated or expressed in collective decision(-making)? What are the limits of specific modes of expression, of representation? To acknowledge these distortions is not only to acknowledge the non-instantiability of the Ideal, but also to acknowledge its importance as a Regulative ideal.

The inexactness of a Regulative Ideal is the seat of its power, the locus of its strength.”

The inexactness of a Regulative Ideal is the seat of its power, the locus of its strength. This is not a case of the tail wagging the dog. When it comes the General Will (and the same holds for the Good), its vagueness or elusiveness makes for public argument, and as such, contributes to political discussion and engagement. On a profound level, what is required in the realm of the ethico-political is a continuous back and forth – a never-ending interlocution – between our ideals and our reality. Regulative Ideals possess the performative force that keeps this back and forth in play, at play. The non-instantiability of Regulative Ideals guards the open-endedness, and thus the continuity of this process. At a fundamental level, the ethicopolitical realm is governed by ideals that, because of their inexactness, keep us engaged and in dialogue – indefinitely. Once this dialogue – or rather: multilogue or manylogue – grinds to a halt, we’ll be in the prickliest of pickles.


Whereas Regulative Ideals might lack an instantiable end-point, this text (thankfully) does not. By way of a final convulsion, it’ll turn to the Nicomachean Ethics, which assures us of the following: 

It is the mark of an educated mind to expect that amount of exactness which the nature of the particular subject admits. It is equally unreasonable to accept merely probable conclusions from a mathematician and to demand strict demonstration from an orator.[26]

Aristotle is spot on. In many – if not all – cases, the desired level of exactness, and thus inexactness, is related to a specific practical standard. Even what we call fact or factuality is structured by these (practical) standards. These standards are apt to change, however. We have grounds to conclude – in Wittgensteinian fashion – that, like meaning, (in)exactness is related to use. But as it turns out, some particular subjects do not only permit or impose inexactness; to a certain extent, they demand it. In order to keep creativity in play, and totalitarian ethicopolitics at bay, some inexactness is exactly what we need.

[1]      “In God we trust, all others must bring data”, American statistician W. Edward Deming once quipped.

[2]      https://www.merriam-webster.com/dictionary/exact

[3]      Feel free to read the word in a spooky-wailing-haunted-house-voice. In fact, feel free to read the rest of this entire piece in whatever tone of voice you desire. There are no rules here. Live a little. The world is your oyster.

[4]      Ludwig Wittgenstein, Philosophical Investigations, translated by G.E.M. Anscombe,
P.M.S. Hacker, and Joachim Schulte (Chichester, West Sussex, U.K: Wiley-Blackwell, 2009), §68, p. 37.

[5]      Wittgenstein, Philosophical Investigations, §88, p. 46.

[6]      Wittgenstein, Philosophical Investigations, §79, p. 42.

[7]      Answering the police-officer’s question ‘do you know how fast you were going?’ with ’95.089.345 millimeters per hour’ is more likely to result in a breathalyzer test than appreciation for your exactness.

[8]      With our teenage years characterized by the ‘almost Xteen’ denotation.

[9]      Wittgenstein, Philosophical Investigations, §105, p. 51.

[10]     Made all the more difficult due to the fact that the name ‘black hole’ is itself, of course, metaphorical.

[11]     Wittgenstein, Philosophical Investigations, §85, p. 44.

[12]     Wittgenstein, Philosophical Investigations, §198, p. 86.

[13]     Wittgenstein, Philosophical Investigations, §87, p. 45.

[14]     Wittgenstein, Philosophical Investigations, §71, p. 38.

[15]     Wittgenstein, Philosophical Investigations, §400, p. 128.

[16]     Dorothy Emmet, The Role of the Unrealisable: A Study in Regulative Ideals (New York: St. Martins Press, 1994), 2.

[17]     Emmet, The Role of the Unrealisable, 8.

[18]     Emmet, The Role of the Unrealisable, 3-5.

[19]     Emmet, The Role of the Unrealisable, 8.

[20]     Emmet, The Role of the Unrealisable, 8.

[21]     The Good might be the arch-Regulative Ideal, going by this criterion, seeing as the other Ideals might still be said to stand in its service.

[22]     Emmet, The Role of the Unrealisable, 34.

[23]     Emmet, The Role of the Unrealisable, 43.

[24]     Emmet, The Role of the Unrealisable, 43.

[25]     Slavoj Žižek and Nadezhda Tolokonnikova, “I would like to conclude with a provocation: Slavoj to Nadya, December 12, 2013,” in Comradely Greetings: The Prison Letters of Nadya and Slavoj (London: Verso, 2014).

[26]     Aristotle, The Nicomachean Ethics, Translated by Harris Rackham and Stephen Watt, (Herts: Wordsworth Editions, 1996), 5.