Wednesday 25 June 2014

Institutional Fear and Academic Performance

I had a discussion with a senior colleague the other day about ways to improve the quality of learning experiences in the institution. Many institutions in the UK and elsewhere have in recent years seen some very unpleasant restructuring programmes which left many out of work, and of those who remained, lower salaries and more work is not uncommon. Ironically, some of that extra work is in attempting to maintain the machinery of "academic quality": such are the strange ways of austerity! The rhetoric to 'ramp-up quality' and to ensure that teachers teach in the most efficient and effective manner, improving student retention  and the quality of learning experiences has become rather more shrill, as these issues are of increasing importance to the viability of the organisation. Such conditions lead to the permanent implicit threat of either "you're out if you're not good enough", or "you're out if your programme doesn't recruit students" or "you're out if overall student recruitment is down", or "we can replace you with someone cheaper". There's still a lot of fear around being fed by interventions beyond the initial trauma of the still-raw restructuring process.

I suggested to my colleague that if the organisation took deliberate steps to reduce the institutional fear, then academic performance would improve. He disagreed quite strongly: "where's your evidence?" he said. I admitted that all I had to go on was having seen frightened teachers teach - frightened teachers who are still excellent colleagues - the results are not very good. Not really compelling evidence, and I'm sure he wasn't convinced by my fluffy appeals for "managing the mood" in the institution. But I've been thinking a bit more about this. It strikes me that since we have really no idea about what is going on in educational processes (particularly when it really works), we should be careful about ruling out any causal connection - particularly one which intuitively at least doesn't seem unreasonable. What might lead us to rule things out are blindnesses which set in through over-attachments to theoretical paradigms, or shadow motivations by senior managers for increasing personal control, or confusion between popular behaviourist assumptions about management with the sociology and psychology of human interactions in classrooms.

The popular theoretical paradigm of teaching and learning is social constructivism. Social constructivism has roots in Soviet (for which one might read 'systemic') psychological theories of Vygotsky and Leonti'ev, american pragmatism (Dewey), and later in the cybernetics of the 1940s. In e-learning, Diana Laurillard established her conversation model as a catch-all for many popular theories of social construction drawing of Vygotsky, Piaget, Kolb, etc - but above all, from Gordon Pask upon whose conversation theory it is based. In recent years, these ideas have been picked up by George Siemens and Stephen Downes as they defended the pedagogical model behind the MOOC.

There are important reasons as to why Social Constructivism might not be right - indeed, why cybernetics itself has some important flaws. Recent work criss-crossing biology, philosophy, sociology, information theory and physics all seems to be pointing in this direction: Terry Deacon's biologically-inspired critique of cybernetics, Katherine Hayles's sociological critique, Alain Badiou's (and Roy Bhasker's) philosophical critique - together with the growth of new movements like Speculative Realism which see the roots of the problem in a post-Kantian inheritance which now needs to be unpicked. This has real implications for our learning theory which at least raises a question mark over the efficacy of social constructivism as a model of teaching and learning, together with all the ideas about 'good practice' which have been grounded in it. The central issue is "what's not there?" - it relates to the ways scientists understand the 'dark matter' of human engagement.

My greatest teacher was Ian Kemp at Manchester University (see He was an unconventional academic: he exemplified a kind of openness, intellectual generosity and endless curiosity not just about music (his field) but everything. Without a PhD he would never have got anywhere in today's universities! What was remarkable about him is his willingness to "tell it how it is", without the fear that less confident members of the department were prone to. He knew what he was about, but he was always willing to "pop his own bubble" (a phrase he was fond of), and he valued particularly highly the views of his students who, in his opinion, had the great advantage of "fresh ears".

I don't think what Ian did was social constructivism, or instructionalism (although you might have diagnosed elements of both in his teaching). He would have viewed any "ism" that was applied to his teaching with suspicion. What he did was engage on a deep human level, entirely comfortable to take risks, and lacking any kind of defensiveness. His own lack of fear gave his students confidence to know that nothing they said might be deemed stupid: the stupid questions were usually the best. What he revealed was, most deeply, himself. This was simply the most open and honest person I had ever met - irrespective of the fact that he was deeply committed to understanding music. It is this encounter with openness, honesty and the willingness to 'reveal oneself' which were the characteristics of my experience and the reasons why it has stayed with me for so long.

