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This article explores some open questions related to the problem of verification of theories in the context of empirical sciences by contrasting three epistemological frameworks. Each of these epistemological frameworks is based on a corresponding central metaphor, namely: (a) Neo-empiricism and the gambling metaphor; (b) Popperian falsificationism and the scientific tribunal metaphor; (c) Cognitive constructivism and the object as eigen-solution metaphor. Each of one of these epistemological frameworks has also historically co-evolved with a certain statistical theory and method for testing scientific hypotheses, respectively: (a) Decision theoretic Bayesian statistics and Bayes factors; (b) Frequentist statistics and

The four quotations that open this article, from [

It is hard to choose a word among verification, confirmation, corroboration, or similar ones, because all of them are already heavily overloaded with very specific meanings. We choose verification for its direct etymological link to the truth bearing status of a statement. We will analyze the verification problem and other related questions from the perspective of Cog-Con—the Cognitive Constructivism epistemological framework. Cog-Con comes equipped with the statistical apparatus of FBST—the Full Bayesian Significance Test. FBST, in turn, defines a statistical support function for sharp hypothesis, namely, the

The FBST solution to the problem of verification is indeed very thin, in the sense that the proposed epistemic support function, the

In the following sections we try to explain and clarify several aspects related to the Cog-Con approach to hypothesis verification, including its central metaphor of objects as token for eigen-solutions. We also contrast this approach with two standard alternatives, neoclassical empiricism and Popperian falsificationism, and their central metaphors, namely, gambling and the scientific tribunal. These two alternative epistemological frameworks are, in turn, associated to two alternative statistical methodologies, namely, frequentist (classical) and Bayesian statistics. In the limited space of this article, we cannot afford to explain any of the aforementioned statistical theories. [

The Cog-Con framework rests upon Heinz von Forster's metaphor of

The notion of autopoietic system is an abstraction aiming to model the most essential properties of a living organism. Autopoiesis can be understood as an operationally based conceptual framework about the systemic nature of living beings. Autopoietic systems are non-equilibrium (dissipative) dynamical systems exhibiting (meta) stable structures, whose organization remains invariant over (long periods of) time, despite the frequent substitution of their components. Moreover, these components are produced by the same structures they regenerate. The regeneration processes in the autopoietic system production network always requires the acquisition of resources such as new materials, energy and neg-entropy (order), from the system's environment. Efficient acquisition of the needed resources demands selective (inter) actions which, in turn, must be based on suitable inferential processes (predictions). Hence, these inferential processes characterize the agent's domain of interaction as a cognitive domain.

In spite of the fact that autopoiesis was a conceptual framework developed to suit the essential characteristics of organic life, the concept of autopoietic system has been applied in the analysis of many other concrete or abstract autonomous systems such as social systems and corporate organizations. In particular, scientific research systems can be seen in this light, see [

The circular (cyclic or recursive) characteristic of autopoietic regenerative processes and their eigen- (auto, equilibrium, fixed, homeostatic, invariant, recurrent, recursive) states, both in concrete and abstract autopoietic systems, are investigated in von Foerster in [

Moreover, von Foerster establishes four essential attributes of eigen-solutions: Eigen-values are ontologically discrete, stable, separable and composable. It is important to realize that, in the sequel, the term “discrete”, used by von Foerster to qualify eigen-solutions in general, should be replaced, depending on the specific context, by terms such as lower-dimensional, precise, sharp, singular

For detailed interpretations of von Foerster's four essential attributes of eigen-solutions, the best references are his original works in [

The gambling metaphor, accompanied by the colorful language of betting odds, is at the core of neoclassical empiricism. In p.152, V.2 of [

“Neoclassical empiricism had a central dogma: the dogma of the identity of (1) probabilities; (2) degree of evidential support (or confirmation); (3) degree of rational belief, and (4) rational betting quotients. This “neoclassical chain of identities” is not implausible. For a true empiricist the only source of rational belief is evidential support: thus he will equate the degree of rationality of a belief with the degree of its evidential support. But rational belief is plausibility measured by rational betting quotients. It was, after all, to determine rational betting quotients that the probability calculus was invented.”

