The very term ‘experiment’ implies uncertainty, because why would one want to conduct an experiment if the outcome is certain? The goal of experimentation is to learn something new about a system, something that is unknown, or only poorly understood. Experiments are a natural outgrowth of models, because a model, whether conceptual, physical or numerical, will always be a simplified representation of a system, and experiments with this model help us to understand its strengths and weaknesses. In terms of consequences, the simplifications embodied in the model may not matter under many circumstances, but then along comes the special situation when the model becomes vulnerable. In this context, models invite experiments that put them to the test, in a process of validation.

Just as model building begins with a concept, a mental image of how something is constructed or functions, so also does experimentation begin with what are called ‘thought experiments’. These experiments are mental forays that explore the consequences of assumptions or possible paths of action. Albert Einstein was a firm advocate of thought experiments; many of his early concepts about relativity stemmed from his attempts to visualize how the universe would appear if he were to hitch a ride on a beam of light.

Experiments essentially pose questions and seek answers. A good experiment provides an unambiguous answer to a well-posed question.

The previous chapter focused on measurements. Here we will talk more about how to extract quantitative information from a collection of measurements. The discussion will lead us into topics as diverse as flood and earthquake frequencies, election polls, and census taking. What can a poll of a small number of registered voters tell us about the likely outcome of a forthcoming election? What can we learn from the past history of flooding along a river that will give some indication of what we might expect in the future? Uncertainties are associated with each topic, uncertainties that arise from different sources and are quantified in different ways.

There are many processes and pathways that lead to ensembles of measurements. One very common source is simply making a number of measurements on a single object – each student in a class at the local elementary school measures the height of their teacher, or all seismograph stations in a region estimate the magnitude of yesterday's earthquake. A second common ensemble comprises one-time measurements of a number of different objects – perhaps the weight of each student in the class on the first morning of the new school term, or the concentration of arsenic in each water well in the county on a given day.

We have walked a long way through the garden of uncertainty and have seen a multitude of flowers and a few weeds, much elegance and a little untidiness. The garden is not a formal garden, laid out geometrically, tended immaculately. It is a garden with many hidden recesses, in places a maze full of surprises, with each plot revealing something not seen before. As we near the end of our tour, we have come to recognize that uncertainty, just like a flower, can be found in many places and presents itself in different colors and intensities.

Throughout this walk through the garden of uncertainty, we encountered many aspects of global climate change: taking Earth's temperature, local trends and global averages, flood probabilities, ozone depletion, science education, industrial propaganda and obfuscation, media confusion, reconstructions of past climate, computer models of the week's weather and the century's climate, and insurance for an uncertain future. In each domain of the garden, the tie to climate change was bundled into a discussion of other natural phenomena and human activity, with uncertainties that paralleled or shared characteristics with the uncertainties of climate change. In this final chapter, I will pull together these components of climate change and address the attendant uncertainties cohesively, as a representation of both the struggles and achievements of climate science, and of the hills yet to climb.

This is a book about uncertainty, particularly the uncertainty we associate with science. Over the years, scientific uncertainty has been addressed by natural scientists, engineers, medical researchers, social scientists, and philosophers. But for all the perspectives that have been laid out in everything from short essays to scholarly monographs, the richness of scientific uncertainty has often been unappreciated and/or misunderstood by the general public, people not regularly engaged in science.

Uncertainty, of course, is not confined to the world of science. It is an everyday fact of ordinary life as well. We regularly face uncertainty in a myriad of ways. Will it rain today? Will Aunt Dorothy's plane arrive on time? Will the stock market tumble? Will an accident snarl the freeway during rush hour? These day-to-day uncertainties come and go, and we move on through life, sometimes preparing for them, but more often just plowing through them.

But uncertainty also colors longer-term concerns. Will my pension program be sufficient two decades from now to enable the full and comfortable life that my wife and I hope for? Will our health allow a free and independent life-style thirty years in the future? These longer-term questions are harder to answer and are cloaked in greater uncertainty. Because we have only one life to live we cannot return to ‘Go’ and take another path.