In such encounters, what's "not there" is important. Quite simply one cannot put one's finger on it. If it was a sound it would be some extraordinarily rich and complex harmony which is analytically unfathomable. (I remember an incident in a workshop with a particular complex passage in a Tippett quartet and Ian asked his academic colleagues "what's going on there then?" - they struggled to impress with clever analyses. Kemp smiled, saying "maybe.... but maybe it's just a nice noise!") If we were to ask about the conditions which allow the 'nice noise' of his teaching to happen, then I think fearlessness would be top of the list. The causal mechanisms around this are obscure - but whatever's going on, the effects are palpable: the encounter with fearlessness is visceral. We can speculate as to why this might be (this is one of my obsessions in this blog) but what is clear is that popular theories are deficient.

Managerial elites inhabit a strangely rigid world where teaching and learning are functions (delivered by functionaries, not real people) whose operations can be scientifically mapped out, and once they have been, they can be reproduced (and the functionaries replaced if they don't come up to scratch). My colleague is particularly keen on the 'scientific' metaphor to defend his idealism about teaching. The chief weapon that such people have for ensuring that everyone adhere's to the plan is fear. It takes courage to tell them they are wrong, and most staff in universities lack that courage to begin with. It takes greater courage still to demonstrate exactly how they are wrong. To do that, you have to become like Ian Kemp.

But they are wrong and the dissociation between power, theory and human reality spells real danger. Institutional fear cannot be dissociated from the business of thinking, learning and knowing. Institutional fear will skew knowledge and with it (if we not alert to the dangers) civil society. 

Tuesday 24 June 2014

Historicism and Institutional Fear: What price 'best practice'?

Karl Popper's "The Poverty of Historicism" carries a dedication:
"In memory of the countless men and women of all creeds or nations or races who fell victim to the fascist and communist belief in Inexorable Laws of Historical Destiny" 
Historicism is a common rhetorical device: sweeping accounts of history are used to provide backing to some crazy (and otherwise indefensible) scheme. But historicism, particularly for Popper, was a deeper trend in the social sciences where accounts of the past are methodically analysed to reveal 'laws of society' which are then used to predict the future. These two aspects, the rhetorical and the methodological, are related. For rhetoric, one need look no further than Hitler's speeches (for example,, but it is also inherent in the kind of TINA (Mrs Thatcher's "There Is No Alternative") formations that typified not only the draconian and self-serving measures of the 1980s, but also the austerity agenda of recent years. Without wanting to compare Thatcher to the Nazis (tempting though it is), the formal similarities are striking.

Rhetorical historicism's danger is that it slips off the tongue so easily and hypnotises an audience into uncritical acceptance. It abstracts everything; it refers to no real people, no real situation; it avoids everything fleshly, embodied and human - yet it is the surest sign that whatever hare-brained scheme the rhetoric of historicism is marshalled to defend, it will spell unpleasant embodied experiences for everyone other than those who promote the enterprise (and it will get them too in the end!). There is no more urgent and telling demand for critique: critique in such instances is the difference between slavery and freedom.

Audrey Watters at the #cetis14 conference at the University of Bolton last week highlighted the abuse of history in education, and the latent historicism in sweeping accounts of the development of learning technologies. These have the same purpose - to paint a picture of the inevitability of the current policy (or in Audrey's case, products) of the person or corporation presenting the account. Audrey's job is to critique this with historical examination.

