To refine search, see subtopics Paradox and Myth and Ritual. For more material on this topic, see subtopic The Philosophy of Mathematics. Laterally related topics: Religion, Time and Space, Mathematics in Recreation, Art, Language and Literature, Music, Measurement, Arithmetic, Mathematics and Mysticism, Geometry, Discrete Mathematics, Optimization, Calculus, Statistics, Social Science, Logic, Computation, Probability, Applied Mathematics (General), Education, Algebra, Number Theory, Optics, Archaeology, Medicine, Creativity, Business, Fractals, and Science.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Hansen, David W. The Dependence of Mathematics on Reality. Mathematics Teacher 64 (1971), 715--19.
Discusses how the greatest mathematicians have been vitally concerned with the real world. Uses Archimedes, Newton, and Gauss as examples. Archimedes did so much applied work that it is hard to see how he fits Plutarch's description of considering mechanical work ignoble and inferior. The case of Newton is of course well known. An interesting example is Gauss, who used the motto "Thou, nature art my goddess;to thy laws/My services are bound" from Shakespeare's King Lear. Newton and Gauss were also very interested in religion. Philosophy was very important to Gauss. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Applied Mathematics (General), Archimedes, Isaac Newton (1642-1727), Karl Friedrich Gauss (1777-1855), and Religion. Also possibly relevant: Literature.
Henle, Jim. Classical mathematics. Baroque mathematics. Romantic mathematics? Mathematics jazz! Also atonal, New Age, minimalist, and punk mathematics. Amer. Math. Monthly 103 (1996), no. 1, 18--29. SC: 01A99 (00A30 00A69), MR: 1 369 148.
Music is often broken into Renaissance, Baroque, Classical, and Romantic periods. This classification is not used so consistently in art and literature, and is rarely applied to mathematics, but the author finds reasonable ways to define these eras for the other disciplines as well. He finds that the periods correspond closely in art and literature, and that they correspond closely in music and mathematics, but that the periods in the latter lag significantly behind the periods in the former. This may suggest some linking between the two fields, the exact nature of which still remains to be determined. The author makes a few good-natured guesses about relationships between mathematics and other types of music as well. Atonalism is associated with formalism, jazz with topology, and, in essence, new age with dynamical systems. A very enjoyable article, and could be a good reading assignment for students in either a History of Mathematics or a Philosophy of Mathematics course. Closely related topics: Music, Art History, and Literature.
Kilmister, C. W. Zeno, Aristotle, Weyl and Shuard: two-and-a-half millenia of worries over number. Math. Gaz. 64 (1980), no. 429, 149--158. (Reviewer: K. E. Hirst.) SC: 01A99 (00A05 03A05), MR: 82i:01075.
Ever since Zeno's paradoxes, mathematicians, philosophers, and logicians have been discussing the nature of the infinite. The author starts by discussing one of Zeno's four paradoxes, the Dichotomy. This leads to a discussion of Aristotle's views of the infinite. Needless to say, philosophical problems remained, and Hermann Weyl made one attempt to rectify them. Weyl advised caution in dealing with impredicative definitions, which he believed could lead to a vicious circle. Unfortunately, as Weyl notes "This vicious circle, which has crept into analysis through the foggy nature of the usual set and function concepts, is not a minor, easily avoided form of error in analysis." And in fact, if impredicative definitions are abandoned entirely, we must also abandon the notion that a bounded infinite set has a least upper bound and of course the related theorem (Bolzano-Wierestrass) that a bounded infinite set has a limit point. As the author notes, "On 9 February 1918, Polya and Weyl made a bet in Zürich, with twelve witnesses (all mathematicians). About [the least upper bound property], Weyl prophesied 'A. Within twenty years, Polya, or a majority of leading mathematicians, will admit that the concepts of number, set and countability involved are completely vague; and that there is no more point in asking about the truth of [the least upper bound property] than of the main assertions of Hegel's physics. B. It will be recognized by Polya, or a majority of leading mathematicians, that in any wording [the least upper bound property] is false...'" When the bet was called, everyone agreed that Polya had won with the single exception of Kurt Gödel. The author notes "if the construction of the real numbers contains subtleties that troubled such an acute intellect as Weyl's as recently as 1917, and still worried Gödel in 1940, it is not to be wondered at that some of our first-year undergraduates find it hard to stomach. Perhaps they are wiser than we are." Closely related topics: Zeno, Aristotle, Hermann Weyl, Infinity, and Paradox.
Kirmser, Philip G. and Hu, Kuo-Kuang. The shape of the ideal column reconsidered. With a reply by Steven Cox. Math. Intelligencer 15 (1993), no. 3, 62--68. (Reviewer: Peeter Müürsepp.) SC: 73K05 (00A69 01A99 49N55 73H05), MR: 94e:73039.
