"No problem can be solved from the same level of consciousness that created it" (Einstein).
"For the solution of great problems it is necessary to get rid of great prejudices" (Dirac).
Problem Solving Strategies
A problem is a matter of interest for which there is believed to be a solution but which is far from obvious from its statement. It poses a challenge to the one trying to solve it and provides satisfaction when solved.
A problem must be clearly defined. A solution to a problem consists of establishing a series of steps that leads from the initial state (the problem statement) to the final state (the solved problem).
General strategies
There are several general strategies for solving problems. A particular problem may require the use of several strategies. The most prominent general strategies are as follows:
Reductionist.
This is the "divide and conquer" philosophy: the recursive decomposition of a problem into simpler and simpler subproblems down to trivial elementary components. It is a top-down process. The solution is built bottom-up from the elementary components found.
Algorithmic (or computational).
It is the application of an algorithm or procedure that leads to the solution of the problem.
Linguistics.
Good notation, using clear, simple and concise language, facilitates the solution of a problem.
Gestalt.
It is the transformation of the context of the problem into another context in which the problem is easier to address.
Heuristics.
Heuristics is the art or science of discovery and invention. It is defined as "the art, technique, or practical procedure for solving problems." Heuristics is a mental "shortcut" and is connected with the intuitive or synthetic consciousness (of the right hemisphere) and allows to save mental resources of lower order of the analytical consciousness (of the left hemisphere). According to the level of abstraction (from higher to lower), there are heuristic principles, heuristic strategies and heuristic rules.
Actually, all problem-solving strategies are heuristic because they involve a raising of the conceptual level.
The term "heuristic" was coined by Einstein in his 1905 paper on the photoelectric effect. The popularization of the concept is due to George Pólya in his book "How to solve it" [Pólya, 1981]. In this book (considered a classic), Pólya describes a set of heuristics in the area of problem solving.
Generic (or deep).
A particular problem, with its corresponding details, is simplified if it is approached from a general point of view, abstracting from irrelevant details. This is the so-called "inventor's paradox" (explained below).
The closer we are to an object or event, the more concrete or specific the thought is, and the farther away we are, the more abstract the thought is. Put another way, the shorter the distance, the more concrete, and the greater the distance, the more abstract. "Distance" can be physical, conceptual, temporal, etc.
Spiral development.
It is a general strategy for the achievement of any type of objective using the consciousness associated with the right hemisphere and the Principle of Descending Causality. Starting from an initial nucleus that represents the objective already achieved, this nucleus is developed (expanded) in a descending process, going from the general to the particular. This strategy is especially useful for software development.
Behaviorist (or superficial).
A method of experimentation based on trial and error. It can be systematic, directed or random. It is to look for patterns, regularities or laws.
Analogical or metaphorical.
The same method is used that has been used successfully in another similar problem or a metaphor is used that has the same formal structure as the problem posed.
Means-ends analysis.
It is based on the space of possible states and the heuristic of progressive difference elimination.
The problem space can be very large, so the idea is not to try all possible states but only those that are closest to the final state. These are "cognitive shortcuts", the heuristics based on "difference reduction". In each operation, the operation that most reduces the difference with the desired goal is selected.
Searches (in order of increasing abstraction) can be blind (random), rational (with a method) or heuristic. Heuristics are usually applied when the solution space is very large, when the problem is complex, or when incomplete information is available.
Means-ends analysis is the method applied by Simon and Newell in GPS (explained below).
Lateral thinking.
It is a problem-solving strategy developed by Edward de Bono [2011]. It is based on generating imaginative, innovative and indirect ideas as possible solutions to problems. It seeks solutions or points of view that are normally ignored by strictly logical thinking, by habitual patterns of thinking.
Particularization.
Is to make the problem more specific and concrete in order to simplify it. Once solved, and with the experience gained, try to solve the whole problem.
Brainstorming.
It is a method of group work in which its components launch all kinds of spontaneous ideas, in a relaxed and informal atmosphere. The main rule is that any idea, in principle, is valid; no idea, no matter how strange it may seem, should be rejected.
Application of conjectures.
A conjecture is a statement that seems reasonable as a hypothesis to solve a problem.
Backward method.
It is to consider that the problem has been solved and try to go back to the initial situation.
Search for symmetries.
