THE ORIGINS OF IMPOSSIBLE PROBLEMS

The Origins of Impossible Problems

Introduction. Impossible problems have always been a part of the landscape of human thought. They arise from various sources, often rooted in cognitive, logical, or structural limitations. Some problems are truly unsolvable due to fundamental constraints, while others only appear impossible because of human limitations in understanding, reasoning, or approach. In many situations, we make difficult problems impossible because of our limitations, psychological and otherwise. It is a curious thought problem to consider what sort of limitations AI will reveal when we give it truly difficult problems to solve. We must hope that we humans have not transferred our complete reliance and dependence to machine-learning tools beforehand. Below are key sources of seemingly impossible problems, along with examples and a few references to philosophical and scientific thought.

Impossible Problems. To explore impossible problems, we must consider our systems for solving problems – to understand the array of tools we can array for an attack.  Tools can be as simple as arithmetic, and as difficult as complex theories. Solutions can be as simple as a number, but as difficult as a hundreds-page program of stages. Before even beginning the details of impossible problems we must first consider both the types and causes of these monsters.

Definition. An impossible problem is a problem that cannot be solved.

Problems in mathematics, science, society, and the world satisfy this definition. In addition, a great variety of personal problems are impossible. Some impossible problems of only a few years ago have become simple today.  Therefore, the definition above is a poor one, simultaneously inadequate yet best possible. This needs further explanation.

‘Impossible’ is a vague word. Yet, we use it all the time.  It has several meanings, and it’s best to be aware of which you are using the next time you evoke the word.

 

·       Impossible currently? – As in a disease uncurable today but maybe not next year. Rabies, polio, tuberculosis, and measles, once impossible, are now cured. It can also be applied to oneself or a team.

·       Impossible logically? - As in some kind of paradox that has no resolution. The barber paradox is just one example. Briefly put on an island, the single barber shaves all those who do not shave themselves. Who shaves the barber? If he does, he doesn’t, and if he doesn’t, he does.

·       Impossible problem? – As in some problems unsolvable and no hint to solve. Many in mathematics and physics are so numerous, it would take a book to explain them. Impossible problems of faith and belief challenge the doubter as much as any with doubts counter-balancing desires, needs, and hopes. Such problems even penetrate the fibers of religion itself.

·       Impossible situation? – As in a real-life situation that cannot occur. How about the flying elephant, unicorn, and all manner of Utopias?

·       Impossible project? – As in a wicked problem that may have multiple solutions. Just try to build a beltway around a large city and you’ll see a zillion problems with no clear starting point and multiple solutions.

·       Impossible forever? – As in some problems, that can never be solved. For example, “What is the origin of the universe?” All we ever get is the next, best – oh, and final model. Each generation does the same. Make the final model. When it fails? On to the next final model. My goodness, do humans have an ego or what?

·       Impossible solution? - Here we note one form of impossible problem, that of pursuing an incorrect solution to a given problem, leading to greater problems, for example in a deep rut from which escape is difficult. Think of the possible consequences of the wrong medical diagnosis and treatment. The patient does not get better, but worse. How many small or large businesses have failed because the principals pursued the wrong solution?

·       Impossible theory? – Any theory that is wrong but we are constrained to use it generates impossible problems. At about the time of Galileo (1564-1642), only Ptolemaic astronomy was allowed, but it became unpredictive, providing increasingly inaccurate answers.

Impossible problems are ever-present and, like the desert stallion, forever wild. They represent what we can’t do and sometimes can’t even think. They are the ultimate unknowns.  Yet, century by century and one by one, impossibilities are tamed. Now, we take up the origins of impossible problems, given in twelve parts.

1. Complexity

Some problems are inherently complex due to the vast number of variables, interdependencies, and unpredictable factors involved. Such problems often appear in fields like artificial intelligence, climate modeling, multiple location scheduling, risk analysis, and quantum mechanics. Complexity is similar to wickedness (below).

For example, pandemics, outbreaks of infectious diseases, and antibiotic resistance pose complex challenges requiring coordinated global efforts – from contagion to treatment. Homelessness involves a complex interplay of factors, including poverty, mental health issues, drug addiction, lack of affordable housing, local politics, and social stigma. Healthcare access and costs include issues of access to quality healthcare, rising costs, and disparities in healthcare outcomes, impacting individuals and communities. Each of these problems, among many more are problems of our time and none have been solved to any measure of agreement. All are very large in scope and  highly complex.