This article criticizes some of the conclusions of Cox, Steven J., The shape of the ideal column, and contains a new derivation of the shape of the "ideal" column. In Cox's view the problem of the ideal column remains far from solved. Cox acknowledges some of the criticism, but in turn objects to the way Kirmser and Hu have had tacitly assumed the existence of a strongest column in order, which he considers far from clear. He says "Faced with their outright contempt for the question of existence of a strongest column, I find solace in L. C. Young's invocation of Perron's paradox." (This paradox starts "Let N be the largest positive integer", and then shows that there exists a larger number.) The mathematics involved is somewhat technical. Closely related topics: The Column, Statics, and Paradox.
Mainzer, Klaus. Symmetry and beauty in arts and mathematical sciences. Physis Riv. Internaz. Storia Sci. (N.S.) 32 (1995), no. 1, 91--103. SC: 01A99 (00A69), MR: 96h:01043.
As this article explains, symmetry appears in a variety of disciplines over a variety of ages. The author begins by briefly discussing the natural and philosophical reasons for studying symmetry (starting in ancient Greek times). He then discusses the appearance of the 7 frieze groups and 17 ornamental groups of the plane and related groups in mathematics and crystallography. Next, he discusses appearances of symmetry and symmetry breaking in modern physics, in the theory of relativity, and in quantum mechanics and superstring theory. He finds that symmetry considerations are important in chemistry and biology as well: "In biochemistry macromolecules (for example L-amino acids or D-sugars) possess a characteristic homochirality ('dissymetry') which is assumed to be caused by parity violations of weak atomic forces." He also explains that "The emergence of pattern structure can be described by symmetry breaking not only in chemistry, but in biology. Since the pioneering work of the famous English logician and mathematician A. Turing on the chemical basis of morphogenesis in biology (1952), there has been an increasing interest in this topic." He then proceeds to discuss "Symmetry and Symmetry Breaking in the Computer World", focusing on dynamical systems. For example, he write, "Nevertheless the Feigenbaum diagram is self-similar. Every part of the tree contains the Feigenbaum diagram infinitely often like Russian dolls. It follows that mathematical chaos can be highly symmetric." He closes with a discussion of modern architecture, where he finds that symmetry concerns are important as well: "But the variety of historical reminiscences and asymmetrical elements in architecture does not mean a movement back to historicism or eclecticism. It is the expression of a sceptic and ironic view of the world which no longer believes in an omnipotent technical rationality and its claim to solve all human problems. It underlines individuality and the importance of accidental details, and has doubts about universal harmony and rationality. So it prefers symmetry breaking as a chance of variety, pluralism, and individual freedom." And this is a theme that nicely rounds of his article: "But variety and pluralism need not be in conflict with unity. It was Leibniz who suggested that the unity of the world can only be experienced by man under special aspects. So his motto was 'unity in variety.' It dates back to the old philosophical idea of Heraclitus that even symmetry breaking is related to a sometimes hidden symmetry." Interesting and thought-provoking article. Closely related topics: Symmetry, Greece, Physics, Chemistry, Biology, Alan Turing, Computation, Fractals, and Architecture.
Rav, Yehuda. On the interplay between logic and philosophy: a historical perspective. Theoria (San Sebastián) (2) 8 (1993), no. 19, 1--21. (Reviewer: Pierre Kerszberg.) SC: 03A05 (01A99 03-03), MR: 95c:03014.
The author discusses some of the connections between philosophy, logic, mathematics, and language. He focuses mainly on the West but also touches slightly on China. The reader should probably have a relatively strong background in philosophy before attempting this article. There is a long bibliography that should be useful for students making further investigations in these areas. Closely related topics: Logic, Language and Linguistics, and China.
Seidenberg, A. The ritual origin of counting. Arch. Hist. Exact Sci. 2 (1962b), 1-40.
It is common to argue that counting and other elementary mathematics arose spontaneously throughout the world in response to a practical, or perhaps psychological, need. Abraham Seidenberg argues instead for a diffusion theory, that counting arose only once, and then spread throughout the world. In fact, many common associations with numbers suggest such a common origin. One such association that Seidenberg is the idea that odd numbers are male and even numbers are female; this is certainly well known from the Pythagoreans, but turns out to be nearly universal. Seidenberg proposes that counting in fact originally arose in a ritual context. Seidenberg draws from a wide variety of anthropological sources for rituals and myths that hint at what this common origin might have been. He finds that counting "was frequently the central feature of a rite, and that participants in ritual were numbered." He focuses more specifically on creation rituals. He suggests that in the enaction of creation myths, men and women may have come onto the scene alternately, easily explaining the odd/male even/female association. He finds that his ideas clarify "pure 2-counting, which is the oldest stratum of counting we can detect." In pure-2 counting, there are separate words for one and two and these are used to form all other number words. He illustrates this with number words from diverse languages such as the Gumulgal of Australia, the Bakairi of South America, and the Bushmen of South Africa. He sheds additional light on his hypothesis with discussions of the possible origin of counting taboos (and connections with ritual sacrifice), of ancient one-one-correspondence "tally" systems (e.g., counting people with stones), of taxation systems, of money, and of gematria. Seidenberg also gives us some fascinating examples of counting in world religions. These include the analogy The Lord : His people = the shepherd : his sheep, the analogy The shepherd : his sheep = the moon : the stars. These two lead one to expect the moon to count the stars; and Seidenberg in fact finds evidence of this in ancient Babylonia. He argues from the equation The Lord's people = the stars of the heaven to The Lord's people = the sand upon the seashore that one would expect to find a ritual counting of sand. In fact, he finds the notion of Counter of the Sands both in Buddhism and among the Ancient Greeks. The equation The Lord = The Counter seems to be confirmed in two of the ninety-nine beautiful names of Allah, namely The Counter and the Reckoner; and there is further confirmation in Chapter's XV and XIX of the Qu'ran. This is a fascinating article, connecting mathematics with a wide variety of disciplines. Closely related topics: Myth and Ritual, Storytelling Traditions, Anthropology, General, Counting, TallySystems, Taxation, Number Words, The Pythagoreans, Gematria, Religion, The Islamic World, and Abraham Seidenberg.