In a game where there is a priori knowledge about the competitors, including perceived differences in strength, skill or other fair or unfair advantages, a score handicap compensation system can be used to equilibrate the winning chances of all the competitors. Gambling, with all its quirks and peculiarities, is the driving metaphor of decision theoretic (orthodox) Bayesian statistics. Several aspects and consequences of using this metaphor are analyzed in [

A modern tribunal follows the principle of

In the tribunal metaphor, a scientific law is (provisionally) accepted to be truthful, until it is refuted or proved wrong by pertinent evidence. In the court of science, pertinent evidence that can be used to refute a theory has the form of empirical observations that disagree with the consequences or predictions made by the theory on trial. Hence, a fair trial in the scientific tribunal can assure the validity of the deduction process leading to a proof a falsehood, but cannot give any positive certification or assurance concerning a theory's validity or good quality.

Empirical sciences, specially the so-called exact sciences, like physics, chemistry or engineering, deal with quantitative entities. Moreover, the standard practice of these sciences also requires the truth content of scientific hypotheses to be handled in a quantitative fashion, that is, to undergo a quantitative form of judgment for accuracy and precision. Furthermore, the Cog-Con framework allows us to model some aspects of the development of science in the context of dynamical systems and evolutionary processes, see [

Given the importance the metrics used to evaluate the scientific statements, and the many roles they play in the practice of science, we must choose these metrics with extreme attention, care and caution, designing their structure and regulating their strength and balance. The standard metrics used in empirical science are based on mathematical statistics. Although many alternative belief calculi have been able to successfully occupy local niches or find special applications, modern statistical data analysis finds no rival for its elegance, robustness, flexibility, computational power, and the generality of its scope of application. Nevertheless, there are many long standing issues and unresolved problems related to the use of statistical metrics in the context of hypothesis verification. This is the topic on the next section.

In order to fully appreciate the Cog-Con+FBST framework, and make further contrasts with other approaches, we use two celebrated examples given in the XIX century by Charles Saunders Peirce. These examples concern the abduction and induction of hypotheses, studying possible procedures to guess, justify and test statements in statistical models. As used in these two examples, Peirce's idea of “induction” would nowadays be called

The statistical work of Charles Saunders Peirce has several aspects that deserve further scrutiny. For example, [

The first example, published by Peirce in 1868, [

Given the English books _{1}, _{2}, … _{k}^{1}, ^{2}, … ^{k} with the frequencies in which every letter in the alphabet occurs in the text. We realize that they all (approximately) agree with with the mean or average frequencies in vector λ^{a}

Given a new English book, _{k+}_{1}, we may, by ^{k}^{+1} (not yet compiled) will also be (approximately) equal to λ^{a}

Given a coded book ^{c}^{a}^{c}^{a}^{c}

A standard formulation for the induction part of this example includes parameter estimation (posterior distribution, likelihood or, at least, a point estimate and confidence interval) in an n-dimensional Dirichlet-Multinomial model, where ^{m}

Peirce's (abductive) hypothesis about the cipher proclaims the “correct” or “true” permutation vector, ^{0}. This hypothesis has an interesting peculiarity: The parameter space, Θ = Λ × ∏, has a continuous sub-space, Λ, and a discrete (actually, finite) sub-space, ∏. However, the hypothesis only (directly) involves the finite part. This peculiarity makes this hypothesis very simple, and amenable to the treatment given by Peirce. However, over-simplification can be a dangerous thing, as shown by an example published by Peirce in 1883 [

“[Kepler] traced out the miscellaneous consequences of the supposition that Mars moved in an ellipse, with the sun at the focus, and showed that both the longitudes and the latitudes resulting from this theory were such as agreed with observation. …The term Hypothesis [means] a proposition believed in because its consequences agree with experience.”

