## Fun with representations V – Maps of the abstract world

Representing information means mapping it into a particular medium –focusing on certain elements of the original data, ignoring the irrelevant ones, and, ideally, simplifying the process of understanding and using it. Unfortunately, our resulting information ‘maps’ are sometimes inappropriate: they may be ambiguous, unintuitive, or downright misleading. To illustrate what I mean, here are some examples of good and bad mappings:

Numbering systems: Our Arabic number system is tremendously effective. It’s not just that there are only ten digits to learn. Assessing magnitudes is easy: a simple glance to a number can tell us if we’re talking about a small (3) or a large (320150297) quantity. And since we have ten fingers in our hands, it’s natural for us to count in base 10.

In comparison, Roman numerals suck (is MXLI less than CCCXLXXI?). They’re only really useful to sound pretentious, as in my title above. And our standard scientific notation, although it uses Arabic numerals, can be very misleading to novices: 3.0×10^1 is humongous when compared to 3.0×10^-10, yet it’s difficult to conceptualize the difference in magnitudes.

Spatial, n-dimensional information: It’s much easier to represent two-dimensional information, such as the Physics problem below, in diagrammatic form rather than as sentences.

The equivalent textual problem statement would have to be several lines long (“There are two masses hanging on a frictionless pulley located at the top of a…”), and it can’t convey the simplicity of the picture. The diagram gives an instantaneous overview of the problem’s information that is perhaps impossible to beat with text. However, things begin to mess up at three dimensions. Is this structure

representing the left or the right cube below?

For complex three-dimensional structures, a two-dimensional representation is rarely satisfying. Incidentally, this three-dimensional ambiguity is the central motive in many optical illusions and Escher drawings such as this one, Belvedere:

Geographical maps are effective at conveying spatial information because, even though they are a two-dimensional representation of a three-dimensional world, data on the height of landmarks and streets is not normally necessary. But we sometimes require three-dimensional information to make sense of our environment, and maps won’t capture this effectively. For example, in Toronto, many people use the CN Tower as a compass of sorts (“the tower is to my right, so I’m facing East”), yet street maps don’t give it the relevance we do.

Can we represent more than three dimensions in a two-dimension diagram? Not really –we can try, but the results (by using colours, animations, etc.) are never entirely satisfactory, even for a small number (four, five) of dimensions. Mapping high-n-dimensional structures in a diagram in two dimensions is impossible by all practical means.

Musical notation systems: We have a notation system to represent musical compositions that is quite effective for Western music. Since Western music is based on a 12-tone scale, with relatively rigid rhythms, the notation makes a pretty good job at recording this information. Unfortunately, it’s useless to describe some kinds of more flexible, traditional Asian and African music. People have attempted to represent these in other systems, with varying degrees of success. For some types of music, repetition and imitation are still the most successful strategies to pass on musical knowledge.

Phonetic notations: One of my early embarrassments when learning English was finding out that recipe is not pronounced as recite. And I was already in Toronto while still calling wheat with the same termination as sweat, not as sweet (so asking, naturally, for wheat bread always led to puzzled looks). Written English is terrible as a phonetic notation. Spanish is much better, but it still doesn’t help me when I try to pronounce non-Spanish sounds, such as those in even the most basic Polish (czesc!) or Chinese (xie xie) words.

In comparison, the International Phonetic Alphabet is precise and comprehensive. Once you learn it (not an easy task!) you can use it to find out, exactly, how to pronounce words in practically any language.

I write about all of these examples because in my very own Software Engineering field, there is a strong community convinced that models and diagrams are the best way to represent software constructs. We have modelling languages to represent almost any software-related concept you can think of: objects, scenarios, states, classes, goals, beliefs, design rationales, threats, risks, you name it. What we don’t have is any real indication that our diagrams map satisfactorily to constructs in the world.

The problem is that we’re dealing with very abstract, very difficult to represent concepts, not with the two-dimensional structures of High School Physics problems. Where did we get the idea that use-case diagrams are an appropriate high-level mapping of human-computer system interactions? What leads us to believe that goal analysis diagrams are an accurate depiction of the real goals of stakeholders? Is a sequence diagram really better than pseudocode to represent the logic of a scenario? Simply put, there is no convincing evidence justifying any of these beliefs.

(Ontological analyses help us address these issues, by pointing out, for instance, that a weak point of entity-relationship diagrams is their difficulty at expressing entities with fuzzy boundaries and non-entities (such as fluids, thoughts and intentions). But ontological analyses do not answer the question of whether a representation appropriately conveys what it should convey to other humans –such as entities in the entity-relationship diagram case.)

I am not claiming that software engineering diagrams are inadequate mappings to the real world. I’m claiming we don’t know, that –considering we’re using these diagrams as communication artifacts– we should know, and that we’re giving too little thought to these matters in our community.

I'm currently a Postdoctoral Fellow at the SEGAL and CHISEL labs in the Department of Computer Science of the University of Victoria.
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### 10 Responses to Fun with representations V – Maps of the abstract world

1. Crystal says:

Hello there Jorge, I was really surprised to see your comment, but I guess that comes with the territory of link tracking these days. =)
I did indeed enjoy your blog a lot, and especially appreciated the bits of cogsci sprinkled throughout – I’m actually a CS + Neurosci doublemajor, weirdly. A funny story, if you’re interested – I found your blog by googling “synatactic sugar” while procrastinating at work. I’ll definately be a regular now =)

2. Jorge says:

I’m happy you liked it! I’ll keep the external cognition bits coming –at some point I’ll get to talk about group/distributed cognition too, which I hope will be interesting.

