Joe Gallian's Interview with Doron Zeilberger February 20, 2007

[Appeared in pp.14-17 of the MAA Focus May/June 2007 issue ]

 JG: How do you describe your research area?

 DZ: I work mainly in combinatorics and the theory of 
     special functions, but in the last 10-15 years I consider myself mainly as an
     experimental mathematician, and combinatorics and the theory of special
     functions are just sources for examples and case-studies 
     of a methodology that its aim is to train the computer to discover conjectures and
     then try to prove them all by itself, without any human intervention.

 JG: Are there now a number of people who are doing experimental

 DZ: Experimental mathematics is a rapidly growing field, both explicitly and implicitly.
     Explicitly, there is a very good journal by that name, an Institute in Simon Fraser
     University, and at the recent annual meeting at New Orleans there was a special
     session dedicated to it. This is a good start, but still at its infancy.
     However, implicitly, more and more mathematicians, even pure ones, use the
     computer daily to formulate and test conjectures, in a mode that George
     Andrews calls "pencil with power-steering". However, more often than not,
     the computer's often crucial contribution is not mentioned, or grossly
     understated. Human beings are such ingrates.
     Also, Jon Borwein and David Bailey wrote a very nice monograph on Experimental Mathematics that 
     I highly recommend. But their emphasis is more on the traditional experimental mathematics that
     has been pursued by all the great, and less-great, mathematicians
     through the centuries, using pencil-and-paper.
     Of course, with computers you can do so much more, and you can be  very systematic, and the great 
     power of today's computers , guided wisely, can take you a very long way.
     However,  their emphasis is still on using computers  to find interesting
     conjectures and phenomena, but not to prove them.
     The  proof itself (when feasible) is still done largely by human beings, although sometimes with
     assistance of computers.

     As for computer-generated proofs, there is  a whole community of Automatic Theorem Proving, that
     is very successful. Their methodology is logic-based, and
     they try to teach the computer formal reasoning, using axioms
     and "laws of deduction". My approach is more akin to what
     the great algebraic geometer Shreeram Abhyankar calls "high-school algebra",
     and Richard Feynman calls "Babylonian mathematics", using
     algorithmic frameworks, that I call "ansatzes". But since
     the algorithms are symbolic rather than numeric, one can hope
     to prove genuinely new general theorems, valid for infinitely many
     cases.  In this kind of research, we try to teach the computer
     to firs make a conjecture, all by itself, and then automatically prove its own conjectures, 
     also all by itself!
     Of course, the only way we can do it, at present, is by focusing
     on  narrowly-defined areas.

JG:  Are you getting some resistance from the traditional math
     community with your approach?

 DZ: Definitely. Many people are not very comfortable with this
     approach. First, they claim that you can't trust computers
     (as though that humans are so trustworthy), and they also
     feel that it's not fun to have a computer do your work, or that it is
     "cheating". For me, teaching the computer how to discover and prove
     new theorems is even more fun than discovering and proving them myself, and as for 
     the cheating part, this is science not sport, and besides we can always
     change the rules of the game.

 JG: The Wilf-Zeilberger algorithmic proof theory is contained in all major
      computer algebra systems. What does it do?

 DZ: It's a collection of algorithms that can discover, and then prove,
     binomial coefficient identities, and identities involving sums or
     integrals of special functions.
     As you know, the binomial coefficient n choose k counts
     the number of k-element subsets of an n-element set.
     Since many objects in combinatorics boil down to
     unions and cartesian products of such choosings, it turns out
     that many enumeration problems can be expressed as such
     binomial coefficient sums, and often we get surprising
     identities. Before W-Z theory, every such identity required
     its own ad-hoc proof, and any such new identity, or even a new
     proof of an old identity, was publishable. But thanks to
     Wilf-Zeilberger theory, any such identity can now be proved
     automatically. No one today would submit a paper
     stating and proving the "theorem" that "ten plus five equals three times five".
     Analogously, an identity like


  first proved by Dixon in 1904, is now completely routinely
  and automatically provable thanks to the Wilf-Zeilberger
  algorithmic proof theory.

 JG: You seem  to have an evangelical-like faith that computers will
     make obsolete the way mathematics has
     been done for over 2000 years. Is that a fair assessment of your view?

 DZ: Yes, but people who know me well know that they should not take me too
     seriously (laughing). In the Talmud it says that if you have
     a talit (garment) and one guy says that it's all his and the
     other guy says that it's all his, then they each get half.
     But if somebody only claims a half and the other guy claims a whole, then
     the guy who claims a half will
     only get a quarter and the other guy gets three quarters.
     So if you want a half, you have to claim a whole. So you have to overstate
     your case. Then again, sometimes overstating can backfire and turn people off.