Instead of formulating Kepler's hypothesis in a contemporary statistical model, we can make use of an equivalent example already at hand, namely, the Hardy-Weinberg model formulated in [

Peirce's idea for testing the cipher hypothesis is a forerunner of modern decision theoretic Bayesian procedures that compute support values like

Peirce's approach to the first example leads, in modern Statistical theory, to a decision theoretic posterior probability for the hypothesis, given the observed data bank. This approach works very well for the Cipher problem. In fact, as the number of observations increase, posterior probabilities will automatically converge, concentrating full support (probability 1) into the true hypothesis. Hence, in this simple problem, we can in fact confuse the problems of induction and abduction. In a context with finite set of alternative hypotheses, one can equivalently speak about the posterior probability of hypothesis _{i}_{i}_{i}_{i}_{i}_{i}_{i}

The posterior probability solution can be adapted for testing hypotheses on continuous parameters if we only consider partitions of the parameter space into a finite number of non-zero measure sets, corresponding to coarse, un-sharp or inexact hypotheses. However, this approach breaks down as soon as we consider sharp hypotheses. The reason for this collapse is the zero-probability trap: A sharp hypothesis has zero-probability measure and, accordingly, zero prior probability. Moreover, the multiplicative nature of probabilistic scaling will never update a zero probability to a non-zero value, see [

The “object as eigen-solution” metaphor implies the sharpness of the corresponding hypotheses;

Hypotheses sharpness implies zero prior probability (in the natural Lebesgue measure);

Zero prior probability implies perpetual null support. This is one way of understanding the following conclusion stated by Lakatos in p. 154, V.2 of [

“But then degrees of evidential support cannot be the same as degrees of probability [of a theory] in the sense of the probability calculus. All this would be trivial if not for the powerful time-honored dogma of what I called the ‘neoclassical chain’ identifying, among other things, rational betting quotients with degrees of evidential support. This dogma confused generations of mathematicians and of philosophers.”

There are two obvious ways out of this conundrum:

Fixing the mathematics to avoid the ZPP; or

Forbidding the use of sharp hypotheses.

(A) Fixing the mathematics in the standard decision theoretic or orthodox Bayesian framework in order to avoid the ZPP is something that is more easily said than done. Modern Bayesian statistics devised several technical maneuvers to circumvent the ZPP. Some of the best known among these techniques are Jeffreys tests, and other handicapped or relative betting odds for scoring competing sharp hypotheses. These techniques provide ad-hoc procedures for practical use, but are plagued by internal inconsistencies, like Lindley's paradox, or by the need of justifying auxiliary ad-hoc assumptions, like the choice of prior betting odds or the design characteristics of artificial prior densities (an obvious oxymoron), see Section 10.3 of [

(B) Forbidding the use of sharp hypotheses may be very tempting from an orthodox decision theoretic (Bayesian) point of view, however, it is unfeasible in statistical practice: Scientists and other customers of statistical science just insist on using sharp hypotheses, as if they were magnetically attracted to them, and demand appropriate statistical methods. From the Cog-Con perspective, these scientist are, of course, just doing the right thing. As a compromise solution, some influential statistical textbooks offer methods like Jeffreys tests, taking however the care of posting a scary caveat emptor, warning the user that he is entering theoretically unsound territory at his own risk, see for example p. 234 in [

“The unacceptability of extreme (sharp) null hypotheses is perfectly well known; it is closely related to the often heard maxim that science disproves, but never proves, hypotheses, The role of extreme (sharp) hypotheses in science and other statistical activities seems to be important but obscure. In particular, though I, like everyone who practice statistics, have often “tested” extreme (sharp) hypotheses, I cannot give a very satisfactory analysis of the process, nor say clearly how it is related to testing as defined in this chapter and other theoretical discussions.”

