First We Build The Tools - Then They Build Us

said Marshall McLuhan. I wrote about this here .

This came into my mind, when I read Society's Changing Needs for Math Debate in Wolfram Blog.

Deeply involved in quant finance, I know Paul Wilmott personally. He is one of the most influencing quants - he and Emanuel Derman released the Financial Modelers Manifesto.

It contains the modelers hippocratic oath:
I remember that I did not make the world and it does not satisfy my equations ...  I will not give the people who will use my models false comfort about is accuracy .....

It is consequent that Paul claims white box learning before black box using or even black box learning.
And I agree and disagree.

In McLuhan's words? - learning arrangements affect the society in which they play a role not by the content delivered but by the learning arrangement itself.

White box learning requires active participation of the learner, abstract patterning and simultaneous comprehension of all parts. Cool.
At the other hand it has the danger of oversimplification. You want closed form solutions and proofs that the formulae are true in infinite many cases but - consequently - they do approximate only small domains.

With black box learning at the other hand you can rely on precision, quantitative analysis and faster time to insight. Hot.
But as Paul says you might miss one link in the innovative spiral that leads you to a complete different even more insightful approach. In the petabyte and petaflop age you can manipulate massive data and transform them into valuable results.

Black box methods become even more important if you surf the waves of some reality and invert them into models. You predict a little into the future to keep you at the surface. See The Black Swan Of Metal treatment. This confronts you with the ill posed inverse problems.
But even here an intimate relationship with the math will help to deeply understand the risky horror of inverse problems.

Here to innovate? If yes, I see a few barriers: tradition, lectures, when not scientific method - engage both sides of the brain instead.
So, it is white AND black box learning.

Simply merge math learning with learning to program?