Mathematics embedded in culture and nature Johan de Klerk School for Computer, Statistical and Mathematical Sciences North-West University (at Potchefstroom) Potchefstroom, 2520 South Africa e-mail: wskjhdk@puk.ac.za
As an introduction Basic questions Is Mathematics embedded in culture and nature? OR: Is it totally divorced from anything else in science and reality? What is meant by the words culture, nature and reality? What is meant by the word science? OR: What is this thing called science? (Chalmers, 1994) Can there be a dialogue between science and religion? OR: Can we talk? (Coyne, 2003) May one ask such questions in a Mathematics class? Is it possible to have / not to have any viewpoint concerning Mathematics?
Remarks In my opinion one may ask such questions in class; bring a certain perspective on Mathematics in one s classes; view mathematics (in a broad sense) and Mathematics (as a subject) as embedded in culture and nature. Two personal points of departure Mathematics as a subject could be viewed in a framework of contexts (for example, such as the context of the history of Mathematics, of mathematical and scientific theories, of society, of nature and of religion). [I will call this the science-in-context view.] Man s life is integral without being compartmentalised into religious and nonreligious parts. Hooykaas (1972): Religion and the rise of modern science. Clouser (1991): The myth of religious neutrality an essay on the hidden role of religious belief in theories.
Overview An elucidation of the science-in-context approach The practical class situation The history of mathematics: positive aspects Mathematics embedded in culture and nature Evaluation and conclusion.
An elucidation of the science-incontext approach The science-in-context approach It provides one with a useful framework for class discussions. It can be used in a continuous, well-planned manner for weekly of biweekly class discussions. It gives one a wider and broader view of science than what is usually regarded as the subject field. It gives the opportunity of discussing some nonmathematical, even philosophical, matters in a structured and planned way. Thus, in my classes I use the following contextual themes: The context of history The context of mathematical theories and relationships The context of science and society The context of nature The context of religion. In what follows I would like to emphasise the context of the history. However, before doing that I would like to give some thematic examples of the other contexts.
The context of mathematical theories and relationships Theorising in Mathematics: Induction and deduction in science. Example: The function f(x) = x 2 +x+41 gives, for x=1, 2,, 39, the prime numbers 43, 47,, 1601, respectively. But f(40)= 41 2, which is not a prime! The context of science and society The power of science and mathematics: The idealisation of science and mathematics. Example: The way in which nonnatural sciences were affected by Mathematics: During the 1630s the political philosopher Thomas Hobbes became a prominent leader in this regard: political economics became the first social science to be mathematised.
The context of nature Mathematical mindscape: To what degree do we discover Mathematics and to what degree do we invent Mathematics? Example: Is there perhaps in a platonic view a mathematical cosmos or mindscape with the mathematical objects waiting for mathematicians to be picked up (Rucker, 1982) (just as the rocks on the moon had been there, even before Neill Armstrong walked on the moon)? The context of religion Dialogue / debate between science and religion: Should there be any dialogue/debate or has the one nothing to say to the other? Example: Coyne asks (Sky & Telescope, 2003: 10): At a time when religious fundamentalism frequently makes headlines, and when astronomical discoveries are being made at a dizzying pace, respectful dialogue about the respective roles of science and religion in our lives takes on a new urgency.
The practical class situation: integrating history and Mathematics A good organisational planning and educational strategy is necessary For example: Didactic aims should be made clear: This can easily be done in a study guide, showing matters like mathematical content, learning outcomes, reading matter, exercises, etc. Contextual background: An introductory description of every context can be given at regular intervals this can be in the form of a short discussion (about 2-3 pages) of each context. My own directives: The discussions should be relatively short and easy, of an informal character, and lectured on a regular basis. An example from Mechanics: In this learning unit on Kinematics of a particle we start with the mathematical formulation of Mechanics. However, to fully understand the development of Mechanics as a subject field, it is necessary to pay attention to its history and development. Special emphasis will be placed on Newton s role in the formalisation of the subject as we know it today.
A good choice of historical material is necessary It is important to thoroughly plan the integration of historical and course material. Consider for example the following two possibilities: Time for only one discussion of a historical nature: One can discuss one important builder of the specific mathematical sub-area, e.g. Newton in Mechanics. Time for more regular discussions of a historical nature: Personal biographical information might be a good starting point. But time should also be spent on the problems, philosophical ideas and paradigms of the particular time period concerned, e.g. Kepler, Halley, etc, and their viewpoints in Celestial Mechanics.
