Problem-Solving - Beyond Impossible

 Problem-Solving -  Beyond Impossible

There is no problem the mind of man can set that the mind of man cannot solve.
― Samuel Johnson

Figure 1. Dr. Samuel Johnson (1709-1784)

Introduction. What can possibly be beyond impossible in problem-solving? It would seem that the “impossible” is as far as we can go. Yet, there is a nether region where the “beyond” modifier finds its place.  

If you recall Johnson’s quote, it may be from the 1946 Sherlock Holmes movie, “Dressed to Kill” starring Basil Rathbone.  It does make you think, but it also gives you hope. It gives us hope we can solve anything we come across.  It gives hope we can ultimately answer every question – though some solutions may take more time than others. Unfortunately, Dr. Johnson was wrong.

Albert Einstein (1879-1955) was known for saying, “You cannot solve a problem with the same mind that created it”. This is similar, after a fashion, to the ancient Greek historian Herodotus who told us “No man ever steps in the same river twice, for it's not the same river and he's not the same man.”  That’s more like it. Of course, by posing the question, one’s mind changes, and then when trying to solve it, a different mind is at work. 

“Asking the right questions is as important as answering them.” This common expression has been echoed by many over the years. This idea is closely related to the Socratic method, where Socrates (d. 399 BCE) emphasized the importance of asking questions to stimulate critical thinking and illuminate ideas. Certainly, until the right questions are asked, where answers are possible, a given situation remains impossible, and possibly beyond.

In this short note, we go beyond even to true questions that no one can answer. We go  beyond the simple questions loaded with vague terms such as “How good is the good boy?”  The reader should note problems discussed herein are complex and not not of the simple school problems variety.

Beyond Impossible.   The problem here is that from the work of mathematicians, particularly Kurt Gödel (1906-1978), we know now there are true propositions that cannot be proved, and can never be proved. Many have already been discovered.  It seems almost paradoxical that we can prove them unprovable but cannot prove them.  The implication is that a problem can be set and then found can never be proved. More on this later. Such problems are beyond unknown unknowns, perhaps unknowable knowns. They are undecidable.  In another language, we can with a slight linguistic stretch prove there will be Black Swan (a vague term meaning unexpected and big) events, but we cannot prove what they will be. Let’s take up a few specific examples of impossible and beyond possible problems, noting that the solution can be so termed.  

Artificial Intelligence. Combine this with AI, now looming ever larger in our lives. The two most serious danger signals of AI are dependency and reliance. It is one thing to rely on AI every day, as a tool or an assistant. However, dependence brings AI to another level, wherein the human cannot survive without AI. One can view AI as a drug or a device. Either way, it is not exactly healthy. It implies a type of co-existence. Moreover, with dependency, there becomes a reluctance toward independent thought. As to problem-solving, the further implication is reliance on AI to solve the problem. And this denies innovation, a feature AI is miles from at this point. With this reliance, problem solutions depend on what’s been done previously, and therefore the same failing solutions could be generated time after time. This, to depend on a “mind” incapable of solving the problem, is certainly indicative of a problem beyond impossible. AI is a completely new form of permanence – to be discusses below.

Medicine. In today’s Coronavirus event, we find even a middle ground for this.  We will find a vaccine for this one, but…  We can agree that another disease will be present that will rock the world, but we can never predict the disease itself, the time it occurs, or where it will arise.  Just a couple of years ago, Ebola announced itself.  This one scared us and still does.

Regarding AI, even now servers and their machine-learning algorithms are digesting as much medical data as they can find.  They have now learned to diagnose medical problems at a truly professional level.  One problem confronting the medical community now is whether to accept such diagnoses as the diagnosis.  This is not a little problem. It is a problem with repercussions across all of medicine, from the school to the courtroom.  Let’s look at a few elementary considerations.

Tools will be put in the hands of the medical practitioner and physician's assistant. The patient may not even qualify to see a doctor until after this “AI-procedure.” For obvious reasons, the doctor contradicting the diagnosis is put at legal risk.  Medical research and new procedures will be underdetermined. On the other hand, if the doctor goes with machine learning, he/she has a legal defense built-in.

