onsdag 9 september 2009

A Critical Analysis of the Ideology of Mathematics

The Ideology of Mathematics as presented in the documents of the previous blogpost can be summarized as:
  • Mathematics has a double character: It is both the most original, complex and beautiful free creation of the human spirit with its own internal standards, and a universal practical tool.  The miraculous double character is described by the Physics Nobel Laurate Eugene Wigner as:
  • 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. 
Mathematics education is based on this mystification, this true miracle:

It allows mathematics education from early grades and on to focus on the language and logic of mathematics learning the words "and", "or", "not", "there is/are", "for some", "for every", "for all", in the safe conviction that this will be most useful to all students in their practical lives. 

It allows mathematics education to focus on axiomatics in the safe conviction that human knowledge can be axiomatized.

It allows mathematics education to focus just about anything in the safe conviction that mathematics is universally applicable.

But there are no miracles, trivialities are trivialities, axiomatization of knowledge is impossible, and no mathematics is universally applicable. To build education on a mystery which is neither understood nor deserved, is not a good idea, because learning is about understanding and students do not profit from free gifts which they do not understand nor deserve and which they cannot use.

For a discussion of how mathematics education can and should be reformed see my blog posts
showing close similarities between mathematics and religion sharing the double character of uplifted divinity and universal practicality. In both cases the idea is to study the language of the Divinity to learn about the World. This is still practiced in Islamic schools, but no longer in Western schools.

The two characters of mathematics clashed in the great war between the logicist/formalist and constructivist schools in 1920-30s. The constructivists won on technical knockout but they were soon cleansed from mathematics departments filled with logicists/formalists still in control and forming the ideology of mathematics today, so well expressed by the Committe on Logical Education. For more war reports, see my knols on mathematics.

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