Response to comments on “Genetic Programming and Emergence”

At the outset, I would like to thank all colleagues who took the time to read my opinion piece ‘‘Genetic Programming and Emergence’’, thought about it, and finally wrote their comments. I am very happy about these comments, as they provide clear evidence that my contribution has been well received by the community. I shall next take up these comments in alphabetic order of their authors, briefly summarize the gist of their argument and provide for each one what I hope is a useful response. Lee Altenberg [1] emphasizes that emergent phenomena in Genetic Programming are produced in a formal system. To bolster this argument, he uses the mathematical structures of a matrix and a vector as an even simpler example of an interacting structure, whose formal manipulation leads to emergent phenomena in the mathematical world examined by ‘‘Linear Algebra’’. As examples he mentions eigenvalues, spectral radii, etc. Thus, he argues, that an entire field of Mathematics occupies itself with emergent phenomena in a formal system, and he rightly is convinced that other areas of Mathematics do the same. Having thus fortified the notion of emergent phenomena and their respectability in science, he states that emergent phenomena in Genetic Programming could also be treated in formal or mathematical terms, and possibly lead to interesting discoveries. He concedes that such an undertaking might be more difficult, due to the complexity of interactions in Genetic Programming that give rise to emergence, but he points to one example from the last decade where something along these lines has been achieved [10]. In his own words: ‘‘But this unruly space is no less formal or no less observable than the simple matrix, and their mathematical properties are simply awaiting our discovery’’. Well spoken.