Out of Context: On Artificial Intelligence (Definitions)

"Artificial intelligence is the science of making machines do things that would require intelligence if done by men." (Marvin Minsky, 1968)

"Artificial intelligence is based on the assumption that the mind can be described as some kind of formal system manipulating symbols that stand for things in the world." (George Johnson, Machinery of the Mind: Inside the New Science of Artificial Intelligence, 1986)

"Artificial intelligence is the mimicking of human thought and cognitive processes to solve complex problems automatically. AI uses techniques for writing computer code to represent and manipulate knowledge." (Radian Belu, "Artificial Intelligence Techniques for Solar Energy and Photovoltaic Applications", 2013)

"Artificial intelligence is defined as the branch of science and technology that is concerned with the study of software and hardware to provide machines the ability to learn insights from data and the environment, and the ability to adapt in changing situations with high precision, accuracy and speed." (Amit Ray, "Compassionate Artificial Intelligence", 2018)

"Artificial Intelligence is not just learning patterns from data, but understanding human emotions and its evolution from its depth and not just fulfilling the surface level human requirements, but sensitivity towards human pain, happiness, mistakes, sufferings and well-being of the society are the parts of the evolving new AI systems." (Amit Ray, "Compassionate Artificial Intelligence", 2018)

"Artificial intelligence is the elucidation of the human learning process, the quantification of the human thinking process, the explication of human behavior, and the understanding of what makes intelligence possible." (Kai-Fu Lee, "AI Superpowers: China, Silicon Valley, and the New World Order", 2018)

"AI is a simulation of human intelligence through the progress of intelligent machines that think and work like humans carrying out such human activities as speech recognition, problem-solving, learning, and planning." (Hari K Kondaveeti et al, "Deep Learning Applications in Agriculture: The Role of Deep Learning in Smart Agriculture", 2021)

On Artificial Intelligence III

"The future offers very little hope for those who expect that our new mechanical slaves will offer us a world in which we may rest from thinking. Help us they may, but at the cost of supreme demands upon our honesty and intelligence. The world of the future will be an ever more demanding struggle against the limitations of our intelligence, not a comfortable hammock in which we can lay down to be waited upon by our robot slaves." (Norbert Wiener, "God and Golem, Inc.: A Comment on Certain Points Where Cybernetics Impinges on Religion", 1964)

"Cybernetics is simultaneously the most important science of the age and the least recognized and understood. It is neither robotics nor freezing dead people. It is not limited to computer applications and it has as much to say about human interactions as it does about machine intelligence. Today’s cybernetics is at the root of major revolutions in biology, artificial intelligence, neural modeling, psychology, education, and mathematics. At last there is a unifying framework that suspends long-held differences between science and art, and between external reality and internal belief." (Paul Pangaro, "New Order From Old: The Rise of Second-Order Cybernetics and Its Implications for Machine Intelligence", 1988)

"The cybernetics phase of cognitive science produced an amazing array of concrete results, in addition to its long-term (often underground) influence: the use of mathematical logic to understand the operation of the nervous system; the invention of information processing machines (as digital computers), thus laying the basis for artificial intelligence; the establishment of the metadiscipline of system theory, which has had an imprint in many branches of science, such as engineering (systems analysis, control theory), biology (regulatory physiology, ecology), social sciences (family therapy, structural anthropology, management, urban studies), and economics (game theory); information theory as a statistical theory of signal and communication channels; the first examples of self-organizing systems. This list is impressive: we tend to consider many of these notions and tools an integrative part of our life […]" (Francisco Varela, "The Embodied Mind", 1991)

"The insight at the root of artificial intelligence was that these 'bits' (manipulated by computers) could just as well stand as symbols for concepts that the machine would combine by the strict rules of logic or the looser associations of psychology." (Daniel Crevier, "AI: The tumultuous history of the search for artificial intelligence", 1993)

"Artificial intelligence comprises methods, tools, and systems for solving problems that normally require the intelligence of humans. The term intelligence is always defined as the ability to learn effectively, to react adaptively, to make proper decisions, to communicate in language or images in a sophisticated way, and to understand." (Nikola K Kasabov, "Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering", 1996)

"But intelligence is not just a matter of acting or behaving intelligently. Behavior is a manifestation of intelligence, but not the central characteristic or primary definition of being intelligent. A moment's reflection proves this: You can be intelligent just lying in the dark, thinking and understanding. Ignoring what goes on in your head and focusing instead on behavior has been a large impediment to understanding intelligence and building intelligent machines." (Jeff Hawkins, "On Intelligence", 2004)

