tisdag 8 september 2009

The Ideology of Mathematics

What is the ideology of mathematics underlying mathematics education as presented by professional mathematicians? Let us seek an answer in the following typical texts to be analyzed in the next blogpost:

  • Mathematics relies on both logic and creativity, and it is pursued both for a variety of practical purposes and for its intrinsic interest. 
  • For some people, and not only professional mathematicians, the essence of mathematics lies in its beauty and its intellectual challenge. 
  • For others, including many scientists and engineers, the chief value of mathematics is how it applies to their own work. 
  • Because mathematics plays such a central role in modern culture, some basic understanding of the nature of mathematics is requisite for scientific literacy. To achieve this, students need to perceive mathematics as part of the scientific endeavor, comprehend the nature of mathematical thinking, and become familiar with key mathematical ideas and skills.
The Committee on Logic Eduction offers the following comprehensive summary:
  • There are creative tensions in mathematics between beauty and utility, abstraction and application, between a search for unity and a desire to treat phenomena comprehensively.
  • Mathematics was originally linked with science and technology; however, it gradually became independent of science and technology, and present-day mathematicians think freely about virtually everything possible.
  • Therefore, mathematics is said to be a free creation of the human spirit. Special characteristics of mathematics are the clarity and precision of definitions, including usuage of words in ways that differ from their use in everyday language, and the certainty of mathematical truth based on deductive mathematical reasoning.
  • Given what Wigner call the "unreasonable effectiveness of mathematics", all students should learn the basic nature of mathematics and mathematical reasoning and its use in organizing and modeling natural phenomena. 
  • In the practice of mathematics, typically some concepts and statements are taken as given. They may be applied or serve as the foundation for the development of further mathematics. Additional concepts can be defined carefully in terms of the given ones. Conjectures can be developed on the basis of experience with examples. Further statements can be proved deductively based on what has been assumed. 
  • This process has been repeated extensively, resulting in mathematics having its own intricate structure, with concepts and areas of specialization that require considerable time and study to grasp. Moreover mathematics is interconnected in many interesting ways. 
  • It may be useful to think of students learning mathematics along the lines of a generalized structure of reasoning: (1) recognition, (2) analysis, (3) informal deduction, (4) formal deduction, (5)  axiomatics. 
  • In early grades, students learn the basic language including the critical logical words "and", "or", "not", "there is/are", "for some", "for every", "for all". They see multidigit numbers being built from single digit numbers. They match the trajectory of a kicked ball with the concept of line. They recognize patterns in sequences of numbers and shapes. In middle grades students develop habits of reasoning "locally", clarifying the assumptions of a particular problems and examining the steps involved in the solution to determine correctness. 
  • For example, one of us recently observed a fifth grade teacher asking her students for the definition of polygon. They knew, for example that triangles, squares and hexagons were polygons. It was exciting to see the students wrestling with abstraction, differentiating polygons from circles, and finally focusing on polygons as figures with sides. 
  • By the end of high school, students should be aware of the global deductive nature of axiomatic mathematics. They should be familiar with the connections between our number systems and algebra, between algebra and geometry. 
  • They should be comfortable reasoning with short sequences of statements, with Venn diagrams and other visual and diagrammatic methods. They should have experience with modeling, recognizing for example that certain natural phenomena obey linear relationships and that linear relationships make prediction so easy that we try to approximate other more complicated phenomena by linear ones. 
  • It is important both to understand how algebraic relationships can describe particular problems and to understand the power derived by working abstractly with the mathematics which applies to many different situations.
  • Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems.
  • As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth.
  • The special role of mathematics in education is a consequence of its universal applicability. The results of mathematics--theorems and theories--are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty.
  • In addition to theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols. Experience with mathematical modes of thought builds mathematical power--a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives. 
  • Mathematics empowers us to understand better the information-laden world in which we live. During the first half of the twentieth century, mathematical growth was stimulated primarily by the power of abstraction and deduction, climaxing more than two centuries of effort to extract full benefit from the mathematical principles of physical science formulated by Isaac Newton. 
  • Now, as the century closes, the historic alliances of mathematics with science are expanding rapidly; the highly developed legacy of classical mathematical theory is being put to broad and often stunning use in a vast mathematical landscape.
  • Several particular events triggered periods of explosive growth. The Second World War forced development of many new and powerful methods of applied mathematics. Postwar government investment in mathematics, fueled by Sputnik, accelerated growth in both education and research. Then the development of electronic computing moved mathematics toward an algorithmic perspective even as it provided mathematicians with a powerful tool for exploring patterns and testing conjectures.
  • At the end of the nineteenth century, the axiomatization of mathematics on a foundation of logic and sets made possible grand theories of algebra, analysis, and topology whose synthesis dominated mathematics research and teaching for the first two thirds of the twentieth century. 
  • These traditional areas have now been supplemented by major developments in other mathematical sciences--in number theory, logic, statistics, operations research, probability, computation, geometry, and combinatorics. In each of these subdisciplines, applications parallel theory. 
  • Even the most esoteric and abstract parts of mathematics--number theory and logic, for example--are now used routinely in applications (for example, in computer science and cryptography). Fifty years ago, the leading British mathematician G.H. Hardy could boast that number theory was the most pure and least useful part of mathematics. Today, Hardy's mathematics is studied as an essential prerequisite to many applications, including control of automated systems, data transmission from remote satellites, protection of financial records, and efficient algorithms for computation.
  • In 1960, at a time when theoretical physics was the central jewel in the crown of applied mathematics, Eugene Wigner wrote about the ``unreasonable effectiveness'' of mathematics in the natural sciences: ``The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.'' 
  • Theoretical physics has continued to adopt (and occasionally invent) increasingly abstract mathematical models as the foundation for current theories. For example, Lie groups and gauge theories--exotic expressions of symmetry--are fundamental tools in the physicist's search for a unified theory of force.During this same period, however, striking applications of mathematics have emerged across the entire landscape of natural, behavioral, and social sciences. 
  • All advances in design, control, and efficiency of modern airliners depend on sophisticated mathematical models that simulate performance before prototypes are built. From medical technology (CAT scanners) to economic planning (input/output models of economic behavior), from genetics (decoding of DNA) to geology (locating oil reserves), mathematics has made an indelible imprint on every part of modern science, even as science itself has stimulated the growth of many branches of mathematics.
  • Applications of one part of mathematics to another--of geometry to analysis, of probability to number theory--provide renewed evidence of the fundamental unity of mathematics. Despite frequent connections among problems in science and mathematics, the constant discovery of new alliances retains a surprising degree of unpredictability and serendipity. 
  • Whether planned or unplanned, the cross-fertilization between science and mathematics in problems, theories, and concepts has rarely been greater than it is now, in this last quarter of the twentieth century.

