Joel states that the law of leaky abstractions is that:

All non-trivial abstractions, to some degree, are leaky.

Of course, that doesn’t make much sense without understanding what abstractions are, and what it means to say that they are leaky. Joel defines an abstraction as a simplification of something which hides the complex details that are occurring under the covers. Protocols such as TCP/IP are a good example of an abstraction in computing. They abstract the details of  communicating over a computer network., allowing you to send messages across wires and guarantee that they will get there, without having to worry about how it actually works. The problem is that sometimes these abstractions leak – that is, some of the hidden complexity bubbles up to the surface. In the TCP/IP example above this might happen because the network cable has physically broken, or because some of the pins have become crossed, or because of faulty network infrastructure elsewhere.

An example from everyday life is how a home heating system abstracts away the details of actually heating your house, and simply replaces it with a dial allowing you to control the temperature. That all works fine until your radiators get air trapped in them (as happens in my Southampton house worryingly frequently), or the heating pump fails, or the water tank overflows.

Now, these kind of abstractions can be seen in science too. Newtonian mechanics is an abstraction of the underlying complexity of motion which works fine most of the time – until you get to speeds near to the speed of light, at which point it stops working. Another example is atomic theory, which states that all matter is composed of individual atoms. In many situations, this theory works very well, but sometimes the underlying complexity of sub-atomic particles (such a neutrons, quarks or gluons) bubbles to the surface.

Abstractions in all fields often come in sets, or pyramids. For example, the abstraction of writing

print "Hello, world!"

to display some text on the screen has a whole pyramid of abstractions below it, a few examples of which are:

• The abstraction that strings can be stored as integers (you only have to look at the difficulty in supporting foreign characters in many programming languages to realise that this abstraction is very leaky)
• The abstraction that programs have an unlimited memory address space to use (performance issues caused by page-faults, and error messages about Windows virtual memory show how this leaks)
• The abstraction of a filesystem containing files and folders (replacing the complexity of tiny magnetic areas on a hard disk)

This type of pyramid is so common that there is even an official standard for it within the field of computer networking: the Open Systems Interconnection model. This has 7 layers, using abstractions from the simple (transmitting one bit over an electrical link) to the complex (sessions and application protocols).

Similarly, there is a pyramid of abstractions in many areas of science. For example, if you abstract away the behaviour of individual electrons you can produce ‘laws’ of electrical circuits. Combining these laws produces the abstractions of electronics (involving complex circuits and logic gates), and these can then be abstracted away again with the creation of integrated circuits, and so on, until a complex electronic device is created. However, every so often the complexity will bubble up again. For example, the fundamental limitation on the speed of modern computer processors is the size of an electron (as the circuits can’t get any smaller than that) – which is a big leak from the bottom abstraction (electron abstracted into the laws of electrical circuits) to the top abstraction (computer processors which run high-level code).

Now, this all raises some interesting questions – some of which I’m hoping to address in future posts:

• Are these pyramids of abstractions present in all areas of science? If not, then why not? I have a suspicion that mathematics will prove to be different…
• Is the process of creating these abstractions part of what defines a field as a science? How does the gradual building of the abstraction pyramid link to theories about the progression of science (such as Kuhn’s paradigms)
• How does the pyramid of abstractions look in a particular field (for example, my field of remote sensing)? Does thinking about the pyramid of abstractions for your field lead to good ideas for research?
• Does ‘good’ (whatever that may mean) research seek to build abstractions, or to demolish abstractions?

Hopefully I’ll get the chance to revisit some of these issues in future posts – but in the meantime please leave your comments.

Sean took his inspiration from an article called The Six Dumbest Ideas in Computer Security, by Marcus Ranum which states that the dumbest idea is Default Accept. Until recently, many computer security systems were designed around the idea of letting everything go through apart from things that were specifically told to be stopped (thus the name: Default Accept). This is easily seen in Windows anti-virus software. Windows (at least before Windows XP) used the Default Accept principle, so anti-virus software was created with huge lists of pieces of software which must be denied. In his article, Marcus Ranum argues that it is far more sensible to base the system on a Default Deny principle, thus only allowing specific software to run.

So far, so good – but what does this have to do with academia? Well, Sean applies this to ideas by asking whether people have Default Deny brains or Default Accept brains, that is: whether they accept any new ideas by default, or whether they immediately reject them until they have studied the evidence and been convinced. He comes to the conclusion that many people have Default Accept brains, but that Default Deny brains would probably be a more healthy situation.