"The brain and its cognitive mental processes are the biological foundation for creating metaphors about the world and oneself. Artificial intelligence, human beings’ attempt to transcend their biology, tries to enter into these scenarios to learn how they function. But there is another metaphor of the world that has its own particular landscapes, inhabitants, and laws. The brain provides the organic structure that is necessary for generating the mind, which in turn is considered a process that results from brain activity." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"From a historical viewpoint, computationalism is a sophisticated version of behaviorism, for it only interpolates the computer program between stimulus and response, and does not regard novel programs as brain creations. [...] The root of computationalism is of course the actual similarity between brains and computers, and correspondingly between natural and artificial intelligence. The two are indeed similar because the artifacts in question have been designed to perform analogs of certain brain functions. And the computationalist program is an example of the strategy of treating similars as identicals." (Mario Bunge, "Matter and Mind: A Philosophical Inquiry", 2010)

"Artificial intelligence is a concept that obscures accountability. Our problem is not machines acting like humans - it's humans acting like machines." (John Twelve Hawks, "Spark", 2014)

"AI failed (at least relative to the hype it had generated), and it’s partly out of embarrassment on behalf of their discipline that the term 'artificial intelligence' is rarely used in computer science circles (although it’s coming back into favor, just without the over-hyping). We are as far away from mimicking human intelligence as we have ever been, partly because the human brain is fantastically more complicated than a mere logic engine." (Field Cady, "The Data Science Handbook", 2017)

On Artificial Intelligence IV

"AI ever allows us to truly understand ourselves, it will not be because these algorithms captured the mechanical essence of the human mind. It will be because they liberated us to forget about optimizations and to instead focus on what truly makes us human: loving and being loved." (Kai-Fu Lee, "AI Superpowers: China, Silicon Valley, and the New World Order", 2018)

"Artificial intelligence is defined as the branch of science and technology that is concerned with the study of software and hardware to provide machines the ability to learn insights from data and the environment, and the ability to adapt in changing situations with high precision, accuracy and speed." (Amit Ray, "Compassionate Artificial Intelligence", 2018)

"Artificial Intelligence is not just learning patterns from data, but understanding human emotions and its evolution from its depth and not just fulfilling the surface level human requirements, but sensitivity towards human pain, happiness, mistakes, sufferings and well-being of the society are the parts of the evolving new AI systems." (Amit Ray, "Compassionate Artificial Intelligence", 2018)

"Artificial intelligence is the elucidation of the human learning process, the quantification of the human thinking process, the explication of human behavior, and the understanding of what makes intelligence possible." (Kai-Fu Lee, "AI Superpowers: China, Silicon Valley, and the New World Order", 2018) 

"AI won‘t be fool proof in the future since it will only as good as the data and information that we give it to learn. It could be the case that simple elementary tricks could fool the AI algorithm and it may serve a complete waste of output as a result." (Zoltan Andrejkovics, "Together: AI and Human. On the Same Side", 2019)

"It is the field of artificial intelligence in which the population is in the form of agents which search in a parallel fashion with multiple initialization points. The swarm intelligence-based algorithms mimic the physical and natural processes for mathematical modeling of the optimization algorithm. They have the properties of information interchange and non-centralized control structure." (Sajad A Rather & P Shanthi Bala, "Analysis of Gravitation-Based Optimization Algorithms for Clustering and Classification", 2020)

"A significant factor missing from any form of artificial intelligence is the inability of machines to learn based on real life experience. Diversity of life experience is the single most powerful characteristic of being human and enhances how we think, how we learn, our ideas and our ability to innovate. Machines exist in a homogeneous ecosystem, which is ok for solving known challenges, however even Artificial General Intelligence will never challenge humanity in being able to acquire the knowledge, creativity and foresight needed to meet the challenges of the unknown." (Tom Golway, 2021)

"Every machine has artificial intelligence. And the more advanced a machine gets, the more advanced artificial intelligence gets as well. But, a machine cannot feel what it is doing. It only follows instructions - our instructions - instructions of the humans. So, artificial intelligence will not destroy the world. Our irresponsibility will destroy the world." (Abhijit Naskar)

On Artificial Intelligence II

"In other words then, if a machine is expected to be infallible, it cannot also be intelligent. There are several theorems which say almost exactly that. But these theorems say nothing about how much intelligence may be displayed if a machine makes no pretense at infallibility." (Alan M Turing, 1946)

"The original question, 'Can machines think?:, I believe too meaningless to deserve discussion. Nevertheless I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted." (Alan M Turing, 1950) 

"The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false. A natural consequence of doing so is that one then assumes that there is no virtue in the mere working out of consequences from data and general principles." (Alan Turing, "Computing Machinery and Intelligence", Mind Vol. 59, 1950)

