Saturday, 16 September 2017

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.

Monday, 11 September 2017

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.

Monday, 4 September 2017

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.

Sunday, 3 September 2017

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.


Thursday, 31 August 2017

Medieval Logic, Cybernetics and the art of D.P. Henry

It's a curious thing that I was talking with a friend about curiosity yesterday, a couple of weeks after visiting the British Library and spotting in the display cases a copy of Boethius's "De Institutione Arithmetica" which contained a beautiful picture of the categorisation of number into arithmetic, geometry and harmony. With no apology, I would say that the "harmony" struck a chord with me! There's something about curiosity and "striking a chord" - or rather, looking for a chord to be struck.

I've recently been immersing myself in physics and symmetry, and was about to attend a conference which included contributions from physicists and cyberneticians. What I wasn't expecting was to be presented with very powerful alignments between medieval logic and cybernetics. The presentation by Dino Buzetti sent me off to look for the common patterns between Scotus, Ockham and George Spencer-Brown. What's the key? It's the obsession with what it is to make a distinction.

Dino's references also led me to seek out the work of D.P. Henry. Henry was one of the leading authorities on medieval logic. The epigraph he chose for his book on "Medieval Logic and Metaphysics" from St. Anselm could have been written by many cyberneticians (and particularly by Bateson):

We ought not to be
held back by the way
in which the improprieties
of speech hide the truth,
but should rather aspire
to the precision of the
truth which lies hidden
under the multiplicity
of ways of talking (from De Casu Diaboli)
I'm still digging into the book, but this statement from Anselm seems to me to also be about curiosity: it is a search for the multiplicity of ways of talking.

Henry is less famous today for medieval philosophy than he is for art. He was a champion of machine-generated art, and produced beautiful images like this one:




Sunday, 27 August 2017

Squander, Universities and George Holmes's Yacht

The recent car-crash interview given by George Holmes of the University of Bolton to the FT, repeated by other mainstream press (e.g. here: http://www.dailymail.co.uk/news/article-4751536/Boss-earns-222k-says-vice-chancellors-not-paid-enough.html) was deeply unsettling and unwelcome to many in the sector (for example, Oxford's David Palfreyman here: http://www.telegraph.co.uk/education/2017/08/03/stop-gravy-train-oxford-bursar-says-attacks-university-vice/. It was hard to work out if Holmes intended to stick two fingers up at the establishment, or he was wanting to parade his Bentley and yacht in front of them as a way of saying, "Well, it may only be in Bolton, but I can cut it with you posh lot!". He seems to flip-flop on this: on the one hand appointing minor royals to ceremonial positions in the University, whilst claiming that the old established universities like Oxford are "dinosaurs waiting to die". The latter comment he made in a TED talk (!) recently at a TED event organised at Bolton at which three senior managers and a few professors (it's very hard to tell the difference between professors and senior management these days) gave the world the benefit of their wisdom. This is worth watching here  https://youtu.be/6vyGqh13J58?t=8570 to gauge the intellectual clarity with which Holmes grasps his mission. It's not, I think, pedantic to note that the Robbins Report on Higher Education was published in 1963. The total self-confidence with which Holmes says it's 1966 and riffs on "route 66" says a lot about him: it's as if he's thinking, "if I say it loudly enough, I can make it true". It would make one question many of his other boasts and indeed his judgement. We see this in the world a lot at the moment.

To the people of Bolton, the obvious point is that Holmes's "success" - his yacht, the Bentley, and his £960,000 house - have been paid for by the university's students (many the children of Bolton) with money they haven't yet earned, and with a debt which will be hanging over them long after his yacht has sunk and the Bentley is at the crushers. He will say (and has), "This is a multi-million pound business". But one might be forgiven for thinking "This is a multi-million pound racket", the product of misguided government policy which turned universities into fiefdoms and put characters like Holmes at the helm without any checks and balances on their behaviour. The fact is, after numerous uncomfortable engagements with the press and previous bad behaviour (see http://www.dailymail.co.uk/news/article-2964002/University-vice-chancellor-handed-1m-bridging-loan-employers-home-47-miles-away-campus.html), HE'S STILL THERE. Is it imaginable that the VC of Oxford would survive this kind of thing? I doubt it. Holmes has been able to arrange things to suit him - at the students' and staff's expense. How long will this last?