Symmetry is a type of invariance: the property that something does not change under a set of transformations. Symmetries must be understood in a general sense, not just geometrically. Symmetries clarify and simplify problems.
Search for dualities.
A duality is a property that establishes a relationship between two different concepts or objects or that are two aspects of the same thing. For example, in quantum physics, the wave-corpuscle duality. In geometry (the Platonic solids), the dual of the cube is the octahedron, and the dual of the icosahedron is the dodecahedron (the tetrahedron is its own dual).
Analysis of limiting cases.
The limiting cases either clarify the problem or prove that there is no solution.
Root cause analysis (RCA).
It is an analysis that attempts to identify the causes and conditions of a previously occurring event. Once the causes and their conditions have been identified, it allows future events to be predicted before they happen. There are many RCA philosophies or paradigms, but they share common principles.
Computer simulation.
A computer program whose purpose is to simulate a given problem or system by means of an abstract model that helps to establish the solution or is the solution itself.
Introduction of auxiliary elements.
To establish links between the problem statement and its solution.
Morphological analysis.
It is a method created in 1969 by the astronomer and physicist Fritz Zwicky. It is based on the conception that any problem or system is composed of a certain set of attributes that have their own identity and can be isolated. The attributes can be physical components, subsystems, parameters, dimensions, etc. The method has 5 phases:
Clearly define the problem to be solved.
Analyze the attributes that compose it. It is necessary to select the relevant attributes, leaving aside those that do not alter the essence of the problem.
Analyze the alternatives, instances or possible configurations of each attribute.
Combine the alternatives of the attributes. The set of all possible combinations is called the "product or morphological space".
Analyzing and evaluating the combinations, eliminating the unfeasible combinations and selecting the most creative ones.
Morphological analysis is really a creative method, generating new ideas.
The IDEAL method.
It is a method proposed by J.D. Brawford and B.S. Stein. It is based on 5 stages linked to the name of the method:
Clear identification of the problem.
Defining, describing and representing the problem accurately.
Exploring possible strategies to solve the problem and choosing one of them.
Acting according to the selected strategy.
Achievements. Whether or not the solution was found, evaluate and learn from the process carried out and look for an alternative solution if there was failure.
Particular strategies
In addition to these general strategies, there are mathematical-logical methods of problem solving:
Deduction.
It is the method of mathematics, born with Euclid, of axiomatic systems, based on axioms and rules of inference to produce theorems.
Reduction to the absurd (Reductio ad absurdum).
It is a logical method of demonstration. It starts from supposing the falsity of a proposition with which a contradiction is reached.
Mathematical induction.
If a proposition Pn, dependent on a natural number n, is satisfied for n=0, and if it is satisfied for < i>Pn, it is also true for Pn+1, then the property is true for all natural numbers.
Loft principle (or Dirichlet's loft).
If mpigeons occupy n nests and m>n, then there is at least one nest with m/n pigeons. Although this principle may seem trivial, it can be used to demonstrate surprising results. For example, in a city of 1 million inhabitants, there are a minimum of 2739 people born on the same day of the year.
Monte Carlo method.
It is a statistical method based on the generation of random numbers applied to a great diversity of mathematical problems (stochastic or deterministic) formalized as computer-implemented algorithms. It was invented by Stanislaw Ulam and John von Neumann.
Paradoxes in the Area of Problem Solving
The inventor's paradox
The inventor's paradox is a paradox formulated by George Pólya in his aforementioned book "How to solve it": "The most ambitious plan is the one most likely to succeed". It is also often described as follows: "It is easier to find a general solution than to find a solution for a specific problem". That is, instead of trying to solve a specific problem, it is easier to approach it from a general point of view and that covers the problem posed as a particular case. It is a paradox because intuitively it would seem that trying to solve a specific problem would be easier than approaching it at a general level.
In general, in a complex problem one usually applies the reductionist (analytical) method: divide the problem into smaller problems that require less effort, But there are problems where this strategy cannot be applied. The solution then is to apply the inverse way, the synthetic way: to generalize it, leaving out irrelevant details.
The philosophy of the inventor's paradox is often illustrated by the problem solved by Gauss (in the 1780s) when at school his teacher posed the problem of adding the first 100 natural numbers. Gauss obtained a solution immediately: 5050. He calculated the sum as follows: 1+...+100 = (1+100) + (2+99) + ... + (50+51) = 50*101 = 5050. In general, the sum of the n first natural numbers is n(n−1)/2.
