Ending the Karl Pearson Controversy (1904): Over the Incomplete Couplets Produced by Mendel’s Fraction-Addition Method

Abstract

Early in the 20th Century, leading mathematicians found a link between Mendel’s Laws and Newton’s Binomial. This enabled multigenerational studies of entire populations. In this regard, K. Pearson in 1904 raised objections to Mendel’s predictions that the ‘pure’ (dominant and recessive) descendants of hybrid ancestors turn out to be incomplete assemblies when using the sum of fractions used by Mendel in his 1866 article “Experiments in Plant Hybridization”. This algorithm is analyzed as a model for the case of just one hereditary characteristic, within an axiomatic framework that necessitates the formulation of a theorem in order to elucidate whether it was, on the one hand, a genuine mistake or, on the other, it is what Mendel, with all conviction and consideration, intended to say. We take into account the contemporary (1850-1870) knowledge of the cell and the structures involved in the transmission of inherited characteristics that this pioneer in the field of genetics would have had available for his deliberations at a time when this discipline was not yet a science. There follows the analysis of an unspecified intermediate member of the sum of fractions (not included in the Mendel’s original paper), which, from a mathematical standpoint, helps us resolve the incomplete assemblies (‘pure’ descendants) enigma.