Mathematica: A system for doing mathematics by computer?

Abstract

The title of my talk coincides with the title of Stephen Wolfram's book on his Mathematica system except that in the title of my talk there is a question mark. The content of my talk is my personal answer to this question.

We start from analyzing what it means to do mathematics and introduce what we call the “creativity spirale in mathematics”: “Doing mathematics”, in my view, is iterating the cycle “observation — proof — programming — application”.

Our main point is that Mathematica supports well three passes of this spirale, namely “observation — programming — application” and it helps a little in some simple forms of proof. However, Mathematica does not yet seem to be the right setting for proving in the broad sense of the word as understood by the mathematics community. We give some arguments for this and develop some ideas how a future system for doing mathematics might look like.