The FT interview was, by any measure, very poorly judged. It was on a Phillip Green/Mike Ashley/Donald Trump level of "bringing the institution into disrepute" behaviour. I suspect Holmes knows this - but for some reason he can't seem to stop himself. The weird thing here is that he knows he can get away with it. The interview was reckless, irresponsible. It was squanderous - as indeed was the Bentley, the yacht, the £100,000 awayday, the £960,000 house, the sacking of the UCU reps, and so on. It was like some drug and alcohol fuelled bender of the kind that would make de Sade blush. It carries the whiff of the thrill it probably gave him as he posed for the centrefold of the FT (ok, it wasn't the centrefold, but it could have been). Playboy next. Dangerous?! "Yeah, but I'll get away with it."

Amid the austerity agenda, squander doesn't get much of a look in. But it's everywhere. George Bataille wrote an entire economic theory around the concept of squander and waste: he argued that in human history, it was the most regular punctuating mark in civilisation: war and destruction, extravagant building, luxurious art, sexual excess, alcohol and drugs, all the way through to the human sacrifice of ancient civilisation. Bataille put it down to humans having absorbed "excess energy" from the sun, and needing to expend it in various ways. He based his ideas on the anthropological theory of Marcel Mauss who explored gift economies and the "potlatch".

There is a weird symmetry between the squander of Holmes and the squander by students going to his (or other) Universities. Since fees were introduced, university funding has quickly revealed itself to be a "Veblen good" - where the demand for something increases with its price. Veblen I suspect subscribed to a similar theory to Bataille - he regarded University as "atavistic", and had a notorious reputation for seducing the wives of Vice-Chancellors. For the students, University is squander which the government encourages. It's actually a form of Keynesianism: what Colin Crouch calls "Privatised Keynesianism". £50,000 of debt for a piece of paper! Wow! That feels fantastic!

All intellectual life has an aspect of squander. At its best it's like the squander of the artist, or perhaps the squander of the priest who lives in self-imposed poverty. As intellectual accomplishment also carries a social status (What Veblen acerbically says is where "the standing of the savant in the mind of the altogether unlettered is in great measure rated in terms of intimacy with the occult forces") there is always a 'marketing' opportunity to sell the "fairy dust". The market has taken all of these forms of squander and turned them into an economic dynamic where student squander is matched by the squander of the likes of Holmes, and the unpleasant corporations making a killing on student accommodation and other services, and the banks who are ramping up interest on student fees.

Catherine Bennett touched on this in her excellent Guardian piece about Holmes, asking what would happen if it all doesn't work:
"What will motivate our young people, supposing we accept the Holmes analysis, if they do not see how good jobs translate into high-end vehicle choices? How else will leaders like him advertise their very successful careers? There can be only one possible compensation for this loss: more money" (see https://www.theguardian.com/commentisfree/2017/aug/05/boys-bragging-about-their-big-cars?CMP=share_btn_fb)
Is this sustainable? Can it carry on growing? Let's see more waste from students, and more outrageous peacock displays by the likes of Holmes! But this is an age of "austerity"! How does any of this make sense?

It's not an age of austerity. It's an age of squander. It's an age where a few like Holmes squander on Bentleys and yachts, whilst students squander on education. In the middle are the teachers and academics - the people with no time, let alone the money, to squander. That's the other side of Holmes's squander: slashing staff, hourly paid contracts, no security. This is the students' future. Bataille might suggest that we've reinvented human sacrifice.


Thursday, 24 August 2017

Uncertainty, Universities and Yacht-owning Vice-Chancellors

One of the greatest sources of confusion concerns the impact of technology on social change. When we survey history, we see patterns which appear to suggest that steam power caused the industrial revolution, and the various social and political transformations which accompanied it.  Obviously, online shopping needed the internet; the collapse of the high street could also be seen to be collateral damage from this, as was highlighted in this interesting (and rather depressing) piece about Bolton this week: https://www.theguardian.com/cities/2017/aug/23/bolton-response-market-city-decline. But this causal connection between technology and social change is what is called "technological determinism" - and it clearly isn't true. Yet because a causal link between technological innovation and social change is constructed, a mindset sets in within institutions like universities and government which sees that the solution to social problems like housing, welfare, employment or health is technological innovation.  But it tends to produce new problems rather than solutions.

