On Convergence IV

"Once more, an invariably-recurring lesson of geological history, at whatever point its study is taken up: the lesson of the almost infinite slowness of the modification of living forms. The lines of the pedigrees of living things break off almost before they begin to converge." (Thomas H Huxley, On the Formation of Coal, 1870)

"Analytic functions are those that can be represented by a power series, convergent within a certain region bounded by the so-called circle of convergence. Outside of this region the analytic function is not regarded as given a priori ; its continuation into wider regions remains a matter of special investigation and may give very different results, according to the particular case considered." (Felix Klein, "Sophus Lie", [lecture] 1893)

"Particular landforms or surface morphologies may be generated, in some cases, by several different processes, sets of environmental controls, or developmental histories. This convergence to similar forms despite variations in processes and controls is called equifinality." (Jonathan Phillips, "Simplexity and the Reinvention of Equifinality", Geographical Analysis Vol. 29 (1), 1997)

"The underlying reason for convergence seems to be that all organisms are under constant scrutiny of natural selection and are also subject to the constraints of the physical and chemical factors that severely limit the action of all inhabitants of the biosphere. Put simply, convergence shows that in a real world not all things are possible." (Simon C Morris, "The Crucible of Creation", 1998)

"Equifinality is the principle which states that morphology alone cannot be used to reconstruct the mode of origin of a landform on the grounds that identical landforms can be produced by a number of alternative processes, process assemblages or process histories. Different processes may lead to an apparent similarity in the forms produced. For example, sea-level change, tectonic uplift, climatic change, change in source of sediment or water or change in storage may all lead to river incision and a convergence of form." (Olav Slaymaker, "Equifinality", 2004)

"Convergence is, in my opinion, not only deeply fascinating but, curiously, it is as often overlooked. More importantly, it hints at the existence of a deeper structure to biology. It helps us to delineate a metaphorical map across which evolution must navigate. In this sense the Darwinian mechanisms and the organic substrate we call life are really a search engine to discover particular solutions, including intelligence and - risky thought - perhaps deeper realities?" (Simon C Morris,  "Aliens like us?", Astronomy and Geophysics Vol. 46 (4), 2005)

"Sometimes the most important fit statistic you can get is ‘convergence not met’ - it can tell you something is wrong with your model." (Oliver Schabenberger, "Applied Statistics in Agriculture Conference", 2006)