"Although it sounds implausible, it might turn out that above a certain level of complexity, a machine ceased to be predictable, even in principle, and started doing things on its own account, or, to use a very revealing phrase, it might begin to have a mind of its own." (John R Lucas, "Minds, Machines and Gödel", 1959)

"Let an ultraintelligent machine be defined as a machine that can far surpass all the intellectual activities of any man however clever. Since the design of machines is one of these intellectual activities, an ultraintelligent machine could design even better machines; there would then unquestionably be an 'intelligence explosion', and the intelligence of man would be left far behind. Thus the first ultraintelligent machine is the last invention that man need ever make." (Irving J Good, "Speculations Concerning the First Ultraintelligent Machine", Advances in Computers Vol. 6, 1965)

"Intelligence has two parts, which we shall call the epistemological and the heuristic. The epistemological part is the representation of the world in such a form that the solution of problems follows from the facts expressed in the representation. The heuristic part is the mechanism that on the basis of the information solves the problem and decides what to do." (John McCarthy & Patrick J Hayes, "Some Philosophical Problems from the Standpoint of Artificial Intelligence", Machine Intelligence 4, 1969)

"There are now machines in the world that think, that learn and create. Moreover, their ability to do these things is going to increase rapidly until - in the visible future - the range of problems they can handle will be coextensive with the range to which the human mind has been applied." (Allen Newell & Herbert A Simon, "Human problem solving", 1976)

"It is essential to realize that a computer is not a mere 'number cruncher', or supercalculating arithmetic machine, although this is how computers are commonly regarded by people having no familiarity with artificial intelligence. Computers do not crunch numbers; they manipulate symbols. [...] Digital computers originally developed with mathematical problems in mind, are in fact general purpose symbol manipulating machines." (Margaret A Boden, "Minds and mechanisms", 1981)

"The basic idea of cognitive science is that intelligent beings are semantic engines - in other words, automatic formal systems with interpretations under which they consistently make sense. We can now see why this includes psychology and artificial intelligence on a more or less equal footing: people and intelligent computers (if and when there are any) turn out to be merely different manifestations of the same underlying phenomenon. Moreover, with universal hardware, any semantic engine can in principle be formally imitated by a computer if only the right program can be found." (John Haugeland, "Semantic Engines: An introduction to mind design", 1981)

"The digital-computer field defined computers as machines that manipulated numbers. The great thing was, adherents said, that everything could be encoded into numbers, even instructions. In contrast, scientists in AI [artificial intelligence] saw computers as machines that manipulated symbols. The great thing was, they said, that everything could be encoded into symbols, even numbers." (Allen Newell, "Intellectual Issues in the History of Artificial Intelligence", 1983)

Statistical Tools VI: Pinball Machines

"Because chaos is deterministic, or nearly so, games of chance should not be expected to provide us with simple examples, but games that appear to involve chance ought to be able to take their place. Among the devices that can produce chaos, the one that is nearest of kin to the coin or the deck of cards may well be the pinball machine. It should be an old-fashioned one, with no flippers or flashing lights, and with nothing but simple pins to disturb the free roll of the ball until it scores or becomes dead." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Dynamical systems that vary continuously, like the pendulum and the rolling rock, and evidently the pinball machine when a ball’s complete motion is considered, are technically known as flows. The mathematical tool for handling a flow is the differential equation. A system of differential equations amounts to a set of formulas that together express the rates at which all of the variables are currently changing, in terms of the current values of the variables." (Edward N Lorenz, "The Essence of Chaos", 1993)

"The pinball machine is one of those rare dynamical systems whose chaotic nature we can deduce by pure qualitative reasoning, with fair confidence that we have not wandered astray. Nevertheless, the angles in the paths of the balls that are introduced whenever a ball strikes a pin and rebounds […] render the system some what inconvenient for detailed quantitative study." (Edward N Lorenz, "The Essence of Chaos", 1993)

"When the pinball game is treated as a flow instead of a mapping, and a simple enough system of differential equations is used as a model, it may be possible to solve the equations. A complete solution will contain expressions that give the values of the variables at any given time in terms of the values at any previous time. When the times are those of consecutive strikes on a pin, the expressions will amount to nothing more than a system of difference equations, which in this case will have been derived by solving the differential equations. Thus a mapping will have been derived from a flow." (Edward N Lorenz, "The Essence of Chaos", 1993)

