Notation, Constraint and Logic

I’ve been doing some experiments with notation in music and video. I’ve got a great music writing app on my tablet called “StaffPad” which does handwriting recognition (so I can handwrite squiggly notes and the software converts them into proper typeset notes) – which is great, if a little bit fiddly. However, it also has a facility for simply drawing on the score. When the score plays back, it scrolls the music, so both drawing and notes appear.


To write a note on a score is to give an instruction. There is a question about whether the instruction is about exactly “what to do” or it is in fact “what not to do”. In other words, does the symbol on the score denote the sound, or does it contribute to the conditions of freedom within which a performer might act freely?

I squiggled some shapes on the score, and then I attempted to “play” it. I should have made my squiggles a bit easier to play! I did this a couple of times. The sound that I produce can be considered as “alternative descriptions” of something. The symbols/squiggles on the score are also descriptions of the same thing. If there is any similarity between these different descriptions it is in the fact that both the graphical description and the sound descriptions have similar entropy: in other words, what counts a surprise in one, counts as a surprise in the other.

Notation is obviously different from a recording. A recording is a faithful description of exactly what is done. Notation is an invitation to create multiple descriptions. The parameters as to what is permissible and what isn’t is contained in the way the notation conveys the flow of entropy over time.

So what about other kinds of marks or notations which we use?

In logic, I can represent the statement “All humans are mortal” as ∀x:human(x) → mortal(x). What’s the difference between these? The variable x is an invitation to generate possibilities – alternative instantiations of the formula. They produce constraints on the imagination bounded by the ways in which the symbols might be manipulated. The meaning is not in the phrase “all humans are mortal”, or even in ∀x:human(x)→mortal(x), the meaning lies in the interplay between the different descriptions which are made in the light of the notation.

We misunderstand formal logic as a denotation of reason. Really it’s an invitation to generate multiple descriptions from which reason is connoted. This mistake is why attempts to prove computer software in formal systems has failed. If we understand the relationship between logic, notation and meaning differently, then we can find new applications for logic. Education is one of these.

Theory, Explanation and Prediction

The word “theory” means different things in different contexts.

Mathematics: Mathematicians use “theory” with reference to things like “number theory”, “set theory”, “group theory”, “category theory”: basically, different kinds of formal system whose properties can be explored and can often be mapped on to other formal systems: for example, category theory (which is much in vogue at the moment) presents ways of accounting for number theory, set theory, etc. Like those systems it accounts for, it is a self-enclosed formal system.

Physics: Physicists use theory to explain and predict physical events like gravitation or quantum entanglement. Physical theories and mathematical theories are closely related: calculus, for example, was developed as a way of describing the motion of planets. There is some argument as to how physical theories are constructed or discovered: classical science sees theory as the constructed result of the observation of event regularities in nature, for which communities of scientists agree causal explanations. Many of the classical arguments for the construction of theory have been challenged by relativity and quantum mechanics where observing becomes part of the scientific/methodological process, and bias and ego of the scientist, or the power dynamics of institutional science feed into theoretical claims.

Social theory: At its origin, social theory followed the classical scientific model: it was assumed that “event regularities" could be established in the social world through statistics. With statistical regularity, the same process of constructing explanations could be established. Today, we call this positivism, and it was in evidence in some of the early industrial improvement processes in Taylorism or Fordism. This has become the root of arguments about method. Contributions from phenomenology (which grew from mathematics through Husserl), psychoanalysis, philosophy and economics has led to conflicting views about the use of statistics in social science (Keynes, Hayek), subjectivity vs. objectivity in observation, value freedom (Weber), intersubjectivity (Husserl, Schutz), Knowledge vs Action (Marx, Lewin), realism vs constructivism (von Glasersfeld, Archer). Education sits (partly) in this theoretical mess.

Psychological Theory: Like early social theory, psychological theory often pursues a classical science model. Experimental conditions are established, experiments are performed, events observed, regularities established through statistical analysis and causal explanations constructed. Like social theory and physics, questions about objectivity, bias, explanation, etc have divided psychologists between those who uphold an empirical model (often working in cognitive science) and those working in social psychology. Education is also caught up in these debates.

Political/Economic theory: Marxist theory presents perhaps the most coherent account of the connection between the material base of existence, social structures and human agency. Its explanatory success is directly connected to the practical effects on the development of social and economic policy from the late 19th century. It remains the best example of the power and importance of theory, and the connection between coherent explanation and social emancipation.

