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Mathematics and Cosmology
There are a surprisingly large number of ways that mathematics has supported our attempts to understand the universe..
If you live near a major urban area as I do, the night sky features the moon, a few bright stars and an occasional glimpse of a planet. But many years ago when I was in South Dakota, on a very clear night I could see virtually to the horizon in all directions. This was my most personal encounter with the cosmos. There are certainly a lot of stars out there, more than books or living near New York City had given me a clue to. Perhaps one of the specks of light I saw was this object, now made more visible by mankind's ingenuity.
Although it may appear that understanding the universe in which we live is far removed from mathematics, this is not true. There are a surprisingly large number of ways that mathematics has supported our attempts to understand the universe. First, there is the question of the nature of the matter which makes up the cosmos. Mathematics has played a big role in understanding the subatomic particles which form the matter of living things, planets and stars. Second, geometers have been working with physicists and astronomers to understand the geometry of the space we live in. Whereas mathematicians might view alternative geometries as "different games" which are fun to investigate because their properties are so varied, physicists are concerned with what is space and its nature.
The origins of the study of cosmology
The history of cosmology has two important threads: the history of the geometry of space as it evolved both in the mathematics and physics communities, and the history of astronomy, broadly interpreted.
Curiosity about the stars was one of the earlier manifestations of early science. A towering figure in this study was Claudius Ptolemy. Ptolemy's work had an enormous effect on the history of astronomy and cosmology. He developed a system that tried to explain the apparent motions of the planets under the assumption that the earth was the center of the universe. This required that the sun and planets' motions had to be explained in terms of the earth's being still, as it seemed to be, and that the sun and planets moved around the earth.
After the decline of Greek culture, the development of interest in astronomy was kept alive in the Islamic world, where mathematics was used in support of observational studies. Some of this concern with astronomy stemmed from an interest in charting the time for prayers, tracking important religious holidays using the lunar calendar, and knowing the direction to Mecca.
Theory and experiment
How has progress in our insights into the cosmos been made? Science proceeds by a delicate interplay of theory and experiment. Sometimes an experimentalist produces careful data over a long period of time which suggests the route for codifying a new theory based on the data. Other times, theory suggests experiments which make it possible to have much greater confidence in that theory. An example of the first type of progress is given in the interaction of the great Danish astronomer Tyco Brahe. It is somewhat ironic that Brahe publicly remained an adherent to his own variant of an earth-centered universe. His concern with the Copernican approach was being able to achieve sufficient verification. His empirical studies laid the foundation for a great leap forward in being able to give confirmation for a heliocentric view.
Towards a modern cosmology
Galileo (1564-1642) played a critical role in the development of a modern cosmology. Although many people think of Galileo as a physicist, in fact, he first earned income by teaching mathematics. Eventually (1589) he took a position at the University of Pisa, where he had studied earlier under the mathematician Filippo Fatoni. In 1592 he moved to Padua to teach mathematics. (His chief duty was to teach the works of Euclid.) The history of cosmology changed dramatically in 1609 when Galileo was informed in a letter from a friend about the invention of a device using a lens which made it possible to view distant objects more clearly. Galileo immediately began making telescopes, and towards the end of 1609 made some dramatic discoveries. He described these discoveries in his book The Starry Messenger, published in 1610. He stated that he had seen mountains on the Moon, stars in the Milky Way, and four objects moving about the planet Jupiter. Galileo's observations immediately showed that the idea that all heavenly bodies orbited the earth was incorrect. Galileo continued his work in observational astronomy, being among the first to make observations of Saturn, though the resolution of his telescopes did not enable him to realize that Saturn was girdled with rings.
(Top: Saturn's rings including some of Saturn's moons; Below: A close-up of Saturn's ring structure. Courtesy of NASA)
When Copernicus suggested that the over 1000-year tradition in explaining the nature of cosmology (the sun moves around the earth) was incorrect, he used a simplified model of circular motion for the earth around the sun. Even though the truth is that the earth's orbit is closer to an elliptical than a circular orbit, this orbit, as ellipses go, is very close to a circle. Yet Ptolemy's model had been so carefully honed over the centuries that it better predicted the motions of the heavenly bodies than Copernicus' new, and more correct, model. Kepler worked very hard on the computational side to demonstrate the accuracy of the heliocentric view of the solar system. He took Brahe's very accurate data and coupled it with an empirically based approach to finding the right curve to fit the data that Brahe had made. Eventually, he realized that an ellipse gave the closest fit for the Mars data, and then, having hit on the idea of elliptical orbits for the planets, verified that the planets other than Mars also move along ellipses. I am sure that Kepler would have been wowed by the following images from Mars. NASA's spacecraft arriving successfully at Mars represents a culmination of work on orbit calculations that Kepler set in motion.
(Images courtesy of NASA)
Yet he retains a small footnote in contributing to mathematics via the attachment of his name to Airy Functions.
It is difficult so say when a modern view of cosmology emerged but one of the landmark figures who made the transition possible was James Clerk Maxwell (1831-1879), without whose famous equations connecting electrical and magnetic phenomena, dramatic progress could not have been made.
Another physicist who captured the world's imagination with his use of mathematical reasoning in support of cosmology is Stephen Hawking (1942- ).
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Bartusiak, M., (ed.), Archives of the Universe: A Treasury of Astronomy's Historic Works of Discovery, Pantheon, New York 2004.
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Cohen, I. and R. Westfall, The Newtonian Revolution: With Illustrations of the Transformation of Scientific Ideas, Cambridge U. Press, Cambridge, 1980.
Daigneault, A. and A. Sangalli, Einstein's static universe: an idea whose time has come back?, Notices Amer. Math. Soc. 48 (2001) 9-16.
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Greene, B., The Fabric of the Cosmos: Space, Time, and the Texture of Reality, Norton, New York, 2004.
Greene, B., The Elegant Universe: Superstrings, Hidden Dimensions and the Quest for the Ultimate Theory, Norton, New York, 1999.
Guth, A., Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Physical Review D 23 (January 15, 1981) 347-56.
Hawking, S. and G. Ellis, The Large-Scale Structure of Space-Time, Cambridge U. Press, Cambridge, 1973.
Hawking, S., A Brief History of Time, Doubleday, New York, 1998.
Hawking , S. and R. Penrose, The Nature of Space and Time, Princeton U. Press, Princeton, 1996.
Kepler, J., Mysterium Cosmographicum, A. Duncan, translator, Abaris, New York, 1981.
Kepler, Epitome of Copernican Astronomy, C. Wallis, translator, Prometheus Books, Amherst, New York, 1995.
López-Corredoria, M., Observational Cosmology: caveats and open questions in the standard model (preprint: July 7, 2004)
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Peebles, P., Principles of Physical Cosmology, Princeton U. Press, Princeton, 1993.
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Smith, G., The newtonian style in Book II of the Principia, In Isaac Newton's Natural Philosophy, J. Buchwald and I. Cohen, (eds.), MIT Press, Cambridge, 2000, p. 249-313.
Tyson, N., Origins, Norton, New York, 2004
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Zebrowski, E., A History of the Circle, Rutgers U. Press, New Brunswick, 1999.
NOTE: Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal, which also provides bibliographic services.
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