Constructivism begins with an emphasis on the constructed world of the knower and the relationship of that world to reality. As von Glaswerfeld observes:

It is necessary to keep in mind the most fundamental trait of constructivist epistemology, that is, that the world which is constructed…makes no claim whatsoever about ‘truth’ in the sense of correspondence with an ontological reality…Among philosophers, Kant usually is recognized as the main person to introduce the notion that the mind does not passively reflect experience but rather actively creates meanings by attributing patterns and regularities that cannot be perceived directly. So, two centuries after Kant’s death, radical constructivists’ preoccupations with naive interpretations of reality seem quaint — especially in a field like mathematics where the notion that mathematics is about truth was abandoned hundreds of years ago — following the discovery of non-Euclidean geometries. In fact, since the discovery of Godel’s Theorem, the Axiom of Choice, and other related issues in mathematics, mathematicians have recognized that they can not even guarantee the internal consistency of any system that is more complex than the integers.

From *Beyond Constructivism, *Lesh and Doerr. I don’t think they’re right about the field of mathematics having discarded truth in light of non-Euclidean geometry. (Quine and Putnam would certainly disagree. Frege wrote after the discovery of non-Euclidean geometry, and was an unabashed realist. Godel too. And so on, many others.)

The critique of radical constructivism is sharp. It applies well to the writings of Constance Kamii:

Piaget’s theory provides the most convincing scientific explanation of how children acquire number concepts. It states, in essence, that logico-mathematical knowledge, including number and arithmetic, is

constructed (created) by each child from within, in interaction with the environment. In other words, logico-mathematical knowledge is not acquired directly from the environment byinternalization.

From *Young Children Reinvent Arithmetic*. This sort of talk either ignores philosophy or rests on a really ungenerous interpretation of Locke, Hume and other empiricists.

More from Lesh and Doerr:

As constructivists surely would agree, the key issue is not whether a theory is true or false, but rather whether it is useful. So, one implication of this policy is that, for researchers whose goals are to test and revise or refine theories, a philosophy that is accepted by nearly everyone is not useful. In other words, constructivism itself ceases to be useful to theory developers precisely when virtually every potentially competing theory claims to adopt this philosophy.

In sum, radical constructivism (and Lesh and Doerr) claim that realism in mathematics is dead. It aint. But to the extent that there is necessarily a gap between reality and our thought, that mismatch is agreed upon by basically everyone and has been since Kant pointed it out, so there’s no real need to shout it from the rooftops today. And since everyone basically does agree on that, radical constructivism is not a particularly useful framework for math educators or researchers.