In the sciences, we first mention Climate change. This encompasses a wide range of interconnected issues, including rising temperatures, extreme weather events, sea-level rise, and the impact on ecosystems and human societies. Another example, similar but different, is predicting long-term weather patterns with absolute accuracy. This is impossible due to the chaotic nature of atmospheric systems, described by Chaos Theory[1].  The reader may be more familiar with the butterfly effect, a concept in chaos theory, which illustrates how minor changes can lead to vastly different outcomes, making certain predictions practically impossible. All this suggests that excellent weather predictions may be impossible forever.

A newer example concerns Artificial Intelligence (AI).  Developing and deploying AI systems responsibly, addressing issues of bias, privacy, and the potential impact on society is highly complex requiring its own special chips and massive electrical power supplies. Even its mode of communication, the LLMs, require millions of variables to function realistically. Of late we see some players spiking the data sources with falsified date. Of course, the AI learning models will accept and then report it as true. AI does not know right from wrong, and relearning is just as costly as learning in the first place. The corruption of data sets may cause AI to create its own sets of impossible problems.

In business, managing complex supply chains is highly complex, particularly for giant retail firms such as Walmart and Amazon. With global supply chains, with multiple suppliers, transportation routes, and potential disruptions, sophisticated planning and risk management is required.

2. Vagueness and Ambiguity

Second in significance to complexity for creating impossibilities are problems that are poorly defined, have vague terms, or ambiguous, lacking clear or even undefinable parameters, making them difficult to solve. Combine this with a team whose knowledge base is weak, vague, or ambiguous, and you have problems poorly stated with a poorly trained team to solve them. Then you have trouble. (Sounds a bit like the government.) As well, some problems with manageable number of parameters, we can soldier through with workable solutions, but when the system must address millions of customers or payments, there become room for ambiguity, fraud, and waste.

Many classical problems, stemming back into antiquity, have vagueness and ambiguity components. For example, “What is the meaning of life?” Or, “What is love?” Both are quintessential examples because vagueness and ambiguity, with both lacking precise definitions and varies based on individual and cultural perspectives. Wittgenstein’s concept of language games[2] suggests that words derive meaning from their usage, implying that certain philosophical questions may lack concrete answers  ̶  indefinitely.

On happiness, Leo Tolstoy[3] instructs us, If you want to be happy, be. First, dismiss the usual forms of happiness, including the hedonic, social, achievement, materialistic, mindfulness, philanthropic, spiritual, health, and creative. These are specific forms from which happiness is sometimes attained. Dismiss as well other aspects, or substates of happiness including joy, contentment, serenity, gratitude, satisfaction, optimism, fulfillment, amusement, bliss, and inner peace.  Then of course, there is deferred happiness, that which is yet to come. So, what is happiness? Impossible because vagueness and ambiguity.

In business, consider the problem, "Develop a new marketing strategy to reach a wider audience." It is vague and ambiguous because it doesn't specify target demographics, preferred marketing channels, or the desired reach increase. The problem may not be impossible if all the definitions are unambiguous and terms have agreed definitions.

Ambiguity and vagueness are found less often in the sciences partly because the demand for unambiguous terms is highly stressed. As well, the clarity of problems is important. It would be difficult to publish a paper in these subjects without clarity of terms. However, when vague goals are suggested, say like “Find the optimal … “ , it is important to clarify what optimal can mean, as it often has different meaning for different metrics. The most classic and expensive mistake on ambiguity was made in 1999 when the Mars Climate Orbiter[4] mission failed due to a metric to imperial unit conversion error, where the spacecraft's navigation software used Imperial units while the ground team provided information in metric units, leading to a catastrophic atmospheric entry at a much lower altitude than intended. 

3. Missing Information or Knowledge Gaps

Some problems appear impossible because we lack the necessary data, tools, or theoretical understanding to solve them. This is most common in all problem-solving areas. For example, before germ theory (Louis Pasteur, 19th century), curing infectious diseases seemed impossible because the role of microorganisms was unknown. The Ptolemaic system of astronomy survived almost two millennia before errors started creeping in. Copernicus introduced the first heliocentric system, but computations were just a difficult. Kepler gave us his three laws, including most notably the elliptical path law.  Finally, Newton invented calculus, the fundamental law of gravitation, and then described how the planets (and star) travel. This was a tour de force of conceptual and analytical thinking.

