"To get a true understanding of the work of mathematicians, and the need for proof, it is important for you to experiment with your own intuitions, to see where they lead, and then to experience the same failures and sense of accomplishment that mathematicians experienced when they obtained the correct results. Through this, it should become clear that, when doing any level of mathematics, the roads to correct solutions are rarely straight, can be quite different, and take patience and persistence to explore." (Alan Sultan & Alice F Artzt,"The Mathematics that every Secondary School Math Teacher Needs to Know", 2011)
"Replication is at the heart of science. If you read a study claiming that toast usually lands buttered side down, you’ll probably want to find at least one replication before you begin to take the result seriously. That’s good scientific practice. A replication experiment inevitably differs a little from the initial experiment - the toast was a little different in shape and dropped in a slightly different way - and so again finding a similar result gives some reassurance that the initial result was not caused by some quirk of the initial experiment." (Geoff Cumming, "Understanding the New Statistics", 2012)
"The ultimate arbiter of truth is experiment, not the comfort one derives from one's a priori beliefs, nor the beauty or elegance one ascribes to one's theoretical models." (Lawrence M Krauss, "A Universe from Nothing: Why There Is Something Rather than Nothing", 2012)
"There are limits on the data we can gather and the kinds of experiments we can perform." (Charles Wheelan, "Naked Statistics: Stripping the Dread from the Data", 2012)
"One good experiment is worth a thousand models […]; but one good model can make a thousand experiments unnecessary." (David Lloyd & Evgenii I Volkov,"The Ultradian Clock: Timekeeping for Intracelular Dynamics" [in"Complexity, Chaos, and Biological Evolution", Ed. by Erik Mosekilde & Lis Mosekilde, 2013)
"Self-selection bias occurs when people choose to be in the data - for example, when people choose to go to college, marry, or have children. […] Self-selection bias is pervasive in 'observational data', where we collect data by observing what people do. Because these people chose to do what they are doing, their choices may reflect who they are. This self-selection bias could be avoided with a controlled experiment in which people are randomly assigned to groups and told what to do. (Gary Smith, "Standard Deviations", 2014)
"We encounter regression in many contexts - pretty much whenever we see an imperfect measure of what we are trying to measure. Standardized tests are obviously an imperfect measure of ability. [...] Each experimental score is an imperfect measure of 'ability', the benefits from the layout. To the extent there is randomness in this experiment - and there surely is - the prospective benefits from the layout that has the highest score are probably closer to the mean than was the score." (Gary Smith, "Standard Deviations", 2014)
"Mathematics is both abstract and concrete, revealing much of the mental experiment, working with unobserved abstractions and objects, and the current scientific progress depended on the ability to operate precisely with abstractions and force of reasoning; […]" (Octavian Stanasila, Metabolism of Mathematics and Computer Science No. 8, 2015)
"The correlational technique known as multiple regression is used frequently in medical and social science research. This technique essentially correlates many independent (or predictor) variables simultaneously with a given dependent variable (outcome or output). It asks, 'Net of the effects of all the other variables, what is the effect of variable A on the dependent variable?' Despite its popularity, the technique is inherently weak and often yields misleading results. The problem is due to self-selection. If we don’t assign cases to a particular treatment, the cases may differ in any number of ways that could be causing them to differ along some dimension related to the dependent variable. We can know that the answer given by a multiple regression analysis is wrong because randomized control experiments, frequently referred to as the gold standard of research techniques, may give answers that are quite different from those obtained by multiple regression analysis." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)
"The work around the complex systems map supported a concentration on causal mechanisms. This enabled poor system responses to be diagnosed as the unanticipated effects of previous policies as well as identification of the drivers of the sector. Understanding the feedback mechanisms in play then allowed experimentation with possible future policies and the creation of a coherent and mutually supporting package of recommendations for change." (David C Lane et al, "Blending systems thinking approaches for organisational analysis: reviewing child protection", 2015)
"[…] people attempt to use highly flexible mathematical structures with large numbers of parameters that can be adjusted to fit the data, the result often being models that fit the data well but lack structural representation of the phenomena and thus are not predictive outside the range of the data. The situation is exacerbated by uncertainty regarding model parameters on account of insufficient data relative to model complexity, which in fact means uncertainty regarding the models themselves. More importantly from the standpoint of epistemology, the amount of available data is often miniscule in comparison to the amount needed for validation. The desire for knowledge has far outstripped experimental/observational capability. We are starved for data." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)
