Technical notes on a theory of simplicity

Recently Samuel Richmond, generalizing Nelson Goodman, has proposed a measure of the simplicity of a theory that takes into account not only the polymorphicity of its models but also their internal homogeneity. By this measure a theory is simple if small subsets of its models exhibit only a few distinct (i.e., non-isomorphic) structures. Richmond shows that his measure, unlike that given by Goodman's theory of simplicity of predicates, orders the order relations in an intuitively satisfactory manner. In this note I formalize his presentation and suggest an improvement designed to overcome certain technical difficulties.     

A factor simplicity index

We propose an index for assessing the degree of factor simplicity in the context of principal components and exploratory factor analysis. The new index, which is called Loading Simplicity, is based on the idea that the communality of each variable should be related to few components, or factors, so that the loadings in each variable are either zero or as far from zero as possible. This index does not depend on the scale of the factors, and its maximum and minimum are only related to the degree of simplicity in the loading matrix. The aim of the index is to enable the degree of simplicity in loading matrices to be compared.The author would like to thank the review team for their insights and recommendations. This work was supported by a grant SEC2001-3821-C05-C02 from the Spanish Ministry of Science and Technology.     

Acts of perceptual inquiry: problems for any stimulus-based simplicity theory

Stimulus-specified simplicity is simply insufficient to predict the appearance of objects' 3D forms and figural shapes, etc., as originally intended. Which information a viewer chooses to attend can determine both what 3D structure is perceived and its attendant perceptual consequences; moreover, a shape's meaningfulness or denotivity [M.A. Peterson, Current Directions in Psychological Science 3 (1994) 105] can overcome simplicity-based figure-ground segregation. In both cases, perceptual consequences, such as subsequent perceived movements, are constrained or primed in ways that can help us corroborate and define what the viewer has perceived; that may help us in studying the underlying events in brain processing; and that should help in designing such perceptual applications as still and animated displays. Demonstrations, theoretical framework, and potential research tools are offered.     

Reflex theory,cautionary tale: misleading simplicity in early neuroscience

Synthese - This paper takes an integrated history and philosophy of science approach to the topic of "simplicity out of complexity". The reflex theory was a framework within early...     

Against simplicity

Sometimes metaphysicians appeal to simplicity as a reason to prefer one metaphysical theory to another, especially when a philosophical dispute has otherwise reached a state of equilibrium. In this paper, I show that given a Quinean conception of metaphysics, several initially plausible justifications for simplicity as a metaphysical criterion do not succeed. If philosophers wish to preserve simplicity as a metaphysical criterion, therefore, they must radically reconceive the project of metaphysics.     

Divine simplicity

According to a doctrine widely held by most medieval philosophers and theologians, whether in the Muslim or Christian world, there are no metaphysical distinctions in God whatsoever. As a result of the compendious theorizing that has been done on this issue, the doctrine, usually called the doctrine of divine simplicity, has been bestowed a prominent status in both Islamic and Christian philosophical theology. In Islamic philosophy some well-known philosophers, such as Ibn Sina (980–1037) and Mulla Sadra (1571–1640), developed this doctrine through a metaphysical approach. In this paper, considering the historical order, I shall first concentrate on Ibn Sina’s view. Then I shall turn to the theory of divine simplicity of Thomas Aquinas (1225?–1274), as the most developed and comprehensive version of the medieval theories in Christian world. Finally, I will return to Islamic philosophy and explore the more complicated and mature account of the doctrine as it was introduced by Mulla Sadra according to his own philosophical principles.     

A simplicity principle in unsupervised human categorization

We address the problem of predicting how people will spontaneously divide into groups a set of novel items. This is a process akin to perceptual organization. We therefore employ the simplicity principle from perceptual organization to propose a simplicity model of unconstrained spontaneous grouping. The simplicity model predicts that people would prefer the categories for a set of novel items that provide the simplest encoding of these items. Classification predictions are derived from the model without information either about the number of categories sought or information about the distributional properties of the objects to be classified. These features of the simplicity model distinguish it from other models in unsupervised categorization (where, for example, the number of categories sought is determined via a free parameter), and we discuss how these computational differences are related to differences in modeling objectives. The predictions of the simplicity model are validated in four experiments. We also discuss the significance of simplicity in cognitive modeling more generally.     

On simplicity of formulas

Simple formula should contain only few quantifiers. In the paper the methods to estimate quantity and quality of quantifiers needed to express a sentence equivalent to given one.     

On the role of simplicity in science

Simple assumptions represent a decisive reason to prefer one theory to another in everyday scientific praxis. But this praxis has little philosophical justification, since there exist many notions of simplicity, and those that can be defined precisely strongly depend on the language in which the theory is formulated. The language dependence is a natural feature—to some extent—but it is also believed to be a fatal problem, because, according to a common general argument, the simplicity of a theory is always trivial in a suitably chosen language. But, this trivialization argument is typically either applied to toy-models of scientific theories or applied with little regard for the empirical content of the theory. This paper shows that the trivialization argument fails, when one considers realistic theories and requires their empirical content to be preserved. In fact, the concepts that enable a very simple formulation, are not necessarily measurable, in general. Moreover, the inspection of a theory describing a chaotic billiard shows that precisely those concepts that naturally make the theory extremely simple are provably not measurable. This suggests that—whenever a theory possesses sufficiently complex consequences—the constraint of measurability prevents too simple formulations in any language. This explains why the scientists often regard their assessments of simplicity as largely unambiguous. In order to reveal a cultural bias in the scientists’ assessment, one should explicitly identify different characterizations of simplicity of the assumptions that lead to different theory selections. General arguments are not sufficient.     

A further simplification of the multiple and partial correlation process

Formulas are derived for simplified computation of partial and multiple correlation coefficients, and generalized ton variables. Time required for computation is compared with other methods.     

An index of factorial simplicity

An index of factorial simplicity, employing the quartimax transformational criteria of Carroll, Wrigley and Neuhaus, and Saunders, is developed. This index is both for each row separately and for a factor pattern matrix as a whole. The index varies between zero and one. The problem of calibrating the index is discussed.The research reported in this paper was supported in part by the Office of Computing Activities, National Science Foundation.     

The improbable simplicity of the fusiform face area

The fusiform face area (FFA) is described as an easily identifiable module on the fusiform gyrus. However, the organization of face-selective regions in ventral temporal cortex (VTC) is more complex than this prevailing view. We highlight methodological factors contributing to these complexities and the extensive variability in how the FFA is identified. We suggest a series of constraints to aid researchers when defining any functionally specialized region with a pleasing realization: anatomy matters.     

A case of extreme simplicity of the core matrix in three-mode principal components analysis

In three-mode Principal Components Analysis, theP ×Q ×R core matrixG can be transformed to simple structure before it is interpreted. It is well-known that, whenP=QR,G can be transformed to the identity matrix, which implies that all elements become equal to values specified a priori. In the present paper it is shown that, whenP=QR − 1,G can be transformed to have nearly all elements equal to values spectified a priori. A cllsed-form solution for this transformation is offered. Theoretical and practical implications of this simple structure transformation ofG are discussed. Constructive comments from anonymous reviewers are gratefully acknowledged.