Education is vulnerable to the abuses of historicism, both in terms of rhetoric and methodology. Because learning is both fundamentally metaphysical and continually concretised for the purposes of the education system, a history of the "development of learning and teaching" can pick and choose its way between narratives (for example, "once we had information transmission... now we have social construction of knowledge"): it all sounds plausible. But we can't see what happens in peoples' heads. So who's to know any better? The propositions are unchallengeable, which is why they appeal rhetorically. The confusion is there to be exploited! I can't imagine that Comenius's ideas of education, or Rousseau, Vygotsky or even Freire's ideas were really that different: teaching and learning doesn't really change because human beings don't really change. Only discourses about what we think teaching and learning is change (usually in the light of developments in the organisation of the education system, not through some blinding scientific insight) - but the discourse is not the same as the act of teaching and learning: the map is not the territory.

This creates problems when we try to determine 'good practice'. It creates more problems if we attempt to punish 'bad practice' with dismissal. The discourse of the 'good' and the 'bad' is framed by the needs of  the education system - its "quality procedures", its organisation, its funding - not the needs of society or the needs of individual learners. Its like trying to punish bad artists: it will come down to the judgement of the powerful as to what is good or bad, and at the very least, they are in a poor position to judge: Constable and Van Gogh find themselves on the rubbish heap, whilst Jack Vettriano is promoted to Dean (for commercial acumen and artistic endeavour)!

What is less clear is that the doctrine of 'good' and 'bad' practice is also historicist. It uses precedent from incommensurable situations, ignoring the incompatibilities and the details whilst pulling out general trends and features of the good or the bad. It abstracts surface features and characteristics which can be easily identified by those with little knowledge, but which tell nothing of deeper processes. Like all historicism, it absents real people, real bodies, real situations. Ironically, whilst education rhetoric rightly extols the virtues of inclusivity and accessibility, the very issues of "acknowledging the whole person" are excluded from the way that teachers themselves are measured.

Historicism as methodology was, for Popper, disastrous precisely because of its failure to describe whole situations: "if we cannot know the whole of the present state of mankind, we cannot know the future of mankind" (an important lesson for those interested in Agent-based modelling in society! - another story...) Furthermore, it is impossible to predict individual human action. And there is an added twist to the managerial imposition of methodologies for determining good and bad practice.

All methodologies in the social sciences interfere with experimental subject: there is no detached observer viewpoint. But nothing interferes with an experimental subject such as a teacher in a university more dramatically than a climate of fear. Whatever laudable aims one might have for determining good or bad practice, the implicit threat of redundancy produces oscillations in the organisation which produce entirely unpredictable results. If the fear extends (as it well might) to those who are in charge of the enterprise, who commissioned the inspection in the first place and who wish to assert their authority, it is likely that rather than admit a terrible mistake, they instead pursue their inspection policy with a zeal that only serves to increase institutional instability. The mechanisms are set in place for institutional disaster. What of teaching and learning then? What price 'best practice'?

Our current enthusiasm for managerialism in institutions rests on historicist doctrines which were once associated with communist and fascist governments. Now the same doctrines are at the heart of our institutions - and particularly those institutions like universities whose purpose is contested. It is up to the leaders of those institutions to read people like Popper and to spot the dangers inherent in their situation. It is up to the rest of us to remind them.

Thursday 19 June 2014

#cetis14: Granting permission to ask questions about education

The #cetis14 conference at the University of Bolton has been a great success. Although run on a self-funding basis for the first time (and consequently using the facilities of its home institution for the first time), it still attracted 100 delegates from the UK HE and FE sectors eager to talk about the impact of interoperability, cloud computing, e-books, systems integration and learning analytics. If anything, the conversation has been more eager, imaginative and focused than in previous years. This was helped by the two keynotes.