We need a new theory which connects technological innovation to social change. I've begun to think that we look at the wrong things if we examine what new tools can do - what psychologists call their "affordances". Rather than doing this, there is a simpler starting point: all new tools provide new ways of doing things. That is, basically, the definition of a tool: it creates a new option for acting.

Today we are continually bombarded with new options for acting: new online communication services (FaceTube), new ways of cutting mobile phone bills (LebaraFone), new ways of getting about (UberLyft), and so on. Our daily conversations often go something like "I use x, it's new - have you tried it? Much better than y".

There are a number of things about options which need to be considered. Any option has to be selected. We use all kinds of techniques for making selections - habit is the most powerful one - but the trouble of deciding on an option is work. David Graeber calls this "imaginative labour". If you give somebody more options, that's more work.

From a more scientific perspective, if the number of options for acting is increased, then the probability that any particular option is selected decreases. If the probability of selecting an option decreases, the chances of somebody else guessing the option you have chosen also decreases. If the other person is not familiar with the scale of options you are selecting from, there is no chance of them guessing. This can make communication more difficult. There is no point in communicating a message to a friend through Twitter if they are not on Twitter.

In order to communicate, it is important that the range of options available to somebody sending a message is the same as the options available to a person receiving a message. If this isn't the case, then there is a chance that something will be selected at one end which cannot be selected at the other. The same thing applies to language: to understand a message, the receiver needs some idea of the inner machinery (psychology) of the person uttering the message so that they can understand how the selection of words was made. These basic principles grow from Ashby's Law of Requisite Variety in cybernetics: that in order to manage the complexity of system a, system b must have an equal or greater amount of complexity. One way of expressing this is through Shannon's information theory which measures the complexity of communications in terms of their "surprisingness", or the degree of uncertainty associated with them.

If adding a new option reduces the probability of selecting an option, and the probability of guessing what option is selected, then uncertainty is increased. Technological innovation does nothing but increases uncertainty.

Because of this, it is incorrect to say that "technology takes people's jobs". Institutions make people redundant, not computers. What causes institutions to do this is the their own reaction to the uncertainties produced by technology and other things in their environment. It is the effect of uncertainty on existential fears of managers, company directors, government ministers and so on which result in the misery of redundancy for staff (and never redundancy for those making the decisions about redundancy!). They tend to respond to it by off-loading their existential crises onto their employees as they seek to defend the structures of their institution in an increasingly uncertain environment.

Present crises of employment, equality, political expression, and institutional corruption are directly the result of institutions struggling to maintain themselves in an increasingly uncertain environment. As institutions seek to defend themselves by reinforcing their structures, sacking staff, and attempting to bend to "market conditions", they feed a growing political crisis which has the effect of ramping-up uncertainty in their environment. It is a positive feedback situation. Political uncertainty leads to new environmental complexities to which the institution has to adapt, sacking staff, reconfiguring courses, etc. Gradually institutions eat themselves.

At the heart of the problem is the way society manages its uncertainty. Traditionally, institutions like churches, universities, hospitals and government have been the means of managing uncertainty, which institutions have done by attenuating the environment and forcing individuals through regulated pathways. This worked because uncertainties could not proliferate because the transformation of means of doing things was a slow process. Computers change that. The transformation of means of doing things is rapid, and the growth of uncertainty is relentless. Institutions in their traditional form are not fit for purpose to manage uncertainty. Indeed, they make the situation worse.

The future of the management of uncertainty in society rests with effective use of technology and self-organisation between individuals. This will not replace universities and hospitals completely. But it will do many of the things which we currently associate with institutions. Meanwhile, the dinosaur institutions will hang on. Vice-chancellors will grab as much as they can to preserve their status and identity whilst things fall apart around them. Some of them will even give unwise interviews to national newspapers about how "well" they are doing. There is no better sign of the crisis we are in than this yacht-owning vice chancellor: http://www.telegraph.co.uk/education/2017/08/01/university-chiefs-should-not-ashamed-salaries-says-yacht-owning/.

My thoughts this week are with those he's just made redundant.