That struck a chord with me, in terms of scientific studies and academia in general. It can easily be said that science must be done with a Default Deny brain (or at least with a Default Deny hat on), but I think there is more to it than this. As a student in secondary school we were expected to work within a Default Accept paradigm. We weren’t expected to question what we were being taught – we were meant to just accept it at face value. That started to change at A-level when we were first introduced to some of the controversy in our subjects. I distinctly remember one of my A-level geography lessons when we were told that scientists still weren’t sure about how drumlins (a type of glacial landform) were formed. I suppose that was the day that I first became really interested in research: I’d found that we didn’t know everything, and I wanted to be one of the people that started to find things out!

However, even though we had been introduced to controversy at A-level we weren’t expected to really question the fundamentals we were taught. When writing an exam essay about the Burgess model I was expected to list a selection of its flaws, but not to question it at a fundamental level, nor to talk about the inherent problems with models.

At the beginning of my undergraduate degree I started to read journal papers: the original places where new scientific ideas are circulated. But, as a new student, I was far too in awe of the authors to criticise their work. After all, how could I, a lowly first year student, find problems in the work of professors and lecturers – especially work which had been accepted by equally experienced peer-reviewers? Gradually, however, I began to realise that it didn’t actually take a huge amount of knowledge to be able to intelligently critique someone’s work. Ignoring the name and affiliation of the author often helped with this. I distinctly remember finding a paper about the Empirical Line Method in a relatively unknown journal (something like the Chinese Journal of Soil Science – a strange place for a remote sensing paper), reading it, and realising that I could have written something better in my sleep! After that they followed thick and fast: I realise that papers in Remote Sensing of Environment are not without their flaws, and my unquestioned acceptance of the Ruddiman hypothesis completely ignored the body of work that presented alternative theories.

The gradual change from a Default Accept paradigm to Default Deny is an important change which occurs during education. Sadly, this doesn’t seem to occur for many people (even students who go on to get good degrees at university), and it should really start far lower down in school (there is no reason why able students should not be questioning the accepted point of view at GCSE level). Science is entirely based around the questioning of facts, and the denying of knowledge that cannot be backed up with evidence. All of my favourite academic papers (and, indeed, most of the important papers ever written in science) came about from the author denying what he has read and heard, and considering alternative points of view. When the change from Default Accept to Default Deny takes place it marks the start of the transition of that individual to a true scientist – and is a mark of increasing academic maturity.

Now, you might think that sand dunes are quite boring things, just hills made out of sand that sit there and do nothing. That couldn’t be further from the truth. As I learnt in my Aeolian Geomorphology and Geomorphic Modelling course, sand dunes are complex, dynamic systems which do interesting things like move, merge, split, and even automatically organise themselves.

I’m doing some research on the self-organisation of sand dunes with Dr Jo Nield, and part of this research involves writing some software to automatically extract the crests of dunes from Digital Elevation Models (DEMs) showing an area of dunes (known as a dunefield), an example of which is shown in the image below), so that pattern metrics (statistics on the organisation of the dunes within the dune field) can be calculated. These DEMs are produced by a computer model which simulates dune field development. This software is being actively developed at the moment, and is available at my Dunes GIS page.

Example Digital Elevation Model (DEM) of a dune field

The issue that I want to write about today is the question what is a sand dune? It turns out that this question, which seems remarkably abstract, becomes very important when trying to write software to extract dune crests automatically – because to extract the crests of dunes you need to work out what is a dune and what isn’t! In fact, this problem often occurs when trying to write computer software to automate tasks that humans can do easily by hand. Academics who are manually tracing the crests of dunes from images on a computer screen have a large body of knowledge they can draw on to decide what is and isn’t a sand dune. Trying to program this knowledge into a computer is difficult, partly because it is often difficult to work out exactly what this body of knowledge contains.

If a human were to try to trace the crests of the dunes shown in the image above then he would struggle, as the image doesn’t show the crests of the dunes very clearly. In fact, the majority of the middle of the dune is shown as a single colour (bright red) – so it is difficult to tell where the actual crest is. Therefore, a sensible analyst would transform the image to look something like the image below.

