Dev Genes Evol (2011) 221:255–279 DOI 10.1007/s00427-011-0378-0 REVIEW Mathematics and biology: a Kantian view on the history of pattern formation theory Siegfried Roth Received: 15 October 2011 / Accepted: 19 October 2011 / Published online: 17 November 2011 # The Author(s) 2011. This article is published with open access at Springerlink.com Abstract Driesch’s statement, made around 1900, that the physics and chemistry of his day were unable to explain self-regulation during embryogenesis was correct and could be extended until the year 1972. The emergence of theories of self-organisation required progress in several areas including chemistry, physics, computing and cybernetics. Two parallel lines of development can be distinguished which both culminated in the early 1970s. Firstly, physico- chemical theories of self-organisation arose from theoretical (Lotka 1910–1920) and experimental work (Bray 1920; Belousov 1951) on chemical oscillations. However, this research area gained broader acceptance only after thermo- dynamics was extended to systems far from equilibrium (1922–1967) and the mechanism of the prime example for a chemical oscillator, the Belousov–Zhabotinski reaction, was deciphered in the early 1970s. Secondly, biological theories of self-organisation were rooted in the intellectual environment of artificial intelligence and cybernetics. Turing wrote his The chemical basis of morphogenesis (1952) after working on the construction of one of the first electronic computers. Likewise, Gierer and Meinhardt’s theory of local activation and lateral inhibition (1972) was influenced by ideas from cybernetics. The Gierer–Mein- hardt theory provided an explanation for the first time of both spontaneous formation of spatial order and of self- regulation that proved to be extremely successful in elucidating a wide range of patterning processes. With the Communicated by R. Sommer S. Roth (*) Institute of Developmental Biology, University of Cologne, Biowissenschaftliches Zentrum, Zülpicher Straße 47b, 50674 Cologne, Germany e-mail: Siegfried.Roth@uni-koeln.de advent of developmental genetics in the 1980s, detailed molecular and functional data became available for com- plex developmental processes, allowing a new generation of data-driven theoretical approaches. Three examples of such approaches will be discussed. The successes and limitations of mathematical pattern formation theory throughout its history suggest a picture of the organism, which has structural similarity to views of the organic world held by the philosopher Immanuel Kant at the end of the eighteenth century. Keywords Chemical oscillations . Spatial patterning . Dissipative structure . Turing pattern . Local activation . Lateral inhibition From Kant to Driesch A hallmark of Immanuel Kant’s philosophy is its openness towards all developments of empirical sciences (Watkins 2001). In one of his earliest works, Kant presented a theory of the formation of the planetary system on the basis of Newton’s laws of gravitation (Universal natural history and the theory of heavens, 1755). He reflected extensively about foundational questions of physics (Metaphysical physical foundation of natural science, 1786). Throughout his life, he absorbed all new developments in chemistry, although he states that the chemistry of his days is still far away from becoming a science. According to Kant, ‘in every discipline of natural sciences there is present only so much real science, as there is mathematics’ (Kant 1900ff Vol 4, 470). Contemporary chemistry clearly was not living up to this ideal. Kant also showed eager interest in biology (Löw 1980; Ginsborg 2001). His essays on the concept of the biological species (1775, 1785) were received as