The book under review tries to link the economic concept of “reward,” or, more accurately, “capture rate,” to the experimental literature of various neuroscientific quantities dealing with motor control. But this reviewer argues that such a linkage requires a richer language of quantification than the book actually affords: a language not just of “greater” or “less,” but of how much greater or less. Without such a methodology, the arguments here cannot be persuasive.

Neural organization attempts to thwart, at least in part, modern neuroscientists' tendency to focus reductionistically on ever smaller microsystems. But although emphasizing higher levels of systems organization, the authors end up enforcing reductionisms of their own, principally the reduction of their domain to the study of invariable normal functioning, without explicit modeling of the deviations that constitute disease states or aging. This reductionism seriously weakens the authors' claims about the truth of their quantitative models.

Punctuated equilibrium and phyletic gradualism are alternative hypotheses that purport to explain the tempo and mode of evolution. We evaluate the two hypotheses, as they apply to the fossil record, on both theoretical and empirical grounds. Hidden randomness in data increases as a function of greater aggregation, and the hypothesis of punctuated equilibrium should not be applied to those examples where randomness is likely to occur. False stasis can result from a sustained pattern of emigration and immigration, and geographic variation must be studied in order to posit an unambiguous case of punctuated equilibrium. We describe a statistical method based on the general linear model for testing the relative fit of the alternative hypotheses to any set of temporally ordered metric data. Our method is hierarchical in the sense that subsets of the total explained variance can themselves be explained. The size of the first molar of the primate Pelycodus and of the condylarth Hyopsodus are analyzed. There are 17 tests in the two data sets, and we discover 12 instances of gradualism, four of punctuation and one of equilibrium.

Over the past quarter-century there has been considerable innovation in methods for assessing the tempo and mode of evolution in paleobiological data sets. The current literature of these methods centers on three competing hypotheses—stasis, random walk, and directional trend—corresponding to an increasing scaling of variance with time interval (unchanging, for stasis; linear, for random walk; quadratic, for trend). For applications to a single trait there are powerful methods for discriminating among these hypotheses; but for multivariate data sets, especially the very high-dimensional multivariate data arising in image-feature-based and morphometric studies, current statistical approaches appear to be of less help. This paper proves that in the limiting case of high-dimensional morphospaces, the principal component or principal coordinate ordination of every sufficiently lengthy isotropic random walk tends to the same geometrical shape, which is not that of an ellipsoid and for which the principal components or coordinates are not independent even though they are uncorrelated. Specifically, the “scatter” of PC1 against PC2 is just a parabolic curve. The quantitative characteristics of this specific shape are not described appropriately by the corresponding “covariance structure” or Gaussian model, and the discrepancy may be pertinent to much of the existing literature of methods for differentiating among those three models of evolutionary multivariate time series. From a close examination of this common geometry of the ideal random walk model as seen in its principal components, I suggest a test for stasis, along with a mixed model illustrated by a reanalysis of some data of Gunz et al., and a related test for directional trend. These comments are intended to apply to all high-dimensional morphospaces, not just those arising in geometric morphometrics. Applications of principal components in this context distort high-dimensional data in ways that have a tendency to mislead; but these distortions can be intercepted so that studies of tempo and mode can nevertheless proceed.

Before one can study evolutionary rates one must reject the null model of symmetric random walk. for which the requisite quantity does not exist. As random walks reliably simulate all the features we find so compelling in the fossil record—jumps, trends, and irregular cycles—rejection of this irritating hypothesis is much more difficult than one might hope. This paper reviews principal theorems from the mathematical literature of random walk and shows how they may be applied to empirical data by scaling net changes according to the square root of elapsed time. The notorious pair of “opposite” findings, equilibrium and anagenesis, may be construed as deviations from random walk in opposite directions. Malmgren's data on Globorotalia tumida, previously interpreted as an example of punctuated anagenesis, are consistent with a random walk showing neither punctuation nor anagenesis, but instead varying in speed over four subsequences.

Many organisms continue to grow their skeletons throughout ontogeny. In the shells of molluscs, protists, and brachiopods and in bovid horns, accretionary spiral growth provides a detailed and continuous growth history. Although a shell may be described as a single static form, the overall morphology is a summation of the ongoing accretionary process. For this reason, an explicitly ontogenetic characterization of form provides insight into the final form achieved. Analysis of landmark transformations offers direct access to major components of morphological variation, both among adult individuals and through an individual's ontogeny. Parameters of preconceived, abstract geometric models can also be used to characterize morphological variation, but there is no guarantee that these parameters will coincide with the major features of shape variation.

In order to locate landmarks at equivalent ontogenetic stages, features that indicate ontogenetic stage of coiled forms must be identified (e.g., growth increments, age, size, whorls). The gastropod Epitonium (Nitidiscala) tinctum exhibits prominent varices that provide landmark locations throughout ontogeny. Recent specimens of this species were obtained from three localities in Baja, Mexico. The morphological variation among individuals, treated as whole shells and within individual ontogenies, was analyzed using shape coordinates of landmark configurations. Deformation of shape is expressed in the uniform and nonuniform shape subspaces. The empirical components of shape variation found are similar to those generated by two parameters of an equiangular spiral: θ, the angle between consecutive varices, and W, the whorl expansion rate. The distribution of individuals is examined within morphospaces constructed from these shape features.

Three scales of analysis are necessary to characterize adequately the shape variation within and among specimens. The smallest scale is equivalent to increment-by-increment changes in θ and W. The middle scale comprises variation equivalent to whorls resulting from systematic changes in θ and W during an individual's ontogeny. Finally, there is the overall ontogenetic trajectory. Mean shape must be a function of initial shape and ontogenetic trajectory in shape. Mean forms that are found to have similar shapes at the same arbitrary growth increment may achieve that shape in different ways.

Cimpian & Salomon (C&S) appear to characterize the inherence heuristic and essentialism as unwise or childish aspects of human reasoning. But actually, these cognitive modes lie at the core of statistical analysis across all of the quantitative sciences, including the developmental cognitive psychology in which the argument here is couched. Their whole argument is as much an example of its topic as an analysis of it.