The 'imidazole alphastat hypothesis' states that intracellular and extracellular pH, partly via buffering by imidazole groups, change with temperature in a way that keeps imidazole and protein ionization constant, thus maintaining cell function and minimizing shifts of base equivalents and total CO(2), while adjustment of P(CO(2)) involves imidazole-based receptors. 'The hypothesis', which is actually several hypotheses, has been variously perceived and judged, but its underlying conceptual framework remains largely valid, and is reformulated using differential equations requiring less information input than their integral equivalents. Their usefulness is illustrated with published data on temperature responses in fish cells and whole tetrapods. Mathematical modelling allows general principles to be explored with less immediate concern for uncertainties in experimental data and other information. In tetrapods, it suggests that warming is followed by a loss of base equivalents from the body, and that this loss is due to metabolic adjustments that are not part of pH homeostasis. Uncertainties include intracellular buffer values, local variations in P(CO(2)) within the body, the possible role of buffering by bone mineral, and the temperature dependence of pK values for CO(2)/HCO(3)(-) and imidazole groups. The equations utilize a single, notional, temperature-dependent pK value for all non-bicarbonate buffers in a given body compartment. This approximates to the 'passive component' of pH adjustment to temperature change as measured by the homogenate technique. Also discussed are the diversity of cell responses within individual animals, relevant aspects of the control of ventilation, metabolism and transmembrane transport, and the basis of optimum pH-temperature relationships.