 JG: Tell me about your frequent co-author Shalosh B. Ekhad

 DZ: It is a charming individual. Of course it is made of silicon, and it is
     not really ONE body,  but it is definitely ONE soul (software). 
     The body has just been reincarnated many time. As we know, computers are very powerful, 
     but their life expectancy is much shorter than that of  humans, 
     since computers get better and faster so quickly, you have to get a new one every three years, 
     but you can always upload all the software from one Shalosh to the next, thereby guaranteeing
     the immortality of its soul.

 JG: Where did you get that particular name?

 DZ: The original Shalosh B. Ekhad was actually a Hebrew translation of the first PC that I owed,
     called AT&T 3B1.  At the time it was a very innovative machine,  the first UNIX PC, that was manufactured by 
     AT&T  in the 80s. The Hebrew translation of 3B1 is Shalosh B. Ekhad.

 JG: Tell me about your book A = B.

 DZ: It was written by Marko Petkovsek, Herb Wilf, and myself,   and is an elementary introduction to so-called 
     Wilf-Zeilberger algorithmic proof theory mentioned above . It also has a long chapter on
     the very important Petkovsek algorithm for deciding whether a sequence is closed-form.

 JG: It was quite a coup to get Donald Knuth to write the preface. How did
     that come about?

 DZ:  First, and foremost, Knuth is a good friend of Herb Wilf. Also
      Knuth loves  binomial coefficient identities. He dedicated
      quite a few pages to them in his classic "Art of Computer
      Programming", which is the "bible" of computer science,
      and  later, in much more detail, in his beautiful book with  
      Ron Graham and Oren Patashnik, "Concrete Mathematics: A Foundation for Computer Science".
      He still loves them. In the latest  issue of American Mathematical
      Monthly (Feb. 2007) there is a problem proposed by him
      that could be easily done using the WZ method. 
      I was a little taken aback. If somebody submitted
      a problem to prove that 11*13= 12^2-1,
      they wouldn't accept it, since there are nowadays (and have been
      for the last 5000 years) algorithms that can routinely prove this.
      Similarly, for Knuth's Monthly problem there are (and have been for
      the last 15 years)  algorithms that can routinely prove Knuth's problem.
      But then again, Knuth is Knuth, if he proposed as a problem
      to prove that 1+1=3-1, it might be accepted. 
      Of course, Knuth is very much aware of WZ, and he
      probably had some hidden agenda in proposing this problem.
      Maybe he meant some cute combinatorial argument.

     JG: I notice that the preface of your book "A = B" begins with Knuth's famous
         quote  "Science is what we understand well enough to explain to 
         a computer. Art is everything else we do."
         Was A = B the first place that quote appeared in print?

     DZ: Yes, I think so.

    JG: I was very surprised that a commercial publisher would agree to
        permitting the book to be downloaded free over the internet.

     DZ:  Herb Wilf is a great advocate of free publishing and A. K. Peters 
          is not a typical publisher. Klaus Peters, and his wife Alice,
          are very good people, who care more about mathematics than making
          a quick buck. Commercially it seems to have been a wash,
          some people who would have bought it, didn't, and vice versa.
     JG: I notice that you prefer Maple over other software. Why is that?

     DZ:  Originally Maple was much cheaper and more accessible  than its
          competitors. Mathematica was  much more commercial.
          Unfortunately, now Maple is very commercial too, and
          Mathematica's prices have gone down, and many people
          prefer the latter for its elegant syntax and beautiful
          graphics. So I use Maple for "old-time's sake" and 
          mainly because I am used to it.

     JG: Describe your Experimental Mathematics course.

     DZ:  I really enjoy teaching it. It is a graduate class (with
          one or two advanced undergraduate students) that is held
          in a "smart" classroom where everyone is connected to  a terminal.
          It is a hands-on course where the main point is to teach students
          how to program in Maple, in order to explore new mathematics.
          The actual topic changes every year, so that I won't get bored,
          and sometimes I even learn a new subject myself, since the best way of 
          learning a new subject is by teaching it. Of course, in this class we
          also learn how to program everything, and an even better way to learn a new
          subject is by teaching it to a computer, i.e. programming. So both myself and
          my students learn the substance of the course very well, and at the same time
          my students learn how to program in Maple.

     JG: You have an Erdos number (2), an Einstein number (3), 
         a Wiles number (3), and a Knuth number (2) but you
         say on your website that you are most proud of your Garfield 
         number (2). Why is that?

  DZ:  Richard Garfield is one of the most talented and creative
       combinatorialists alive, but he used his talents in a non-standard way. He is also a very nice guy,
       and I am fortunate to have known him, although only briefly.
       During the mid nineties, he was a teenage idol
       thanks to his innovative and lucrative "Magic: the Gathering"
       card game. He designed the game while he was Herb Wilf's
       PhD student at Penn in the late eighties and early nineties.
       He did good research, but not as much as he could have, because
       he was so busy developing and testing his card game. After his PhD,
       he got a one-year job in some small college in Washington State, with a salary of about
       22K. When the year was almost up, and it was not clear what
       his prospect for the future in academia would be, his card game
       caught on, and the rest is history.

