problem

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A problem ( Greek πρόβλημα próblema , German “the accused, the presented” , “that which was presented [for the solution], cliff, obstacle”) is called a task or issue , the solution of which is connected with difficulties. Problems represent obstacles that have to be overcome or circumvented in order to get from an unsatisfactory starting situation to a more satisfying target situation. Problems occur in various forms in all areas of life and in all sciences . In order to be able to solve a problem, it can make sense to break it down into simpler subtasks or to trace it back to a problem that has already been solved, or to look at the initial situation in an unfamiliar way. Several problems with a superordinate causal connection can be described as a problem .

features

In dealing with problems on a day-to-day basis, numerous sciences have worked out characteristics that can be used to characterize problems. Individual problems can thus be grouped into problem classes. The similar expression of the problems of a class suggests that solution methods for these problems are equally good or bad.

Solvability

Not all problems are solvable. In many problems, the apparent unsolvability is due to a lack of well-defined definition: the initial situation, obstacle and target situation are not formulated clearly enough to enable a solution. But even for properly formulated problems in environments with clearly defined rules it can be shown that a task is unsolvable, such as B. the squaring of the circle , which has become proverbial for unsolvable problems. Putting effort into solving proven unsolvable problems does not make sense. In this case, one can workaround (Engl. Workaround ) help the problem. The targeted goal is then changed in such a way that the problem no longer arises.

The unsolvability of a problem can also be due to trying to achieve multiple conflicting goals at the same time. In this case, there is a conflict of interest that can possibly be resolved through a compromise . In technical contexts, this is also referred to as optimization . However, compromises can create new problems. If a compromise is impossible due to the underlying rule system, one speaks of aporia .

Dismantling

If a problem can be broken down into several sub-problems, it is called separable or hierarchical . Real sub-problems are easier and require less work ( divide et impera ). They can offer a more compact and tangible description of the initial situation that suggests a solution in and of itself.

If the decomposition itself is a difficult problem, if a problem cannot be subdivided at all, or if the sub-problems that arise for each possible decomposition are just as complex as the initial problem, then the problem is called non-decomposable or elementary. A special statement about the decomposability makes this distinction:

  • Insight problems are problems that cannot be broken down because they only require a single, inseparable transformation step, which is, however, very difficult because it requires a completely new view of things.
  • Transformation , however, problems can be divided into a number of working steps disassemble ( computer science ) who want to be properly matched.

Kinship

Some problems are so closely related in nature that one problem solves another problem at the same time. In this case, the starting point and the target situation are the same for both problems, even if they are usually formulated completely differently. However, one of the problems can be converted into the other problem; especially the complexity theory speaks here of it to reduce a problem to another. In this way, entire problem classes can be found whose problems are unsolved. However, it is known that solving one problem would solve all other problems in the class at the same time. If a problem cannot be traced back to other problems, it forms a problem class of its own and may require an entirely new insight .

Solution effort

You can judge problems according to the effort required to solve them: The solution to a problem can be short and sweet, but it can also be so complex that the achievable goal is not worth the effort. Extremely complex problems can even require unlimited solutions. A problem can be theoretically solvable, but still count as "unsolvable" in practice.

The effort required to solve a problem depends on its complexity and the capabilities of those involved. The term performance here includes various factors - from the intelligence of a person to the computing power of a computer. Different initial situations also influence the effort required for solutions, since the availability of raw materials and tools ( resources ) or simply other knowledge will vary.

subjectivity

Everyday problems in particular are subject to the subjectivity of the people involved. Differing goals mean that the difficulty and complexity of problems are assessed differently. Certain problems are unsolvable for those involved, but manageable for outsiders or even easy: It is impossible to determine the time of one's own death. Sometimes changing one's point of view creates a better understanding of the problem, for example when emotions are involved and the problem can only be grasped through empathy .

Special problem terms

Some sciences have developed special problem terms in order to make problems more tangible and accessible to formal attempts at solutions and quantitative measurements. Technology and economics perceive problems as difficulties in converting an existing current state into a desired target state . During the transfer, a barrier must be overcome with the help of a solution process. Problems differ from tasks in which there is also a barrier, but the solution is known from the start. Technology is particularly interested in structured approaches to problem solving, as these promise faster success than blind trial and error. Economics, on the other hand, is interested in how the effort required for a solution can be estimated and assessed in the form of financial values.

