Introduction

In any sophisticated system of inquiry, we expect to find several features intimately tied to questions of error and error avoidance. Above all, we want a system (1) that produces relatively few erroneous beliefs (without resort to the skeptical gimmick of avoiding error by refusing to believe anything) and (2) that, when it does make mistakes, commits errors that tend to be of the less egregious kind than of the more egregious kind (supposing that we can identify some error types as more serious than others). Finally, (3) we want to have mechanisms in place with the capacity to eventually identify the errors that we have made and to tell us how to correct for them. In short, we want to be able to reduce errors, to distribute those errors that do occur according to our preferences, and to have a self-correction device for identifying and revising our erroneous beliefs. This is, of course, an unabashedly Peircean view of the nature of inquiry, although one need not be (as I confess to being) a card-carrying pragmatist to find it congenial.

Most of the papers in this volume deal with the problem of error as it arises in the context of scientific research. That approach is fair enough because most of us are philosophers of science.

In this chapter I address what seems to be a sharp difference of opinion between myself and Mayo concerning a fundamental problem in the theory of confirmation. Not surprisingly, I argue that I am right and she is (interestingly) wrong. But first I need to outline the background carefully – because seeing clearly what the problem is (and what it is not) takes us a good way towards its correct solution.

The Duhem Problem and the “UN” Charter

So far as the issue about confirmation that I want to raise here is concerned: in the beginning was the “Duhem problem.” But this problem has often been misrepresented. No sensible argument exists in Duhem (or elsewhere) to the effect that the “whole of our knowledge” is involved in any attempt to test any part of our knowledge. Indeed, I doubt that that claim makes any sense. No sensible argument exists in Duhem (or elsewhere) to the effect that we can never test any particular part of some overall theory or theoretical system, only the “whole” of it. If, for example, a theory falls “naturally” into five axioms, then there is – and can be – no reason why it should be impos-sible that some directly testable consequence follows from, say, four of those axioms – in which case only those four axioms and not the whole of the theory are what is tested.

Introduction

Deborah Mayo's “error-statistical” account of science and its mode of progress is an attempt to codify and capitalise on the “new experimentalist” approach to science that has made its appearance in the past few decades as an alternative to “theory-dominated” accounts of science. Rather than understanding scientific progress in terms of the replacement of one large-scale theory by another in the light of experiments designed to test them, new experimentalists view progress in terms of the accumulation of experimental discoveries and capabilities established independently of high-level theory. The new experimentalists aspire to an account of science and its mode of progress that evades problems associated with the theory dependence of observation, the underdetermination of theories by evidence, the Duhem problem and incommensurability that have beset the theory-dominant approach. Here is how Mayo (this volume, p. 28) herself characterizes the situation:

Central to Mayo's version of the new experimentalism is her notion of a severe test.

Although I have offered some criticisms of her views on evidence and testing (Achinstein, 2001, pp. 132–40), I very much admire Deborah Mayo's book (1996) and her other work on evidence. As she herself notes in the course of showing how misguided my criticism is, we actually agree on two important points. We agree that whether e, if true, is evidence that h, in the most important sense of “evidence,” is an objective fact, not a subjective one of the sort many Bayesians have in mind. And we agree that it is an empirical fact, not an a priori one of the sort Carnap has in mind. Here I will take a broader and more historical approach than I have done previously and raise some general questions about her philosophy of evidence, while looking at a few simple examples in terms of which to raise those questions. It is my hope that, in addition to being of some historical interest, this chapter will help clarify differences between us.

Mill under Siege

One of Mayo's heroes is Charles Peirce. Chapter 12 of Mayo's major work, which we are honoring, is called “Error Statistics and Peircean Error Correction.” She has some very convincing quotes from Peirce suggesting that he was a model error-statistical philosopher. Now I would not have the Popperian boldness to say that Mayo is mistaken about Peirce; that is not my aim here.

The Issues

There are at least two aspects to understanding: comprehensibility and veracity. Good explanations are supposed to aid the first and to give grounds for believing the second – to provide insight into the “hidden springs” that produce phenomena and to warrant that our estimates of such structures are correct enough. For Copernicus, Kepler, Newton, Dalton, and Einstein, the virtues of explanations were guides to discovery, to sorting among hypotheses and forming beliefs. The virtues were variously named: simplicity, harmony, unity, elegance, and determinacy. These explanatory virtues have never been well articulated, but what makes them contributions to perspicacity is sometimes apparent, while what makes them a guide to truth is obscure.