Seidenberg, A. The ritual origin of the circle and square. Arch. Hist. Exact Sci. 25 (1981), no. 4, 269--327. (Reviewer: M. P. Closs.) SC: 01A10 (51-03), MR: 83h:01008.
Abraham Seidenberg advances a theory that the circle first arose in the context of the ritual enactment of a creation myth. In many cases, stars seem to play an important role in these myths. Seidenberg's research suggests that participants in these myths generally moved in a circle in imitation of the stars in the heavens. It is interesting that individuals in these societies often move in the same direction as the stars, and movement in the opposite direction is often considered unlucky. The fact that the Aztec god Tezcatlipoca is missing is right foot, forcing him to walk clockwise in a circle may be related. Seidenberg suggests that the creation myth is the origin for the dance around the may pole, which is for example observed near the summer solstice in northern Scandinavia today. Analogous rituals may play (or have played) a role in a wide variety of other cultures as well; examples are found in the Aztecs, ancient Indians, American Indians, and Greeks. (Spinning tops may have a ritual significance as well.) Special support is given to Seidenberg's these through the fact that in some cases, a pole may have been set up at an angle so as to point towards the pole star. Seidenberg notes that the moon might have motivated the circle rather than the stars, but the sun is unlikely to. His investigations tend to confirm this, and also suggest that lunar culture is older than solar culture. Seidenberg believes that the square arose from the circle, through the process of dividing a group into a dual organization, where for example members of one group marry someone in the other group and also (as he notes) play complementary roles in ritual. If a society divides a second time, one can think of it dividing the tribal circle into four parts. He finds some evidence of this as well. The four parts naturally define a square. His theory therefore implies that the circle arose first and that the square arose as a dual form of the circle; there is some other evidence (e.g., architectural) that may tend to confirm this. Seidenberg mentions several interesting dualities involving the circle and the square. The Altar of Heaven in Peking, for example, exhibits the equations Heaven : Earth = circle : square = three : two = South : North = White : Yellow. In Sinhalese art he finds the equation circle : square = standing : sitting. In the Omaha tribe he finds the equations that Sky : Earth = superior : inferior = one : two. He also notes the equations Heaven : Earth = Male : Female and Male : Female = one : two. The former is well known, and the latter is extensively discussed in Seidenberg, A., The ritual origin of counting The ancient Egyptians appear to be an exception as they associated the square with the earth and the circle with the sky. A fascinating paper. Closely related topics: Myth and Ritual, Religion, Anthropology, General, The Circle, Kinship Systems, The Square, and Abraham Seidenberg.
Seidenberg, A. and Casey, J. The ritual origin of the balance. Arch. Hist. Exact Sci. 23 (1980/81), no. 3, 179--226. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82j:01008.
The author's trace the beginnings of the balance back to a rituals where principals contended against each other on a kind of see-saw (somewhat similar sports are of course known from medieval times). The grain-crusher and water-lifter are similar, and perhaps derived from, the see-saw; the fact that one stands on these suggested to the authors that the contestants may have been standing on the see-saw. The authors note that in ancient Egypt, one's heart was believed to be weighed against a feather in order to decide whether one would be able to enter the afterlife. Other parts of the body, such as hair, can be used to represent an individual, and in other instances these may have been weighed instead; the authors give examples of rites where hair is weighed. An interesting use of the balance in Greece is from the Iliad where Zeus weighs Achilles and Hector on pans of a balance. "That of Hector sinks toward Hades and Hector falls, slain by Achilles." An even more interesting weighing ritual was once common in the far east, where a ruler was balanced against a quantity of a precious substance such as gold, and gave that substance (and thereby symbolically himself) to his people. The authors found many other interesting examples in a wide variety of cultures and world religions. The authors believe that only items of ritual significance were weighed at first, and that widespread commercial use came much later. Although the authors don't focus greatly on this, they also briefly discuss the different kinds of balances (and the balance-like instrument used to carry loads on the shoulders) and the weight multiples that were used on balances. Closely related topics: The Balance and the Measurement of Weight, Myth and Ritual, Religion, Sports, Ancient Egypt, Greece, The Islamic World, and Abraham Seidenberg.