The complexity theory of theoretical computer science is based on a mathematically anchored problem concept. The basis here are decision-making problems , where the task is always the same: Decide whether this input is accepted or not. A problem is basically the same as a formal language , where the question is: decide whether this word belongs to this language or not. The advantages of this highly structured concept of a problem are that it is understandable for people and machines alike, the correctness of a solution approach can be proven and the number of steps required to solve a problem - its complexity - can be determined mathematically. Decision problems are only apparently too simple to investigate complex questions with; in fact, they can be reformulated into more natural optimization problems or search problems .

Complexity theory makes another important separation in that it distinguishes problems from problem instances. Instances are special cases of a generalized problem and specify, for example, concrete numbers or words, where the general problem speaks of any variable or character strings that can be assigned as required. The aim is always to solve the general case, problem instances are only used for developing ideas and manually checking experiments.

In the game of chess there is the problem of chess as an artistic form of expression.

First World Problem is pejorative for blown-up problems of the First World .

Famous problems

Some problems kept people preoccupied or had a major impact because groundbreaking new knowledge was revealed during their investigation. The following selection represents only a few mathematical and thus strongly pre-structured problems. In fact, the solution to a problem can be assumed behind every major technical, scientific or social breakthrough.

Squaring the circle
The geometric problem of creating a square of the same area from a circle using only compasses and rulers has preoccupied mankind since ancient times. The problem was "solved" in 1882 by Ferdinand von Lindemann , who proved that a precise solution is impossible.
Königsberg bridge problem
The aim of this topological problem was to find a (circular) path over the seven bridges of the city of Königsberg that only uses each bridge once. In 1736 Leonhard Euler showed that such a path does not exist and that the problem cannot be solved. The investigation of the more general Euler's circle problem , however, had lasting effects on complexity theory .
Hamilton cycle problem
With this problem of graph theory, a path through a graph is to be found that contains every node exactly once. Although the problem is similar to the Königsberg bridge problem, it turned out to be much more complex. It is related to the traveling salesman problem , which comes in countless variations in a variety of use cases.
Satisfiability problem of propositional logic
This problem of determining whether a propositional formula is satisfiable led to the concept of NP-completeness in 1971 with Cook's Theorem . The NP-complete problems to which it belongs - like the Hamilton cycle problem - form a class of difficult problems in computer science, all of which are closely related to one another; If one of these problems could be solved efficiently, it would be shown that all problems in NP can be solved efficiently, and P = NP would be proved. Currently only probabilistic and heuristic solution methods are known for these difficult problems , for example a large variety of optimization methods .
Goat problem
This probability problem was about providing advice to a game show contestant on choosing between three doors that concealed a prize and two rivets (goats). When the astonishing and, for some, common sense solution, delivered back in 1889, was described in a newspaper by Marilyn vos Savant - the “most intelligent woman in the world” - in 1990 , mathematicians all over the world fell out over the concept of conditional probability .

Unsolvable problems often arise in questions of epistemology and logic when two equally true principles contradict each other in the form of the aporia or the paradox (or the antinomy in logic). Well-known examples of these hopeless puzzles are the sentence “This sentence is wrong” and the question “Can God create a stone that he himself cannot lift?” If one assumes God's omnipotence, this results in an irresolvable contradiction.

Cultural perspective

Different cultures have different philosophical concepts for problems. In Judaism z. B. there is the term Mitzrayim ( Hebrew מצרים, "Strait", "serious emergency", actually the name for the country of Egypt ), which as a collective term describes the difficulties, problems and challenges that a person has to face in the course of his life and with which he grows. The focus of the theological discourse on this term are the spiritual limitations that the individual must overcome in order to find God and the values ​​of the Torah .

literature

Web links

Wikiquote: Problem  - Quotes
Wiktionary: Problem  - explanations of meanings, word origins, synonyms, translations

Individual evidence

  1. ↑ Ursula Hermann, Knaurs etymological dictionary , 1983, p. 391
  2. ↑ Leaving Egypt ( Memento from March 16, 2006 in the Internet Archive )