We know that under various assumptions there is a connection between testing – doing something that could reveal the falsity of various claims – and the second aspect of coming to understand. Various forms of testing can be stages in strategies that reliably converge on the truth, and some cases even provide probabilistic guarantees that the truth is not too far away. But how can explaining be any kind of reliable guide to any truth worth knowing? What structure or content of thoughts or acts distinguishes explanations, and in virtue of what, if anything, are explanations indications of truth that should prompt belief in their content, or provide a guide in inquiry?

Philosophy of Science: Problems and Prospects

Methodological discussions in science have become increasingly common since the 1990s, particularly in fields such as economics, ecology, psychology, epidemiology, and several interdisciplinary domains – indeed in areas most faced with limited data, error, and noise. Contributors to collections on research methods, at least at some point, try to ponder, grapple with, or reflect on general issues of knowledge, inductive inference, or method. To varying degrees, such work may allude to philosophies of theory testing and theory change and philosophies of confirmation and testing (e.g., Popper, Carnap, Kuhn, Lakatos, Mill, Peirce, Fisher, Neyman-Pearson, and Bayesian statistics). However, the different philosophical “schools” tend to be regarded as static systems whose connections to the day-to-day questions about how to obtain reliable knowledge are largely metaphorical. Scientists might “sign up for” some thesis of Popper or Mill or Lakatos or others, but none of these classic philosophical approaches – at least as they are typically presented – provides an appropriate framework to address the numerous questions about the legitimacy of an approach or method.

Methodological discussions in science have also become increasingly sophisticated; and the more sophisticated they have become, the more they have encountered the problems of and challenges to traditional philosophical positions. The unintended consequence is that the influence of philosophy of science on methodological practice has been largely negative.

Introduction

For a domain of inquiry to live up to standards of scientific objectivity it is generally required that its theories be tested against empirical data. The central philosophical and methodological problems of economics may be traced to the unique character of both economic theory and its nonexperimental (observational) data. Alternative ways of dealing with these problems are reflected in rival methodologies of economics. My goal here is not to promote any one such methodology at the expense of its rivals so much as to set the stage for understanding and making progress on certain crucial conundrums in the methodology and philosophy of economics. This goal, I maintain, requires an understanding of the changing roles of theory and data in the development of economic thought alongside the shifting philosophies of science, which explicitly or implicitly find their way into economic theorizing and econometric practice. Given that this requires both economists and philosophers of science to stand outside their usual practice and reflect on their own assumptions, it is not surprising that this goal has been rather elusive.

The Preeminence of Theory in Economic Modeling

Historically, theory has generally held the preeminent role in economics, and data have been given the subordinate role of “quantifying theories” presumed to be true. In this conception, whether in the classical (nineteenth-century) or neoclassical (twentieth-century) historical period or even in contemporary textbook econometrics, data do not so much test as allow instantiating theories: sophisticated econometric methods enable elaborate ways “to beat data into line” (as Kuhn would say) to accord with an assumed theory.

Statistics and Inductive Philosophy

What Is the Philosophy of Statistics?

The philosophical foundations of statistics may be regarded as the study of the epistemological, conceptual, and logical problems revolving around the use and interpretation of statistical methods, broadly conceived. As with other domains of philosophy of science, work in statistical science progresses largely without worrying about “philosophical foundations.” Nevertheless, even in statistical practice, debates about the different approaches to statistical analysis may influence and be influenced by general issues of the nature of inductive-statistical inference, and thus are concerned with foundational or philosophical matters. Even those who are largely concerned with applications are often interested in identifying general principles that underlie and justify the procedures they have come to value on relatively pragmatic grounds. At one level of analysis at least, statisticians and philosophers of science ask many of the same questions.

• What should be observed and what may justifiably be inferred from the resulting data?

• How well do data confirm or fit a model?

• What is a good test?

• Does failure to reject a hypothesis H constitute evidence confirming H?

• How can it be determined whether an apparent anomaly is genuine? How can blame for an anomaly be assigned correctly?

• Is it relevant to the relation between data and a hypothesis if looking at the data influences the hypothesis to be examined?

• How can spurious relationships be distinguished from genuine regularities?

• How can a causal explanation and hypothesis be justified and tested?

• […]

Critical Rationalism, Explanation, and Severe Tests

This chapter has three parts. First, I explain the version of critical rationalism that I defend. Second, I discuss explanation and defend critical rationalist versions of inference to the best explanation and its meta-instance, the Miracle Argument for Realism. Third, I ask whether critical rationalism is compatible with Deborah Mayo's account of severe testing. I answer that it is, contrary to Mayo's own view. I argue, further, that Mayo needs to become a critical rationalist – as do Chalmers and Laudan.