"In a complex society, individuals, organizations, and states require a high degree of confidence - even if it is misplaced - in the short-term future and a reasonable degree of confidence about the longer term. In its absence they could not commit themselves to decisions, investments, and policies. Like nudging the frame of a pinball machine to influence the path of the ball, we cope with the dilemma of uncertainty by doing what we can to make our expectations of the future self-fulfilling. We seek to control the social and physical worlds not only to make them more predictable but to reduce the likelihood of disruptive and damaging shocks (e.g., floods, epidemics, stock market crashes, foreign attacks). Our fallback strategy is denial." (Richard N Lebow, "Forbidden Fruit: Counterfactuals and International Relations", 2010)

Central Limit Theorem

"I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the ‘Law of Frequency of Error’. The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along." (Sir Francis Galton, "Natural Inheritance", 1889)

"The central limit theorem […] states that regardless of the shape of the curve of the original population, if you repeatedly randomly sample a large segment of your group of interest and take the average result, the set of averages will follow a normal curve." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"The central limit theorem says that, under conditions almost always satisfied in the real world of experimentation, the distribution of such a linear function of errors will tend to normality as the number of its components becomes large. The tendency to normality occurs almost regardless of the individual distributions of the component errors. An important proviso is that several sources of error must make important contributions to the overall error and that no particular source of error dominate the rest." (George E P Box et al, "Statistics for Experimenters: Design, discovery, and innovation" 2nd Ed., 2005)

"Two things explain the importance of the normal distribution: (1) The central limit effect that produces a tendency for real error distributions to be 'normal like'. (2) The robustness to nonnormality of some common statistical procedures, where 'robustness' means insensitivity to deviations from theoretical normality." (George E P Box et al, "Statistics for Experimenters: Design, discovery, and innovation" 2nd Ed., 2005)

"The central limit theorem differs from laws of large numbers because random variables vary and so they differ from constants such as population means. The central limit theorem says that certain independent random effects converge not to a constant population value such as the mean rate of unemployment but rather they converge to a random variable that has its own Gaussian bell-curve description." (Bart Kosko, "Noise", 2006)

"Normally distributed variables are everywhere, and most classical statistical methods use this distribution. The explanation for the normal distribution’s ubiquity is the Central Limit Theorem, which says that if you add a large number of independent samples from the same distribution the distribution of the sum will be approximately normal." (Ben Bolker, "Ecological Models and Data in R", 2007)

"[…] the Central Limit Theorem says that if we take any sequence of small independent random quantities, then in the limit their sum (or average) will be distributed according to the normal distribution. In other words, any quantity that can be viewed as the sum of many small independent random effects. will be well approximated by the normal distribution. Thus, for example, if one performs repeated measurements of a fixed physical quantity, and if the variations in the measurements across trials are the cumulative result of many independent sources of error in each trial, then the distribution of measured values should be approximately normal." (David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010)

"These days the term 'central limit theorem' is associated with a multitude of statements having to do with the convergence of probability distributions of functions of an increasing number of one- or multi-dimensional random variables or even more general random elements (with values in Banach spaces or more general spaces) to a normal distribution3 (or related distributions). In an effort to reduce ambiguity - and in view of historic developments - the denotation “central limit theorem” in the present examination will usually refer only to the 'classical' case, which deals with the asymptotic equality of distributions of sums of independent or weakly dependent random variables and of a normal distribution." (Hans Fischer, "A History of the Central Limit Theorem: From Classical to Modern Probability Theory", 2011)

"Statistical inference is really just the marriage of two concepts that we’ve already discussed: data and probability (with a little help from the central limit theorem)." (Charles Wheelan, "Naked Statistics: Stripping the Dread from the Data", 2012)

"The central limit theorem tells us that in repeated samples, the difference between the two means will be distributed roughly as a normal distribution." (Charles Wheelan, "Naked Statistics: Stripping the Dread from the Data", 2012)

"According to the central limit theorem, it doesn’t matter what the raw data look like, the sample variance should be proportional to the number of observations and if I have enough of them, the sample mean should be normal." (Kristin H Jarman, "The Art of Data Analysis: How to answer almost any question using basic statistics", 2013)

"At very small time scales, the motion of a particle is more like a random walk, as it gets jostled about by discrete collisions with water molecules. But virtually any random movement on small time scales will give rise to Brownian motion on large time scales, just so long as the motion is unbiased. This is because of the Central Limit Theorem, which tells us that the aggregate of many small, independent motions will be normally distributed." (Field Cady, "The Data Science Handbook", 2017)

"The central limit conjecture states that most errors are the result of many small errors and, as such, have a normal distribution. The assumption of a normal distribution for error has many advantages and has often been made in applications of statistical models." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)

