Relationism and Impredicativity

Whatever criticisms people may have of the relationism vs positionism debate, it has clearly sparked many necessary conversations. 

Football is obviously more analysed now than at any time in history. Every match is filmed from dozens of angles, wearable tech lets us track every millimetre of movement to the millisecond, and probabilistic models seem to be able to tell us what should have happened at any given moment. We see that this is having some impact on the game, but we also sense that the datafication of the game has missed something important. Relationism has tapped into the desire many have to express exactly what it is that has been left out. 

I make the case here that the concept of impredicativity is central to what has slipped through the cracks of positionism. 

Let’s start with Russell and predicativity. Russell wanted to ground mathematics in set theory. To avoid paradoxes, he devised the following plan:

“An analysis of the paradoxes to be avoided shows that they all result from a kind of vicious circle. The vicious circles in question arise from supposing that a collection of objects may contain members which can only be defined by means of the collection as a whole” (Russell & Whitehead, 1910, p. 39).

So Russell began to separate the good (predicatively defined) sets, which didn’t contain themselves, from the bad (impredicatively defined) sets, which did contain themselves. This is where he ran into his great eponymous paradox. The set of all the good, clean sets could not be placed in either group, it appeared. The set of all the acceptable sets - all the sets which did not contain themselves as members - contains itself as a member. 

Not long after, Kurt Godel dropped the famous incompleteness theorem and impredicativities were cemented as a necessary part of the natural world we live in. It is also worth noting that a complex system is one that has irremovable impredicativities. A definition which is impredicative is essentially one that refers to an embedding context rather than a lower-level component or a context-free unit (e.g. metres, kilograms etc). Stating Sarah’s height in centimetres is predicative. Saying that Sarah is the second-tallest person in the room is impredicative because it requires us to know the context (room) which Sarah is in. It is only definable as a self-referential loop between the part and the whole. The fact that Sarah is the second-tallest person in the room is not subjective, however. It is a relationship between Sarah (member), and the room (set that she is in). If she moves to a new set, her height in cm will not change, but her height-order in the new room likely will. Impredicativities are context-dependent. To put it in football terms, the second-to-last defender that sets the offside line is also defined impredicatively. 

Impredicativies are also what led William James (1912) to call experience ontologically “double-barrelled”, and unable to be defined as either subjective or objective. As with Sarah’s height-order, we perceive something that is neither subjective nor objective (in the classic sense of context-free). Why does this matter? Because in performance contexts like sport, the experience of the performer must refer to the external physical world, but with the performer’s action capabilities as units of scale.

In other words, if we consider what perception must refer to in order to support successful action, we are led directly to impredicativity. Performers must perceive the real physical environment, but with reference to their own action capabilities. A performer has little need to determine how many centimetres a gap is. They must determine how the gap refers to their abilities. Is it jump-across-able? Step-across-able? etc. Performance is successful when it is able to identify these relationships. They are what Gibson (1979) called affordances.

Impredicativity also has implications for the concept of space. In the classic “coastline problem” it was shown that the length of the coastline of the UK cannot be given without implicating the unit of measurement. A long measuring stick, say 10km lines, will give a short measurement because it doesn’t pick up on small coves and bays. As the measuring stick gets shorter, the measured length of the coastline gets longer. It wasn’t until the middle of the twentieth century that maverick mathematician Benoit Mandelbrot took on this monstrous problem of roughness and imperfection. What he realised was that these structures broke the rules of Euclidean geometry. In neat, clean Euclidean space, things spread out in integer dimensions. One dimensional lines, two dimensional planes, three dimensional volumes etc. The imperfect natural objects often seen in the natural world - like the coastline of the UK - are between these clean integer dimensions Euclid just assumed space consisted of. The term fractal comes from the latin “fractus” for broken, and refers to these fractional, non-integer, dimensions. Euclidean space has been considered an a priori in many philosophical systems. The fact that it was incomplete, and that other geometrical systems existed sent great shockwaves through philosophy.  

   

Fractals are the geometry of self-reference. A measurement of the coastline of the UK is true in a self-referential way, but it is still true. It is a real relationship that exists between the coastline and the length of the measuring unit. 

It is this non-euclidean, impredicative, definition of space that I think relationism wants to get at. 

I see relationism as being about the impredicative, relational units of the pitch. The pitch in terms of what it affords a group of players. But should this really come as a surprise? Where do the dimensions of the pitch come from? Why should an official pitch be roughly 120x80m? The dimensions of the pitch force  us to see that the pitch is already defined (scaled) to our action capabilities. Whatever the predicative measurement (metres) happens to be is incidental. For example, an 80m wide pitch is just about pass-across-able for high-level players. It is clear that something along the lines of this ability is the relevant unit of scale, and however that relates to the context-free unit is secondary.

These impredicative relationships are what underwrite the conditions of possibility in sport. Direct perception is the understanding that perception does not refer to objective, context-free units of measurement like metres, but neither does it refer to a subjective copy of the environment inside the performer. The performer sees the real, physical world out there, but in units that are scaled to their own action. No other story can support the precise relationship to the physical environment that we see in all skilled performance. The relational pitch is neither subjective, nor objective. It consists of real relations. Real because action must conform to the world, and relational because it must conform to the world as measured by the performer’s abilities. 

Empirically, we know that the structure of interactions in sport are well-characterised by fractal geometry. We see relations of self-reference at many scales. Scales rippling and flowing far beyond the pitch. This suggests that the conversations around relationism are not just hipsters arguing over aesthetics. To the extent that much of tactics discourse and data analysis tries to characterise the game predicatively, a very real (relational) part of the game is dismissed. 

This is why I continue to bet big on multifractal approaches in my own research and for the future of football analysis, by the way. 

Sources:

Gibson, J. J. (2014). The ecological approach to visual perception: classic edition. Psychology press.  (Original work published 1979) 

James, W. (1976). Essays in radical empiricism. Cambridge, MA: Harvard University 

Press. (Original work published 1912) 

 Russell, B., & Whitehead, A. N. (1910). Principia Mathematica Vol. I.