Some philosophers have quipped that science advanced grave by grave, with the old theories discarded in favor of the better one. One could also say science advances by filling in the holes, the gaps, and finding the unknowns.

Sometimes gaps are filled by new knowledge as discussed. Other times gaps are filled by filtering the noise from the signal. This is the subject of Nate Silver’s book[5]. Silver examines the world of prediction, investigating how we can distinguish a true signal from a universe of noisy data. Most predictions fail, often at great cost to society, because most of us have a poor understanding of probability and uncertainty. A more basic type of filtering is a television signal, which all televisions now have. Other gaps appear suddenly and have the unusual name Black Swans, as given by Talib[6]. Black swans are extremely unlikely events that usually occur without warning, though some foolishly like to predict them. You see, if it can be predicted, then it can’t be a black swan. For example, one of the most recent black swan events was the 2008-2009 financial crisis known as the Great Recession.

4. Wicked Problems

A wicked problem[7] is one that is difficult to define, has no clear solution, and involves complex interdependencies. However, they have been discussed earlier. They rarely have a single solution, there is no good or bad, best or second best. There seems to be no clear solution method. Examples:  climate change, global poverty, and political corruption are wicked problems because they involve multiple stakeholders with conflicting interests and no definitive resolution. For a quick summary of wicked problems, see the Appendix or for a lengthy introduction, see
https://used-ideas.blogspot.com/2025/02/the-character-of-wicked-problems.html

5. Lack of Understanding

If a person lacks the necessary background knowledge or expertise, a problem may seem impossible. Lack of understanding can lead us to using some ideas, formulas, and frameworks beyond their capacity, domain, applicability, or law. For example, a judge may make a ruling based on an incorrect interpretation of some law. Student typically make “new” discoveries by using formulae improperly – all based on a lack of understanding.

 Lack of understanding and the resulting self-consciousness can cause emotional reactions and inner fantasies that trigger different actions (such as curiosity, loneliness, anxiety, rejection, depression, mood swings), depending on how much or little we understand, what or whom we do not understand, and what is at stake if we do not understand. Lowered feeling of self-worth makes difficult problems more so, and often impossible. [Basic advice to students. Learn all you can now while to have the time.]

So, concomitant with the lack of understanding is that totally wrong solutions may be obtained, and moreover they can have psychological impact on the problem-solver.

6. Internal Conflicts in the Problem or Solver

Some problems contain contradictions or paradoxes, making them unsolvable. Additionally, cognitive biases and emotional conflicts in the solver can prevent resolution. For example, the Liar Paradox ("This statement is false.") creates a logical contradiction that makes it impossible to determine its truth value. As well, Zeno’s Paradoxes (5th century BCE) argue that motion and change are logically impossible, though they occur in reality. This problem combined the conflicts of thinking finitely in the face of infinite processes[8], and was unresolved completely until the invention of calculus two millennia later. However, it is important to note that without an expansion of thought toward infinite processes, Zeno’s paradoxes would remain insoluble to this day.

7. Wrong Mindset or Techniques

Using the wrong approach or perspective can make a problem seem impossible when it is actually solvable. Early attempts at human flight failed because engineers tried to mimic bird wings rather than studying aerodynamics. That is, one can build something that looks like a wing, but without the study of the wing in flight will miss the general principle of an airfoil. The Wright brothers succeeded by so noticing and changing their approach. In some cases, it took more than observation to solve something impossible. It took an entire paradigm shift[9] in thought. Relativity (Einstein) is the canonical modern example. Uncertainty principles are others. The integrated circuit made high speed computers possible. The newly discovered prion as another form of disease (mad cow disease) was accepted slowly, but eventually. As well, plate tectonics was another originally rejected idea counter to the mainstream. It gives a greater understanding of earthquake and volcano formation. The wrong mindset or techniques is not unlike a rut in which an investigator or entire team may find itself permanently separated from correct solutions.

8. Lack of Expertise or Creativity

Creativity and expertise are crucial for solving complex problems. Without them, even solvable problems may appear impossible. The first well-known example was the creation (now called invention) of the wheel to help transport objects such as crops. While Pasteur’s application of germ theory to develop vaccines was truly creatives, so also was use Alexander Fleming’s 1929 use of molds to create penicillin. Do not forget Carnegie’s creativity in using steel for the first time in the construction of long span bridges (Mississippi). More generally, instead of directly addressing a problem, a creative approach might involve re-framing it as an opportunity. For example, if a company is facing declining sales, they might reframe it as an opportunity to explore new markets or develop new products.