"Theoretically, the normal distribution is most famous because many distributions converge to it, if you sample from them enough times and average the results. This applies to the binomial distribution, Poisson distribution and pretty much any other distribution you’re likely to encounter (technically, any one for which the mean and standard deviation are finite)." (Field Cady, "The Data Science Handbook", 2017)

"The central limit theorem in statistics states that, given a sufficiently large sample size, the sampling distribution of the mean for a variable will approximate a normal distribution regardless of that variable’s distribution in the population." (Jim Frost)

Igor Aleksander - Collected Quotes

"Neural computing is the study of cellular networks that have a natural property for storing experimental knowledge. Such systems bear a resemblance to the brain in the sense that knowledge is acquired through training rather than programming and is retained due to changes in node functions. The knowledge takes the form of stable states or cycles of states in the operation of the net. A central property of such nets is to recall these states or cycles in response to the presentation of cues." (Igor Aleksander & Helen Morton, "Neural computing architectures: the design of brain-like machines", 1989)

"A neural network is a massively parallel distributed processor that has a natural propensity for storing experiential knowledge and making it available for use. It resembles the brain in two respects: 1. Knowledge is acquired by the network through a learning process. 2. Interneuron connection strengths known as synaptic weights are used to store the knowledge." (Igor Aleksander & Helen Morton, "An Introduction to Neural Computing", 1990) 

"Neural Computing is the study of networks of adaptable nodes which through a process of learning from task examples, store experiential knowledge and make it available for use." (Igor Aleksander & Helen Morton, "An Introduction to Neural Computing", 1990)

"For a machine, the mark of consciousness is the ability (possessed by organisms) to know in some detail where it currently is, to understand where it comes from, and to have its own drives to make decisions. It must therefore have a detailed representation of its current position in its world, some knowledge of its own makeup, and a great deal of knowledge about how it might interact with humans." (Igor Aleksander, "How to Build a Mind: toward machines with imagination", 2001)

"One of the factors that distinguishes engineering from science is that the engineer builds complex systems from simple bits, whereas the scientist breaks complex systems into hopefully comprehensible components. The first is called understanding by synthesis and the second is understanding by analysis." (Igor Aleksander, "How to Build a Mind: toward machines with imagination", 2001)

"People talk far too glibly about 'recognizing' things and then build machines that simply label patterns. There is a vast difference between recognizing patterns by labeling them correctly and knowing the objects that are perceived. Such knowledge is a happy resonance between imagination and perception, possessed neither by WISARD nor by the many neural pattern-recognition machines built over the last fifteen or so years. Something extra is required: yes, inner states are necessary, but they cannot be just any old inner states." (Igor Aleksander, "How to Build a Mind: toward machines with imagination", 2001)

"Yes, learning and adaptation seem to constitute one of the dividing lines between list processing and brains. Another seems to be that the brain is a highly structured piece of engineering in which most of what happens is determined by its specialized structure. The engineering of a computer is such as to be as general as possible to let the programmer write his list-processing programs: so, the hardware of the brain does matter in letting it do what it does. In the brain it creates specific overall aptitudes, but in computers it is carefully made neutral so as to keep them as general as possible." (Igor Aleksander, "How to Build a Mind: toward machines with imagination", 2001)

"Machine consciousness refers to attempts by those who design and analyse informational machines to apply their methods to various ways of understanding consciousness and to examine the possible role of consciousness in informational machines." (Igor Aleksander, "Machine consciousness", Scholarpedia, 3(2), 2008)

Ludwig Wittgenstein - Collected Quotes

"If a fact is to be a picture, it must have something in common with what it depicts. […] What a picture must have in common with reality, in order to be able to depict it correctly or incorrectly - in the way it does, is its pictorial form. […] What any picture, of whatever form, must have in common with reality, in order to be able to depict it - correctly or incorrectly in any way at all, is logical form, i.e., the form of reality. […] Logical pictures can depict the world." (Ludwig Wittgenstein, "Tractatus Logico-Philosophicus", 1922)

"The logical picture of the facts is the thought. […] A picture is a model of reality. In a picture objects have the elements of the picture corresponding to them. The fact that the elements of a picture are related to one another in a determinate way represents that things are related to one another in the same way." (Ludwig Wittgenstein, "Tractatus Logico-Philosophicus", 1922)

"The process of induction is the process of assuming the simplest law that can be made to harmonize with our experience. This process, however, has no logical foundation but only a psychological one. It is clear that there are no grounds for believing that the  simplest course of events will really happen." (Ludwig Wittgenstein, "Tractatus Logico-Philosophicus", 1922)

