Happy Silver Anniversary

I started using Mathematica in my classes in 1989 because I believed then that this tool can help produce much better-trained scientists and engineers. I was never disappointed and it is simply amazing to see how much Mathematica has matured, from deep underlying mathematical routines to a beautiful and more and more useful front end. Michael Trott’s treatment of Riemann surfaces with Mathematica would have impressed Riemann! I have often wondered where Euler could have taken us with Mathematica (but Mathematica wouldn’t be possible without Euler). Almost every class that I have taught in the last 25 years has had some Mathematica—most had a lot!

During the last few years, Mathematica has been an important tool in my new-found research in human vision. It helped Mark Nawrot of North Dakota State University and me understand and settle an important open problem about human visual perception of depth from motion. The math that grew out of that suggested many new experiments and our report on “integration time” for the two brain signals contains the first .cdf to appear in a scientific journal. I have submitted an “Optic Flow” animation as my 25th anniversary present. Watch the “retina” translate in space and see the retinal image.

I love calculus and helping students learn the subject because I believe it truly is “The Language of Change” (at least continuous change. NKS is a whole other interesting thing.) I believe calculus is one of the greatest achievements of the human mind—right up there with Shakespeare, for example. It can help many people solve important problems in their lives—if they can master the technical difficulties—and that’s where Mathematica can help.

To my mind, there were three important advances in calculus in the 20th century: differential algebra and the Risch algorithm, Robinson’s theory of infinitesimals, and “computer algebra” systems, of which Mathematica is the most well-developed system. These are not unrelated. One reason to teach beginning calculus students with Mathematica is because the vexing problem of the distance traveled by a planet, solved by Louisville 150 years after Newton, is trivial with Mathematica (and “non-elementary” as the Risch algorithm would tell us.) Our students should have access to these powerful tools, because it can advance science. Computing in general and Mathematica in particular open many ways to “condense” very difficult previous scientific learning into tools usable by many people.

Infinitesimals and Mathematica have never had a marriage. Neither modern infinitesimals nor Mathematica have developed the interest they deserve amongst mathematicians. Goldbring used infinitesimals to solve Hilbert’s fifth problem for local groups in 2010 (the way Lie would have stated it!) on about the 50th anniversary of Robinson’s discovery. I’ve dreamed of developing an infinitesimal “data type” in Mathematica, but haven’t done it. Maybe I’ll have it ready for Mathematica‘s 50th.

But Mathematica is a much more important tool for educators to develop (than infinitesimals, which are quite intuitive). Students are way ahead of most teachers with Wolfram|Alpha, but they are casting about on their own and serious teachers should help them use Mathematica to advance their understanding. Each new generation of teachers should be sifting and winnowing current knowledge to advance science, not holding fast to Thomas’ 60-year-old dogma. I’m working on a new edition of my “Engineering Math 2: Multivariable Calculus” materials. I hope it will use the latest advances in Mathematica—if I can keep up! Each new advance in Mathematica brings us closer to the dream of advancing science for more people with powerful and accessible computing.

Happy 25th, Mathematica. The leadership of Stephen and Theo, evident at the start, has continued undiminished for 25 years. I hope I’m here to help celebrate the 50th. I’m sure it will be even more amazing.