Jejudo, South Korea
I am woefully inadequate to address these issues so I hope someone else with more experience in this field than I have will take it on.
My first thought was that it reads pretty much like the complaints against English departments and literary theory. My second thought was that the problems discussed are the same in any field that I've actually done anything in (literature, Asian Studies, EFL/ESL education).
A few stand out quotes for me:
Set theory as presented to young people simply doesn't make sense, and the resultant approach to real numbers is in fact a joke! You heard it correctly---and I will try to explain shortly. The point here is that these logically dubious topics are slipped into the curriculum in an off-hand way when students are already overworked and awed by all the other material before them. There is not the time to ruminate and discuss the uncertainties of generations gone by. With a slick enough presentation, the whole thing goes down just like any other of the subjects they are struggling to learn. From then on till their retirement years, mathematicians have a busy schedule ahead of them, ensuring that few get around to critically examining the subject matter of their student days.
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!
So, is it time to rethink how we teach mathematics and, if Wildberger is right, perhaps open the field to more and better mathematicians?