The first keynote was from Phil Richards, Chief Innovation Officer at JISC. As Phil acknowledged, this was always going to be a slightly tetchy affair, since government reorganisation and JISC restructuring has meant a marked reduction of support for projects and services (like CETIS) in institutions about which people are understandably upset. Phil presented the new vision for the future of JISC innovation first by critiquing past mistakes. Quite simply: too many projects didn't go anywhere, the benefit to the taxpayer was not clear, the bidding process obscure... Stuff which I and many others have commented on before. But at the same time, I thought, the value of JISC projects was that they gave participants permission to think about education, in circumstances where this would otherwise have been impossible. It was this business of 'asking questions about education' which seemed curiously absent from the vision of the 'new JISC': it seemed that the new JISC vision is to think about keeping JISC going, not thinking about education. When explicitly asked about who in JISC was asking the 'big questions', the response given was "people above my pay grade". The old JISC was good at getting everyone asking those questions, and the conclusion is that the movement from old to new JISC is a movement from what was a 'committee' to something rather more autocratic (which kind of mirrors what's happened in our universities!). For CETIS and its diaspora, asking the big questions about education has always been the key thing. It's ironic that the movement from committee to autocracy has been facilitated by technology!

The second keynote, from Audrey Watters, was the kind of wake-up call that should send all those who think about education and technology into the new academic year with renewed determination to take on the self-serving forces which are currently carving-up education. Her focus was on the 'edupreneurs', Pearson, Blackboard and co, whose chief weapon is not spectacular software or innovation (because their products are neither spectacular or innovative), but the promotion of cultural amnesia and the re-writing of history. Spot on. Audrey wants us to really think about education and technology, about how we got here, about where we are in history, about where we are going.

In between the two keynotes, technically-focused workshops looked at developments in technologies and standards that will affect us all. I attended the analytics strand and helped with the session on cloud computing and systems integration led by Scott Wilson. I was sorry to have missed the session on e-books, because I'm beginning to think there's something really important in the whole issue of bounded, authored content which the messy web on its own cannot deliver. But all of these provided a permission to ask questions. Technology's value is often not in implementation, but in illumination: it provides a torch where we re-inspect ancient practices.

Andrew Feenberg (who gave the CETIS conference keynote a few years ago) has commented on the need to situate technology as part of the democratic process.
“Technology can deliver more than one type of technological civilization. We have not yet exhausted its democratic potential” ("Between Reason and Experience: Essays on Technology and Modernity", p29)
The question is how we go about doing this - particularly at a time when technocracy sets in to serve the interests of managerial elites. There's something important about the technical discussion in CETIS: it provides a clear focus for debate which seems strangely absent from the discourse in education more generally. One delegate commented on the difference between the discussion at CETIS and the discussions at the Association for Learning Technologists (ALT) conference, which is far bigger: "it's too general - nothing gets carried forwards. CETIS gives you new tools and ideas you can think with." So focused technical discussion does much more than address the issues of the technology. The standards discussion is particularly interesting in this regard, because standards are not about tools; they are about the 'in-between tools'.

As education screws itself deeper into the ground - facilitated by the technologies it has commandeered - we should hope that the critical debate about those technologies, their implementation and development serves to give us permission to ask the questions about education that urgently need to be asked.

Monday 16 June 2014

More on Social Networks and Category Theory

Recently I've been thinking a lot about learning analytics, big data and all those pretty pictures of social networks that people are producing in tools like Gephi and R. There's something important about this stuff. Maybe the most important thing is it is something which fascinates us, but which is also very difficult to pin down. Is this just a variety of visualised statistics? But then, statistics has the hold it does over the social sciences because it produces something like 'empirical regularity' by averaging events, and then inferences can be made from the statistical regularities (we have understood for a long time a statistical correlation between smoking and cancer, although the causal connection has only very recently been discovered). The issue for social network analysis is that there doesn't tend to be an emphasis on establishing regularities. In which case, the visualisations do not seem to be part of an empirical effort. That means it's about something else.

In place of regularity, we have visualised 'patterns', 'clusters', etc. The identification of these leads to questions about similarities between different situations which pertain to each cluster and pattern. The Triple Helix people, for example, on looking at clusters of industry and universities, will point to the discourse dynamics in those regions and the intermixing of academic labour with industrial innovation. But I wonder what Victorian biologists would make of this as scientific procedure. They too looked at pattern and cluster (for example, on a butterfly's wing), but their efforts were to establish the regularity of occurrence of pattern, identify its genus, species, etc. Unfortunately social networks are much bigger than butterflies, and the pattern of a network tends to be a one-off. However, this is not to rule out the possibility of a typology of social networks - but I don't see it happening anywhere - certainly not in the learning analytics world (and maybe it's not a good idea!). Typologies would produce a different kind of regularity.

