The committee is confident that deepening interactions between mathematics and biology will transform the biosciences. Of equal interest is the possibility that the areas of mathematics that interact most strongly with biology will themselves be altered by these interactions. Indeed, much of modern mathematics was shaped by four centuries of intimate interaction with the physical sciences and engineering. As the prominence of the biosciences increases—and as they interact more intensively with mathematics—a similar dynamic may be expected to occur. As discussed above, biological processes have different characteristics than the processes commonly encountered in engineering and physical science. In comparison with scientists involved in materials science, plasma physics, or cosmology, bioscientists work on muddier problems. The vast scales of time and space that characterize the world of biology are complemented by nonquantitative, organizational features that are so extraordinarily complex. The number of different interacting components is huge (ranging up to millions or even billions of entities), and they can all possess individual characteristics and contingent properties and be influenced by historical events. The systems are typically far from equilibrium or even stable steady states. High-order interactions between the components are the rule: The amount of feedback regulation in the simplest cell greatly exceeds that presently incorporated into devices designed by humans. (Indeed, it is this reliance on feedback regulation that accounts for the robustness of living systems.) Small events at one spatial or temporal scale often have large effects at another very different scale. These generalizations apply to cells, whose components are molecules, and also to ecosystems, whose components are commonly taken to be populations of individual members of many species. Calculus, the mathematical properties of continuous, very small elements, has been the essential language for describing the physical world and the language employed in the physical sciences, but biology has discrete elements, and the quantitative language of the computational and information sciences appears far more suited to be the language of biology. As a consequence of these many ways in which biology differs from the physical sciences, the committee looks forward to its many influences on mathematics, including some explicitly new mathematics.
An important goal in developing this report was to illustrate, in diverse contexts, these distinctive characteristics of biological systems. They may appear intimidating to nonbiologists at first, but on closer inspection, it is apparent that there has been great progress in dealing with them in the past and that this process is expanding as more mathematicians address biologically motivated problems. Historically, some new math-