Critical Rationalism

Critical rationalism claims that the best method for trying to understand the world and our place in it is a critical method – propose views and try to criticise them. What do critical methods tell us about truth and belief? If we criticize a view and show it to be false, then obviously we should not believe it. But what if we try but fail to show that a view is false? That does not show it to be true. So should we still not believe it?

Notoriously, the term “belief” is ambiguous between the act of believing something (the believing) and the thing believed (the belief). Talk of “reasons for beliefs” inherits this ambiguity – do we mean reasons for believings or reasons for beliefs? Critical rationalists think we mean the former. They think there are reasons for believings that are not reasons for beliefs.

Background to the Discussion

The goal of this chapter is to take up the aforementioned challenge as it is posed by Alan Chalmers (1999, 2002), John Earman (1992), Larry Laudan (1997), and other philosophers of science. It may be seen as a first step in taking up some unfinished business noted a decade ago: “How far experimental knowledge can take us in understanding theoretical entities and processes is not something that should be decided before exploring this approach much further” (Mayo, 1996, p. 13). We begin with a sketch of the resources and limitations of the “new experimentalist” philosophy.

Learning from evidence, in this experimentalist philosophy, depends not on appraising large-scale theories but on local experimental tasks of estimating backgrounds, modeling data, distinguishing experimental effects, and discriminating signals from noise. The growth of knowledge has not to do with replacing or confirming or probabilifying or “rationally accepting” large-scale theories, but with testing specific hypotheses in such a way that there is a good chance of learning something – whatever theory it winds up as part of.

A central question of interest to both scientists and philosophers of science is, How can we obtain reliable knowledge about the world in the face of error, uncertainty, and limited data? The philosopher tackling this question considers a host of general problems: What makes an inquiry scientific? When are we warranted in generalizing from data? Are there uniform patterns of reasoning for inductive inference or explanation? What is the role of probability in uncertain inference? Scientific practitioners, by and large, just get on with the job, with a handful of favored methods and well-honed rules of proceeding. They may seek general principles, but largely they take for granted that their methods “work” and have little patience for unresolved questions of “whether the sun will rise tomorrow” or “whether the possibility of an evil demon giving us sensations of the real world should make skeptics of us all.” Still, in their own problems of method, and clearly in the cluster of courses under various headings related to “scientific research methods,” practitioners are confronted with basic questions of scientific inquiry that are analogous to those of the philosopher.

Nevertheless, there are several reasons for a gap between work in philosophy of science and foundational problems in methodological practice. First, philosophers of science tend to look retrospectively at full-blown theories from the historical record, whereas work on research methods asks how to set sail on inquiries and pose local questions.

Abstract

We show that a choice of Young function, for quantum states given by density operators, leads to an Orlicz norm such that the set of states of Cramér class becomes a Banach manifold. A comparison is made with the case studied by Pistone and Sempi, which arises in the theory of non-parametric estimation in classical statistics.

The work of Pistone and Sempi

The work of (Pistone and Sempi 1995) arises as a generalisation to infinitely many parameters of the theory of the best estimation of parameters of a probability distribution, using the data obtained by sampling. It is also sometimes called ‘nonparametric estimation’. In 1995, Pistone and Sempi obtained a notable formalism, making use of an Orlicz space. From the point of view of quantum mechanics, the classical case corresponds to the special case where all observables generate an abelian algebra. The quantum case of a finite-dimensional Hilbert space leads to the theory of quantum information, but does not involve delicate questions of topology; this is because all norms on a space of finite dimension are equivalent. The question arises, whether we can imitate the use of an Orlicz norm in the infinite-dimensional case. We here show that this is possible, by completing the outline made earlier (Streater 2004a). We must start with a brief review of the classical case. We follow (Streater 2004a), with minor corrections.

(Pistone and Sempi 1995) develop a theory of best estimators (of minimum variance) among all locally unbiased estimators, in classical statistical theory.

Abstract

In this chapter algebraic statistics methods are used for design of experiments generation. In particular the class of Gerechte designs, that includes the game of sudoku, has been studied.

The first part provides a review of the algebraic theory of indicator functions of fractional factorial designs. Then, a system of polynomial equations whose solutions are the coefficients of the indicator functions of all the sudoku fractions is given for the general p2×p2 case (p integer). The subclass of symmetric sudoku is also studied. The 4×4 case has been solved using CoCoA. In the second part the concept of move between sudoku has been investigated. The polynomial form of some types of moves between sudoku grids has been constructed.

Finally, the key points of a future research on the link between sudoku, contingency tables and Markov basis are summarised.

Introduction

Sudoku is currently a very popular game. Every day many newspapers all over the world propose such puzzles to their readers. From wikipedia we read:

This description refers to the standard game but also 4 × 4, 6 × 6, 12 × 12 and 16 × 16 grids are played.