Thursday, March 19, 2009


The difficult thing here is not to dig down to the ground; no, it is to recognize the ground that lies before us as the ground.

For the ground keeps giving us the illusory image of a greater depth, and when we seek to reach this, we keep on finding ourselves on the old level.

Our disease is one of wanting to explain.

Ludwig Wittgenstein, Remarks on the Foundations of Mathematics, section VI-31

So the logic of explicaiton calls for the principle of a regression ad infinitum: there is no reason for the redoubling of reasonings ever to stop. 
. . .
Explication is not necessary to remedy an incapacity to understand. On the contrary, that very incapacity provides the structuring fiction of the explicative conception of the world. It is the explicator who needs the incapable and not the other way around; it is he who constitutes the incapable as such. To explain something to someone is first of all to show him that he cannot understand it by himself. Before being the act of the pedagogue, explication is the myth of pedagogy, the parable of a world divided into knowing minds and ignorant ones, ripe minds, and immature ones, the capable and the incapable, the intelligent and the stupid.

Jaques Ranciere, The Ignorant Schoolmaster: Five Lessons in Intellectual Emancipation 

Mathematics educators generally spend a lot of their time explaining. The standard view holds that the best educators are the ones who are good at unpacking and breaking down difficult concepts. However, as the quotes above suggest, there are those who think that an emphasis on explaining is a problem both for mathematics and for education. For Wittgenstien, the perpetual search for explanations in mathematics leads towards a foundational morass and away from the reality of mathematical practice. Ranciere identifies the process of explanation with one of stultification that leads away from intellectual emancipation. This contrarian view suggests that mathematics itself, and the learning of mathematics, is experienced and revealed only in the act of doing mathematics.