Category theory tells us about a typology of connection - and maybe that's a starting point. My understanding of it is only beginning to emerge (I may have to revisit what I say now, but I have to start somewhere!) To begin with, we have the difference between a monomorphism and an epimorphism as the differences between types of connection. I see these two mappings as being essentially about the differences between the contexts within which a person might say there is a connection. There are some connections where there is a mapping to a single point from a number of points: it is a point of 'focus' - perhaps it's like a 'resolution' - for example, the resolution of a musical dissonance. Mapping onto, often termed 'surjective', is related to the concept of epimorphism (although not equivalent to it). A 'monomorphic' mapping is different from epimorphism in the sense that monomorphism maps points uniquely onto other points (for example, mapping chairs onto students). Monomorphism and Epimorphism can be expressed using an arrow notation which in turn reveals the algebra. Monomorphism is written:

Essentially, this means that the map i¡f = i¡g --> f = g. Badiou argues that a monomorphism preserves difference. That is to say that if f and g are different, this difference is preserved in i. Monomorphic, or one-to-one mappings are related to what set theory calls 'injective mappings'.

The dual of the monomorphism is the epimorphism. In an Epimorphism, we start with an identity (a single arrow) and it is followed by the possibility of difference immanent in that identity (2 arrows).
It seems that the possibility of introducing difference and preserving identity in transformations between objects begins to open out an expressive power in the category theory notation that goes far beyond the simple 'connecting' of points. Thinking in this way introduces its own algebra, but an algebra which embraces the degrees of uncertainty and absence which get lost in our fascination with social networks. 

The algebra begins to appear when we start to think about how the mappings between objects might be viewed from other objects. For example, it is a common situation in a social network for a node to have a number of points pointing at it. In category theory, we can speculate on the objects which sees all these other objects. Such a situation where there is a central object which can see the other objects and which presents the internal logic between those objects is a kind of universe called a 'Topos'.

If we look at a social network graph as a Topos, then we examine its topology asking "who can see what?", "what is it like to be an inhabitant of this topos?". But more importantly, we cannot ask those questions without asking of ourselves, "where am I in this topos?". That, to me, is the really important question - and the reason why category theory may provide a solution to the mystery of why we are so fascinated by these topologies.

Wednesday 11 June 2014

Category Theory, Society and Technology

Category theory is a mathematical formalism that can be used to represent many aspects of mathematics within a coherent framework that uses 'mappings' (i.e. arrows) between objects (i.e. points). It was formulated by Saunders Mac Lane in the 1940s - that is, at the same time as other remarkable mathematical and technological developments were taking place which led to the development of Cybernetics (from the Macy conferences in 1948) and the succeeding developments that span out of Von Neumann, Wiener, McCulloch, Shannon, Mead, Bateson and Ashby in economics, computer science, telecommunications, psychology, psychiatry, anthropology, philosophy, education, art, management science, and so on. The metaphor of 'mapping' would have found particular resonance with the functional transformations and feedback loops of computers, and indeed, Category Theory has its strongest foothold in computer science departments as an important formalism for thinking about algorithms.

My first encounter with it was when trying to understand some fascinating work by Louis Kauffman on "Time" at the American Society for Cybernetics. Kauffman presented a way in which time could be seen to be immanent in processes of perception, producing an elegant description that showed how the square root of minus one could be seen to be a "clock". Powerful stuff but quite challenging in the depth of its implications and importance (I am convinced that 'time' is one of the biggest problems in cybernetics - no mechanism works outside time! - and I'm not entirely convinced by Kauffman's explanation, but no doubt my understanding will deepen). My reticence in accepting Kauffman's  description of time partly revolved around my suspicion that cybernetics has not been entirely successful in dealing with 'absence'. Bateson wrote well about absence, but I still felt that the whole business of cybernetic 'difference' is too 'singular'. Differences to me are never 'points' - they are always surrounded by a context, much of which we can't see. Differences have dark matter.

