Representing Problems Symbolically - Part 1

Hello loyal and valued readers! Many thank-yous are needed to all of you for your continued support and readership of this newsletter. Writing about problem solving is truly a passion of mine because I believe that many opportunities are just problems in disguise, to paraphrase a common expression. But there is real truth in this expression in that problems offer us opportunities to grow, to increase our knowledge, to enrich our lives, to make the world a better place.
However, problem solving is a specialized topic, normally considered a sub-topic or field under psychology, critical thinking, logic, and math. Problem solving is often seen as a tool, as applicable methods and techniques to make advancements or progress in other fields. Its utility is undeniable in helping to solve complex questions. But problem solving is also a daily part of our lives whether we recognize or acknowledge it. When confronted with obstacles we often turn to tried-and-true methods. Most assuredly that is a valuable way to approach problems. But in our problem solving capacity there is always room for growth and new ways to approach daily problems. That is what I am offering here in this newsletter: A new way to help you solve problems.
0. READ THIS IF NOTHING ELSE
Problems, whether mathematical, scientific, business, career, or personal, are representations of a larger system that is not operating optimally or not at all. Once a problem develops, identify its structure so that you can represent it in a form where potential solutions can be derived, developed, decided. Give your problem form so that the functional deficiencies that led to it can be operated and acted upon.
1. THE BIG IDEA
What if the way you solve math problems could help you solve personal problems? Problem solvers benefit when they represent non-mathematical problems symbolically. This includes such representations as equations, graphs, diagrams, and several other methods.
2. KEEPING THINGS NON-ABSTRACT
The material in this newsletter and subsequent editions on this topic may seem a little abstract to those who may abhor math, equations, and diagrams, but let me reassure you I will keep them relatively simple. The explanations will be clear, the examples plentiful, and the discussion of great practicality. Once I have described this method I can guarantee you that you will find it extremely useful!
3. BACKGROUND
We live in a world of symbols. Written language consists of symbols. It is the way we understand the world, how we communicate, how we develop ideas, and so much more. Sure, we could (and originally did) grunt out noises and communicate somewhat, but of course that was and would continue to be very limiting. The development of formal language and a symbolic way to express it is one of the crowning achievements of humankind. Both animals and insects communicate and have an informal language, but human beings are the only ones to have created a symbolic representation of our thoughts. This has led to tremendous advancements for our own species that would not have been possible otherwise.
By the way, in the scholarly world the field of semiotics explores, examines, and explains the uses and meanings of signs and symbols. It is a fascinating subject for those interested in a deeper dive to understand their vast use, meaning, and importance in human communication.
The true power of symbols is that they compact meaning in an abbreviated form, hence their utility in communication. Language, or more accurately communicating thoughts in verbal form, can be a lengthy process. However, one symbol can carry the meaning of hundreds of words. Combining symbols vastly expands their power. Indeed, mathematics is built on symbols.
Mathematical problems are part of the larger system of mathematics and are written down in symbolic form; but interestingly, true mathematical understanding consists of understanding both the symbols and the underlying verbal explanation of what the mathematics is describing. Physical reality, what we understand that to be presently, can be understood through the language and symbols of mathematics. This is a wonderful achievement in compacting enormously complex ideas about how the world operates into a few powerful symbols.
4. GETTING REAL WITH GRAVITY
As a concrete example of how symbols can encapsulate ideas, the force of gravity can be described mathematically, not just as something we experience as Earth bound beings (for the most part). The force of gravity on Earth (g) has a number assigned (-32 ft/sec^2) that has been measured. But we also understand, without numbers, that things fall when dropped, generally speaking. These two different ways of understanding physical concepts are approaches that complement each other. Mathematics is a rigorous description where symbols can be manipulated to arrive at further conclusions. Verbal descriptions help to enrich understanding by painting a picture.
Correspondingly, non-mathematical problems benefit by symbolic/graphical representation as a means to better understand the variables involved. The truth is that almost anything can be made mathematical, or described mathematically. It may seem like a stretch that we can model non-mathematical problems in quasi-mathematical form using symbols to represent situations, events, feelings, change, etc., but this technique works because problem solvers can gain insight into the problem by manipulating these symbols, assigning values to traditionally non-quantitative variables, and using equations or diagrams to derive surprising results verbal representation might not allow.
5. ASSUMPTIONS
If physical reality can be modeled through symbolic abstraction, why assume human systems cannot be modeled the same way?
Of course, a basic assumption here is that many non-mathematical problems can be understood dynamically as evolving systems that are undergoing rates of change. Obviously, static problems with non-moving parts, so to speak, are easier to represent visually than dynamic problems. The delta symbol (∆) is a symbolic representation for how situations develop over time. Understanding and measuring rates of change is a crucial concept to understand in using this technique. One can assign a quantity (number), which can describe these kinds of changes. After all, that is really the goal: to model events, situations, feelings, etc. so that we can understand the underlying dynamics of the problem.
6. PROBLEMS RARELY EXIST IN ISOLATION
Problems emerge from systems or networks of variables interacting over time. When a system functions properly, outcomes are stable and predictable. When the system fails, problems appear. The task of the Problem Solver is not merely to react to the symptom, but to model the system by identifying the failing component and restoring its functionality.