Medical schools will teach doctors to rely on the software.  This could undermine their diagnostic discipline, making them the tools of the software.  Sure, they will offer comfort and prescribe the recommended medicine. But their self-confidence will be undermined.  They will relax, losing basic sharpness with their fundamental skills. The drug companies will become fierce competitors to make their new drugs acceptable for AI-approved treatment recommendations. One consequence is that innovations in medicine will be diminished.  Another is that such engines will strongly enhance the "abilities" of poor doctors. (This is actually good.)  The poorest doctors will get better; the better doctors will get poorer, or at best extremely more cautious. Resolving such homogenization of medical skills, effectively removing humans from the loop, seems to be a future problem and possibly beyond impossible.

Language. Vagueness is the bane of pragmatic people and philosophers alike. Can you build a highway that is generally safe, and allows brisk traffic in all but the worst conditions? Is Peter a good boy or a bad boy?  Such questions as these are loaded with vague terms and can be considered and then answered only conditionally, and even not then.

To be more formal, the French philosopher Jacques Derrida (1930-2004) argues that there are no self-sufficient units of meaning in a text, because individual words or sentences in a text can only be properly understood in terms of how they fit into the larger structure of the text and language itself. Derrida is most celebrated as the principal exponent of deconstruction, a term he coined for the critical examination of the fundamental conceptual distinctions, or “oppositions,” inherent in Western philosophy since the time of the ancient Greeks.

The key argument in deconstruction is that meaning is unreliable as the language that communicates meaning is itself unreliable. The Polish mathematician Alfred Tarski (1901-1983) recognized this in his work in the 1930s on Truth*, where he posited truth in a given language must more-or-less be certified within another. Derrida also posits many inconsistencies arise by using the analogy. The analogy is a powerful tool to convey understanding, though it is often riddled with errors and misconceptions. All of us use analogies to explain new ideas. In particular, most analogies, to be relatable are simplified renditions of the new idea or problem but are often inaccurate and misleading. Moreover, they reveal only a single facet of the problem. This incompleteness often leads to a misconstruction of it, which in turn leads to forms of incorrect and impossible solutions. This occurs more in teaching than problem-solving per se but plays a role in problem-solving as well.

We make a final observation about problems with conflicting and or competing values, particularly when one of the values is vague and the other is precise. For example, in the Sorites Paradox (alt. Heap Paradox), where starting with a pile of sand, a single grain of sand is added, one by one until there is a heap of sand. Exactly, at what grain does the transition take place? This is beyond impossible to answer because the terms “heap” and “pile” are vague. Therefore there can be no clear answer to the problem. When clarity is lost, so also is the solution, making the original problem impossible. Similar terms are good-bad, tall-short, weak-powerful.

Fear. Fear can play a significant role in problem-solving, influencing both the process and the outcome. It is a complex emotion that can manifest in various ways, potentially either hindering or enhancing an individual's ability to solve problems effectively. Fear can be and is a hindrance to problem-solving in some situations, as is its cousin, self-confidence. 

Paralysis by Analysis: Fear can lead to overthinking and excessive analysis, which may result in "paralysis by analysis." When someone is afraid of making the wrong decision, they may hesitate to take any action at all, leading to stagnation and inaction. This penetrates into the psychology of problem-solving, the key factor in giving solutions, as opposed to the problem itself.

As well fear causes uncertainty, and this generates more uncertainty and indecision about the solutions suggested and certainly hinders solving the problem in the first place. Often this results in a restructuring of the problem or reluctance to accept the solution, making the problem even more difficult, to the point where no solution can achieve a consensus among stakeholders. The feedback goes to the problem, moving from tractable to intractable. Fear and uncertainty render indecision, and the problem thereby becomes impossible or beyond.

The story of fear in problem-solving is supported by extensive literature, and only this single snapshot of its effect is given here.

Computing and Mathematics. In terms of computing, we can say a problem is decidable if we can construct a Turing machine (an abstract computer) that will halt in a finite amount of time for every input and give the answer ‘yes’ or ‘no’. A problem is undecidable if this is not so. And there are many such problems. One classic is Hilbert’s Tenth Problem: Is there an algorithm that can determine (in finite time) if every polynomial equation with integer coefficients has an integer solution?  It was proved undecidable in 1972 by Russian mathematician Yuri Matiyasevich  (1947- ). 