The mathematician A N Whitehead, who wrote the bible of the logicist school Principia Mathematica together with Bertrand Russell, explains to us in Mathematics in the History of Thought:
  • The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit. 
  • The originality of mathematics consists in the fact that in mathematical science connections between things are exhibited which, apart from the agency of human reason, are extremely unobvious. 
  • Suppose we project our imagination backwards through many thousands of years, and endeavour to realise the simple-mindedness of even the greatest intellects in those early societies. Abstract ideas which to us are immediately obvious must have been, for them, matters only of the most dim apprehension. For example take the question of number.
  • We think of the number 'five' as applying to appropriate groups of any entities whatsoever-to five fishes, five children, five apples, five days. Thus in considering the relations of the number 'five' to the number 'three,' we are thinking of two groups of things, one with five members and the other with three members. But we are entirely abstracting from any consideration of any particular entities, or even of any particular sorts of entities, which go to make up the membership of either of the two groups. We are merely thinking of those relationships between those two groups which are entirely independent of the individual essences of any of the members of either group. This is a very remarkable feat of abstraction; and it must have taken ages for the human race to rise to it. 
  • During a long period, groups of fishes will have been compared to each other in respect to their multiplicity, and groups of days to each other. But the first man who noticed the analogy between a group of seven fishes and a group of seven days made a notable advance in the history of thought. He was the first man who entertained a concept belonging to the science of pure mathematics. At that moment it must have been impossible for him to divine the complexity and subtlety of these abstract mathematical ideas which were waiting for discovery. Nor could he have guessed that these notions would exert a widespread fascination in each succeeding generation.
  • The tremendous future effect of mathematical knowledge on the lives of men, on their daily avocations, on their habitual thoughts, on the organization of society, must have been even more completely shrouded from the foresight of those early thinkers. Even now there is a very wavering grasp of the true position of mathematics as an element in the history of thought. 
  • When we think of mathematics, we have in our mind a science devoted to the exploration of number, quantity, geometry, and in modern times also including investigation into yet more abstract concepts of order, and into analogous types of purely logical relations. The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. 
  • All you assert is, that reason insists on the admission that, if any entities whatever have any relations which satisfy such-and-such purely abstract conditions, then they must have other relations which satisfy other purely abstract conditions.
  • Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about. So far is this view of mathematics from being obvious, that we can easily assure ourselves that it is not, even now, generally understood.

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