A shaded relief version of the DEM

This transformation can be accomplished with one command in most image analysis software (if you’re interested I produced the above image in ENVI using the Topographic -> Topographic Modelling command) and shades the image so that it looks like a sun is shining on it from a certain direction. This makes the image look more ‘lifelike’, producing an image which is similar to the view you would get from an aircraft with the sun low in the sky) and therefore easier to interpret. The crests of the dunes can easily be seen, and therefore can be traced with a high accuracy.

When implementing my automated method for tracing dunes I attempted to mirror the approach a human would use. Instead of using a shaded relief version of the DEM I used a similar version in which the values of each cell have a real-world meaning: the aspect map (shown below). Creating this image involves a simple process where each cell is assigned the direction that the slope of that cell faces (that is, its aspect) in degrees.

An aspect image created from the DEM above

As you can see, it is relatively easy to see where the dune crests are on this image. Therefore, it is relatively simple to write an algorithm to extract the dunes, by looking for the changes in colour on the image above (which are represented quantitatively by the shifting of the aspect value from less than 180 degrees to greater than 180 degrees).

However, this leads to a very messy set of crest lines, as much of the noise in the image is identified as a separate dune. When a human starts to trace dunes from this image their eye will be automatically drawn to some of the large dunes, and they will, almost subconsciously, ignore much of the noise which is visible in the image. A computer won’t do this, so therefore a number of filtering processes are used to both remove the noise before the image is processed, and remove the crests of ‘invalid’ dunes (for example, extremely short dunes) after the crests have been extracted.

But now the big question: are these crests that have been extracted actually the crests of real dunes? If we’re going to use this automated method to get statistics which can be used to develop scientific theories about the behaviour of sand dunes, then we’d better be pretty sure that the crests we’ve extracted actually belong to real dunes!

So, this all leads back to the question what is a sand dune? The answer, unsurprisingly, is not simple. I haven’t yet investigated all of the literature, but I remember learning in my aeolian geomorphology course that most models of dune formation require the dune to have three key characteristics before they can be said to be fully-developed sand dunes:

1. A gently sloping windward slope
2. A steeply sloping lee slope
3. Fully-developed grainflow and grainfall processes leading to dune movement

Note that the definition above mentions neither the existence of crests nor a minimum length for dunes. The existence of crests can be inferred from characteristics 1 and 2, as if these are present then there will be a crest. However, the presence of a crest does not necessarily mean that one slope is gentle and the other is steep. Also, the processes of dune movement are not, and in fact cannot, be identified from a single DEM. Therefore, it could be said that the existence of a crest is a necessary condition but not a sufficient condition for a dune to exist.

It can be seen from the above that the question what is a dune? is most definitely important when studying dunes, and also very difficult to answer. It also shows the limitations of using imperfect data to study the natural world. With access to the right data dunes could be identified with a very high confidence, but sadly this data is frequently unavailable. Therefore, information derived from imperfect data, using methods with unavoidable problems, is often used to develop scientific theories. This is obviously a problem, but is sadly often (or maybe always?) unavoidable.

At first the definition seems very simple, rather obvious in fact (at least to me). Of course Van Gogh’s The Sunflowers can’t be described “in other words”, in fact, it cannot really be described in words at all. It definitely can’t be described in other blobs of paint (at risk of sounding insulting to artists…) as then it wouldn’t be the piece of art that it is. But how about other types of art?

Van Gogh's Sunflowers

Can Beethoven’s Fifth Symphony be described “in other words”, or even “in other sounds”? Well no, I would say that a description of the symphony in terms of the notes, the chords and the key changes is not a work of art: instead, it is a reduction of the art of composition to a simple listing (“stamp collecting”, in the words of Rutherford). I was introduced to this concept at a young age (I think I must have been around eight at the time) when I composed my first piece of music (a very simple piece for piano). I wrote it down very carefully on manuscript paper and took it to show my Dad. “I don’t want to see it!” he said, “I want to hear it – it’s music, it needs to be heard to be appreciated” (even though my Dad can read music and hear it in his head easily).

Moving away from the visual arts…how about a poem? Can that be described “in other words”? One of the few poems I know relatively well (through studying it at GCSE) is Wilfred Owen’s Dulce et decorum est, an excerpt of which is below:

Bent double, like old beggars under sacks,
Knock-kneed, coughing like hags, we cursed through sludge,
Till on the haunting flares we turned our backs
And towards our distant rest began to trudge.

[...]

