For better and for worse, in ways both obvious and subtle, the work of scientists has helped to shape the world around us. The obvious impacts of science on our lives include the technologies that depend on scientific research. Our approaches to communicating, eating, becoming or staying healthy, moving from one place to another, reproducing, entertaining ourselves – all of these have been changed by the findings of scientific research. Perhaps less obvious, but not less important, are changes in how we think about ourselves and our world. The concepts and ways of thinking that scientists employ as they engage in research and seek to explain the phenomena they uncover tend to migrate out of the specialist settings in which they originate. Though they tend to be transformed in the process, these concepts and methods turn up in the broader culture as themes we draw upon to describe and explain the world.

Clearly, science is a big deal. This book is not just about science, though, it is about the philosophy of science. Even if you agree that one should care about science, you might wonder whether you should care about the philosophy of science. The point of this preface is to begin to persuade you that you should care about it.

Introduction

We have already discussed how the logical empiricists sought to develop a scientific philosophy that would enable investigators to reason in a manner that would be independent of their commitments to ideologies or political causes. Philosophers sometimes refer to this aspect of the logical empiricists' project as the ideal of a value-free science. But what are values?

Humans spend a lot of effort evaluating things, whether they be sneakers, songs, or senators. We can evaluate how honest a person is, how tasty a doughnut is, or how probable it is that an asteroid will collide with the Earth. One way in which such evaluations differ concerns what we value and how we value it. We value both honesty and tastiness, but in different ways. Valuing the continued survival of ourselves and other species, we might prefer that an asteroid not hit the Earth, but it would be odd to say that we value the probability of this happening: that would be something that we simply want to know. We might think of the category of things we value as those things that we take to be good, and the features of good things that give us reasons to value them (i.e., to regard them as good) we might call values.

Introduction

“…a little brainwashing will go a long way in making the history of science duller, simpler, more uniform, more ‘objective’ and more easily accessible to treatment by strict and unchangeable rules” (Feyerabend, 1988, 11). In the introduction to his book Against method, Paul Feyerabend thus expresses his views about the project of the methodology of science pursued by most philosophers (as he understands them). He not only disagrees with their pursuit of this project, he considers it harmful.

What is the philosophical project that he rejects? Why does he reject it? What does he offer in its place? In this chapter we will seek answers to these questions. First, however, a word of caution: Feyerabend's willingness to mix what we might think of as reasoned philosophical arguments with polemics makes for entertaining and stimulating reading, but it can also complicate the search for his underlying argument. Moreover, for systematic reasons that we will discuss, Feyerabend sometimes plays ‘devil's advocate,’ defending a position that is not really his own, and it is not always obvious when he does so.

Like his friend Imre Lakatos, Paul Feyerabend came of age in the midst of the turmoil enveloping central Europe in the 1930s and 1940s. Born in Vienna on January 13, 1924, he was drafted into the German army in 1942, four years after Germany's annexation of Austria.

Introduction

Our discussion of logical empiricism focused on efforts to articulate a ‘scientific philosophy’ centered upon the verification principle. Vienna Circle philosophers and scientists also pursued the project of unified science. To advance this project, the logical empiricists undertook to assemble the International Encyclopedia of Unified Science. As Otto Neurath, a social scientist, philosopher, and socialist reformer, wrote in the first volume of the Encyclopedia, “To further all kinds of scientific synthesis is one of the most important purposes of the unity of science movement, which is bringing together scientists in different fields and in different countries, as well as persons who have some interest in science or hope that science will help to ameliorate personal and social life” (Neurath, 1938/1955, 1).

In this chapter we will consider the philosophy of Thomas Kuhn, much of whose impact upon philosophy of science resulted from work that first appeared under the aegis of the unified science movement, yet whose views came to be seen as an emphatic rebuttal of at least some of the primary logical empiricist commitments of that movement.

Between 1938 and 1970, University of Chicago Press published the first two volumes of the Encyclopedia under the series title Foundations of the Unity of Science, with individual articles appearing as monographs. Volume II, number 2 appeared in 1962.