High level theories in Education: Observation of regularities in social life has led to various high-level categories of causal mechanisms in education. Buzzwords emerge whose definitions are often woolly: sociomateriality, semiotics, critical pedagogy, transformative learning, constructivism, etc are high level constructs whose provenance is obscure. Despite lack of clarity (and maybe because of it) these terms get discussed a lot in the literature. Because of intrinsic rewards of the journal system for helping academics establish their impact and job security, popular terms tend to persist since it leads to citations.

Summary
So at one level (e.g. maths) theory is well-defined. For most of physics it remains so, but where physics concerns very small, very fast, or very far-away things, theory bifurcates. In education the theoretical picture is very confused. Added to this is the fact that data analysis is now seen as a viable alternative to theory: prediction, which is one of the principal features of theory, can be achieved from simply crunching numbers (i.e. counting). In this process, explanation is deemed less important.
Having said all this, theory – or the building of explanations – is not something which only occurs in turgid textbooks. Everybody does it. We cannot not theorize. To deny the importance of theory is itself a theory. But it is a theory which doesn’t explain or predict very much, so it is not very good. Holding to multiple inconsistent or bad theories renders us confused.

The quest for a coherent theory of educational technology is a response to a range of questions:

1. Can we explain (and predict?) the reaction of institutions and individuals to technologies? 
2. Can we explain (and predict??) the development of students whose demonstrable skill increases with educational engagement? 
3. Can we explain the reticence of some individuals, or the enthusiasm of others, to engage in technology? 
4. Can we explain why so many learners (and teachers) seem to prefer face-to-face communication over online? 
5. Can we explain how we feel when we engage in learning online? 
6. Can we explain why status, accreditation, certification seem so important in education and society? 
7. Can we explain why our existing explanations/theories do not explain much of what happens in education? 
8. Can we explain the difference between university higher learning, school and kindergarten? 
9. Can we explain curiosity? 
10. Can we explain why YouTube is fab? Or why there’s so much porn on the internet? 
11. Can we explain why so many are addicted to social media? 
12. Can we explain why teachers want to teach? 
13. Can we explain why scientists continue to publish their work in journals? 
14. Can we explain why institutions exist? (and why dogs don’t have universities?) 

Because all these things are connected, many different and inconsistent descriptions can only produce confusion which can not only be imprisoning, but confound our ability to develop technologies which make society better: the unforeseen consequences of technical development might take us to self-destruction through lack of critical inquiry.

It is worth noting that those political forces which demonstrate antipathy to deep critical inquiry are those now in control in the US, Turkey, Russia, North Korea and the UK. We need to think our way out of a very dangerous situation.

Vice Chancellors' and Footballers' salaries compared: HOT NEWS! VC Transfer Window Closing soon - Who'll get the Lukaku treatment?

The establishment is closing ranks on VC pay. After the crass "bling display" of George Holmes saying students want to be taught by rich professors (https://www.thesun.co.uk/news/4147562/uni-boss-who-owns-a-bentley-and-a-yacht-insists-he-deserves-his-cash-because-students-want-rich-professors/), the VC of Oxford, Louise Richardson, has blamed politicians for stirring-up the pay issue: https://www.theguardian.com/education/2017/sep/04/oxford-vice-chancellor-louise-richardson-attacks-tawdry-politicians-university-pay-row using many of the same arguments as Holmes! He'll be flattered, I'm sure.

Interestingly, these high calibre and highly sought-after people can't seem to engage with the press without shooting themselves in the foot. Richardson has quite needlessly done this by defending homophobic lecturers: https://twitter.com/lottelydia/status/904702882603294720 - a gaffe which is in the same league as Holmes's miscalculation. What this all really tells us is that these people are just as confused about education as the rest of us. They try to defend their salaries by pretending that they are not confused by education, but then do or say something which reveals the crassness of their own intellectual position. There is no head of any university anywhere who is not hiding their confusion behind an enormous pay packet.

