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statements of the sort most other people do. Scientists aren’t necessarily always the most articulate, but they aim to state precisely what they do and don’t know or understand, at least when speaking about their field of expertise. So they rarely just say yes or no, since such an answer doesn’t accurately reflect the full range of possibilities. Instead, they speak in terms of probabilities or qualified statements. Ironically, this difference in language frequently leads people to misinterpret or underplay scientists’ claims. Despite the improved precision that scientists aim for, nonexperts don’t necessarily know how to weigh their statements—since anyone other than a scientist with as much evidence in support of their thesis wouldn’t hesitate to say something more definite. But scientists’ lack of 100 percent certainty doesn’t reflect an absence of knowledge. It’s simply a consequence of the uncertainties intrinsic to any measurement—a topic we’ll now explore. Probabilistic thinking helps clarify the meaning of data and facts, and allows for better-informed decisions. In this chapter, we’ll reflect on what measurements tell us and explore why probabilistic statements more accurately reflect the state of knowledge—scientific or otherwise—at any given time.

SCIENTIFIC UNCERTAINTY

Harvard recently completed a curricular review to try and determine the essential elements of a liberal education. One of the categories the faculty considered and discussed as part of a science requirement was “empirical reasoning.” The teaching proposal suggested the university’s purpose should be to “teach how to gather and assess empirical data, weigh evidence, understand estimates of probabilities, draw inferences from the data when available [so far, so good], and also to recognize when an issue cannot be settled on the basis of the available evidence.”

The proposed wording of the teaching requirement—later clarified—was well intentioned, but it belied a fundamental misunderstanding of how measurements work. Science generally settles issues with some degree of probability. Of course we can achieve high confidence in any particular idea or observation and use science to make sound judgments. But only infrequently can anyone absolutely settle an issue—scientific or otherwise—on the basis of evidence. We can collect enough data to trust causal relationships and even to make incredibly precise predictions, but we can generally do it only probabilistically. As Chapter 1 discussed, uncertainty—however small—allows for the potential existence of interesting new phenomena that remain to be discovered. Rarely is anything 100 percent certain, and no theory or hypotheses will be guaranteed to apply under conditions where tests have not yet been performed.

Phenomena can only ever be demonstrated with a certain degree of precision in a set domain of validity where they can be tested. Measurements always have some probabilistic component. Many science measurements rely on the assumption that an underlying reality exists that we can uncover with sufficiently precise and accurate measurements. We use measurements to find this underlying reality as well as we can (or as well as necessary for our purposes). This then permits statements such as that an interval centered on a collection of measurements contains the true value with 95 percent probability. In that case, we might colloquially say we are confident with 95 percent confidence. Such probabilities tell us the reliability of any particular measurement and the full range of possibilities and implications. You can’t fully understand a measurement without knowing and evaluating its associated uncertainties.

One source of uncertainty is the absence of infinitely precise measuring instruments. Such a precise measurement would require a device calibrated with an infinite number of decimal places. The measured value would have an infinite number of carefully measured numbers after the decimal place. Experimenters can‘t make such measurements—they can only calibrate their tools to make them as accurate