"Limitations on experimentation can result in limitations on the complexity or details of a theory. To be validated, a theory cannot exceed the experimentalist’s ability to conceive and perform appropriate experiments. With the uncertainty theory, modern physics appears to have brought us beyond the situation where limitations on observation result only from insufficient experimental apparatus to a point where limitations are unsurpassable in principle." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)
"Scientific knowledge is worldly knowledge in the sense that it points into the future by making predictions about events that have yet to take place. Scientific knowledge is contingent, always awaiting the possibility of its invalidation. Its truth or falsity lies in the verity of its predictions and, since these predictions depend upon the outcomes of experiments, ultimately the validity of scientific knowledge is relative to the methodology of verification." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)
"The foundations of a discipline are inseparable from the rules of its game, without which there is no discipline, just idle talk. The foundations of science reside in its epistemology, meaning that they lie in the mathematical formulation of knowledge, structured experimentation, and statistical characterization of validity. Rules impose limitations. These may be unpleasant, but they arise from the need to link ideas in the mind to natural phenomena. The mature scientist must overcome the desire for intuitive understanding and certainty, and must live with stringent limitations and radical uncertainty." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)
"A good estimator has to be more than just consistent. It also should be one whose variance is less than that of any other estimator. This property is called minimum variance. This means that if we run the experiment several times, the 'answers' we get will be closer to one another than 'answers' based on some other estimator." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)
"The crucial concept that brings all of this together is one that is perhaps as rich and suggestive as that of a paradigm: the concept of a model. Some models are concrete, others are abstract. Certain models are fairly rigid; others are left somewhat unspecified. Some models are fully integrated into larger theories; others, or so the story goes, have a life of their own. Models of experiment, models of data, models in simulations, archeological modeling, diagrammatic reasoning, abductive inferences; it is difficult to imagine an area of scientific investigation, or established strategies of research, in which models are not present in some form or another. However, models are ultimately understood, there is no doubt that they play key roles in multiple areas of the sciences, engineering, and mathematics, just as models are central to our understanding of the practices of these fields, their history and the plethora of philosophical, conceptual, logical, and cognitive issues they raise. "(Otávio Bueno, [in" Springer Handbook of Model-Based Science", Ed. by Lorenzo Magnani & Tommaso Bertolotti, 2017])
"How do you know when a correlation indicates causation? One way is to conduct a controlled experiment. Another is to apply logic. But be careful - it’s easy to get bogged down in semantics." (Daniel J Levitin, "Weaponized Lies", 2017)
"One kind of probability - classic probability - is based on the idea of symmetry and equal likelihood […] In the classic case, we know the parameters of the system and thus can calculate the probabilities for the events each system will generate. […] A second kind of probability arises because in daily life we often want to know something about the likelihood of other events occurring […]. In this second case, we need to estimate the parameters of the system because we don’t know what those parameters are. […] A third kind of probability differs from these first two because it’s not obtained from an experiment or a replicable event - rather, it expresses an opinion or degree of belief about how likely a particular event is to occur. This is called subjective probability […]." (Daniel J Levitin, "Weaponized Lies", 2017)
"The crucial concept that brings all of this together is one that is perhaps as rich and suggestive as that of a paradigm: the concept of a model. Some models are concrete, others are abstract. Certain models are fairly rigid; others are left somewhat unspecified. Some models are fully integrated into larger theories; others, or so the story goes, have a life of their own. Models of experiment, models of data, models in simulations, archeological modeling, diagrammatic reasoning, abductive inferences; it is difficult to imagine an area of scientific investigation, or established strategies of research, in which models are not present in some form or another. However, models are ultimately understood, there is no doubt that they play key roles in multiple areas of the sciences, engineering, and mathematics, just as models are central to our understanding of the practices of these fields, their history and the plethora of philosophical, conceptual, logical, and cognitive issues they raise." (Otávio Bueno, [in" Springer Handbook of Model-Based Science", Ed. by Lorenzo Magnani & Tommaso Bertolotti, 2017])
"The mental model is the arena where imagination takes place. It enables us to experiment with different scenarios by making local alterations to the model. […] To speak of causality, we must have a mental model of the real world. […] Our shared mental models bind us together into communities." (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)
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