A "Problem" Defined
With a little extra thrown in...
“It isn’t that they cannot find the solution. It is that they cannot see the problem.” – G.K Chesterton
This newsletter is a companion piece to my prior newsletter “IF everything is a problem, THEN nothing is a problem.” That newsletter discussed “10 Problem Attributes” and how the word “problem” is often used to denote a situation that is not actually a problem. A key aspect of problem solving is comprehending the nature of a problem. Understanding first what constitutes a problem, and then how problems are defined, conceptualized, and formulated, is essential to being able to use the right tools to solve them. Definitions matter.
Basic though it may be (and possibly unnecessary or redundant if you are already a good problem solver), this information sets the stage for more complicated ideas and lessons. Framing more complex material to be introduced later, these introductory newsletters form the structural basis to provide clear ideas about the field of problem solving. Whatever level of problem solver you currently are, I believe this information will help you become better at whatever you do in your life, education, and career!
Problem solving is an activity, but as with any other activity, learning the fundamentals is crucial to eventual mastery.
Why Study Problem Solving?
I have pointed out that we are all problem solvers by nature and experience. Also, I emphasized that we solve problems every day, whether as part of our jobs, in our education, or in our personal lives. Being able to utilize the best problem-solving techniques gives us an advantage when we (or others) are faced with really tough problems. We can then deploy our problem-solving toolkit to maximum benefit. Today, working collaboratively as part of a team in developing solutions to problems is common, as often there exists problems we cannot solve alone. This is especially true in business and academia.
There are many benefits to becoming a proficient problem solver. Solving tough problems, as many entrepreneurs do, can make you rich and famous.
Scientists and researchers advance in their careers when they are great problem solvers.
People who work in business, finance, engineering, medicine, or others fields often are rewarded the better they are at solving problems.
Furthermore, great problem solving has a major impact in our society. Scientists work to solve problems that have a transformative impact on our knowledge and understanding of the world. Research problems in medicine, biology, and genetics lead the direction in which many new advances will help us live better and longer. Mathematicians every day solve important problems in math that expand and enrich that field. Physicists solve problems that help us better understand the universe. The list of important achievements is exceptionally long, thanks to great problem solvers
A Problem-Solving Extraordinaire
To me, a life of problem solving is a noble pursuit, especially where that includes the search for knowledge or helping the world. One could spend much of their entire lives just solving difficult math problems, much as the legendarily brilliant Hungarian mathematician Paul Erdős (1918-1996) did over the course of his lifetime. Problems WERE this mathematician’s life! He is one of the most fascinating thinkers of the 20th century, not only for his genius, but also for the way he lived. Erdős was an itinerant mathematician who spent the majority of his professional life couch surfing. The author of over 1500 papers, and collaborator with over 500 other mathematicians, he was one of the most important problem solvers of the 20th century, and even has a number named after him: the Erdős number, similar to the idea of six degrees of Kevin Bacon [the actor]. This idea of being linked together in a “social network” is explored dramatically in the movie Six Degrees of Separation (1993).
DEFINITION AND DISCUSSION
Universal agreement does not exist on what defines a problem or even what constitutes problem solving. Indeed, there are some notable and surprising differences in how certain researchers and academics define problems, and therefore problem solving as an activity. However, what I want to do here is provide the best and most accepted definition of what a problem is, from authoritative sources, along with my own ideas about “problems” and how those affect the way I approach teaching problem solving methods.
None of this definitional material is set in stone, of course!
A problem is what YOU consider it to be. If you FEEL you have a problem, then it is a problem for you.
Any formal definition of a problem or problem solving is just a starting point for ordinary understanding; there are subtleties and nuances, with individual viewpoints that are just as important as any traditional or current definition.
My writing reflects my understanding of the topic; I have read a great deal of material on problem solving; however, my own thoughts evolve and are influenced by other writers, as I uncover and discover more about the subject.
A Note On Language and Writing
As both writers and readers know, words matter. Correctly understanding the definition of a word leads to clarity, not only about how that word should be understood, but also about how it should be used. Words are processed conceptually; in fact, words are concepts that through the tool of language (whatever language that may be) represent some quality or state of existence about the world. Context also matters where it comes to using specific words, that is to say, using the correct words in the right place at the right time, what linguists refer to as semantics.