The inventor's paradox is a phenomenon that occurs mainly in mathematics, logic and computer science. In computer science it is especially useful, because it facilitates the development of generic (usually parameterized) programs, covering many particular cases. In addition, the programs are shorter, more understandable and easier to maintain.
Variants of the inventor's paradox are:
Knowing a lot about a given subject is less effective and creative than possessing only general knowledge.
It is easier to pay off all debts than to pay them off one at a time.
It is easier to save by buying a lot at once than by buying a little at a time.
It is easier to make money by diversifying investments than by making money in one investment.
"If you want something done, entrust it to a busy person" (Chinese proverb).
The paradox of simplicity
Simplicity is a philosophical principle, a psychological principle, a problem-solving strategy, a way of life, an aesthetic criterion, a design strategy, a state of consciousness, a mental clarity, ...
"Simplicity is the ultimate sophistication" (Leonardo da Vinci).
"Simplicity is the best design" (Ken Segall).
"If you can't explain it simply, you don't understand it well enough" (Einstein).
Simplicity is not the same as simplification. Simplicity is richness, freedom and higher consciousness. Simplification is impoverishment, limitation and lower consciousness.
As a concept, simplicity is simple, so it might seem that achieving simplicity is an easy task. It is just the opposite, just as it is not easy to raise consciousness. The paradox of simplicity is that it is difficult to achieve, but obvious when it is achieved.
Simplicity is the culmination of a process of reflection, maturation, experimentation, trial and error. When simplicity is achieved in a problem or discipline, perfect knowledge, maximum creativity and maximum effectiveness are achieved.
Simplicity is not utopian. It can be achieved, but the road to it is not easy.
"Simplicity doesn't come quickly, and it can be a lot of work to make something simple" (Nigel Holmes).
"How complex it is to achieve simplicity" (Carlos Guyot).
"Since I don't have time to write a short letter, he writes you a long one" (Mark Twain).
When simplicity is taken to the limit, to supreme conceptual simplicity, simplicity becomes a Theory of Everything, a universal paradigm, the philosopher's stone, the key to wisdom and consciousness. Once achieved, supreme simplicity is the master key that opens all doors.
The paradox of choice
We live in an increasingly complex world, with more and more possibilities of choice (books, movies, TV channels, technologies, products, leisure, etc.). Paradoxically, although our freedom of choice is greater, these situations create confusion, anxiety, mental paralysis and greater indecision.
To get out of this situation, the best thing to do is to apply simplicity: limit the existing possibilities to only a few according to our needs, that is to say, to what is really important and essential.
Every decision implies a restriction, a renunciation, a descent from the general to the particular level. This implies a descent in the level of consciousness, because consciousness is associated with freedom, with possible alternatives. That is why the famous donkey of Buridan was unable to decide between only two possibilities (two piles of hay) and died of starvation.
Barry Schwartz is an American psychologist and author of "The Paradox of Choice: Why Mores is Less". This author's thesis is that having no choice makes us unhappy, having some choices makes us happy, but having too many choices makes us unhappy.
In problem solving, it is better to have few strategies than many. We must choose the most important strategies.
The General Problem Solver
"General Problem Solver" (GPS) is a computer program created in 1957 by Herbert Simon, Allen Newell and John Clifford Shaw to try to solve all kinds of particular problems by a general procedure. GPS was implemented in a language created for this purpose called IPL (Information Processing Language).
GPS could be applied to a wide variety of problems: proof of theorems, playing chess, recursive problems (such as the towers of Hanoi), algebraic identities, cryptoarithmetic, geometric problems, etc.
GPS was one of the great milestones of artificial intelligence. It separated the knowledge of the problem (the "what") from the strategy to solve it (the "how"), always using the same general reasoning mechanism, regardless of the problem to be solved.
The knowledge of the problem was expressed (in a formal symbolic language) by objects and the operations that could be performed between objects to give rise to other objects. Operations could be restricted to apply only to certain classes of objects. An operation could produce more than one object as a result. In GPS, the "problem space theory" strategy of Newell and Simon in their 1972 book "Human Problem Solving" was applied.
