Washington, DC, USA. Research in biology and psychology used to require minimal mathematics. This is quite unlike the situation today: the life sciences and math take their historical exchange and collaboration to entirely new levels of sophistication. Today, mathematics is often a required component of effective interdisciplinary teamwork.

 

Mathematicians have been challenged by vital new disciplines, such as neuroscience, that explore fundamental questions in very different ways from past practice. This poses the need to describe the underlying mechanisms — and large, complex sets of data — with ever-greater precision and reproducibilty. The exchange and collaboration between mathematics and the life sciences (as in other areas of physical science) has led to the development of new mathematical, computational, and modeling tools and theories.

 

There is an extensive history behind today's interdisciplinary collaborations. Before slide rules, calculators and computers, people would resort to mathematical tables. They listed numbers, the standardized results of calculations with varied arguments to provide a broad range of results. Common versions included multiplication tables, a strong memory for people of a certain age. However, tables go back much further. Hipparchus (Gr. Ἵππαρχος; ca. 190 BC – ca. 120 BC) used tables of trigonometric functions to speed up his calculations.

 

Some recent events are worth a closer look to gain additional perspective on the past while exploring future opportunities. The milestones discussed below are exemplary of the progress so far, but by no means exhaustive.

 

Mathematical Handbook

 

Milton Abramowitz and Irene Stegun edited the first publication of a 1,000 page mathematical reference in 1964. It was called the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables and became the standard reference in the field. [C1] Abramowitz and Stegun were with the US National Bureau of Standards (now the US National Institute of Standards and Technology - NIST).

 

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Handbook Specimen Page

 

A page from the Handbook of Mathematical Functions by Abramowitz and Stegun shows a table of common logarithms (first edition, first printing, US GPO, 1964, p. 97).

 

Image courtesy of Wikipedia.

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Following initial publication, the Mathematical Handbook (also known as Abramowitz and Stegun) became an essential resource for practitioners. The notation used in the Handbook is a de facto standard for much of applied mathematics.

 

The Handbook's popularity speaks to its influence on science and engineering. NIST says it may be the most widely distributed and most cited NIST technical publication of all time. The US government has sold over 150,000 copies, with over 450,000 reprinted and sold by commercial publishers.

 

Even today, the Handbook is one of the most comprehensive sources of information on special functions, a particularly important reference area. This functions are considered special because they occur very frequently in mathematical modeling of physical phenomena, from atomic physics to optics and water waves. These functions have also found applications in many other areas; for example, cryptography and signal analysis.

 

Special functions have generally established names and notations due to their importance. The Handbook contains definitions, identities, approximations, plots, and tables of values of numerous functions used in virtually all fields of applied mathematics.

 

Nowadays, computer systems have replaced the printed function tables, but the Handbook remains an important reference source. NIST has a project underway to develop a successor volume in digital format (Cf. below).

 

The Interface Between Mathematics And Biology

 

A conference in 1992 explored the interface between mathematics and biology. The conference, Mathematics and biology: The interface. Challenges and Opportunities, explored what were perceived as the challenges and opportunities posed by enhanced collaboration. [C2]

 

Many of the insights and visionary proposals promoted at the conference have come to pass, while others wait for further technical developments. In any case, the original conference record is well worth review.

 

Mathematical-Biological Linkages

 

The US National Science Foundation (NSF) and the National Institutes of Health (NIH) jointly sponsored a workshop called Accelerating mathematical-biological linkages in 2003. [C3] It examined the interface that had been defined between mathematics and biology to that date, then prepared a road map to exploit the opportunities and meet the intellectual challenges involved. The outcome was a challenge to remove the barriers — whether cultural, educational, and institutional — to forming interdiscipinary partnerships.

 

The symposium featured a plenary address by Dr. Joel E. Cohen that resonates to this day. He explored the cutting edge of joint mathematical and biological work in three key areas:

 

The US National Institute of Standards and Technology (NIST) has been working on the much-anticipated online Digital Library of Mathematical Functions (DLMF). [C4] The DLMF is designed to be a modern successor to the 1964 Handbook of Mathematical Functions. [C1]

 

NIST has released a DLMF preview for comment. As with the earlier Handbook, the DLMF is designed to be the definitive reference work on the special functions of applied mathematics. The DLMF provides basic information needed to use special functions in practice, such as their precise definitions, alternate ways to represent them mathematically, illustrations of how the functions behave with extreme values and relationships between functions.

 

The DLMF provides references to or hints for the proofs of all mathematical statements. It also provides advice on methods for computing mathematical functions and pointers to available software.

 

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 Example 3D Visualization

 

This is a DLMF example from Chapter 5: Gamma Function showing the Complex Argument (5.3.5).

 

Both the height and color correspond to the absolute value of the function.

 

Image courtesy of NIST.

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The current DLMF preview is a fully functional beta-level release of five of the eventual 36 chapters. It provides various visual aids to provide qualitative information on the behavior of mathematical functions, including interactive Web-based tools for rotating and zooming in on three-dimensional representations. These 3-D visualizations can be explored with free browsers and plugins designed to work in virtual reality markup language (VRML). In the online version, users will be able to mouse over any mathematical function and the DLMF provides a description of what it is. If users click on it and the DLMF will go to an entire page on the function.

 

The complete DLMF, with 31 additional chapters providing information on mathematical functions from Airy to Zeta, is expected to be released in early 2009. With over 9,000 equations and more than 500 figures, it will have about twice the amount of technical material of the 1964 Handbook. An approximately 1,000-page print edition that covers all of the mathematical information available online also will be published.

 

The DLMF received initial seed money from the US National Science Foundation (NSF) and resulted from contributions of more than 50 subject-area experts worldwide. The NIST editors for the DLMF are Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert and Charles W. Clark.

 

[C1] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Milton Abramowitz and Irene A. Stegun (Eds. National Bureau of Standards Applied Mathematics Series. U.S. Government Printing Office, Washington, D.C. 1964.

Additional history on the composition and publication of the original Mathematical Handbook is availabe via a TS-Si.org download.  [ Download PDF ]

Corrections appeared in later printings up to the 10th Printing in December, 1972. Reproductions by other publishers have been available, in whole or part, since 1965 in common editions.

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Milton Abramowitz and Irene A. Stegun (Eds.).  New York: Dover; ninth Dover printing, tenth GPO printing, 1964. ISBN: 0-486-61272-4.

[C2] Mathematics and biology: The interface. Challenges and Opportunities. Levin S, editor. Lawrence Berkeley Laboratory Pub-701 Berkeley (California): University of California. 1992. Available at an archive site via the internet.

[C3] Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better. Joel E. Cohen. PLoS Biology 2(12) e439. doi: 10.1371 / journal.pbio.0020439  [ Download PDF ]

Excerpt

Although mathematics has long been intertwined with the biological sciences, an explosive synergy between biology and mathematics seems poised to enrich and extend both fields greatly in the coming decades (Levin 1992; Murray 1993; Jungck 1997; Hastings et al. 2003; Palmer et al. 2003; Hastings and Palmer 2003). Biology will increasingly stimulate the creation of qualitatively new realms of mathematics. Why? In biology, ensemble properties emerge at each level of organization from the interactions of heterogeneous biological units at that level and at lower and higher levels of organization (larger and smaller physical scales, faster and slower temporal scales). New mathematics will be required to cope with these ensemble properties and with the heterogeneity of the biological units that compose ensembles at each level.

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.

[C4] Digital Library of Mathematical Functions: Preview Release. US National Institute of Standards and Technology (NIST).

 

 

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