"The so-called law of induction cannot possibly be a law of logic, since it is obviously a proposition with a sense. - Nor, therefore, can it be an a priori law." (Ludwig Wittgenstein, "Tractatus Logico Philosophicus", 1922)

"For a large class of cases - though not for all - in which we employ the word 'meaning' it can be defined thus: the meaning of a word is its use in language." (Ludwig Wittgenstein, "Philosophical investigations", 1953)

"Like everything metaphysical the harmony between thought and reality is to be found in the grammar of the language." (Ludwig Wittgenstein, "Philosophical Investigations", 1953)

"All right: the concept of number is defined for you as the logical sum of these individual interrelated concepts: cardinal numbers, rational numbers, real numbers etc.; and, in the same way the concept of a game is the logical sum of a corresponding set of sub-concepts." - It need not be so. For I can give the concept 'number' rigid limits in this way, that is, use the word "number" for a rigidly limited concept, but I can also use it so that the extension of the concept is not closed by a frontier. And this is how we use the word "game". For how is the concept of a game bounded?" (Ludwig Wittgenstein, "Philosophical Investigations", 1953)

"The problems are solved, not by giving new information, but by arranging what we have known since long." (Ludwig Wittgenstein, "Philosophical Investigations", 1953)

"Why do we call something a 'number'? Well, perhaps because it has a direct relationship with several things that have hitherto been called number; and this can be said to give it an indirect relationship to other things we call the same name. And we extend our concept of number as in spinning a thread we twist fibre on fibre. And the strength of the thread does not reside in the fact that some one fibre runs through its whole length, but in the overlapping of many fibres. But if someone wished to say: 'There is something common to all these constructions - namely the disjunction of all their common properties' - I should reply: 'Now you are only playing with words. One might as well say: 'Something runs through the whole thread - namely the continuous overlapping of those fibres.' " (Ludwig Wittgenstein, "Philosophical Investigations", 1953)

"To convince someone of the truth, it is not enough to state it, but rather one must find the path from error to truth." (Ludwig Wittgenstein, "Philosophical Occasions", 1953)

"Our craving for generality has [as one] source […] our preoccupation with the method of science. I mean the method the method of reducing the explanation of natural phenomena to the smallest possible number of primitive natural laws; and, in mathematics, of unifying the treatment of different topics by using a generalization. Philosophers constantly see the method of science before their eyes, and are irresistibly tempted to ask and answer in the way science does. This tendency is the real source of metaphysics, and leads the philosopher into complete darkness. I want to say here that it can never be our job to reduce anything to anything, or to explain anything. Philosophy really is ‘purely descriptive’." (Ludwig Wittgenstein, "The Blue and Brown Books", 1958)

"Images tell us nothing, either right or wrong, about the external world. […] It is just because forming images is a voluntary activity that it does not instruct us about the external world. […] When we form an image of something we are not observing. The coming and going of the pictures is not something that happens to us. We are not surprised by these pictures, saying ‘Look!’"  (Ludwig Wittgenstein, "Zettel", 1967)

"All testing, all confirmation and disconfirmation of a hypothesis takes place already within a system. And this system is not a more or less arbitrary and doubtful point of departure for all our arguments; no it belongs to the essence of what we call an argument. The system is not so much the point of departure, as the element in which our arguments have their life." (Ludwig Wittgenstein, "On Certainty", 1969)

"People are deeply imbedded in philosophical, i.e., grammatical confusions. And to free them presupposes pulling them out of the immensely manifold connections they are caught up in." (Ludwig Wittgenstein, "Philosophical Occasions 1912-1951", 1993)

On Problem Solving 11: On Problem Solvers

"The future mathematician should be a clever problem-solver; but to be a clever problem-solver is not enough. In due time, he should solve significant mathematical problems; and first he should find out for which kind of problems his native gift is particularly suited." (George Pólya, "How to solve it", 1945)

"The intelligent problem-solver tries first of all to understand the problem as fully and as clearly as he can. Yet understanding alone is not enough; he must concentrate upon the problem, he must desire earnestly to obtain its solution. If he cannot summon up real desire for solving the problem he would do better to leave it alone. The open secret of real success is to throw your whole personality into your problem." (George Pólya, "How to Solve It", 1945)

"The mathematical experience of the student is incomplete if he never had an opportunity to solve a problem invented by himself." (George Pólya, "How to Solve It", 1945)

"As long as we try and patiently do our best to solve the problem, although we may not get the answer we are looking for, we always get something - even if it is only the valuable experience." (Charles Kettering, "Short Stories of Science and Invention: A Collection of Radio Talks", 1954)

"An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity." (Howard W Eves, "In Mathematical Circles", 1969)