Is it really essential that the public understand science? Why not let scientists do their thing, and let the rest of the world get on with their business too? Unfortunately, in the modern world, that is a path we can ill afford to follow. Whether we realize it or not, science is too much a part of the fabric of our lives to be shunted aside as a curious sideshow. The economy, national defense, environment, and our health are more than ever before dependent on scientific progress. The emergent role of information technology in our economic productivity, the feasibility of a ballistic missile defense shield, the human contribution to climate change through the combustion of fossil fuels, the implications of the newly mapped human genome all should be reminders that we cannot divorce ourselves from science, even if we might like to. And yet for all of the obvious relevance of science to our daily lives, many people remain ill equipped to assimilate much beyond the rudiments of science.

If, as I have argued in the previous chapter, our schools have generally failed to develop an awareness and appreciation of science, one can envision a second line of defense against scientific illiteracy: scientists working closely with the mass media to inform and educate the public.

The quantification of measurements through statistical analysis, the discovery of a trend in temperature over time, or observation of a pattern in water pollution data displayed on a map – all motivate scientists to begin thinking and formulating ideas about what process lies behind the relationships they are observing. These ideas are initially simple and rudimentary, and they may lead to later testing through experimentation. In the next two chapters, we will explore the world of conceptualization and experimentation and gain insight into how uncertainty promotes creativity.

Scientists, indeed everyone, always operate with simplified concepts of the way things work. We call these simplified representations ‘models’, and they come in many forms: conceptual, physical, numerical. We receive imperfect guidance in model building from the real world, through incomplete, sometimes inaccurate, and occasionally conflicting measurements or observations about the phenomenon or system we are trying to understand. There is a continuous interaction between models and observation, with each undergoing adjustment in the face of the other. New observations lead to revision of a concept, and a new concept, in turn, suggests new experiments or observations to be made that will again put the concept to a test. It is this iterative back-and-forth interplay that generally improves understanding of a system, and which under some circumstances can reduce the uncertainty associated with system behavior.

When we experience things in the course of our lives, we become familiar with them, perhaps understand them, and come to accept them as a normal part of life. But when we first encounter something that we have not previously experienced, something we are unfamiliar with, there is a natural tendency toward caution. And if we are presented with an abstraction, something totally outside of our experience, skepticism or even disbelief is not an unnatural reaction.

In this context, uncerainty goes hand in hand with unfamiliarity. What we are unfamiliar with, we are uncertain of. And much of science is unfamiliar ground for many people. Although Albert Einstein thought otherwise, science is really not just ‘common sense’. If it were, no one with a modicum of common sense would be puzzled or baffled by it. Science requires a certain amount of abstraction, and the placing of observations into a context or framework. When that framework is one's immediate environment, familiarity and understanding come readily. But when the spatial framework is much smaller, as in particle physics, or much larger as in astronomy, it takes a willing mind to explore this unfamiliar, uncertain terrain. Similarly, there are processes that operate at time scales vastly different from those of everyday human experience. The apparently instantaneous completion of a chemical reaction or the inordinately slow pace of geological change both require an intellectual stretch.

What are science's powers and limits? That is, where is the boundary between what science is and is not able to discover? The American Association for the Advancement of Science has identified that issue as a critical component of science literacy: “Being liberally educated requires an awareness not only of the power of scientific knowledge but also of its limitations,” and learning science's limits “should be a goal in all science courses” (AAAS 1990:20–21; also see p. xv). As Caldin (1949:vii) warned, “The prestige of science to-day is very great; scientists have therefore a certain responsibility to consider carefully the scope and limitations of their professional activities.”

This question about science's reach has elicited great interest. For instance, Horgan (1996) interviewed dozens of leading scientists regarding the limits of scientific knowledge, and his book became a runaway best-seller and made the front page of the New York Times book-review section (30 June 1996). Likewise, the end-of-the-millennium special issue of Scientific American offered fascinating reading on “What Science Will Know in 2050” (December 1999).

People's motivations for exploring the limits of science can easily be misconstrued, so they should be made abundantly clear from the outset. Unfortunately, for some authors writing about science's limits, the motivation has been to exaggerate the limitations in order to cut science down, support anti-scientific sentiments, or make more room for philosophy or religion.