But then a curious coincidence was encountering Alain Badiou's work. There's much in Badiou which is consistent with Critical Realism, which I have been interested in for a long time (but which also has some problems). Most importantly for me, Badiou too has dealt with absence. Equally important, he has taken on the problem of 'difference' - or what he prefers to call the 'event'. But the really cool thing about Badiou is he knows his maths - and in recent years, the branch of maths he has done most of his work is Category Theory. It's always when I get the same thing from two different places that I start to take a serious interest.

Since then I've been reading Badiou's recently published "Mathematics of the Transcendental", whilst glancing at Lawvere's "Sets for Mathematics" and other texts. Understanding emerges only slowly with this stuff. But the DIAGRAMS are fascinating. Here we have networks of lines and algebraic interpretations which say things about absence, and being. In fact, Lawvere has even written recently about Category Theory being able to characterise "becoming": phenomenology has always suffered from a lack of formalism (although Husserl's wrote some fascinating things about geometry).

Why are the diagrams fascinating? Because they look like social networks! In the early days of category theory and computer science, I guess the network that mattered was the wiring between components and the transformations that occurred. It seems to be that the archetypal image of computing now is the diagram of electronic utterances between people connected by a network, or the diagram of document terms in a corpus: the diagram of the relation of ideas. On the whole we are poor at thinking about the meaning of these topologies. Gordon Pask complained years ago that the digital computer fascinated us like a 'magic lantern' and actually stopped us thinking, rather than being a tool for thought. Von Foerster made a similar comment (in 1971!) that
"we have, hopefully only temporarily, relinquished our responsibility to ask for a technology that will solve existent problems. Instead we have allowed existent technology to create problems it can solve."
That was before the internet. Things are worse now.

The question is how to think about our existent problems. The problem is that technology is part of our existent problems. Social network diagrams and our poor understanding of what they mean are part of our existent problems. We need better ways of reasoning about the world, and by extension, reasoning about technology. That means using our analytical and critical capacities to dig into the underlying logic of what is happening to us. Maybe Category Theory can provide a way of doing that.

Thursday 5 June 2014

Surreal and Transcendental Social Networks

With apologies to RenĂ© Magritte and @paulhollins (who gave me the idea) I think it is useful to be reminded of the distinction between reality and pictures. We are currently investing certain pictures (not unlike the one below) with mystical properties which we believe grant privileged access to an 'objective reality'. This was Magritte's target when he painted his pipe: but for him it was a clever joke about mimesis and representation. For us, and for the pictures we now confuse with reality, the consequences are more serious: policies are made by looking at such pictures, people will get hurt, others will get rich and some will get elected. 

Having said this, it is not quite the same joke as Magritte's, because the pictures of social networks can lead to actions whose consequences appear to be consistent with the logic of the picture. This is to say that a social network picture is a 'logical representation': one whose formal characteristics are isomorphic to lived experience. The question about the nature of a social network picture is a question about the relationship between reality and its logical representation. It is analogous to the relationship between geometry and the natural world. 

Descartes saw geometry precisely in this way: that activity of the mind which connected with the body to produce relations which were isomorphic to transcendental reality. I think it is awareness of the connection to the transcendental which is missing in the contemporary obsession with social network graphs. We have forgotten about our bodies and we have forgotten about what we cannot know. 

The connection between the transcendent and measurement is beautifully expressed by Blake's "The Ancient of Days". I would suggest that instead of measuring the heavens, Blake's Urizen (the bearded man) is now measuring pictorial depictions of reality. That's what the social network analysts are doing. The irony of this is that Urizen represents rational man. Instead of rational man encountering the heavens and constraining them with his compass, he now encounters the artefacts of his own technological making, similarly and vainly trying to constrain them too. We appear to have gone up a level of recursion!