More specifically, many mathematical proofs, such as Fermat’s Last Theorem, remained unsolved for centuries until Andrew Wiles used modern number theory in 1994. (Note. It wasn’t just using new techniques. Wiles worked seven years solid. Then, a logical error was found, and it took another two years to fix it.) In fact, all of the great discoveries in every science have a hugely creative component. All solved important, previously impossible problems. Henri Poincaré[10] argued that creativity is essential in mathematical discovery and problem-solving. While on the subject, we recall Georg Cantor[11] (1845-1918) while working on an entirely different topic, the convergence of Fourier series, constructed/invented set theory and an entire family of infinities. This led to the structure of modern mathematics and to much of modern analytical science, particularly cosmology.

9. Logical Errors and Cognitive Biases

Faulty reasoning and biases can make problems seem insurmountable. Certainly, drawing conclusions from the many logical fallacies can lead to contradictions and incongruities. Everyone using logic in their arguments, scientists, engineers, lawyers, and all, have made logical errors. They can be heartbreaking when discovered, and embarrassing when others find them. That is, not counting on those made intentionally.

Confirmation bias causes people to reject evidence that contradicts their beliefs, preventing them from finding solutions. One form is Daniel Kahneman’s[12] work on cognitive biases. It demonstrates how human thinking is prone to errors that can create artificial roadblocks. For example, the Dunning-Kruger Effect is the tendency for individuals with low competence in a particular area to overestimate their abilities, while those with high competence may underestimate their abilities. Hindsight Bias is the tendency to believe, after an event has occurred, that one would have predicted it, even if one did not. For a third example, the Halo effect is one’s tendency to use their overall impression of someone when making judgments on their character or perhaps their test papers.

10. Conceptual Gaps and Unidentified Unknowns

Some problems are impossible simply because the necessary concepts have not yet been developed. For example, causes of disease persisted throughout most of history simply because people were ignorant of what it was. The four humors (blood, phlegm, yellow bile, and black bile) didn’t work, nor did the general description of disease as a malaise. It wasn’t until Pasteur applied germ theory to manufacture a vaccine prove the nature of disease. Also, before the concept of zero was introduced in mathematics (Brahmagupta, 7th century CE), certain calculations were much harder. Such are among the known and unknown unknowns in Rumsfeld terminology[13].  Similar was the previously impossible problems of long division, which at one time was taught in a graduate course available only in Germany, so difficult it was.

Sometimes we know something is missing, but the situation degrades when we don’t even know where or what is missing. Unknown unknown problems are particularly interesting among all impossible problems. They must have two stage solutions, the first being their discovery in the first place and the second actually solving them. Of course, no one seriously works on unknown unknowns for the simple reason you can work on what you don’t know to work on.

11. Mental Fixation

Being stuck in a particular way of thinking can prevent someone from seeing alternative solutions. That is, a person might get stuck using a specific strategy to solve a puzzle, even when a different strategy is more efficient. For example, the Nine-Dot Puzzle requires drawing outside the assumed boundaries to solve it, illustrating how mental fixation can make easy problems seem impossible. Also, and very common is a fixation on a prior problem-solving approach, even when a more efficient solution exists. (Einstellung Effect). These examples are included within Gestalt psychology (Köhler, Wertheimer), which explores how problem-solving often requires a shift in perception. With a mental fixation, one can make a problem impossible if only for himself. Another type of mental fixation comes with the attempt to apply a personal truth that is incorrect. Such problems, in Rumsfeld’s terms, are the result of unknown knowns.

12. Fundamental Limitations and Physical Constraints

Some problems are truly impossible due to physical laws or mathematical constraints. For example, A perpetual motion machine is impossible because it violates the laws of thermodynamics. This may be more comprehendible from an energy principle, in that the perpetual motion machine simply creates new energy out of nothing, violating the conservation of energy law. The Heisenberg Uncertainty Principle (1927) shows that certain aspects of reality (e.g., position and momentum) cannot be simultaneously known, placing fundamental limits on measurement. These days, uncertainty problems are a major concern in almost all facets of modern life. One method of attack is to convert them into probabilistic or statistical problems.  Also, engineers often face uncertainty when designing systems and products, as they must account for potential variations in materials, manufacturing processes, and environmental conditions. In medicine, doctors often face uncertainty when diagnosing illnesses, as symptoms can be ambiguous, and tests may not provide definitive answers. In diagnosis, AI seems to be displacing doctors.

