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 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.     

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.     

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.     

The simplicity principle in human concept learning

How do we learn concepts and categories from examples? Part of the answer might be that we induce the simplest category consistent with a given set of example objects. This seemingly obvious idea, akin to simplicity principles in many fields, plays surprisingly little role in contemporary theories of concept learning, which are mostly based on the storage of exemplars, and avoid summarization or overt abstraction of any kind. This article reviews some evidence that complexity minimization does indeed play a central role in human concept learning. The chief finding is that subjects' ability to learn concepts depends heavily on the concepts' intrinsic complexity; more complex concepts are more difficult to learn. This pervasive effect suggests, contrary to exemplar theories, that concept learning critically involves the extraction of a simplified or abstracted generalization from examples.     

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.     

Factor simplicity index and transformations

A scale-invariant index of factorial simplicity is proposed as a summary statistic for principal components and factor analysis. The index ranges from zero to one, and attains its maximum when all variables are simple rather than factorially complex. A factor scale-free oblique transformation method is developed to maximize the index. In addition, a new orthogonal rotation procedure is developed. These factor transformation methods are implemented using rapidly convergent computer programs. Observed results indicate that the procedures produce meaningfully simple factor pattern solutions.This investigation was supported in part by a Research Scientist Development Award (K02-DA00017) and research grants (MH24149 and DA01070) from the U. S. Public Health Service. The assistance of Andrew L. Comrey, Henry F. Kaiser, Bonnie Barron, Marion Hee, and several anonymous reviewers is gratefully acknowledged.     

SEM with simplicity and accuracy

Professor Iacobucci has provided a useful introduction to the computer program LISREL, as well as to several technical topics in structural equation modeling (SEM). However, SEM has not been synonymous with LISREL for several decades, and focusing on LISREL's 13 Greek matrices and vectors is not the most intuitive way to learn SEM. It is possible today to do model specification via a path diagram without any need for filling in matrix elements. The simplest alternative is based on the Bentler–Weeks model, whose basic concepts are reviewed. Selected additional SEM topics are discussed, including some recent developments and their practical implications. New simulation results on model fit under null and alternative hypotheses are also presented that are consistent with statistical theory but in part seem to contradict those reported by Iacobucci.     

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.     

Inductive Incompleteness

Nelson Goodman cast the ‘problem of induction’ as the task of articulating the principles and standards by which to distinguish valid from invalid inductive inferences. This paper explores some logical bounds on the ability of a rational reasoner to accomplish this task. By a simple argument, either an inductive inference method cannot admit its own fallibility, or there exists some non-inferable hypothesis whose non-inferability the method cannot infer (violating the principle of ‘negative introspection’). The paper discusses some implications of this limited self-knowledge for the justifiability of inductive inferences, auto-epistemic logic, and the epistemic foundations of game theory.     

Inductive countersupport

The basic idea by means of which Popper and Miller proved the non-existence of inductive probabilistic support in 1983/1985/1987, is used to prove that inductive probabilistic countersupport does exist. So it seems that after falsification has won over verification on the deductive side of science, countersupport wins over support on the inductive side.     

Transforming three-way arrays to maximal simplicity

Transforming the core array in Tucker three-way component analysis to simplicity is an intriguing way of revealing structures in between standard Tucker three-way PCA, where the core array is unconstrained, and CANDECOMP/PARAFAC, where the core array has a generalized diagonal form. For certain classes of arrays, transformations to simplicity, that is, transformations that produce a large number of zeros, can be obtained explicitly by solving sets of linear equations. The present paper extends these results. First, a method is offered to simplifyJ ×J × 2 arrays. Next, it is shown that the transformation that simplifies anI ×J ×K array can be used to also simplify the (complementary) arrays of order (JK −I) ×J ×K, of orderI × (IK −J) ×K and of orderI ×J × (IJ −K). Finally, the question of what constitutes the maximal simplicity for arrays (the maximal number of zero elements) will be considered. It is shown that cases of extreme simplicity, considered in the past, are, in fact, cases of maximal simplicity.