The principle of parsimony recommends that from among theories fitting the data equally well, scientists choose the simplest theory. Thus, the fit of the data is not the only criterion bearing on theory choice. Additional criteria include parsimony, predictive accuracy, explanatory power, testability, fruitfulness in generating new insights and knowledge, coherence with other scientific and philosophical beliefs, and repeatability of results. The principle of parsimony has four common names, also being called the principle of simplicity, the principle of economy, and Ockham's razor (with Ockham sometimes Latinized as Occam).

Parsimony is an important principle of the scientific method for two reasons. First and most fundamentally, parsimony is important because the entire scientific enterprise has never produced, and never will produce, a single conclusion without invoking parsimony. Parsimony is absolutely essential and pervasive.

Second and more practically, parsimonious models of scientific data can facilitate insight, improve accuracy, and increase efficiency. Remarkably, parsimonious models can be more accurate than their data. Or, in other terms, parsimonious models can be extremely efficient, requiring considerably less data collection than do more complicated models to achieve the same accuracy and results. These advantages of accuracy and efficiency are important because scientists want to find the truth, but they also want to spend no more time and money finding the truth than is necessary.

This chapter addresses science's evidence, the “E” portion of the PEL model.

Probability and statistics are deductive and inductive tools, respectively, that are used to deal with uncertainty. Probability concepts and values are used in both deductive and inductive reasoning, and inductive problems often contain deductive subproblems, so probability theory is needed to handle diverse contexts. Probability is important because in many cases the best answers that science can deliver are more or less probable conclusions, rather than absolute certainties. Daily life and scientific research are full of unavoidable practical decisions that must be made on the basis of the best available information.

Correct probability reasoning is not easy. Indeed, probability blunders are among the most common kinds of blunders, even in professional, refereed scientific journals. Regulatory agencies and law courts can perpetuate such blunders, as they often rely on scientific findings, thereby inflicting wrongful losses and injustices. The challenge in probability reasoning is that it requires some precise distinctions that are critical for getting the correct results, but those distinctions are not intuitively obvious, and few scientists learn or teach them. Also, there are conflicting paradigms that can seriously affect research efficiency and technological progress, but precious few scientists understand the issues. So there is much to be gained from a proper understanding of probability and statistics, as pursued in this and the next chapters.

This chapter cannot possibly do what an entire book on probability would do: present a comprehensive treatment.

Science education is absolutely crucial for the scientific enterprise. Without education and culture, we might today be celebrating the discovery of fire and invention of the wheel, rather than exploring space and interpreting genomes. Also, most scientists employed by universities are called upon to be educators as well as researchers. Accordingly, this chapter will consider the interactions between scientists and educators, especially as regards a fundamental element in the science curriculum, namely, scientific method, or, somewhat more broadly, the nature of science.

Science educators typically place the topic of scientific method within a broader context: “For science educators the phrase ‘nature of science,’ is used to describe the intersection of issues addressed by the philosophy, history, sociology, and psychology of science as they apply to and potentially impact science teaching and learning” (William F. McComas et al., in McComas 1998:5). Figure 11.1 depicts these four disciplines that contribute to our description and understanding of science. The various sizes of these circles represent the relative importance of these sources.

Educators and scientists alike perceive the centrality of the nature of science (NOS) in the science curriculum. Indeed, the NOS constitutes the very first chapter or topic in most documents specifying national standards for science education, because the NOS is seen as “a vital foundation for all future science learning” (William F. McComas and Joanne K. Olson, in McComas 1998:51).

The thesis of this book, as set forth in Chapter 1, is that there exist general principles of scientific method that are applicable to all of the sciences, but excessive specialization often causes scientists to neglect the study of these general principles, even though they undergird science's rationality and greatly influence science's efficiency and productivity. These general methodological principles involve the use of deductive and inductive logic, probability, parsimony, and hypothesis testing. Neither specialized techniques nor general principles can substitute for one another, but rather the winning combination for scientists is mastery of both.