In further understanding the usefulness of this technique, Systemic Problem Solving (SPS) posits that most problems develop not in isolation, but rather as part of a larger system which has various operational and functional components. These components and their connections can be better understood when given symbolic or graphical form. Problems develop when the system malfunctions or is not operating properly. Representing the system where a problem develops by using symbols helps the Problem Solver see connections among components of the problem.
7. MOST PROBLEMS HAVE ROOT CAUSES
Furthermore, nearly every problem has a root cause. Problems may be discovered but almost always have an antecedent. While some maybe believe that certain problems arise organically or "out of the blue", most problems have well-defined causes. In an attempt to solve a problem, one must fully comprehend what caused it. If you are unaware of what has caused a problem, how can you possibly approach a solution that addresses the various factors resulting in a situation that is far below optimal? Developing a symbolic/graphical representation  may also help trace the beginning of a problem.
8. SOLUTIONS THROUGH SYMBOLS
As a result, by attempting to represent certain non-mathematical problems in a symbolic fashion, either through mathematical type notation or graphically, one can see approaches to how a problem might be solved. Some kind of visual representation is a crucial factor in a deeper comprehension of a problem. This technique might possibly help you solve your problems more easily by giving you insight or ideas on how the problem "works" or interacts within the problem "space."
9. FAULTY SYSTEMS LEAD TO PROBLEMS
The usefulness of this technique is that the problem can then be explored and modeled to gain further insight into how the "faulty" system is operating non-optimally and where the failure may reside. Developing an understanding of any problem often requires visualizing the relationships that exist among operational parts or variables. Functional diagrams (how the system is functioning over time) can help problem solvers develop a clearer sense of how to arrive at a solution. Also, since many problems develop as a result of an "operational fault" or "broken function" of some system, a first step in problem solving is to isolate what part (variable/s) of that previously working system has/have "malfunctioned."
10. PERSONAL PROBLEMS DEVELOP WITHIN FAULTY SYSTEMS
This idea may initially seem just theoretical, but when you think about it, many personal problems originate with a possibly unknown fault in the way a person thinks, behaves, or acts. Similarly, many unsolved problems in mathematics arise because of a lack of clarity of how certain functions work in conjunction with one another.
EXAMPLE:
As a project manager, you are having a problem at work with your project team. The team is undergoing a lot of stress because there is a lack of communication among the team about the project’s goals and progress. You might understand or model the situation, or system, by this basic equation:
T = A - M
T = Tensions and unanswered questions among the group
A = Assumptions made about the project's status
M = Number of project clarification meetings per week
[This is a very simple model, with much room for additional symbols, say for the efficiency rating for each meeting in terms of questions answered.]
Suppose five major assumptions circulate about the project, but the team only meets once per week and can only address one assumption per meeting. Assumptions will then accumulate faster than any clarification occurs. Tensions rise among team members because of their lack of clarity on the project's status. This is a substantial problem for you as manager since it may disrupt the long term success of the project.
What kind of understanding was accomplished by representing this situation symbolically? A vague feeling (“this team is stressed”) became a structural insight. The problem is no longer emotional fog; rather, it can be seen an imbalance between variables that can be fixed. A solution thus presents itself through this simple model. Once a problem becomes structural, it becomes actionable. Vagueness becomes structure. Complexity becomes a little clearer.
Another example is this: You say, "My day is getting worse by the minute!” Well, that can be represented as -∆m. It really doesn't matter the particular symbol, ONLY WHAT IT MEANS TO YOU and how you can then use that symbol to communicate further information or manipulate that symbol to derive some further conclusion. That's the real power of this technique!
11. PONDER THIS
Why the heck are emojis so popular? The answer, or at least one answer, is that emojis compact feelings elegantly, graphically, visually; that is, symbolically. Sometimes they can express more, and more concisely/compactly, than the language they represent.
TRY IT YOURSELF
Try modeling a problem you have symbolically. The goal should not explicitly be to solve some personal problem or equation, but to gain insight and understanding. Comment below if you need some help in this. Remember, be creative but consistent in what your symbols mean and how they might be "manipulated" to derive some solution.
Two final questions can be helpful to consider:
1. What problem in your life feels vague right now and could benefit from translating it into symbolic or graphical form?
2. What variables might be hiding inside the problem that can be understood by giving them a symbolic or numerical form?
Start with one problem. Not the biggest one—just one that feels unclear.
Define the variables. Assign symbols. Map the relationships.
You are not solving the problem yet. You are learning how it behaves.
And once you understand how it behaves, you are already closer to solving it.
Remember, this is not about turning life into math. It is about turning confusion into structure.
12. NEXT STEPS
In the next part of this newsletter, I will dive deeper into what this technique is, how it works, why it works, develop further examples, explain advanced concepts, and raise certain caveats about its practical usage. Also, I will link how understanding problems as systems relates to the power of using symbols to represent and solve non-mathematical problems. Since life is a system of operations and functions, one can further understand dynamic human problems through the use of symbols.
Happy Problem Solving!
Evan