 JG: There must be a story behind the "Who you gonna call" t-shirt.

 DZ: This is a T-shirt that I am wearing in my picture on my website.
     It features a certain binomial-coefficient identity, with the
     caption "who you gonna call". The back of that T-shirt
     has the few-lines of Macsyma code needed to prove it, and the caption
     "the binomial-coefficient-identity-busters".
     My kids told me that it is an allusion to "ghost busters".
     This cute design was made my Herb Wilf's son, David, who is
     a lawyer by profession, and Herb game me one of them.

 JG: Something that I found surprising is that the narratives for your
     grant proposals are posted at your website.
     That certainly helps people learn about your work.

 DZ: I think that it's a waste to write something just for 5 or 6 people
     and for no one else to be able to see it.
     I think that everything should be public; I don't like secrecy.
     I also hate the tradition  of anonymous refereeing. I think that there should
     be open refereeing. Also people should post their PhD theses, especially
     the introduction, that often gives a very good overview of a field.

  JG: Another thing that surprises me is that you slip a bit of humor in
      your grant proposals. One is ``There is a
      delicate balance and trade-off between the general and the specific, the
      abstract and the concrete, the strategical and
      the tactical, the sacred and the profane."  Another is the line ``My
      particular shtick is experimental mathematics." A third is
     ``The computer is a powerful tool, go forth and use it!" Your proposals
     are less formal than I would expect.

  DZ: That's my style, I don't like to be too serious. But I think that it has
     cost me some funding.  In my last grant
     proposal, I was hoping to get co-funded by the computer science division.
     I had no problem getting funding from the
     mathematics division, but of course mathematics doesn't have much funds,
     so I was hoping to get some additional funding
     from computer science.  So I also sent my grant proposal to the computer
     science division. That was a disaster. One of the panelists
     gave me a lecture, saying this might make a good essay, but it's
     not what one would expect in a grant proposal, and
     they refused to give me funding. 

 JG: You are known for your opinions on your web page, some of which are a
     bit over-the-top, especially those on April 1.  What changes--if
     any--are you hoping to encourage in the mathematics community by
     posting your opinions online?

 DZ: I'm not trying to change people's views, at least not consciously.
     I just like to express my opinions. It's only the Internet,
     which is a free-for-all, and one doesn't have to be too uptight.
     So I don't really worry about whether or not my pieces always make sense.
     Hopefully they do most, or least some, of the time.

 JG:  You have been known to celebrate both Valentine's Day and
      April Fools day in the classes you teach. What is the most memorable thing
      you have done to mathematically celebrate a holiday?

 DZ: In my Calculus class, I assign a homework "project", due Feb. 14, to graph the
     parametric equation for a cardiodid. Some
     people came with very strange shapes, but some people realized what was
     going on and just drew a regular heart shape.
     But they made a pointy one, which is wrong, because a cardioid is more
     rounded.  It was nice to get 150 valentines (some rounded some not),
     and I could brag to my wife how much my students love me.
     In my Computer Algebra class, I gave my students some extremely complicated differential equations that they had
     to use Maple to solve. The solution happened
     to be a cardioid. Then they had to draw it (using the Maple plot program), and
     cut it out.

 JG:  You have been known to spontaneously hand out money to
      students for solving problems or pointing out a correction in class.
      What is the largest prize you have ever awarded, and what was it for?

 DZ:   For calculation  errors the most I've given out is $1. But for
       conceptual  errors in a graduate classes, I think I have once given $10.  
       I also offer prizes for really challenging problems.

 JG:  Do you have any concluding words?

 DZ:  Yes, I believe that teaching is at least as important as research, and I put lots
      of effort into my teaching , and take great pride when I do a good job.
      Many research mathematicians dislike teaching and view it as an unavoidable chore,
      but they are wrong. First, teaching is great fun, and secondly, it is very important,
      since this is the future!

     So already today teaching is at least as important as research. But in years to come,
     when more and more original mathematical research will be conducted by computers,
     the importance of the human research mathematician will diminish, and the importance
     of teaching, at all levels, will increase tremendously. Also for a long time to come,
     we still need humans to program the computer, but what is programming?, it is
     teaching computers, and I am sure that being a good programmer and being a good teacher
     (for humans) are strongly correlated. One of the reasons that I believe that
     I am a good teacher is that I do so much programming, and am used to spelling
     out each and every step. In short, the future of mathematics is in good teaching,
     both to machines and to humans.

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