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Although mathematics has long been intertwined with the biological sciences, an explosive …

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Box 1
 Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better
Box 1. Challenges
Here are five biological challenges that could stimulate, and benefit from, major innovations in mathematics.

Understand cells, their diversity within and between organisms, and their interactions with the biotic and abiotic environments. The complex networks of gene interactions, proteins, and signaling between the cell and other cells and the abiotic environment is probably incomprehensible without some mathematical structure perhaps yet to be invented.

Understand the brain, behavior, and emotion. This, too, is a system problem. A practical test of the depth of our understanding is this simple question: Can we understand why people choose to have children or choose not to have children (assuming they are physiologically able to do so)?

Replace the tree of life with a network or tapestry to represent lateral transfers of heritable features such as genes, genomes, and prions (Delwiche and Palmer 1996; Delwiche 1999, 2000a, 2000b; Li and Lindquist 2000; Margulis and Sagan 2002; Liu et al. 2002; http://www.life.umd.edu/labs/Delwiche/pubs/endosymbiosis.gif).

Couple atmospheric, terrestrial, and aquatic biospheres with global physicochemical processes.

Monitor living systems to detect large deviations such as natural or induced epidemics or physiological or ecological pathologies.
Here are five mathematical challenges that would contribute to the progress of biology.

Understand computation. Find more effective ways to gain insight and prove theorems from numerical or symbolic computations and agentbased models. We recall Hamming: “The purpose of computing is insight, not numbers” (Hamming 1971, p. 31).

Find better ways to model multilevel systems, for example, cells within organs within people in human communities in physical, chemical, and biotic ecologies.

Understand probability, risk, and uncertainty. Despite three centuries of great progress, we are still at the very beginning of a true understanding. Can we understand uncertainty and risk better by integrating frequentist, Bayesian, subjective, fuzzy, and other theories of probability, or is an entirely new approach required?

Understand data mining, simultaneous inference, and statistical deidentification (Miller 1981). Are practical users of simultaneous statistical inference doomed to numerical simulations in each case, or can general theory be improved? What are the complementary limits of data mining and statistical deidentification in large linked databases with personal information?

Set standards for clarity, performance, publication and permanence of software and computational results.
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