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Response to Boojatta (Original post)

Sun Feb 5, 2012, 06:31 PM

5. The title of the OP is wildly unrelated to the content discussed in Wildberger's article....

The article that is excerpted from in the OP is a sophisticated consideration of the basis of mathematics and questions many things that form said basis - ZF set theory, etc.

To the question presented in the OP:

Of course, any group that does not understand high school mathematics could organize a reform movement. However, without further training, said group is likely not going to be able to understand the reform any better than they understand the high school mathematics that initially defined them as a group.

Take Wildberger's example of Schwartz's distribution theory versus the Dirac delta function for instance. An intuitive statement is replaced by a much more formal theory. Hence, the reformulation of mathematics that is requested would likely not actually improve the understandability of the mathematics that is taught in high school. It would shift the foundation of said mathematics to something else which may be more complicated and probably less intuitive.

Set Theory: Should You Believe?
N J Wildberger
Elementary mathematics needs to be understood in the right way, and the entire subject needs to be rebuilt so that it makes complete sense right from the beginning, without any use of dubious philosophical assumptions about infinite sets or procedures. Show me one fact about the real world (i.e. applied maths, physics, chemistry, biology, economics etc.) that truly requires mathematics involving `infinite sets'! Mathematics was always really about, and always will be about, finite collections, patterns and algorithms. All those theories, arguments and daydreams involving `infinite sets' need to be recast into a precise finite framework or relegated to philosophy. Sure it's more work, just as developing Schwartz's theory of distributions is more work than talking about the delta function as `a function with total integral one that is zero everywhere except at one point where it is infinite'. But Schwartz's clarification inevitably led to important new applications and insights.

If such an approach had been taken in the twentieth century, then (at the very least) two important consequences would have ensued. First of all, mathematicians would by now have arrived at a reasonable consensus of how to formulate elementary and high school mathematics in the right way. The benefits to mathematics education would have been profound. We would have strong positions and reasoned arguments from which to encourage educators to adopt certain approaches and avoid others, and students would have a much more sensible, uniform and digestible subject.

The second benefit would have been that our ties to computer science would be much stronger than they currently are. If we are ever going to get serious about understanding the continuum---and I strongly feel we should---then we must address the critical issue of how to specify and handle the computational procedures that determine points (i.e. decimal expansions). There is no avoiding the development of an appropriate theory of algorithms. How sad that mathematics lost the interesting and important subdiscipline of computer science largely because we preferred convenience to precision!

But let's not cry overlong about missed opportunities. Instead, let's get out of our dreamy feather beds, smell the coffee, and make complete sense of mathematics.


Here is some background information in a PDF on L. Schwartz and distribution theory:

from SIAM News, Volume 34, Number 9
The Joy and Suffering of Research

It’s been said (jocularly) that “Mathematicians and physicists used to live in productive sin with
the delta function and its derivatives. Schwartz provided them with a marriage certificate, which they take out of the closet
occasionally to consult and show around.”



All in all, the article is interesting and is worth reading, if one likes mathematical philosophy. The article seems to harken back to the situation that existed in the 19th century when Kronecker stated "God created the integers, all else is the work of man."

Leopold Kronecker
We have already indicated that Kronecker's primary contributions were in the theory of equations and higher algebra, with his major contributions in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers. However the topics he studied were restricted by the fact that he believed in the reduction of all mathematics to arguments involving only the integers and a finite number of steps. Kronecker is well known for his remark:-

God created the integers, all else is the work of man.

Kronecker believed that mathematics should deal only with finite numbers and with a finite number of operations. He was the first to doubt the significance of non-constructive existence proofs. It appears that, from the early 1870s, Kronecker was opposed to the use of irrational numbers, upper and lower limits, and the Bolzano-Weierstrass theorem, because of their non-constructive nature. Another consequence of his philosophy of mathematics was that to Kronecker transcendental numbers could not exist.


The level of mathematics that is being considered by the article in the OP is beyond what would be reasonably expected of high school teachers unless the teachers had at least a MS in mathematics - I suspect.

For further discussion of the topic, Professor Wildberger is on YouTube: http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=plcp

Hopefully, anyone considering this OP now has adequate information to begin to assess the questions in Prof. Wildberger's article. Enjoy.

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Boojatta Feb 2012 OP
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mzteris Feb 2012 #3
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