Saturday, 26 July 2014

Mathematical Quantification and the Order of Education

We have grown accustomed to almost all empirical investigations in either the physical or the social sciences as involving some level of quantification: x% this, y% that, and so on. The foundation for the fiduciary qualities of this kind of work sits on an ontology of numbers where the truth becomes associated with the higher number (100%!). Behind this lies the view that numbers exist on a continuous scale. Yet mathematical work in analysis and number theory questions this. From Dedekind to Cantor, the fundamental issue with number is not continuity and quantity, but ordering and the way that numbers exist within limits. Indeed, quality is something that may lie within the mathematical ontology, rather than something to be deduced through processes of quantification. There's so much that's troubling with qualitative research: not least that almost always, qualities apparently only make themselves amenable for analysis through quantification; indeed, technologies have become instrumental in the industry of the transformation of qualities into quantities. Work done in this way - for the benefit of evidence-based policy (which as Hugh Willmott pointed out today is really 'policy-based evidence') - has real and often negative impacts on the lives of real people. Mathematics is beautiful, and its distortion which produces these effects demands that for all these reasons, it may be important to look again at number and mathematical ontology.

I've found myself studying the Category Theory of Mac Lane, Goldblatt, Badiou (who's taken much from Goldblatt) and Lawvere (who writes particularly beautifully for the uninitiated). Category theory is a development of set theory which works on the principle of describing processes of transformation between different states of constitution (my term - basically it's a set), where a "state of constitution" might be called an 'object', and a transformation might be called a 'mapping'. Most importantly, Category Theory gives us a way of describing ordering without numbers.

My educational empiricism is a concerted effort to study the ordering of education. That means looking at the logical structure of the relations between people, objects, institutional structures and so on. The relations between people, objects structures are (as John Searle and Tony Lawson independently insist upon) networks of rights, responsibilities, obligations, commitments, and duties. We can empirically discover some aspects of the structure simply by asking people questions like "who tells you to do x?", "who are you doing it for?", "what happens if you don't do it?", and so on. At the same time, a structure ought to make the distinction between what a 'right' is, what a 'responsibility' is, what an 'obligation' is, and so on. In each case we see a different form or geometry of the ordering, but where for each form, there is a central idea of a 'limit' with which a particular obligation or commitment might be identified.

Category theory has a number of basic forms which might map onto different kinds of relations.
The picture above is the Category Theoretical 'epi-morphism' which is characterised by the two arrows from B to C. I look at this and think of those situations where people seem to acknowledge that 'C is the case', although they might do so for different reasons (hence the separate arrows). In each case, the two arrows stem from an initial single arrow f which is the beginning for each of h and i. But you can also stand outside the whole situation with arrow from a position that 'sees' A, B and C (although I haven't drawn the arrow to C in this case). An obligation, in this sense, appears to me to be a limit where the arrows h and i are acknowledged to be the same and that the declaration of equality between them is a statement of compliance with each others norms. This limit, a unique position of balance, is indicated by arrow k. This might be the politician's stance: the obligation to coordinate education that conforms to the norms of stakeholders in society.

Turning this diagram around, we get the category theoretical 'monomorphism' which starts from a diverse position to become a single point. The starting point of arrow g  indicates the viewpoint of being able to 'see everything (B, C and A). The limit line here, k, might be seen to be the point at which an identification of the difference between h  and i must be made. Is this a moment of 'responsibility taking' - the moment at which someone has to make a judgement the things that matter bearing in mind the different perspectives feeding into it?

There are also structures where two lines focus on a particular object (like the lines from D and C lead to E), where there is a point B that can see these relationships, but where is a point A that sees the whole situation including the observer B, and where A determines a limit on B. Category theory allows us to move up hierarchies like this (the diagram above is called a 'pull-back'), at each point challenging us to think about the limits imposed. Maybe 'rights' are limits imposed by the super-structure on the sub-structure? But then again, there is a difference between 'you have a right..." and "we demand our rights!". But to demand rights is to demand that A changes its limits. 

Perhaps that will do for now. The point here is that behind each of these diagrams is an inherent logic and ordering. The strength of category theory is that within this inherent logic are particular orientations towards truth, falsity and absence where in each case, the structural relationship between absence (say) and limits may be explored.

The absence bit really interests me because in addition to asking people about their rights, obligations, commitments, and so on, we can also observe their redundancies. I only have a hunch that absence is the same as redundancy (I gather Lacan - Badiou's teacher - held a similar view), or that Hume's regularity theory is really a redundancy theory (Tony Lawson rightly challenged me on this today), but I am also mindful of the work on pattern, figure and ground which people like Ernst Gombrich conducted in his "A Sense of Order": we are so 'figure oriented' in our approach to empiricism across all the sciences.

There is a discoverable order to education. There are practical steps we can take to unpicking it. This is not learning analytics! That (analytics), along with all our technologies, must now be considered as part of the contemporary "order of education".

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