Peirce's intuition for testing sharp hypotheses in his second example leads to the ^{0}, is a specific value, ^{1}. The ^{0} ≠ ^{1}, and has uniform limit distribution if the hypothesis is true, ^{0} = ^{1}. These properties are very convenient, since they can be used to obtain numerical approximations relatively easy to compute. Nowadays it is hard to appreciate the importance of these properties in a world where digital computers were not (easily) available, and statistical modeling had to be done using tools like slide-rules, numerical tables and graphical charts, see [

Lakatos, in pp. 31–32, V.2 of [

“Since the difficulties with induction had been known for a long time, it is remarkable that independently and nearly simultaneously Neyman and Popper found a revolutionary way to finesse the issue by replacing inductive reasoning with a deductive process of hypothesis testing. They then proceeded to develop this shared central idea in different directions, with Popper pursuing it philosophically while Neyman (in his joint work with Pearson) showed how to implement it in scientific practice.”

However useful as a practical technique, even in the case of point hypotheses, the ^{0} = ^{*}, where ^{*} is the maximum likelihood (or MAP—maximum a posteriori) estimator under the original hypothesis, given the observed data. But the maximum likelihood auxiliary hypothesis is post-hoc, and therefore cannot adequately represent the original hypothesis. Other alternatives consider a priori reductions or projections of the composite hypothesis into a point hypotheses by “nuisance parameter elimination” procedures. An excellent survey, containing more than 10 different techniques with this purpose, is given in [

Decision theoretic (Bayesian) posterior probabilities and classical statistics

The FBST solution for testing sharp hypotheses can be seen as a “dual” of the

In the last sections we saw that the Cog-Con was able to tame the ZPP. This process involved taking three basic conceptual steps:

Adopting the standard Bayesian statistical model setup, including the posterior probability measure in the parameter space.

Making a clear distinction between the parameter and the hypothesis spaces.

Defining the

The

The use of the _{n}_{n}

The definition of the

Moreover, the

These properties allow the

Solve the ZPP for sharp hypothesis , and

Work like a bridge, harmonizing probability—the underlying logic of statistical inference, the paradigm of belief calculus for empirical science—and classical logic—the prototypical rule of deductive inference.

Step 7 represents an absolution. Sharp hypotheses are freed from the zero-support syndrome, and admitted as full citizens in the hypothesis space. However, Step 7 does not warrant that there will ever be a sharp hypothesis in an empirical science with good support. In fact, considering the original ZPP, finding such an outstanding (sharp) hypothesis should be really surprising, the scientific equivalent of a miracle! What else should we call showing possible what is almost surely (in the probability measure) infeasible? Nevertheless, we know that miracles do exist. (Non-believers are encouraged to take some good classes in experimental physics, including a fair amount of laboratory work.)

In the Cog-Con framework, the certification of a sharp hypothesis by

From this perspective, the Cog-Con framework not only redeems sharp or precise hypotheses from statistical damnation, but places them at the center stage of scientific activity. (The star role of any exact science will always be played by eigen-solutions represented by a statement called

Mathematics is the common language used for the expression and manipulation of symbolic entities associated with the quantities of interest pertinent to the scope of each particular empirical science. Hence, we are particularly interested in the nature of mathematical language. In this section we will argue that, in the Cog-Con framework, mathematics can be regarded as a quasi-empirical science, an idea developed at length by the philosopher Imre Lakatos. The key of our arguments is Step 8 in the last Section. Step 8 constitutes a bridge from physics to mathematics, from empirical to quasi-empirical science. From this perspective, mathematics is seen as an idealized world of absolutely verified theories populated by hypotheses with full (or null) support.

Ontologies are controlled languages used in the practice of science. They are developed as tools for scientific communication. This communication has typical external and internal aspects: we need language in order to communicate with others and with ourselves. We use language as a tool for effective coordination of action and as a tool for efficient structuring of understanding. Equipped with the appropriate ontologies, scientists are supposed to build models capable of providing reliable predictions and insightful explanations. Moreover, at least in the domain of exact sciences, these models are required to have a formal and quantitative nature. Hence, the approach we follow naturally highlights the special role played by formal or mathematical languages, our main interest in this section.

