A Formula is Worth a Thousand Pictures: Dijkstra vs Buzan’s Mind-Maps

“A formula is worth a thousand pictures”!

Here I detail how I became convinced that graphs and pictorial representations such as Mind-Maps are just a first cut and a very raw representation of the complex ideas that we have in our minds. Limitations of such pictorial representations start becoming evident the moment you start labeling the concepts, then start labeling the arrows (relationships) that connect them, then start identifying all the many different kinds of relationships that may exist between these concepts, and then discovering cyclical structures that exist between them, and then start adding more and more variations of the concepts and their relationships and their siblings and descendants, and you soon find out that these diagrams become so cluttered, and the bubbles become so numerous, and arrows become so confusing that the initial understanding and excitement actually starts evaporating away. Add to this the number of different colors you are forced to use to manage these and the hierarchical layers you are compelled to introduce that you soon realize that diagrams are no longer helpful but are becoming more of a nuisance than a help, more of a chain that is pulling you behind, and less of a spring-board. You, then begin to appreciate the need for more refined methods for solving the complexity that exists in real life situations beyond the few bubbles, arrows and colors with which a mind map initially excited you. 

The hierarchy of representations is shown as a pyramid with physical models at the bottom, followed by the verbal models, followed by pictorial (or graphical) models with the top of pyramid containing mathematical models (also called formulas). 

A model or a representation of reality simply captures the features considered essential for the purpose at hand and ignores other numerous details. Physical models/representations are like the models one sees in ads who capture outer (physical) features of some typical stereotypes, or the dinky cars with which children play with, and which capture the physical (outer) physical features of some specific cars. Physical models often have the greatest amount of detail, but are inflexible as they can describe only a specific instance of reality. Next in flexibility are the verbal representations which are often expressed in natural languages such as presentations, prose or poetry. However, graphical and pictorial models have even greater brevity as represented by the adage “A picture is worth a thousand words”. But the greatest brevity and preciseness is, of course, captured by mathematical models or formula that lie at the very top of the pyramid. To appreciate just visualize the immensity of power captured in such a short formula as E = mc². This small formula captures the energy unleashed by a tiny atom in an atomic explosion and witnessed at Hiroshima. It can’t get more simpler and direct than this! 


Hence, Dijkstra’s comment “A formula is worth thousands of pictures”. His camp believes that you resort to pictures when you are vague and unclear and have not yet crystalized your ideas. But, this realization is not easy to make by those who are just embarking on this journey of managing complexity and trying to identify simple, beautiful and precise expression. This is the quest of special people who think beauty is their business. See my post Beauty is Our Business: Mathematics, Excellence and the Great Dijkstra.


I now relate my personal journey as I transited from the pictorial models to the formula level of the pyramid above. 


My first exposure to pictorial graphs was the use of flowcharts in the 1980s to understand and document the flow of program control. During my internship at IBM Karachi in 1986, I programmed an Inventory Management System using BASIC. My supervisor at IBM was quite impressed with what I had made and asked me to document the program, which I did by profusely illustrating the program flow with many flow charts. [Ten years later when I met this supervisor, I was quite surprised to learn that he not only remembered my internship project but also the flow charts of the documentation!]. 

Spaghetti Code

1970s and 80s were the days in which “spaghetti” code was quite common, so called because the control flow moved from one part of the program to another in a mished-mashed manner resembling cooked spaghetti. This type of crisscrossing of the control flow could only be understood through flow chart diagrams that tried to explain pictorially what was happening. [Although anyone who has seen some huge flow charts containing thousands of elements know their limitation]. Programming languages in the 1980s were still using the notorious Goto Statement that was responsible for such mish-mash spaghetti type of code writing, and were slowly and painstakingly trying to do away with its “unstructuredness”. This was despite Dijkstra’s classic paper on “Goto Statement Considered Harmful” in 1968 that laid the foundation for the elimination of spaghetti code. This paper heralded the era of “structured” programming which would eventually make the use of flow charts irrelevant. Dijkstra was a firm believer of elegance and beauty which could only be represented by brevity and preciseness [often called “less is more”]. Another of Dijkstra’s epigram about languages like COBOL that allow such shoddy thinking and its expression: 

“The use of COBOL cripples the mind; its teaching should, therefore, be regarded as a criminal offense.”

Please note that unfortunately COBOL is still in use! Bad habits die hard. Another of Dijkstra’s epigrams: “The prisoner falls in love with his chains.” Our familiarity and expertise of an inadequate tool can imprison us in our inefficiency. Often, we can not tear out of the prison of our incompetence because of the steep learning curve required to make the break with the chains of our past. 

“Being abstract is something profoundly different from being vague.” — Dijkstra

Now when I look at diagrams and use of pictures for solving problems I am reminded that: 

“… if 10 years from now, when you are doing something quick and dirty, you suddenly visualize that I am looking over your shoulders and say to yourself, Dijkstra would not have liked this, well that would be enough immortality for me.” — Dijkstra

All Dijkstra quotes are from IN MEMORIAM: EDSGER WYBE DIJKSTRA.

See Also:

• Fairness in Grading: A Lesson by the Great Dijkstra

• “How can we explain Edsger W. Dijkstra to those who didn’t know him?”
• Dijkstra and his contributions

• Myth: Impact Factor Measures Real Impact
• Myth: We are Backward because we Lag Behind in Science and Technology
• Beauty is Our Business – Mathematics, Excellence and the Great Dijkstra
• 5 Myths of Higher Education in Pakistan