Sunday, June 30, 2013

Grouping Theories

Natasha Vally

Drawing on Jane Guyer's lecture "Is Confusion A Form", Natasha Vally unpacks mathematical group theory as a way of engaging varieties of form and their relationality.
The mathematics department at Wits is somewhere called “the annex” on the third floor of central block “close to Sociology”. It was only in my third year of a mathematics degree that I actually found the department, by chance, and when it really wasn’t needed. The sense of confusion and of the need to find somewhere and something at a relevant time but it being stubbornly out of reach was how mathematics always felt to me.
This blog post won’t deal in any systematic way with today’s not-lecture on “Confusion as a Form”. Perhaps like the speakers, I think that to detail in a linear way the content of a lecture on confusion would be dishonest and ill in formed. There is however reference to and shameless phrase-borrowing from some of the themes, terms and forms that were presented in this morning’s session.
Jane Guyer invoked mathematics and a need to engage with the ways in which disciplines that we are less familiar with encounter confusion. I luxuriated in Guyer’s mention of Boolean groups knowing very well that the smugness of understanding something needed to get me through the many anomalies and unknown categories of confusing future lectures. Groups, though, there is a place I can fit in.
Group theory is the conductor in the orchestra of mathematics. It is strictly taught as a method for analysing abstract and physical systems.[i] You spend years solving for x - listening to the music through your headphones – and then you realise that there are organising principles which, baton-wielding, conduct and order the possibilities of x. The x you were solving for could not be any value that fit nicely, instead the possibilities of its existence were bounded. Neo Muyanga, a leitmotif of the Workshop, has an album – Dipalo – where the track names are mathematical equations. He reminds us of the order in the tunes we drift away to and the drifting away in the numbers we are attuned to. The reason many people tell you they like mathematics is that a clear answer is possible. It’s a lie. The axiomatic assumptions and background work allow for the illusion of an unambiguous answer and it is in group theory where some of the foundations which format what is and isn’t possible unfold.
There is something romantic about Mobius strips which several of the talks on confusion elegantly knotted into their analyses. Geometry is a more obviously tangible and visceral mathematics. The lack of beginnings and ends appeals to our attraction to the undoing of binaries. It also makes for the writing of good papers and art because we like punctuation and lists of words: beginnings/ends, rise/fall, day/night, (dis)order. But we should also keep an eye on the beginnings and ends, partly because they are created and movable and thus open to subversion. This too has been a theme of the workshop – what are the possibilities of using the discomfort of anomalous forms to politically intervene to shift meanings and action? These are social and material considerations. 
Source: cdninstructables.com
Avoid the temptation to shut down when you see the letters and symbols below which float outside of words and vocabularies that you may be familiar with. Group theory invokes what Filip de Boeck, in his paper read by Guyer, calls amalgamation, where the theory tries to knot equations into the meta-discourse of the group. There are four rules to qualify something as a group. They display many of the concepts and themes raised when discussing confusion and order. What the overview of group theory is intended to do is to foreground the disciplinary similarities in the categorisations which we use to express belonging and exclusion in numerous fields. It is not in any way mathematically rigorous.
If you can remember, take for example the equation
2 + x = 3
The task is to figure out what x is. Because we need to do the same to either side of the equation (to maintain the equivalence), we get
2 + x -2 = 3 -2
So, x = 1
But this was based on a presumption that we were letting x be a positive number. If x only belonged to the category of negative numbers then there would be no x to satisfy the equation: we couldn’t subtract anything from 2 to give us 3.
What needs to be taken away from this is that a decision is made (provided/accepted/imposed) as to what “things” x can and cannot be.
Now, to generalise this
Say you have a • x = b
Then in group theory you ask these questions: What objects are a and b? To what class of objects is x allowed to belong? What is the operation 
symbolized by the dot (•)?
The four “rules” that a mathematical sentence need to obey to be a group are:
  1. CLOSURE: If a and b are in the group then a • b is also in the group.
If two elements are part of a form and you perform an action on these elements then the result is part of the form
  1. ASSOCIATIVITY: If a, b and c are in the group then (a • b) • c = a • (b • c).
If you perform an action on elements constituting a form then as long as the sequence of elements remain the same, the order of their grouping does not affect their belonging to that form
  1. IDENTITY: There is an element e of the group such that for any element a of the group
    a • e = e • a = a.
There is something in a form that when acting on an element gives you the result of that element itself
  1. INVERSES: For any element a of the group there is an element a-1 such that
    • a • a-1 = e
    • a-1 • a = e
The point should be apparent though. These are some of the sorts of questions we ask when we think through forms, their content, their thingness and thinghood, their political and material ramifications and the important question of what happens when they are not actually an acceptable group, what happens to the thing x then? Struggling with different groups of theories may allow for a new way of getting to know the lives of forms we’ve been introduced to so far.

[i] Group theory is an abstraction of symmetry, the notion that an object of study may look the same from different points of view. While it is relevant here, it may involve more mathematics than I can remember and more symbols than Word easily makes available.
Natasha Vally is a PhD student at WiSER, University of the Witwatersrand

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