In his most recent talks, Media for Thinking the Unthinkable, Stop Drawing Dead Fish and Drawing Dynamic Visualizations, Bret Victor very eloquently makes the case for using computers as tools to explore systems in an entirely new way. His ideas struck a chord with me, not only because I have a passing interest in interaction design–after all, almost everything I touch is in some way shaped by it–but also because I have spent the last year and a half studying physics.
At its very core, this process involves two things: Getting a handle on the maths and developing an intuitive understanding of the principles involved. Those are not separate approaches, however. Understanding the maths behind something gives you a more robust intuition about it, and a better intuition gives you better ideas for the mathematical approaches you can take.
I firmly believe that be best way to grasp a new concept intuitively is the one humans have used for millennia: Play with it. But an equation (or an entire theorem) is not a rubber ball, you can’t just take it and bounce it around with your hand. It is a fundamentally dynamic, non-physical object.
The obvious approach to dealing with this problem is visualisation: Draw a few sketches, run a few calculations and hope for the best. That kind of sucks, because you’re stuck to static images.
Victor points that there is a better tool available: A computer, the ultimate dynamic drawing machine. The only thing missing is the right software (don’t get me started on Mathematica), and I think that Bret Victor’s prototypes are an incredibly exciting first step in the right direction. Those tools encourage playing, and they not only might enable us to think the yet unthinkable, they also make it easier to understand what others have thought before us.