Using some ideas, formulas, and frameworks beyond their capacity, domain, applicability, or law is more than common. For example, a judge may make a ruling based on an incorrect interpretation of some law. Another similar error is made by scientists is to use results for which their data or conditions do not apply. This is especially prevalent in the use of statistics. Finally, many applied mathematicians, economists, and finance experts discuss the consequences of various models using variables that cannot be computed or even evaluated. As an example, consider using self-declared valuations of opinions on a number scale, say from one to ten. The results may not be valid owing to the fact that there is no link between the criteria used by individuals. Likert scales have suffered this problem for generations. For example, consider the performance-type survey we see all too often in politics. Everyone uses their own criteria for what the terms mean. Yet, the results lead the news almost every day.

□	Strongly Approve
□	Approve
□	Neutral
□	Disapprove
□	Strongly Disapprove

Conclusions. It is important to note we have addressed the origin of impossible problems for you, for us, and for everybody. If you don’t understand something, that can be quite different from anyone else not understanding it. With that in mind, many problems that seem impossible are actually solvable with the right knowledge, perspective, or approach. Others remain unsolvable due to fundamental constraints in logic, physics, or human cognition. By identifying the root causes of difficulty, we can distinguish between truly impossible problems and those that are simply awaiting a breakthrough. A short video on impossible problems can be found at https://www.youtube.com/watch?v=c43AsHoYppA

Appendix. Wicked Problems – in brief. The scope of wicked problems by characteristics follows. However, every person in the business uses their own, often similar list. A problem with only a handful of these still qualifies as wicked, making the definition of a wicked problem a wicked problem itself.

·       They do not have a definitive formulation.

·       They do not have a “stopping rule.” In other words, these problems lack an inherent logic that signals when they are solved.

·       Their solutions are not true or false, only good or bad.

·       There is no way to test the solution to a wicked problem.

·       They cannot be studied through trial and error. Their solutions are irreversible so, as Rittel and Webber put it, “every trial counts.”

·       There is no end to the number of solutions or approaches to a wicked problem.

·       All wicked problems are essentially unique.

·       Wicked problems can always be described as the symptom of other problems.

·       The way a wicked problem is described determines its possible solutions.

·       Planners, that is those who present solutions to these problems, have no right to be wrong. Unlike mathematicians, “planners are liable for the consequences of the solutions they generate; the effects can matter a great deal to the people who are touched by those actions.”

See, https://www.stonybrook.edu/commcms/wicked-problem/about/What-is-a-wicked-problem

©2025 G Donald Allen

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[1] Lorenz, E. N. (1993). The Essence of Chaos. University of Washington Press, Seattle.

[2] Wittgenstein, L. (1953). Philosophical Investigations. Translated by G.E.M. Anscombe. Oxford: Blackwell. 

[3] Leo Tolstoy, Russian - Novelist September 9, 1828 - November 20, 1910

[4] https://en.wikipedia.org/wiki/Mars_Climate_Orbiter

[5] Silver, Nate. The Signal and the Noise: Why So Many Predictions Fail – but Some Don't. W. W. Norton & Company, 2012.

[6] Taleb, Nassim Nicholas (2007), The Black Swan : the Impact of the Highly Improbable. New York :Random House.

[7] Rittel, H. W., & Webber, M. M. (1973). "Dilemmas in a General Theory of Planning." Policy sciences, 4(2), 155-169.

[8] Here we are tempted to attribute infinite processes to Issac Newton, inventor of calculus. However, it was Simon Stevin (1548-1620), the Flemish mathematician and engineer who used limiting ideas to discuss pressures versus depth against a dam. One could say Stevin broke a conceptual dam of thought.

[9] Kuhn, T. S. (1962). The Structure of Scientific Revolutions. University of Chicago Press, Chicago.

[10] Poincaré, Henri. (1908). Science and Method. Translated by Francis Maitland (1914). London: T. Nelson & Sons.

[11] J W Dauben (1979), Georg Cantor: His Mathematics and Philosophy of the Infinite, Cambridge, Mass.

[12] Kahneman, Daniel. (2011). Thinking, fast and slow. Farrar, Straus and Giroux.

[13]  Rumsfeld, Donald (2011). Known and Unknown: A Memoir. New York: Penguin Group. p. xiv. ISBN 9781101502495.