Different pedagogical approaches might be useful in class discussions Different approaches: Biographic approach, anecdotal approach, philosophical approach, religious approach. Teaching of Mathematics according to its historical development: Hairer & Wanner (Analysis by its history): we attempt to restore the historical order Edwards (The historical development of the calculus): What is involved here is the difference between the mere discovery of an important fact, and the recognition that it is important. Teaching of Mechanics according to its historical development?: Mathematics: Diversions in order sine and cosine formulae can still be used for the purpose of multiplication and division (as has been done in the days before logarithms). Mechanics: Diversions not necessarily in order the idea of impetus (the viewpoint that something pushes a body in its flight through the air) has no real sense anymore.
The history of mathematics: positive aspects The value of integrating the history of a subject with the subject History can add to the student s interest: Stillwell: Biographical notes have been inserted... partly to add human interest but also to help trace the transmission of ideas from one mathematician to another. History can help to illuminate the subject: Edwards: For example, the gradual unfolding of the integral concept... cannot fail to promote a more mature appreciation of modern theories of integration. History can bring forward something of the cultural flavour of mathematics: Chabert: This book... is not intended as a text book, but to provide a cultural context, a sort of source book for the history of mathematics. History can motivate students and humanise the subject: Matthews: Such a study... can humanize the sciences and connect them to personal, ethical, cultural and political concerns. There is evidence that this makes programs more attractive to many students, and particularly girls, who currently reject them.
Mathematics embedded in culture and nature Background Culture and nature: Almost everything in reality can be classified as either culture or nature: Intricately woven weaver bird nest nature. Flimsy woven hessian for a dog culture. Natural sciences: Astrophysics nature. Chemistry, Botany, Zoology culture/nature. Applied Mathematics and Mathematics: Applied Mathematics nature/culture: o Study of the motion of planets nature. o Study of the motion of projectiles culture. Mathematics nature and/or culture? o In developing theories: first stage: led by problems from nature and/or culture. o Later stages: perhaps only culture. o Conclusion: Rooted in both nature and culture (also symbols, logic, language, etc) Mathematics rooted in nature and culture: This aspect can only come to its full right when studying the history of the subject and the broader contexts.
Example: From nature to a mathematical theory Problem: To find the position of an object on the surface of the earth (e.g., a ship at sea). First solution of problem (from culture): Latitude: Easy it can be done by measuring the height of the pole star. Longitude: Difficult it can be done by finding the time difference between the object s position and that of a fixed base point. The difference can be converted to degrees. Second solution of problem (from nature): Latitude: Easy as above. Longitude: Difficult but it can be done by knowing the moon s position as a function of time (like having a timekeeper that is visible almost every night/day). Lunar theory: This problem led to a lunar theory: Important names: Euler, Clairaut, D Alembert and Newton. Difficulties in developing theory: Perturbations (evection, annual equation, variation). Euler s theory especially outstanding and used by Mayer in 1755 to set up tables. Later: GW Hill and EW Brown. 20000 pound prize: 3000 to Mayer s widow and 300 to Euler in 1765.
Evaluation and conclusion Final remarks One s viewpoint can indeed play a role in one s scientific work. Kepler is a good example with his belief that God constructed the universe according to a geometrical scheme. Kozhamthadam (1994): The discovery of Kepler s laws the interaction of science, philosophy and religion. One can counter the viewpoint that a subject is a real and complete entity, divorced from reality, by studying the history of the subject. Hooykaas (1994): The teaching of science is more than technical training. If we restrict ourselves to the latter, the psychological effect will be that the scientific world picture is taken to be the real and full one, representing all that can be said with certainty about the universe and mankind. One can avoid educating intelligent specialists with no knowledge of the frameworks of thought and paradigms underlying their subject. Du Plessis (2000): A bad university therefore is a university where we train specialists without foundational knowledge, specialists who lack knowledge of the thought systems and paradigms of their subject fields.
Final conclusion If only a few of my students have, in some sense, been positively influenced by studying the contextual topics mentioned in this discussion (with special emphasis on the history of mathematics), then, in my view, such a study has served a good purpose and has resulted in a positive, value added, outcome.
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