In the real world, for example, suppose in some business venture, it is determined the risks of Solution A and Solution B are enormous. Yet, one solution must be chosen. This brings the firm to a type of undecidable impossibility if it is known that maintaining the status quo may result in a malingering decline in profitability leading to bankruptcy. This is but one real-world example of the notion of undecidability, a topic to my knowledge that has not been studied.

An age-old problem in mathematics was this, “Can we prove every true proposition?” This would mean Mathematics is a complete system. It should be true. Don’t you think? Well, around 1930 this problem was solved in the negative by Kurt Gödel (1906-1978). This means there are true propositions we can never prove, no matter how smart we become. Mathematics is incomplete. The proof is not difficult but a little tricky. It came as a complete shock to the Mathematics world, and now many old unsolved math problems have been proven to be undecidable. No proof ever! Such issues seem beyond impossible, and it signals many problems well known to undergraduates may be of this variety, not solvable ever, even in the real world. Mercifully, we’ll not pursue this topic further.

Permanence. The Permanence of theories going forward, is now with us more than ever. Complexity is one reason. Quality is another. So, established theories become so entrenched they cannot be breached or changed. The classic example is  Ptolemy's astronomy from ancient Greece, where the planets revolve around the Earth along epicycles. It worked well for quite a while. Then errors were noticed. But with the Catholic church supporting it, it was nearly impossible to change to a heliocentric model. Even the great Galileo (1564-1642) was arrested and sentenced by the Papacy to a lifetime of house arrest for suggesting otherwise. The notion of bloodletting to cure diseases persisted for decades after it was debunked. More recently, the belief that vaccines cause autism persists, all owing to a paper, later debunked, that so indicated.   Most theories do not inch along, getting better and better. Theories experience revolutions, as per the philosopher of science Thomas S. Kuhn (1922-1996). Even today, the Standard Model in cosmology is at risk as new findings that dark energy is being depleted are the cause for fueling the expansion of the universe. Yet, careers have been built on the Standard Model, and advocates will not give it up gently.  Old, failing theories persist because institutions or communities want them to.

Evolution. Does the human brain have the horsepower to solve all these problems, or at least come to terms with them? We need another quote, from one of the exemplars of modern analytical thinking.

“The imagination of nature is far, far greater than the imagination of man.”
--- Richard P. Feynman (1918-1988)

Important it is to observe that the current human brain evolved for success at hunting, gathering, aggression,  and all-around survival. Yet, nature in its munificence evolved a brain fully capable of critical thinking, language, innovation, and even abstraction. This alone is difficult (or impossible?) to understand. But Feynman suggests understanding nature may take a further evolutionary step in humans to understand and answer and answer deeper questions, questions we don’t even know enough to ask let alone understand, even if we stumbled upon one.

Figure 2. Richard P. Feynman

By way of analogy, I once wrote an essay titled, “Teaching an Ant How to Read.” In it, I explained it just can’t be done, simply because the ant simply hasn’t the mental capacity. At a higher level, the same situation may be ours. Perhaps one day, it will be proved that a mind in a single container, such as a human, can only evolve to a certain limit no matter how the atoms are arranged. Is this a question beyond impossible or undecidable? (See, https://used-ideas.blogspot.com/2012/05/how-to-teach-ant-how-to-read.html)  Another of these types of problems is, “Does knowing something change it?” Yes, it can happen.

Conclusion. It is simple to say that Samuel Johnson was wrong. He was. Many important people have been wrong but remain luminaries in our world. But in a deeper meaning about the cosmos, it indicates we may have one day two entirely different fully compatible explanations of the universe but will never be able to determine which, if either, is correct.  Being more theosophical, we may never be able to decide the merely impossible question of whether or not there is a God, maintaining the age-old feeling that God is a matter of faith. ■

 

* Truth, by the way, is one of those primitive concepts that have for millennia never really been defined to the satisfaction of all or even most. Sometimes new definitions are offered that seem at first to be exciting and attractive, but within a generation are deconstructed, often to vagueness, and lose adherents. With postmodernism created in the 1960s, where emotion is allowed on equal footing with critical thinking, Truth has become even more vague – and impossible.

 

 

©2024 G Donald Allen