For example:
In the 8-puzzle or 15-puzzle game, the objects are the different configurations of the pieces. The operations are the possible moves that make you move from one object to another.
In the game of chess, objects are the different arrangements of the pieces on the board (board states). Operations are the legal moves that give rise to new board states.
In mathematics, the objects are the axioms and theorems. The operations are the rules of inference. To prove a theorem is to transform some initial objects (the axioms) into a previously specified object (the theorem).
Constructing a computer program. The objects are the possible contents of the memory. Operations are the programming language (or native computer language) instructions that alter the contents of memory. A program is a sequence of operations that transforms one state of memory into another. The problem in programming is to find a sequence of instructions that leads from an initial state of memory to a final state.
GPS heuristics
GPS used two heuristics:
Means-ends analysis.
Means are the objects and the operations between the objects to achieve the goal. Means-ends analysis is based on selecting the objects and operations that come closest to the objective.
Planning.
Planning makes it possible to construct a solution in general terms, before considering the details. This is done by omitting certain details of objects and operations to simplify the problem, ignoring non-significant differences between objects and between operations. Once an objective state is reached, the details are considered.
In general, the process for reaching a goal is recursive, and is as follows:
Create a list with the initial object.
Go through all possible operations to obtain new objects, selecting those objects that are closest to the final object, adding them to the list.
Remove each expanded object from the list.
If one of the new objects is the target, finish the process successfully.
If there are no more possible new states, end the process with failure. Otherwise, go to point 2.
The structure of possible objects can be traversed in breadth or depth:
In breadth, new objects are obtained by horizontally expanding all objects of the previous level. The subobjects (the new objects) are added at the end of the list.
In depth, new objects are obtained by expanding vertically the objects of the previous level.
In both cases, the subobjects (the new objects) replace the expanded object in the list.
MENTAL as General Problem Solver
The most fundamental problem solved by MENTAL is the problem of having a universal formal language for science.
MENTAL is not a general problem solver in the sense of a computer program (like GPS), but it is a facilitator or simplifier of all kinds of problems. It is a GPS in a higher sense: generic and universal. From the deep level all problems are seen as the same problem.
All problems are simplified because it starts from the supreme simplicity (the universal semantic primitives) to achieve complexity. All problems have a mental root. By transcending the mind as much as possible, that is, by reaching the primary archetypes, problems disappear or become simpler.
MENTAL has had two phases:
An ascending one, of supreme abstraction and discovery of the primitives. This phase has been realized only once. MENTAL is the supreme abstraction because its basic concepts are universal semantic primitives, philosophical categories and primary archetypes.
Another descending one, of practical application by applying the Principle of Descending Causality. This second phase is performed each time it is applied.
All problems have a mental root and derive from duality, from the existence of pairs of opposites. There is only one problem: the one produced by the existence of opposites. According to Jung, "We are crucified between opposites and we free ourselves from this torture when the reconciling third party takes shape". MENTAL is the third mediator between all opposites.
According to the first Wittgenstein (that of the Tractatus), an ideal language would make all philosophical problems disappear. MENTAL, as a universal formal language for science, solves or facilitates the resolution of many problems in multiple fields.
The general strategies
Many of the general problem-solving strategies mentioned above come together in MENTAL:
Linguistics.
MENTAL is a formal language that facilitates the representation of the problem and its possible solution.
Algorithmic.
In addition to being a descriptive language, MENTAL is an operational language with which algorithms and procedures can be represented.
Generalization.
To better understand a problem and try to solve it, the best strategy is to look at it from a higher point of view. From that point of view, the problem disappears or becomes simpler.
Problems must be looked at from a higher, general, cognitive and abstract point of view. Then the Principle of Descending Causality is applied to get into the details.
"The order of our thoughts must always go from the simplest to the most composite" (Descartes, Discourse on Method).
Reductionist.
It is a general reductionist approach because it reduces formal concepts and operations to a few primitives. It is the simplest possible language capable of representing all types of problems. At the same time, it is a holistic language.
Heuristics.
MENTAL is the supreme heuristic (or the supreme heuristic principle), the simplest and most efficient heuristic, as a foundation for problem solving. Heuristics of a lower level of abstraction may also be used to solve particular problems.
Creativity.
Because of its unlimited combinatorial capacity and its ability to apply different paradigms, MENTAL is the supreme creativity, where everything can be related to everything.
