INSTRUMENTATION by Ralph H. Müller
The Role of Mathematics in Science "It is a pleasant surprise to the mathematician and an added problem if he finds that the arts can use his calculations, or that the senses can verify them, much as if a composer found that the sailors could heave better when singing his songs." —George Santayana
W "Mathematics—Queen
H E N ERIC TEMPLE BELL speaks of
and Servant of Science" we recognize the latter role the more easily: often the royal edict passes us unnoticed. For a long time scientists have heeded Lord Kelvin's injunction—"When you can express in numbers that, of which you speak, you have the beginning of a science. Until that time your knowledge is meager and unsatisfactory." The long history of Science has witnessed the gradual assimilation and ordering of countless facts and observations. From time to time reasonably accurate mathematical equations were applied to give some semblance of order, regularity, and predictability to the several phenomena. Our present day insistence upon mathematical rigor and the mathematical approach to science and technology is not only commendable, it is the mark of an enlightened age. One fears, however, that in many instances it has engendered in our students the notion that most of the laws of Nature are directly predictable by mathematical reasoning, or were handed down by Moses. We have long wished that someone would write a treatise putting the role of mathematics in science in its proper perspective. If the score were written down, we would note the countless routine examples of empirical formulas which lent some degree of respectability to hitherto poorly correlated data. The score would also contain some scintillating examples of mathematics in its Queenly role. We think of Planck's quantum theory, Einstein's equivalence of mass and energy, Bohr's model of the hydrogen atom, Fermi's prediction of the neutrino and Dirac postulating the
existence of the positron. Our overenthusiastic student would note that Einstein and Dirac were engaged in speculations which led to new physical discoveries, and while the mathematical skill and genius of the others were of equal caliber, the original incentives were to explain poorly understood phenomena. Planck wished a precise description of the energy distribution in a black-body radiator, Bohr's quantitative model gave a precise prediction of the Balmer series of lines in the hydrogen spectrum. Fermi sought an explanation for the energy distribution among beta particles and his theoretical treatment, yielding perfect agreement, required the existence of a particle with no charge and negligible mass. IMPLICATIONS OF TOPOLOGY
A recent discovery of considerable practical importance derives from the relatively queer subject of topology. "Topology is the geometry of distortion —it raises questions and propositions that are childishly obvious (until you try to prove them) or so difficult and abstract that not even a topologist can explain their intuitive meaning." (Newman). Much solemn and serious study has gone into linear figures, surfaces and solids, shapes like a pretzel, knots, networks, maps, etc. Is the hole "inside" or "outside" the doughnut? Can one construct a surface with just one side or make a bottle with no edges, no inside, and no outside? The recent practical discovery is a direct consequence of one of the commonest topological examples—a surface with only one side, the Mobius strip or loop. This is formed by tak-
ing a long rectangular strip of paper and pasting its two ends together after giving one a half-twist. As Courant, has said—"a bug crawling along this surface, keeping always to the middle of the strip, will return to its original position upside down." Anyone who contracts to paint one side of a Mobius strip could do just as well by dipping the whole strip into a bucket of paint. The Mobius strip has only one edge; if it is cut down the center, it remains in one piece. If the resultant surface is once more cut down the middle, two separate but intertwined strips are formed. Recently, physicist Richard L. Davis at Sandia Corporation in Albuquerque was seeking still newer and better methods of producing noninductive resistors. He constructed a Mobius loop using a metalized strip of plastic. When connections were made to opposite sides of the loop and pulses were applied, the current divided, flowed in both directions through the conducting foil, and passed itself twice. The observed inductance was extremely low. We like to think that this inspired experiment carries much greater impact than a possible improvement over bifilar windings and other well known techniques for minimizing self-inductance and distributed capacitance. We like to ask what are the implications of topology and its queer systems as applied to optics, magnetic phenomena, and circuit networks. Is any chemist prepared to predict reaction kinetics in a Klein bottle which has no edges, no inside, or no outside? The question may be childish, but like many questions in pure topology or analysis situs, the answers are tough.
VOL 36, NO. 12, NOVEMBER 1964
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