But geometry and its relation to the transcendental haven't gone away. When we look at a picture - particularly a picture of labelled nodes and arcs - we are examining a logical structure and piecing together its inherent logic. In so doing, we re-encounter something fundamental about the relation between logic and the natural world, and in this case about the relation between geometry and the natural world. Geometry is literally "earth-measurement": that means it is precisely at the point of encounter between reason and experience. Whatever reason can bring to bear on the geometer's constructions, experience and the body moves in harmony with it. Geometry, fundamentally, is an art of harmony. Is it so fanciful to suppose that the earliest geometers (and the ones whose work most closely resembles our social network analysts) were the astrologers for whom the 'music of the spheres' played such an important role as they examined the starry heavens?

When we take those social network geometries and we apply the logical reasoning of formal techniques like category theory, we may be able to uncover something of the territory of our experience of those graphs and pretty pictures. At a time when we are so confused about the import of the miraculous and enchanting pretty pictures that our technology presents to us, revealing the deeper logic behind those pictures as the logic of our experience might be extremely valuable.

Sunday 1 June 2014

Social Network Analysis and Category Theory: Rethinking the logic of connection

At an illuminating presentation by David Knoke at the Triple Helix Workshop ( last week, I found myself staring at his diagrams of nodes and arcs and asking myself "but what is this about?". David is very much alive to the question (the current trend for big data pictures can be frustrating and confusing) but "what is this about?" has remained one of the biggest questions as I think about the impact of technology on education. The problem is that these diagrams have a veil of empiricism about them - as if they in some way grant an objectivity to social processes which we have traditionally regarded as subjective. But empiricism is about event regularities. Where are the regularities?

Something David said then prompted what felt like a light-bulb moment. "Here we have a social structure" he said pointing to a diagram. That was it. "It's not a social structure. It's a logical structure. The meaning we attribute to it arises from our exploration of the logic.". "But I think there there are relationships expressed in those diagrams," said another attendee. "No, I said. This is a relationship between you and me now. That's a picture." I don't know what David thinks, but I found the exchange to be the highlight of the conference.

There is something fascinating about the Triple Helix dynamics between Universities, Government and Industry. At the conference I found people on the whole rather too receptive to simplistic explanations for complex phenomena (such as the clustering of industry and universities), and unwilling to engage in deeper critique (a point made by economist Mark Casson who laid quite a few powerful blows to the Triple Helix idea). But on reflection, Triple Helix embraces my own concerns of the relationship between society and eduction, together with the prime importance of economics. But it needs more self-critique.

The fundamental question is, What is the nature of the critical inquiry? Is it an analytical inquiry? Is it an empirical inquiry? Is it political (there was a distinct absence of politics - which bothered me)? Is it phenomenological?

This led me to think about Badiou's fundamental question about the relationship between the analytical, critical (i.e. Marxist) and phenomenological. But more interesting is Badiou's method which is to turn to a logical investigation using mathematics. In his book Logics of Worlds, he turns to Category Theory as a way of looking at the ways that events relate to thought and action. I've come across Category Theory in other contexts, not least in Cybernetics, where it is the favoured tool of Louis Kauffman as he explores the deeper recesses of cognition and perception. Category Theory has also been influential in computer science, and become a popular way of formalising algorithmic structures.

Badiou's take on Category Theory is, I think, the most profound. It is also the most accessible, since he has recently published his mathematics introduction as "The mathematics of the Transcendental". As I turn the pages of this remarkable book and examine the diagrams of function mappings, pushouts and pullbacks, etc, I couldn't help thinking about David Knoke's diagrams. I don't fully understand any of this yet. But if there is an understanding of social network analysis to be had, I am more convinced of the importance of the ways in which we read logical structures. Fundamentally, it's all a bit Cartesian. Of course, that tends to be taken in a derogatory way these days. But Descartes was doing something very important in the "Meditations" and the "Discourse on Method"; and the relationship between geometry and thought is perhaps something that we should be thinking about more carefully as we get swept up in pretty diagrams.