From axiomatic systems to the Dogmatic gene and beyond

Abstract

The positivistic views that dominated the early debate on the foundations of mathematics, at the beginning of the 20th century, survived the “negative results” that have shown the limits of the axiomatic approach since the 1930s. Rigour, abstraction and symbolism have been confused with formalism, based on finite strings of signs, pre-given axioms, and potentially mechanisable rewriting rules. This contributed to major clarifications in the mathematical praxes but obscured the limits of formalisms due to the exclusion of the historical creation of sense proper to any science. We expand on this sometimes fruitful confusion with some case studies. We then hint to the historical creation of sense as a component of an epistemology of mathematics. We continue with an analogy with genocentric approaches in biology, as similar positivistic views resurfaced there fifty years later. Finite sequences of letters in the DNA would completely determine ontogenesis and phylogenesis, according to the Central Dogma of molecular biology. Limits and “negative evidence” have been disregarded while searching for the “gene for” everything. Alternative perspectives require a reconstruction of the sense of history as locus for the constitution of any object of biological knowledge. In particular, the historicity of biological evolution will be understood in terms of changing phase spaces and of the role of rare events in all phylogenetic trajectories. The analysis of the evolutionary production of variability, adaptivity and ecosystemic diversity is a key component of the project we hint to, as part of a renewed relation to the biological environment.