Sometimes writers can be led astray in looking up synonyms as substitutes for other similar words. Everyone makes mistakes; we must allow for unintentional errors from ourselves and others, forgive, and fix those errors when brought to our attention! There are subtleties in words that are frequently overlooked or under appreciated in our hectic world; reading the dictionary is not for the faint of heart! One word generally is not a perfect replacement for another, despite similarities in definitions. This confusion is often understandable, especially in a modern context. Therefore, I find myself referring to authoritative dictionaries, such as the Oxford English Dictionary (OED), to check my understanding of words I have become so used to, and therefore familiar with, that I sometimes take them for granted.
Over time the usage of a certain word changes, language evolves; as a writer, it is particularly important to say what you mean, so that readers have an accurate picture (as a mental representation) of what is being communicated.
I believe clarity is the key to communication, and a writer cannot communicate if the tools (words) used are not the proper ones.
There is another very interesting thing can happen when I revisit a word’s meaning: occasionally, what I originally thought or currently believe a word means is not entirely accurate, or there is a subtlety that brings me to a clearer (or even new) understanding and appreciation of that particular word. I am sure that this happens to other writers. Much as listening to a piece of music again, or watching a movie a second time, one might experience the understanding of a word differently over time or in a new context.
THE BEGINNING TO THE PRESENT
A History of the Word “Problem”
Interestingly, the word “problem” has a reasonably long history, and its meaning has evolved and grown over time. I have already extensively covered what I consider to be a problem, so it is time to consider other perspectives and definitions.
The word problem has its origins in Greek and Latin, but its meaning and usage has shifted, like many words, all the way to its current and colloquial definition. What is curious about the present usage of the word is that it has so many sub-meanings and associations.
First, looking back in history for a moment, the ancient Greek definition of problem comes from “próblēma” which means an obstacle, a projection, a barrier; or, that which is proposed or (thrown) put forward, such as a situation or question. One could say there might be an interesting synthesis between these ideas: “throwing” is a state of action…a problem could be considered as a situation where action (physical or mental) is required. Próblēma derives from “proballein” [before to throw]. Through Latin it became problēma.
Let us stop here for a minute and consider these really early “problem” definitions. Already, there is a split meaning between what might be considered a material definition, i.e., a barrier or obstruction, from the more conceptual definition, that which is proposed or put forward. Only the synthesis of these two (which may be implied within these definitions, but not explicitly stated) would be superior to both definitions individually.
The first definition is not only better, but also encapsulates the uniqueness of problems specifically. Problems are obstacles, that in order to “get around” a “path” must be discovered, i.e., a solution. However, there may or may not be a way around that obstacle, a “solution” that is satisfactory, only that one could exist. It is also interesting to note that in climbing, a “boulder problem” has the meaning of finding a sequence of moves to the top.
In contrast, to propose or put forth is similar to the stating of a proposition, question, or speculation. As I have noted earlier, questions are meant to elicit answers…problems require, or shall I say, “gravitate” towards solutions.
We humans may speculate on a lot of issues, but those speculations are not necessarily problems. Only when a solution is required to move forward does a situation meet the criteria for being a problem. That’s when the fun begins: PROBLEM-SOLVING.
To be fair to those that adopt the latter definition of problems as propositions, I can see the appeal: The word “problem” is a great catchall to capture just about anything we may not like, disagree, understand, or know the answer. I call this the “Fix It” aspect of problems. “I have a problem…my kid is getting D’s in Calculus. I guess I should get them a tutor” or ‘They have a problem attitude” or “My engine is having a problem!” or “Why is my teacher giving me a problem?” or “The problem is I just don’t understand quantum particle spins!” One could ask: “What aren’t problems?” in this expansive definition.
Moving forward, although I recognize the utility of seeing problems as “proposals” or “speculations” related to the idea of asking questions, this will be considered only a secondary, and therefore a less accurate definition.
However, since personal problems have UNIQUE properties to them, I will discuss, though differentiate, problems that have more of a “mathematical” or “scientific” aspect to them, as opposed to problems with personal/cultural/societal characteristics. Both are equally valid problems to those who face them!
Second, returning to the definition, in the 15th century the French word “problème” meant a difficulty proposed for discussion or solution; a riddle; a scientific topic for investigation.