Introduction

In the last chapter, we explored that aspect of probability ideas concerned with “the degree of belief warranted by evidence.” Now we turn to the other aspect: “the tendency, displayed by some chance devices, to produce stable relative frequencies” (Hacking, 1975, 1). Our use of examples involving games of chance to introduce the mathematics of probability at the beginning of the previous chapter relied implicitly on this idea. But are dice games relevant to science? As we will see, thinking of scientific experiments or measuring procedures as chance mechanisms with tendencies “to produce stable relative frequencies” holds the key to the potential value of relative frequency ideas for philosophy of science.

Frequentism understands probability statements as statements about the relative frequency with which a certain outcome would occur under repeated execution of some process. Frequentist ideas have been central to the development of theoretical and applied statistics, and thus have had a significant influence on how scientists analyze data and report their results. Some of these statistical practices have been subjected to considerable criticism, however, and philosophers of science have tended to regard Bayesianism as based on a more coherent set of principles. Recently, the error-statistical philosophy has attempted to respond to philosophical criticisms of frequentist statistics, while remaining founded on the idea of experiments as chance mechanisms.

Introduction

That we use scientific theories to explain things is a matter of broad (though incomplete) agreement. But what is it to explain something? Intuitively, explanations enable us to understand the phenomena we observe, where understanding involves something more than merely knowing that something occurs. Intuitively, again, that something more seems to involve a knowing why or knowing how the phenomenon occurs. But none of these intuitive ideas helps very much with an analysis of what constitutes an explanation (or what constitutes a good explanation). For what constitutes understanding? and what does it take for an explanation to help us achieve it?

This is a very old question in philosophy. Aristotle insisted that in order to “grasp the ‘why’ of something” one should understand each of its four different kinds of cause: what it is made of, its form, its source of change or stability, and its end or purpose.

Aristotle's four causes are really four kinds of explanation, and they reflect his own approach to explaining natural phenomena. The discussion of explanation within contemporary philosophy of science likewise draws motivation from the explanatory practices of contemporary science; it really begins with Carl Hempel and Paul Oppenheim's Deductive-Nomological (D-N) model of explanation, which constitutes the core of the Covering Law model. We will therefore begin with the Covering Law account and objections to it before turning to several prominent alternative accounts.

Introduction

To the lay reader, much of what is written by scientists can seem barely comprehensible. Even to someone who has had some science courses in school, a sentence like “The M2 proton channel … has a 40-residue region that interacts with membranes consisting of a transmembrane helix (which mediates tetramerization, drug-binding, and channel activity), followed by a basic amphiphilic helix important for budding of the virus from cellular membranes and release (scission)” will seem as though it has been written in a language not quite identical to English (Fiorin, Carnevale, & DeGrado, 2010). As a result, the non-specialist may find much of what is accomplished by scientists mysterious and esoteric.

Philosophers of science have sometimes attempted to “humanize” scientific work by portraying scientific methods as extensions of our ordinary ways of knowing things. It is true that scientists use technologies both material and mathematical to make observations and draw conclusions that we could never achieve otherwise. Nonetheless, they observe, conjecture, infer, and decide just as we all do, if perhaps in a more systematic and sophisticated way. Although a certain kind of training is needed to understand some of the language scientists use to report their findings, those findings are not the result of magic or trickery, but of an especially refined and disciplined application of the cognitive resources we enjoy as a species.

Semidefinite programming (SDP) is an optimization model with vector or matrix variables, where the objective to be minimized is linear, and the constraints involve affine combinations of symmetric matrices that are required to be positive (or negative) semidefinite. SDPs include as special cases LPs, QCQPs, and SOCPs; they are perhaps the most powerful class of convex optimization models with specific structure, for which efficient and well-developed numerical solution algorithms are currently available.

SDPs arise in a wide range of applications. For example, they can be used as sophisticated relaxations (approximations) of non-convex problems, such as Boolean problems with quadratic objective, or rank-constrained problems. They are useful in the context of stability analysis or, more generally, in control design for linear dynamical systems. They are also used, to mention just a few, in geometric problems, in system identification, in algebraic geometry, and in matrix completion problems under sparsity constraints.

11.1 From linear to conic models

In the late 1980s, researchers were trying to generalize linear programming. At that time, LP was known to be solvable efficiently, in time roughly cubic in the number of variables or constraints. The new interior-point methods for LP had just become available, and their excellent practical performance matched the theoretical complexity bounds. It seemed, however, that, beyond linear problems, one encountered a wall.