A Transition to Proof: An Introduction to Advanced Mathematics describes writing proofs as a creative process. There is a lot that goes into creating a mathematical proof before writing it. Ample discussion of how to figure out the "nuts and bolts'" of the proof takes place: thought processes, scratch work and ways to attack problems. Readers will learn not just how to write mathematics but also how to do mathematics. They will then learn to communicate mathematics effectively. The text emphasizes (...) the creativity, intuition, and correct mathematical exposition as it prepares students for courses beyond the calculus sequence. The author urges readers to work to define their mathematical voices. This is done with style tips and strict "mathematical do’s and don’ts", which are presented in eye-catching "text-boxes" throughout the text. The end result enables readers to fully understand the fundamentals of proof. Features: The text is aimed at transition courses preparing students to take analysis Promotes creativity, intuition, and accuracy in exposition The language of proof is established in the first two chapters, which cover logic and set theory Includes chapters on cardinality and introductory topology. (shrink)

Real causal systems are complicated. Despite this, causal learning research has traditionally emphasized how causal relations can be induced on the basis of idealized events, i.e. those that have been mapped to binary variables and abstracted from time. For example, participants may be asked to assess the efficacy of a headache-relief pill on the basis of multiple patients who take the pill (or not) and find their headache relieved (or not). In contrast, the current study examines learning via interactions with (...) continuous dynamic systems, systems that include continuous variables that interact over time (and that can be continuously observed in real time by the learner). To explore such systems, we develop a new framework that represents a causal system as a network of stationary Gauss--Markov (``\OU'') processes and show how such \textit{OU networks} can express complex dynamic phenomena such as feedback loops and oscillations. To assess adult's abilities to learn such systems, we conducted an experiment in which participants were asked to identify the causal relationships of a number of OU networks, potentially carrying out multiple, temporally-extended interventions. We compared their judgments to a normative model for learning OU networks as well as a range of alternative and heuristic learning models from the literature. We found that, although participants exhibited substantial learning of such systems, they committed certain systematic errors. These successes and failures were best accounted for by a model that describes people as focusing on pairs of variables, rather than evaluating the evidence with respect to the full space of possible structural models. We argue that our approach provides both a principled framework for exploring the space of dynamic learning environments as well as new algorithmic insights into how people interact successfully with a continuous causal world. (shrink)

The Democratic Republic of the Congo is facing two crises: a potentially explosive Ebola epidemic and a major insurgency. But they are not wholly distinct from each other: the first is intertwined with the second, and public mistrust and political violence add a dangerous dimension to the Ebola epidemic. The World Health Organization and other health emergency responders will increasingly find themselves fighting outbreaks in insecure, misgoverned or ungoverned zones, possibly experiencing active conflict. Yet the WHO has neither the mission (...) nor the capabilities to navigate these security threats. We cannot expect that the usual public health strategy will succeed when health workers’ lives are directly imperiled and community resistance runs deep. Tackling health emergencies amidst complex humanitarian crises requires fresh thinking. Here, we offer a blueprint for fighting diseases in complex humanitarian emergencies. The building blocks of security and trust include high‐level political support, street‐level diplomacy, community engagement, enhanced funding, and protection of health professionals working in conflict or disaster zones. (shrink)

Over-flexibility in the definition of Friston blankets obscures a key distinction between observational and interventional inference. The latter requires cognizers form not just a causal representation of the world but also of their own boundary and relationship with it, in order to diagnose the consequences of their actions. We suggest this locates the blanket in the eye of the beholder.

The Patient Self-Determination Act (PSDA) requires hospitals to ask patients upon admission whether they have an advance directive. Although the PSDA has received extensive criticism, little attention has been paid to the cost of the law, either during its legislative course or following its implementation. Nonetheless, several tangible and intangible costs are associated with the PSDA. Such costs may be incurred by different parties. This paper examines the costs and benefits of the PSDA and illustrates the extent of some of (...) its tangible costs. The incremental start-up cost for one institution's response to the PSDA is estimated to be $49,304 ($1.31 per admission) and the total implementation cost of the program to be $114,528. In addition, the national incremental start-up cost for hospitals to implement the PSDA is estimated to be between $43,625,114 and $101,569,922. Finally, the potential implications of the PSDA for future governmental health care ethics regulation are discussed. (shrink)

Generalization does not come from repeatedly observing phenomena in numerous settings, but from theories explaining what is general in those phenomena. Expecting future behavior to look like past observations is especially problematic in psychology, where behaviors change when people's knowledge changes. Psychology should thus focus on theories of people's capacity to create and apply new representations of their environments.