"By a variable we will mean an attribute, measurement or inquiry that may take on one of several possible outcomes, or values, from a specified domain. If we have beliefs (i.e., probabilities) attached to the possible values that a variable may attain, we will call that variable a random variable." (Judea Pearl, "Causality: Models, Reasoning, and Inference", 2000)
"Of course, the information systems governing the feedback we receive can change as we learn. They are part of the feedback structure of our systems. Through our mental models we define constructs such as GDP or scientific research, create metrics for these ideas, and design information systems to evaluate and report them. These then condition the perceptions we form. Changes in our mental models are constrained by what we previously chose to define, measure, and attend to. Seeing is believing and believing is seeing. They feed back on one another." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)
"Agent subroutines may pass information back and forth, but subroutines are not changed as a result of the interaction, as people are. In real social interaction, information is exchanged, but also something else, perhaps more important: individuals exchange rules, tips, beliefs about how to process the information. Thus a social interaction typically results in a change in the thinking processes - not just the contents - of the participants." (James F Kennedy et al, "Swarm Intelligence", 2001)
"Probability is not about the odds, but about the belief in the existence of an alternative outcome, cause, or motive." (Nassim N Taleb, "Fooled by Randomness", 2001)
"The danger arises when a culture takes its own story as the absolute truth, and seeks to impose this truth on others as the yardstick of all knowledge and belief." (F David Peat, "From Certainty to Uncertainty", 2002)
"There are endless examples of elaborate structures and apparently complex processes being generated through simple repetitive rules, all of which can be easily simulated on a computer. It is therefore tempting to believe that, because many complex patterns can be generated out of a simple algorithmic rule, all complexity is created in this way." (F David Peat, "From Certainty to Uncertainty", 2002)
"If 'memory' is a misleading metaphor for recording devices, so is the epithet 'problem solver' for our computing machines. Of course, they are no problem solvers, because they do not have any problems in the first place. It is our problems they help us solve like any other useful tool, say, a hammer which may be dubbed a 'problem solver' for driving nails into a board. The danger in this subtle semantic twist by which the responsibility for action is shifted from man to a machine lies in making us lose sight of the problem of cognition. By making us believe that the issue is how to find solutions to some well defined problems, we may forget to ask first what constitutes a 'problem', what is its 'solution', and - when a problem is identified - what makes us want to solve it." (Heinz von Foerster, "Understanding Understanding: Essays on Cybernetics and Cognition", 2003)
"Randomness is a difficult notion for people to accept. When events come in clusters and streaks, people look for explanations and patterns. They refuse to believe that such patterns - which frequently occur in random data - could equally well be derived from tossing a coin. So it is in the stock market as well." (Didier Sornette, "Why Stock Markets Crash: Critical events in complex financial systems", 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)
"A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalize it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently." (Michael F Atiyah, 2004)
"Contrary to popular belief, mathematics is not a universal language. Rather, mathematics is based on a strict set of definitions and rules that have been instated and to which meaning has been given." (Christopher Tremblay,"Mathematics for Game Developers", 2004)
"The Bayesian approach is based on the following postulates: (B1) Probability describes degree of belief, not limiting frequency. As such, we can make probability statements about lots of things, not just data which are subject to random variation. […] (B2) We can make probability statements about parameters, even though they are fixed constants. (B3) We make inferences about a parameter θ by producing a probability distribution for θ. Inferences, such as point estimates and interval estimates, may then be extracted from this distribution. (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)
"The important thing is to understand that frequentist and Bayesian methods are answering different questions. To combine prior beliefs with data in a principled way, use Bayesian inference. To construct procedures with guaranteed long run performance, such as confidence intervals, use frequentist methods. Generally, Bayesian methods run into problems when the parameter space is high dimensional." (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)
"We must begin by distinguishing between visual mental imagery and visual perception: Visual perception occurs while a stimulus is being viewed, and includes functions such as visual recognition (i. e., registering that a stimulus is familiar) and identification (i. e., recalling the name, context, or other information associated with the object). Two types of mechanisms are used in visual perception: ‘bottom-up’ mechanisms are driven by the input from the eyes; in contrast, ‘top-down’ mechanisms make use of stored information (such as knowledge, belief, expectations, and goals). Visual mental imagery is a set of representations that gives rise to the experience of viewing a stimulus in the absence of appropriate sensory input. In this case, information in memory underlies the internal events that produce the experience. Unlike afterimages, mental images are relatively prolonged." (Stephen M Kosslyn, "Mental images and the brain", Cognitive Neuropsychology 22, 2005)
"Context is not as simple as being in a different space [...] context includes elements like our emotions, recent experiences, beliefs, and the surrounding environment - each element possesses attributes, that when considered in a certain light, informs what is possible in the discussion." (George Siemens, "Knowing Knowledge", 2006)
"In science, for a theory to be believed, it must make a prediction - different from those made by previous theories - for an experiment not yet done. For the experiment to be meaningful, we must be able to get an answer that disagrees with that prediction. When this is the case, we say that a theory is falsifiable - vulnerable to being shown false. The theory also has to be confirmable, it must be possible to verify a new prediction that only this theory makes. Only when a theory has been tested and the results agree with the theory do we advance the statement to the rank of a true scientific theory." (Lee Smolin, "The Trouble with Physics", 2006)
"Often our mental models serve a very useful and practical purpose by helping us to make very quick sense of our experiences and interactions. However, the danger for us is that our mental models do not always reflect the truth, i.e. the way things really are. Often they reflect what we believe to be true, and sometimes we get it wrong." (John Middleton, "Upgrade Your Brain: 52 brilliant ideas for everyday genius", 2006)
"Our generational perspective contributes to the mental models we hold about ourselves, the world, and the way things ‘should’ be. These beliefs create blind spots that can become our undoing as we pursue our values and seek to accomplish our goals. Likewise, they can have a powerful effect on our culture." (Deborah Gilburg,"Empowering Multigenerational Collaboration in the Workplace", The Systems Thinker Vol. 18 No. 4, 2007)
"[…] a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it." (Steven G Krantz, "The Proof is in the Pudding", 2007)
"Mental models reflect the beliefs, values, and assumptions that we personally hold, and they underlie our reasons for doing things the way we do." (Kambiz E Maani & Robert Y Cavana,"Systems Methodology", The Systems Thinker Vol. 18 No. 8, 2007)
"Abstraction is a mental process we use when trying to discern what is essential or relevant to a problem; it does not require a belief in abstract entities." (Tom G Palmer, "Realizing Freedom: Libertarian Theory, History, and Practice", 2009)
"Gaining awareness about how the system is built up and how it works can also help us to avoid solutions that only treat the symptoms of an underlying problem without curing the problem itself. System thinking is powerful because it helps us to see our own mental models and how these models color our perception of the world. In many cases, it is difficult for us to alter our mental models. There are always some beliefs or viewpoints that we are not willing to change, no matter what evidence is presented against it. This causes a certain resistance to new concepts. Problems can occur, however, when a rigid mental model stands in the way of a solution that might solve a problem. In such situations, adherence to mental models can be dangerous to the health of the organization." (Akhilesh Bajaj & Stanisław Wrycza, "Systems Analysis and Design for Advanced Modeling Methods: Best Practices", 2009)
"Knowledge about things beyond our immediate environment may be acquired through deduction, if the initial premises are believed to be correct." (Nayef Al-Rodhan, "Sustainable History and the Dignity of Man", 2009)
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