such as "Introduction", "Conclusion"..etc
Joel E. Cohen
Laboratory of Populations, Rockefeller and Columbia Universities, New York, New York, United States of America
PLoS Biol 2(12): e439. This is an open-access article distributed under the terms of the Creative Commons Attribution License.
The discovery of the microscope in the late 17th century caused a revolution in biology by revealing otherwise invisible and previously unsuspected worlds. Western cosmology from classical times through the end of the Renaissance envisioned a system with three types of spheres: the sphere of man, exemplified by his imperfectly round head; the sphere of the world, exemplified by the imperfectly spherical earth; and the eight perfect spheres of the universe, in which the seven (then known) planets moved and the outer stars were fixed (Nicolson 1960). The discovery of a microbial world too small to be seen by the naked eye challenged the completeness of this cosmology and unequivocally demonstrated the existence of living creatures unknown to the Scriptures of Old World religions.
Mathematics broadly interpreted is a more general microscope. It can reveal otherwise invisible worlds in all kinds of data, not only optical. For example, computed tomography can reveal a cross-section of a human head from the density of X-ray beams without ever opening the head, by using the Radon transform to infer the densities of materials at each location within the head (Hsieh 2003). Charles Darwin was right when he wrote that people with an understanding “of the great leading principles of mathematics… seem to have an extra sense” (F. Darwin 1905). Today's biologists increasingly recognize that appropriate mathematics can help interpret any kind of data. In this sense, mathematics is biology's next microscope, only better.
Conversely, mathematics will benefit increasingly from its involvement with biology, just as mathematics has already benefited and will continue to benefit from its historic involvement with physical problems. In classical times, physics, as first an applied then a basic science, stimulated enormous advances in mathematics. For example, geometry reveals by its very etymology (geometry) its origin in the needs to survey the lands and waters of Earth. Geometry was used to lay out fields in Egypt after the flooding of the Nile, to aid navigation, to aid city planning. The inventions of the calculus by Isaac Newton and Gottfried Leibniz in the later 17th century were stimulated by physical problems such as planetary orbits and optical calculations.
In the coming century, biology will stimulate the creation of entirely new realms of mathematics. In this sense, biology is mathematics' next physics, only better. Biology will stimulate fundamentally new mathematics because living nature is qualitatively more heterogeneous than non-living nature. For example, it is estimated that there are 2,000–5,000 species of rocks and minerals in the earth's crust, generated from the hundred or so naturally occurring elements (Shipman et al. 2003; chapter 21 estimates 2,000 minerals in Earth's crust). By contrast, there are probably between 3 million and 100 million biological species on Earth, generated from a small fraction of the naturally occurring elements. If species of rocks and minerals may validly be compared with species of living organisms, the living world has at least a thousand times the diversity of the non-living. This comparison omits the enormous evolutionary importance of individual variability within species. Coping with the hyper-diversity of life at every scale of spatial and temporal organization will require fundamental conceptual advances in mathematics.
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