Here's a quote from Richardson's interview:

"My own salary is £350,000. That’s a very high salary compared to our academics who I think are, junior academics especially, very lowly paid. Compared to a footballer, it looks very different; compared to a banker if looks very different. But actually, we operate, as I keep saying, in a global marketplace,"

Three points to make about this (thanks to Oleg for much of this)

1. Her "lowly paid academics" are lowly paid because she decides they should be.
2. Footballers in the premier league earn vast sums of money. Footballers in League 2 earn about £40,000. Oxford is a premier league university. George Holmes's Bolton isn't. So why are all VCs paid the same? It looks like a cartel, doesn't it?
3. And finally, the Marketplace. What's that, exactly? Is she saying there is a market for Vice Chancellors in the same way there is a market for footballers, or (more appropriately) football managers?

Universities, encouraged by the government, have convinced themselves that the environment in which they operate is a "market". What this means - certainly for places like Bolton - is that obeying the "will of the student who pays their fees" is the essential criterion for success. But then Richardson, who would argue that Oxford "competes" for the brightest students, then says to students uncomfortable about homophobic professors,

"I'm sorry, but my job isn't to make you feel comfortable. Education is not about being comfortable. I'm interested in making you uncomfortable"

Weird market, eh?! The confusion here is that "the market" cannot possibly be the environment of the University; education's environment is society at large - past, present and future - not the "will of the student".  Universities are in trouble because they don't know what environment they are really working in or have to adapt to. Misunderstanding their environment is leading to cruel managerial interventions (such as those at Manchester and the OU at the moment) and this current pay scandal which is stirring-up greater political threats for them in the real environment. The mixed messages and confusion is compounded by the enormous sums of money these people lay claim to (and not to mention their enormous pensions which will bleed an already bleeding university pension system dry).

Our VCs think they are worth £220,000 or £350,000??? Let's put them in the "VC transfer market" and see what happens! Who is the Lukaku or Alex Ferguson of Vice-Chancellors? Holmes? Not likely! Richardson? Well, Oxford's a great "club" - comes top of the league tables... but.. is that because of her? Did she score all the goals? Did she win the research contracts? But let's say she is really great - money talks, so the post that's currently held by Michael Crow at the University of Arizona, which pays him $1,554,058 (see https://www.forbes.com/sites/laurensonnenberg/2017/07/17/the-top-paid-public-university-presidents/#2ca8c8b7114c) ought to be attractive to her. I'm sure they'd be willing to make an offer. So why doesn't she go?

And then, just for fun, who is the Alan Ball, described by the Guardian as a "ruthlessly efficient relegation machine" (https://www.theguardian.com/football/2008/may/04/sportfeatures.footballmanagers). Well, Bolton isn't exactly at the top of the table. But Ball was sacked. Holmes is still there!

These people are having a laugh at society's expense. They are not, however, as guilty as the bankers, who Richardson also mentions. We must deal with them both.

What is notation? (in Maths and Music)

When we are taught notations, whether in music or mathematics, we are always taught what certain symbols denote. In music, students have to work out which note is meant to be played when, and all of these "instructions" are contained in the way the note is written. In maths, we are taught which symbol means what and how they can be combined, and how strings of symbols can be manipulated and related to other symbols. So we learn that 2 + 3 = 5 is a legitimate use of the symbols 2, 3, 5, +, =, but 2 + 3 = 6 is not. When we learn maths, we are conditioned to mistake the symbols from the meaning. The meaning we only learn by playing with the symbols and working out what is legitimate and what isn't. What does "legitimate" mean? It must be some kind of social expectation: mathematicians coordinated their "dances with symbols" with the dances of other mathematicians. Without this coordination, there is really no maths at all.

It's the same with music. In any notated music, we are told which notes to play and in what order, and in what time. There is much that we are not told. The symbols are really an attempt to convey the constraints within which one might express oneself freely in music to be coherent with others expectations. The notation tells us what not to do.

Is the flow of logic a flow of constraint? What not to do at time t1 is not the same as what not to do at time t2. When solving a mathematical problem, or doing some kind of formal logical proof, there is a fluctuation in what not to do. To indicate these is to coordinate a common set of constraints between mathematicians.

Notation indicates constraints. But it produces its own constraints. Whatever is the reality of number lies in the common lifeworld which is experienced between people manipulating representations of number. But if notation is assumed to be real of itself, it will produce unexpected results which lead to confusion.

An example (from Lou Kauffman): if Euler's identity is:

then

which then means

How can an imaginary number equal a real number?

Musicians don't get caught in this. They coordinate their expectations at a deeper level. The mathematical example is produced because there is a double-layer constraint (much like a double-bind). There is constraint between the coordination of expectations about number at one level, and coordination of expectations about the notation at another.