The primary purpose of this book is to help scientists become better scientists, more creative and more productive, by fostering a deeper understanding of the general principles of scientific method. For instance, parsimonious models often can lead to greater accuracy and thereby improve decision-making, accelerate progress, and increase returns on research investments. Also, scientists can improve the statistical analyses of their data by understanding how the Bayesian and frequentist paradigms relate to different research questions and technological objectives.

The secondary purpose is to help scientists gain perspective on science's rationality and role. Every conclusion of science, when fully disclosed, involves components of three kinds: presuppositions, evidence, and logic. Accordingly, an explanation of scientific method amounts to disclosing and securing these three inputs. Also, clearly understood methods beget realistic expectations and legitimate claims.

Approximately halfway through her Ph. D. program in the biological sciences, my first doctoral student requested that we meet to review progress toward the degree. Knowing that both the research and course work of this student were progressing very nicely, I entered the appointment confident of a glowing report both for graduate student and major professor. However, the discussion took an unanticipated twist as we finished talking about the topics on my agenda.

When and where, this student asked, do we get to the more philosophical part of this Doctorate of Philosophy degree in science? Was it really true that this program that we guidance-committee members had so carefully designed for her was not going to include even one advanced course in philosophy or history of science? If not in formal course work, when would we as major professor and graduate student deal with the logical underpinnings and processes of science at the level of basic principles? The final question had the greatest unintended sting – something to the effect of “Will I graduate feeling worthy of more than a technical degree?”

Stunned and somewhat befuddled, I sent this student on her way with lame explanations: There simply wasn't sufficient time in a modern science education for students to become renaissance scholars as well as well-published researchers capable of competing successfully for federal grant dollars.

This book explores the general principles of scientific method that pervade all of the sciences, focusing on practical aspects. The implicit contrast is with specialized techniques for research that are used in only certain sciences. The structure of science's methodology envisioned here is depicted in Figure 1.1, which shows individual sciences, such as astronomy and chemistry, as being partly similar and partly dissimilar in methodology. What they share is a core of the general principles of scientific method. This common core includes such topics as hypothesis generation and testing, deductive and inductive logic, parsimony, and science's presuppositions, domain, and limits. Beyond methodology as such, some practical issues are shared broadly across the sciences, such as relating the scientific enterprise to the humanities and implementing effective science education.

The general principles that are this book's topics are shown in greater detail in Figure 1.2. These principles are of three kinds: (1) Some principles are relatively distinctive of science itself. For instance, the ideas about Ockham's hill that are developed in Chapter 8 on parsimony have a distinctively scientific character. If occasionally lawyers or historians happen to use those ideas, they will not be reprimanded. Nevertheless, clearly those ideas are used primarily by scientists and technologists. (2) Other principles are shared broadly among all forms of rational inquiry. For example, deductive logic is squarely in the province of scientists, and it is explored in Chapter 5. But deductions are also important in nearly all undertakings.

This is the first of five chapters directed mainly at this book's primary goal of increasing productivity (the others being Chapters 6–9). Logic is the science of correct reasoning and proof, distinguishing good reasoning from bad. Logic sorts out the relationships that are fundamental in science, the relationships between hypotheses and evidence, between premises and conclusions.

In the context of logic, an “argument” is not a nasty dispute, but rather is a structured set of statements in which some statements, the premises, are offered to support or prove others, the conclusions. A deductive argument is valid if the truth of its premises entails the truth of its conclusions, and is invalid otherwise. Many deductive systems, including arithmetic and geometry, are developed on a foundation of predicate logic in the modern and unified vision of mathematics.

“The most immediate benefit derived from the study of logic is the skill needed to construct sound arguments of one's own and to evaluate the arguments of others” (Hurley 1994:ix). Every college offers courses in logic from its philosophy or mathematics faculty, and many fine introductory books on logic are available (such as Kearns 1988 and Hurley 1994). Unfortunately, most scientists never take such a course or read such a book.

Of course, given the simple premises that “All men are mortal” and “Socrates is a man,” one trusts scientists to reach the valid conclusion that “Socrates is mortal,” even without formal study of logic.