[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
mathematics?
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
Sum((-1)^k*binomial(2*n,n+k)^3,k=-n..n)=(3*n)!/n!^3,
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.