"Deep in the human nature there is an almost irresistible tendency to concentrate physical and mental energy on attempts at solving problems that seem to be unsolvable." (Ragnar Frisch, "From Utopian Theory to Practical Applications", [Nobel lecture] 1970)

"The definition of a problem and the action taken to solve it largely depend on the view which the individuals or groups that discovered the problem have of the system to which it refers. A problem may thus find itself defined as a badly interpreted output, or as a faulty output of a faulty output device, or as a faulty output due to a malfunction in an otherwise faultless system, or as a correct but undesired output from a faultless and thus undesirable system. All definitions but the last suggest corrective action; only the last definition suggests change, and so presents an unsolvable problem to anyone opposed to change." (Herbert Brün, "Technology and the Composer", 1971)

"As our ability to solve problems expands. the scale of the problems attacked themselves seems to expand at a similar rate. As a result there always exist over the horizon new categories of problems of greater size to tackle." (David M Himmelbau, "Decomposition Methods", 1977) 

"The problem solver needs to stand back and examine problem contexts in the light of different 'Ws' (Weltanschauungen). Perhaps he can then decide which 'W' seems to capture the essence of the particular problem context he is faced with. This whole process needs formalizing if it is to be carried out successfully. The problem solver needs to be aware of different paradigms in the social sciences, and he must be prepared to view the problem context through each of these paradigms." (Michael C Jackson, "Towards a System of Systems Methodologies", 1984)

"A problem exists when there is a discrepancy between an initial state and a goal state, and there is no ready-made solution for the problem solver." (John D Bransford & Barry S Stein, "A Guide for Improving Thinking, Learning, and Creativity" 2nd Ed., 1993)

"Our ability to solve problems is not simply equivalent to a set of general problem-solving skills. One implication of this conclusion is that the same individual may be both good and poor at problem solving, depending on the nature of the problem." (John D Bransford & Barry S Stein, "A Guide for Improving Thinking, Learning, and Creativity" 2nd Ed., 1993)

"The term mental model refers to knowledge structures utilized in the solving of problems. Mental models are causal and thus may be functionally defined in the sense that they allow a problem solver to engage in description, explanation, and prediction. Mental models may also be defined in a structural sense as consisting of objects, states that those objects exist in, and processes that are responsible for those objects’ changing states." (Robert Hafner & Jim Stewart, "Revising Explanatory Models to Accommodate Anomalous Genetic Phenomena: Problem Solving in the ‘Context of Discovery’", Science Education 79 (2), 1995)

"Often a successful problem-solver is one who creates a new context in which to view the problem. This can often be done by directing one's attention away from the distracting details of the difficulty. From a detached perspective, we may examine the situation in a new or different light and, after exploring information and options, choose an appropriate course of action." (John Templeton, "Wisdom From World Religions: Pathways Toward Heaven on Earth", 2002)

"We are constantly using old knowledge in new situations. When a solver can successfully use a solution procedure used in the past to solve a target problem this is known as positive transfer. […] However, it is also the case that a procedure learned in the past can impede one's learning of a new procedure. This is known as negative transfer. In this case what you have learned prevents you from solving a new problem or at least prevents you from seeing an optimal solution." (S Ian Robertson, "Problem Solving", 2001)

"Mathematics is not a matter of 'anything goes', and every mathematician is guided by explicit or unspoken assumptions as to what counts as legitimate – whether we choose to view these assumptions as the product of birth, experience, indoctrination, tradition, or philosophy. At the same time, mathematicians are primarily problem solvers and theory builders, and answer first and foremost to the internal exigencies of their subject." (Jeremy Avigad, "Methodology and Metaphysics in the Development of Dedekind’s Theory of Ideals", 2006)

"A good problem solver must also be a conceptual mathematician, with a good intuitive grasp of structures. But structures remain tools for the problem solver, instead of the main object of study." (David Ruelle, "The Mathematician's Brain", 2007)

"The fledgling problem solver invariably rushes in with solutions before taking time to define the problem being solved. Even experienced solvers, when subjected to social pressure, yield to this demand for haste. When they do, many solutions are found, but not necessarily to the problem at hand." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)

"Under normal conditions the research scientist is not an innovator but a solver of puzzles, and the puzzles upon which he concentrates are just those which he believes can be both stated and solved within the existing scientific tradition." (Thomas S Kuhn, "The Essential Tension: Selected Studies in Scientific Tradition and Change", 2011)

"Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. All good problem-solvers know that hard problems become easier when they’re split into chunks. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme - all the way out to infinity." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"Diverse groups of problem solvers outperformed the groups of the best individuals at solving complex problems. The reason: the diverse groups got stuck less often than the smart individuals, who tended to think similarly." (Scott E Page)