Formal or “abstract” mathematics, including several of its less formal or popular dialects, is the common language used for the expression and manipulation of symbolic entities associated with the quantities of interest pertinent to the scope of each particular empirical science. Hence, we will be particularly interested in the nature of mathematical language. In fact, we will argue that mathematics should be regarded as an ontology for a class of concepts relevant to all exact sciences, namely, those related to the intuitive ideas of counting, symmetry, number, infinity, measure, dimension, and the continuum. From this point of view, mathematics can be regarded as a quasi-empirical, as opposed to an Euclidean science, according to the classical distinction defined by by Lakatos in p.40,V.2 of [

“Whether a deductive system is Euclidean or quasi-empirical is decided by the pattern of truth value flow in the system. The system is Euclidean if the characteristic flow is the transmission of truth from the set of axioms “downwards” to the rest of the system—logic here is the organon of proof; it is quasi-empirical if the characteristic flow is “upwards” towards the “hypothesis”—logic here is an organon of criticism. We may speak (even more generally) of Euclidean vs. quasi-empirical theories independently of what flows in the logical channels: certain or fallible truth or falsehoods, probability or improbability, moral desirability or undesirability, etc. It is the how of the flow that is decisive.”

Of course, in the Cog-Con framework it is verification (or not), measured by

“As far as the statements of mathematics refer to actual truth, they are not certain; and as far as they are certain, they do not refer to actual truth”.

(Kappa:) “If you want mathematics to be meaningful, you must resign of certainty. If you want certainty, get rid of meaning. You cannot have both.”

Lakatos' view of mathematics as a quasi-empirical science is not the most widespread current opinion, even though a few authors explore similar ideas, see for example [

For example, in the beginning of Greek mathematics, the use of the word

Following Szabó arguments, it is possible answer his famous interrogation: How did mathematics become a deductive science? Nevertheless, in this paper, we are more interested in a closely related question, not how, but why: Why did mathematics become a deductive science? Why was that transformation of the mathematical ideal, from down to earth science to aethereal philosophy, even possible?

Once again, the miraculous or wonderful nature of a well supported eigen-solutions, discussed in the previous sections, can offer a positive answer to the last questions. After all, it is only natural to believe that miraculous theorems are born in heaven. I will not venture into the discussion of whether or not good mathematics comes from heaven or “straight from The Book”, as Pál Erdős used to say. I will only celebrate the revelation of this mystery. It represents the ultimate transmutation of the ZPP, from bad omen of confusion, to good augury of universal knowledge.

The history of mathematics provides many interesting themes of study that we hope to explore in forthcoming papers. For example, modern logic and set-theory seem to have followed some trends that converge to the Cog-Con perspective. As an illustration, in p. 27, V.2 of [

“the role of the alleged ‘foundation’ is rather comparable to the function discharged, in physical theory, by explanatory hypotheses… the actual function of axioms is to explain the phenomena described by the theorems of this system rather than to provide a genuine ‘foundation’ for such theorems.”

Eugen Wigner, [

The author is grateful for the support of the Department of Applied Mathematics of the Institute of Mathematics and Statistics of the University of São Paulo, FAPESP—Fundação de Amparo à Pesquisa do Estado de São Paulo, and CNPq—Conselho Nacional de Desenvolvimento Científico e Tecnológico (grant PQ-306318-2008-3). This paper was first presented at EBL-2011, the XVI Brazilian Logic Conference, held on May 9–13 at LNCC—Laboratório Nacional de Computação Cienífica, Petrópolis, Brazil, and also presented at COBAL-2011, the III Latin American Meeting on Bayesian Statistics, held on October 23–27 at UFRO—Universidad de La Frontera, Pucón, Araucaía, Chile. The author is grateful for the advice received from anonymous referees, and also for helpful discussions with several of his colleagues at the Bayesian research group at University of São Paulo, especially its head, Carlos Alberto de Bragança Pereira, who is always poking and probing our ideas on the foundations of probability and statistics. Finally, the author is grateful for some comments concerning Lakatos' late works made by Gábor Kutrovátz of Loránd Eötvös Budapest University.