And yes, if you were googling for “syntactic sugar” I diagnose procrastination 😀 Thanks for your comment!

3. Yoni says:

Jorge, I knew a long time ago that you are bothered by whether the de-facto standards in our industry (e.g., use-case diagrams) are actually proven to be the best, or do we just think/assume that they are the best.
Even if you do some empirical research and reach some decisive conclusion, where do we take it from there? I think that this problem is universal, in the sense that it exists in other domains, and also that it existed a long time ago, long before com-sci started.

You are asking where did we get the idea that use-case diagrams are an appropriate high-level mapping of human-computer system interactions? Similarly, I can ask where did we get the idea that a 2-dimensional diagram is the best way to describe physics problems? I know, time is one your side on this one, and you can probably quote some past empirical research that supports it.

I think that 2D diagrams were proven best in this domain (you use the phrase “easier to represent”) because they were successful in teaching, i.e., in conveying the problems to children learning physics. They were also quite successful in conveying the problems fast, sometimes with the order of minutes (e.g., during exam)

Is ‘teaching’, and ‘fast’, the purpose of use-cases and sequence-diagrams? or is it more likely that the producer and consumers of those diagrams are already educated and know the way of the force (hmm, I mean the way of UML)?

Also, 2D diagrams don’t scale to other areas of physics. I don’t think that you can properly explain Einstein’s theory of relativity during a high-school class with 2D diagrams.

In addition to scalability, there’s the issue of usage. 2D diagrams are good for teaching, but are hard to manipulate. A bunch of equations are easier to manipulate, for example, if we are trying to cast one type of problem into another, in order to solve it.

(A recent blog post that I read comes to mind. Hmmm, where did I read it … 🙂

Btw, I liked what you said about phonetic notation. So true …

4. Jorge says:

Excellent points, Yoni. You’re right in reminding me that even when one representation is good for something (“teaching”), it doesn’t mean it’s the best for everything else, nor that other alternatives should be used for the same thing.

What I tried to say was that, once the contextual elements are set (we know what are we using the representation for, who uses it, etc.), there is still the problem of whether the representation maps the domain appropriately or not. To use the phonetic notation example, say we define the problem as that of someone learning a new language (English) on her own non-English speaking country. The IPA will give her a one to one correspondence between symbols and sounds, and so it’s convenient to use it to learn pronunciation. This doesn’t mean the IPA is *better* -we still need to factor in the cost of learning it, and the fact that very few people know about it. An alternative solution is to write the pronunciation of English words the way you’d pronounce them in your own language (for Spanish, “hello” becomes “jelou”), which has the advantages that you don’t need to learn a funky IPA notation and that everyone who speaks your own language can help you with it, but the disadvantage that you lose precision.

There is a lot of research on Cognitive Science and Education on the convenience of diagrams in Classic Physics problems –the one everyone goes back to is Larkin and Simon’s “Why a Diagram Is (Sometimes) Better than Ten Thousand Words”. I am not aware of similar studies for phonetic or musical notations, but that’s because I haven’t searched for them.

Finally, suppose we go on to empirically prove that UML sequence diagrams are inadequate for a particular task and type of user (say, validating with stakeholders). You rightly ask, where do we take it from there? I’d say, if this is all our study tells us, it was almost a waste of time. We need to design studies so we find out *which parts* of the representation fail, and which succeed, in order to amplify the strengths and eliminate the weaknesses. Other domains, such as map making, have had less of a need to do empirical studies because we’ve been basically experimenting through trial and error for thousands of years. Since conceptual modelling has been around for far less time, we need faster feedback loops to the designers so their creations can reach an acceptable level in our lifespans 🙂

And I challenge you to teach Einstein’s theory of relativity to a high-school class, using *any* type of diagrams you want! 😉

5. Yoni says:

I think you confused ‘lifespan’ with ‘the duration of my phd’ 🙂

Btw, I have read quite a fow books about Einstein’s theory and physics in general. The best explanation that I have ever read for the theory of relativity actually comes from a person who is not a scientist (i.e., he is not a geek). His name his Gary Zukav, and he wrote “Dancing Wu Li Masters: An Overview of the New Physics”. His explanation for the theory of relativity involves no formulae and no diagrams. It’s a great read 🙂

6. leo says:

How about representing with a 2D diagram how the duration of a PhD is not part of a life span? 🙂

Jorge, I have been enjoying all your posts and all your replies to the comments – some of which could’ve been posts on their own – It has been really interesting to see how much you think that the representation of a problem contributes to its solution. You know that I agree with that but I wanted to point out that the way you write about it is very articulate. In other words, you are proving your points, about how helpful a good representation is, by being so eloquent when writing about it.

7. Jorge says:

Yoni,
that book by Zukav sounds pretty interesting –I’ll check it out.

Leo,
thanks a lot for your comments. I guess it would be very inconsistent for me to talk about good representations, and to research comprehensibility, while making no sense at all in my own writing 🙂

8. Pingback: Tamer Hosny

9. Jorge says:

Glad you liked it, Tamer, hope to see you around soon 🙂