Previous <<||>> Next

On Problem Solving 14: On Representation I

"This diagrammatic method has, however, serious inconveniences as a method for solving logical problems. It does not show how the data are exhibited by cancelling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i.e., 'throwing of the problem into an equation' and the transformation of the premises, nor with the subsequent steps, i.e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form." (Louis Couturat, "The Algebra of Logic", 1914)

"Graphics is the visual means of resolving logical problems." (Jacques Bertin, "Graphics and Graphic Information Processing", 2011)

"A mental model is a knowledge structure that incorporates both declarative knowledge (e.g., device models) and procedural knowledge (e.g., procedures for determining distributions of voltages within a circuit), and a control structure that determines how the procedural and declarative knowledge are used in solving problems (e.g., mentally simulating the behavior of a circuit)." (Barbara Y White & John R Frederiksen, "Causal Model Progressions as a Foundation for Intelligent Learning Environments", Artificial Intelligence 42, 1990)

"An important symptom of an emerging understanding is the capacity to represent a problem in a number of different ways and to approach its solution from varied vantage points; a single, rigid representation is unlikely to suffice." (Howard Gardner, “The Unschooled Mind”, 1991)

"The term mental model refers to knowledge structures utilized in the solving of problems. Mental models are causal and thus may be functionally defined in the sense that they allow a problem solver to engage in description, explanation, and prediction. Mental models may also be defined in a structural sense as consisting of objects, states that those objects exist in, and processes that are responsible for those objects’ changing states." (Robert Hafner & Jim Stewart, "Revising Explanatory Models to Accommodate Anomalous Genetic Phenomena: Problem Solving in the ‘Context of Discovery’", Science Education 79 (2), 1995)

"The purpose of a conceptual model is to provide a vocabulary of terms and concepts that can be used to describe problems and/or solutions of design. It is not the purpose of a model to address specific problems, and even less to propose solutions for them. Drawing an analogy with linguistics, a conceptual model is analogous to a language, while design patterns are analogous to rhetorical figures, which are predefined templates of language usages, suited particularly to specific problems." (Peter P Chen [Ed.], "Advances in Conceptual Modeling", 1999)

"What it means for a mental model to be a structural analog is that it embodies a representation of the spatial and temporal relations among, and the causal structures connecting the events and entities depicted and whatever other information that is relevant to the problem-solving talks. […] The essential points are that a mental model can be nonlinguistic in form and the mental mechanisms are such that they can satisfy the model-building and simulative constraints necessary for the activity of mental modeling." (Nancy J Nersessian, "Model-based reasoning in conceptual change", 1999)

"A model is an imitation of reality and a mathematical model is a particular form of representation. We should never forget this and get so distracted by the model that we forget the real application which is driving the modelling. In the process of model building we are translating our real world problem into an equivalent mathematical problem which we solve and then attempt to interpret. We do this to gain insight into the original real world situation or to use the model for control, optimization or possibly safety studies." (Ian T Cameron & Katalin Hangos, "Process Modelling and Model Analysis", 2001)

"Understanding a problem means building some kind of representation of the problem in one's mind, based on what the situation is or what the problem statement says and on one's prior knowledge. It is then possible to reason about the problem within this mental representation. Generating a useful mental representation is therefore the most important single factor for successful problem solving." (S Ian Robertson, "Problem Solving", 2001)

"The key role of representation in thinking is often downplayed because of an ideal of rationality that dictates that whenever two statements are mathematically or logically the same, representing them in different forms should not matter. Evidence that it does matter is regarded as a sign of human irrationality. This view ignores the fact that finding a good representation is an indispensable part of problem solving and that playing with different representations is a tool of creative thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Mathematical modeling is as much ‘art’ as ‘science’: it requires the practitioner to (i) identify a so-called ‘real world’ problem (whatever the context may be); (ii) formulate it in mathematical terms (the ‘word problem’ so beloved of undergraduates); (iii) solve the problem thus formulated (if possible; perhaps approximate solutions will suffice, especially if the complete problem is intractable); and (iv) interpret the solution in the context of the original problem." (John A Adam, "Mathematics in Nature", 2003)

"What is a mathematical model? One basic answer is that it is the formulation in mathematical terms of the assumptions and their consequences believed to underlie a particular ‘real world’ problem. The aim of mathematical modeling is the practical application of mathematics to help unravel the underlying mechanisms involved in, for example, economic, physical, biological, or other systems and processes." (John A Adam, "Mathematics in Nature", 2003)

Previous <<||>> Next