My friend, you would not tell with such high zest
To children ardent for some desperate glory,
The old Lie; Dulce et Decorum est
Pro patria mori

That can’t be described in other words can it? The little excerpt doesn’t do it justice – read the whole thing, and think of the emotions it’s showing, and the powerful view on war expressed in the last stanza. Could I express that “in other words”?

Well, maybe I could. I could write something like the following:

It’s not nice being in the trenches. Seeing your friends die is horrible. In general, war is really horrible, so please don’t tell people it’s a good, honourable thing to do.

But that’s not art – that’s just a description of art, and it doesn’t do it justice at all. Reading what I just wrote makes you think “oh yeah, war’s not very nice really”, reading Owen’s poem gives you a very different feeling.

The same could be said for other literary works. West Side Story is a very good retelling of Shakespeare’s Romeo and Juliet, but I’m sure Bernstein wouldn’t suggest that he’s trying to recreate Shakespeare “in other words”. Yes, it is derived from Shakespeare, but it is not Shakespeare “in other words”

So, I’ve argued above that work produced in various fields generally taken to be “art” (various types of visual art and literature) cannot be described “in other words”. But how about some other fields?

Take computer programming for example (see, everything in my life comes back to computers somewhere!). Can you describe a computing algorithm “in other words”? Yes, of course you can represent it differently, in different programming languages or using iteration instead of recursion (or vice versa), but surely it is still the same algorithm? Take the Quicksort algorithm – I can’t describe that in other words. Can you?

How about Fermat’s Last Theorem? Can you describe the question Fermat posed in any other words? No, at least not within our current mathematical system. In maths, and to a lesser extent in science, there is one way to describe something. Pythagoras’ Theorem has multiple proofs, but only one statement of what it is. Indeed, mathematical proofs should always be as simple as possible, based on other simple proofs, which in turn are based on more proofs which eventually lead back to axioms.

So, what does this mean?
Is this a simple test as to whether something is an art? Well, maybe…

Computer programming is not generally considered an art, yet one of the pioneers of the field, Donald Knuth, wrote (and is indeed still writing) a series of books called The Art of Computer Programming. He obviously thinks it is, at least partially, an art.

How about maths? Although it is based entirely on simple logical steps, there is definitely an art to doing maths, and certain pieces of maths are definitely “works of art”. Andrew Wiles’ proof of Fermat’s Last Theorem (described so well in Simon Singh’s book) is, in my opinion, just as much a “work of art” as a painting by Monet, or a novel by Jane Austen.

This is why I feel there is a problem with the black and white distinction between “arts” and “sciences”. You find this a lot in education, at A-Level or University particularly. For example, most universities offer a BSc Geography degree (science-based) and a BA Geography degree (arts-based). However, is there not a significant amount of art in my BSc Geography degree? My (very efficient) code to calculate the Getis-Ord statistic on satellite images is, I believe, an artistic achievement.

I want to end this essay by asking you – what do you think? Do you think that the “in other words” test is a good test for whether something is an art? Is there a distinction between arts and sciences? How fuzzy is it? Can the “in other words” test be combined with a test for whether something is a science or not (Popper’s falsificationism comes to mind) to enable distinction between arts and sciences. If so, what happens when something can’t be described in other words, but is also falsifiable? Is it an art or a science? Or a ScArt (no, not the cable you use to connect your DVD player to your TV) or an Arence?

Comments are most definitely enabled on this post, and I would be very interested to hear what you’ve got to say.

“I hope that posterity will judge me kindly, not only as to the things which I have explained but also as to those which I have intentionally omitted so as to leave to others the pleasure of discovery.” (Descartes)

“I am none the wiser, but I am much better informed.” (Queen Victoria)

So basically what he’s saying is that (1) He’s deliberately left some things out so that you can have fun discovering them and (2) The book won’t make you any wiser, but it might inform you. As well as being amusing these also seem to have deeper truths for me. No academic book could have all the information the author knows in it, and anyway to be spoon-fed all of the information for a course is pointless: it is far more fun (and more educational) to discover it yourself. Also, the quote about widsom reminds me that when analysing remotely sensed images (which is what this book is about) one can know all the technical information about them that it is possible to know, but one still has use wisdom to decide exactly which processing operations to apply to them to get the best results.

In an interesting little addendum, my father pointed out that according to this website the source of the second quote was not Queen Victoria (as stated in the book) but Lord Birkenhead.