**VIII.5 Mathematics: An Experimental Science**

*Herbert S. Wilf*

**1 The Mathematician’s Telescope**

Albert Einstein once said, You can confirm a theory with experiment, but no path leads from experiment to theory.” But that was before computers. In mathematical research now, there’s a very clear path of that kind. It begins with wondering what a particular situation looks like in detail; it continues with some computer experiments to show the structure of that situation for a selection of small values of the parameters of the problem; and then comes the human part: the mathematician gazes at the computer output, attempting to see and to codify some patterns. If this seems fruitful, then the final step requires the mathematician to prove that the apparent ...