The following are quotes are from 1. (A mathematical Platonist)
"As Gertrude Himmelfarb alerts us in her essay on “Academic Advocates” in Commentary:
The animating spirit of postmodernism is a radical skepticism and relativism that rejects any idea of truth, knowledge, reason, or objectivity. More important, it refuses even to aspire to such ideas, on the ground that they are not only unattainable but undesirable—that they are, by their very nature, authoritarian and oppressive”[167]
This traditional view is today being deconstructed by some mathematicians and by many mathematic educators.[169] The notion of mathematics as objective and eternal is today being replaced, among mathematics educators, by the postmodernist notion of “social constructivism.” According to “social constructivism,” knowledge is subjective, not objective; rather than being found by careful investigation of an actually existing external world, it is “constructed” (i.e., created) by each individual, according to his unique needs and social setting. Absolutism is deliberately replaced by cultural relativism, as if 2 + 2 = 5 were correct as long as one’s personal situation or perspective required it to be correct.
When much of the proposed pedagogy is driven by educators who are not themselves mathematicians, or who perhaps are seeking to politicize mathematics, then the inadequacy of the training for the next generation of mathematicians becomes suspect. I believe that realistic, but conceptually and numerically cumbersome, applications are better left to their specialty disciplines, as they hinder and even obscure the mathematical tools being developed in mathematics courses.
The increasing emphasis on inductive reasoning, with the concomitant de-emphasis of deductive reasoning, might not be the best way to prepare careful thinkers. Instead, I detect here the specter of the postmodernist rejection of rational thought. The “definition-theorem-proof” format that has survived scrutiny since Euclid, and stands as the model of mankind’s intellectual potential and achievement, is now under such an attack that, without resistance from its supporters, it might soon vanish entirely from the high school and calculus curricula. “Writing to learn” and classroom discourse can be effective pedagogically, but if carried to excess, they threaten to distract from precision of thought. To what extent do the “rule of three” (numerical, graphic, and symbolic approaches) and redefining mathematics as a laboratory discipline make pedagogical sense? To what extent do they inject socio-politics into our discipline?
Reformers would have us avoid problems having just one correct solution. Surely this is postmodernist relativism asserting itself again."
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Study pages 11 in this document for a very quick introduction to Postmodernism.