Spiral development.
All development must take place from the general to the particular. It is the application of the Principle of Descending Causality.
Dualities.
MENTAL integrates all opposites or dualities with their basic concepts (the semantic primitives).
Root Cause Analysis (RCA).
MENTAL facilitates the analysis of the causes of problems and their representation.
Metaphors.
Simplicity and complexity correspond to two different ways of looking at the same problem. This is what Max Tegmark calls "the perspectives of the frog and the bird". For the frog, everything is very difficult and complex because it moves on the surface, horizontally, in the particular, in the detail. For the bird, from a higher perspective, everything is much easier because it sees obvious relationships and connections, which are not possible to perceive from the ground from the frog's perspective.
The strategy to be used must always be that of the bird, always advancing (or descending) from the universal or general to the particular, always deriving particular truths from general or universal principles.
The paradoxes
Paradox of the inventor.
MENTAL is the paradigm of this paradox. It is simpler to address the ambitious issue of seeking a universal language for science than to seek a particular language for a particular domain or problem.
Paradox of simplicity.
MENTAL is also the paradigm of this paradox. Finding the universal semantic primitives has been far from easy, despite their simplicity. Getting to establish the primitives has been a task of trial and error and maturation until finally reaching a limit level where no further level of abstraction was possible. Once found the primitives turn out to be totally obvious.
Paradox of choice.
With MENTAL the paradox of choice disappears because the possible choices (the primitives) are reduced to the maximum. These are the degrees of freedom.
The problems
With MENTAL many problems are simplified, solved or clarified. We can highlight the following:
The identification of the archetypes of the consciousness and the neutral language that Jung and Pauli were looking for to unify physics and psychology.
The question of the nature of mathematics, its foundation and its language.
The harmonization of the schools of foundations of mathematics and that of the formal sciences in general. It clarifies definitively the relation between logic and mathematics: logic (represented by the primitive "Condition" is a dimension of reality and an indissoluble part of mathematics. Logic does not found mathematics (logicism) and does not derive from mathematics (intuitionism).
Hilbert's dream of a formalist meta-mathematics postulated as the foundation of mathematics.
The question of considering types to eliminate paradoxes. Type theory is not necessary, and it also limits expressiveness and creativity.
The true nature of Gödel's incompleteness theorem.
The innate universal grammar postulated by Chomsky. MENTAL is not only universal grammar but also universal language. Both are the same thing.
The formalization of semantics.
This problem is solved by the union of opposites, the union of syntax and semantics, as the two sides of the same coin, establishing a biunivocal correspondence between the two: given the syntax one knows the semantics, and given the semantics one obtains the syntax.
Wigner's question.
The problem of logical paradoxes.
The problem of logical implication.
The problem of identity.
Galileo's paradox of infinity.
The continuum problem and the continuum hypothesis.
The question of transfinite numbers (the hierarchy of infinities).
The nature of infinitesimal numbers.
The true nature of the imaginary unit.
The limits of artificial intelligence.
The frame problem.
The problem of logical truth.
The quantum enigma and its relation to the two modes of consciousness.
Addenda
More about GPS
GPS used to be complex software. Today, the core functionalities of GPS can be rewritten by simple code using a high-level programming language, and even more easily using an artificial intelligence language.
GPS was very limited. It was not general and did not solve all problems. Also the search process was not very efficient. It required too much time to solve complex problems (like the game of chess) because of the problem of the "combinatorial explosion" of possible states.
GPS was intended to be a computational model of a unified theory of cognition.
GPS was an extension of an earlier program called "The Logic Theorist Machine", which performed demonstrations of theorems of propositional logic from Russell and Whitehead's Principia Mathematica.
The GPS paradigm evolved into the symbolic-cognitive architecture SOAR (State Operator And Results), a model of cognition for problem solving and learning.
Subsequently, expert systems emerged in artificial intelligence, where the knowledge of an expert is specified by means of rules, making it possible to solve problems in specific domains. In 1967, the first expert system was released: Dendral, a system that helped chemists to identify unknown organic molecules. In 1974, Mycin, an expert system for medical diagnostics, was released. Starting in the 1980s, artificial intelligence languages such as Lisp and Prolog began to be developed. Prolog separates (like GPS) the knowledge of a problem (the "what") from the "how" to solve the possible problems that may arise.
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