Research roadmaps in computational systems biology, autonomic computing and communications target the enrichment of knowledge and technology transfer between (analytic) life sciences and (synthetic) engineering sciences. However, we claim that it is impossible to make significant progress in this transdisciplinary field without a breakthrough paradigm change towards biologically driven mathematics and computation. Turing Machines used in biology and elsewhere in science today are Newtonian in a broad sense because they deal exclusively with syntax and inference rules based on discrete logic in absolute space and time to deliver predictable behaviour. Despite this approach being extraordinarily useful in engineering human processes, the interactions within the real world has proven to be vague and relational in many ways. A profoundly new understanding of the role of biology in natural and engineering sciences needs to be set out. Our driving argument is that living systems have fundamentally different notions of self-organization from those in engineering sciences. We therefore propose a research programme to investigate the imperatives of computation in a cardinal new way by comprehending the fundamental principles of emergence, development and evolution in biology. The goal will be a set of novel mathematical formalisms capable of addressing the multiple facets of an integral model and a general theory of biocomputation within an adequate frame of relevance. Its base will be a long-term fundamental research programme in mathematics, systems biology and computation that we call Integral Biomathics. The proposed support action has the goal of identifying, consolidating and organizing transdisciplinary research in Europe around this focus with the objectives: i) to devise a set of challenges for future FET research at proactive initiative level; ii) to produce a position paper motivating the need for action and reflecting the impact on science, technology & society.