Here we begin to see the modern conception of the word. In some regards this definition is related to the idea of answering that which is not known, or which is puzzling. I would like to interject that when writing or speaking, clarity and specificity can make the difference between being understood or not; in this case, whether you have a problem or not. A riddle is a riddle. A question is a question. Speculation is speculation. They all have well-defined meanings. Synonyms are not exact substitutes for other words.
The proxy that the word “problem” has become for anything challenging robs the word of any real meaning and reduces almost anything, including life itself, to a problem. I believe that life cannot be, nor should not be a problem to solve; there is no “solution” to the life “problem” and even if there was, what fun would that be.
Third, in England in the 1560’s (approximately) the word “probleme” takes on the meaning of a proposition that requires some operation to be performed, such as an exercise. However, implicit in that definition is that the answer to an unknown problem requires operations (mathematical in many cases) for a solution to be reached.
This definition may at first seem to contradict my assertion that “problem sets” in mathematics, physics, engineering are not problems in the truest sense, but rather drills or exercises. However, what this definition suggests is that a problem is not just mere speculation; problems require “operations” to be performed, leading to the idea of “exercising” the mind. This definition does not suggest that problems are simple exercises to be worked through.
As a side note, the world was a far different place in the mid-1500’s, and there were many genuine practical problems to be solved then, that is, many problems with unknown solutions that we now discovered. The mathematics and science that existed five centuries ago bears little if any resemblance to those fields presently. Today we take for granted the mind-boggling technology that 500 years ago would have seemed unimaginable, except perhaps for a genius like Leonardo da Vinci.
The gist of all this is this: Real problem-solving is an inventive activity, for the most part. Discovery of solutions by invention itself is necessary to solve most important problems, or the inventive juxtaposition of already existing ideas or techniques.
Finally, we get to the modern (and all inclusive) definition of a problem, including the kitchen sink:
Something that is difficult to achieve or accomplish
A person, matter, or situation to be dealt with that is challenging
A question arising from given conditions to demonstrate a proposition, law, theory, or principle
Something requiring completion or a specified result by which one has construct a process or demonstrate a method, procedure, or technique
An inquiry or proposition for consideration or action
A puzzling circumstance
Formally, I believe in a kind of hybrid definition of a problem that incorporates part of our modern day understanding of the term with the way many academic researchers view problems (see below). What I would like to see deleted from the definition of the word “problem” is the idea that problems are questions, speculations, general difficulties, drills…
Many, but not all, academic writers and researchers on problem solving consider a problem as a question or issue that remains uncertain, unanswered, unsolved. I agree to a certain extent with this definition because it is very inclusive and captures much of what is essential about any problem. However, that is not enough. As I wrote about before, questions are not problems. Just because a question is unanswered or there is uncertainty about the answer or that it is un(re)solved does not make it a problem. Seeing an open question as a problem misses something very crucial about problems and confuses two very related, but different concepts. There are infinitely many unanswered and unsolvable questions (given our current technology, methods, and mathematics) …theoretically, one could pose unsolved mathematical questions (which may or may not have solutions) until the end of time, as the answer to one question raises many more, in a kind of question and answer loop. Strictly speaking, problems have potential “solution spaces” that can be defined: that is, the set of all theoretically possible solutions to that problem. Without a possible answer, a question is mere considered speculation, not a problem!
Moreover, I would like to further add (in subsequent newsletters) to my reader’s general understanding of problems being conceived as “functions” of one or several variables (based on a set/sets of conditions). The “problem space” defines all the essential properties of a problem. Transforming certain problems into functions of variables allows better conceptualization, ordering, systematizing, and solutions to them. There is much value in “seeing” some problems through the “lens” of functions, because it will not only sharpen the focus of the problem, but will also strip away many unnecessary details. Surprisingly, this method works very well on personal problems, or on problems of a less quantitative measure, such as societal or cultural problems. I look forward to addressing this fascinating reframing of certain kinds problems as functions later.
Well, enough said about what I and others view as a “problem.” Whatever definition or understanding of the word suits the reader, we must move on to the problem-solving sections of these newsletters. I will return at points to reference certain aspects of problems and their definitions, but the foundation has been laid.
One final thought:
The search for solutions to a problem is a journey of discovery, of excitement, of the unknown. Imagine yourself when problem solving as an explorer searching for treasure. There is treasure out there…solve your problems to find it.
Happy Problem Solving!
If you have enjoyed reading this newsletter, please let others know by sharing it with them!