Doron Zeilberger

It's worth reading Doron Zeilberger's many opinions about proof/rigor and other things. https://sites.math.rutgers.edu/~zeilberg/OPINIONS.html

Apart from his interesting ideas about infinity, Zeilberger has things to say about math contests. See http://sites.math.rutgers.edu/~zeilberg/Opinion71.html I'd love to find a modern equivalent of the "gilyonot lematematika" he mentions.

An Applied Mathematician's Apology by Nick Trefethen[1] has a chapter (a couple of pages) titled "Cleve Moler and Matlab". > I first met Cleve Moler when I was a graduate student and he visited Stanford, where his loud and friendly voice reverberated around Serra House. Moler is the antithesis of a European, and as a transatlantic soul, I love both Europeans and their antitheses. A room with Moler in it is a no-nonsense zone. He has no interest in showing you how your problem is connected with the theory of pseudodifferential operators. He just wants to get things done computationally, and nobody has done it better. Moler is about the same age as Knuth, and while Knuth was writing his great books on the analysis of discrete algorithms, Moler was creating the modern era of numerical software. He was an author of both of the foundational software packages of the 1970s, EISPACK and LINPACK, and he also published two influential software-based numerical analysis textbooks. And then, in around 1977 in the Computer Science department at the University of New Mexico, he invented Matlab, which changed the world. I never liked using Matlab (the little I used it), but after reading this I understood better what its innovation was: “All the right algorithms would be invoked in all the right places, without the user needing to know the details.” A footnote I found interesting: > For me `eig(A)` epitomizes the successful contribution of numerical analysis to our technological world.…

https://sites.math.rutgers.edu/~zeilberg/Opinion8.html "[...]I was shocked the other day, when I was browsing in the Web, that the AMS journals are not available freely for downloading, but charge subscription fees. Some `non-profit' organization! Let me remind you that there are several excellent electronic journals, that are viewable free of charge, for example New York Journal of Mathematics and the Electronic J. of Combinatorics. If you publish your papers there, you would be accessible to the ten millions people who surf the Web regularly. Of course, if you are already a tenured full professor, it is even pointless to publish there, since a publication reachable from your Home Page is equally accessible, although the journals might reach more browsers. "Let's hope that the new medium will destroy the old organizations with their corrupt paper mentality and elitism, or at least reform them. After all, thanks to the Web and E-mail, both journals and conferences are becoming increasingly pointless. Then again, institutions have a very strong inertia, look at established religion, so perhaps, unfortunately, it is too soon to rejoice in the devil's death." Opinion 8: Organized Mathematics = Organized Crime, Dec. 1, 1995

>How much precedence is there for machines or tools getting an author credit in research? For a datum of one, the mathematician Doron Zeilberger give credit to his computer Shalosh B. Ekhad on select papers. https://medium.com/@miodragpetkovic_24196/the-computer-a-mysterious-co… https://sites.math.rutgers.edu/~zeilberg/akherim/EkhadCredit.pdf https://sites.math.rutgers.edu/~zeilberg/pj.html

Here is a paper on just how a serious ultrafinitist copes with that https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf The short answer is that they deal with such things symbolically.

There's a short proof of a 2D version of the paradox on page 684 of the Princeton Companion to Mathematics: https://sites.math.rutgers.edu/~zeilberg/akherim/PCM.pdf

I majored in statistics undergraduate and being able to derive and prove the gamblers ruin problem is one of the requirements. At 50/50 it's your money over your plus house's money. And it gets drastically worse as soon as the house gets even a slight edge. I wasn't the casino's target anyways as I saw it was a waste of time and I loathe all the time wasted walking past the gaming area to even see music at a done here like a Hard Rock. Edit, my school is the top link when I searched for a reference, I guess they included it after the casinos went into Atlantic City. Generalized Gambler's Ruin Problem - Rutgers Math https://sites.math.rutgers.edu/~zeilberg/EM20/GamblersRuin.pdf

https://sites.math.rutgers.edu/~zeilberg/Opinion158.html " So brace yourself. In two years, you would open-up an AMS journal, and read an abstract that looks like this: "Using Theory T1014, and Theorems Th100154, Th54135, and Th87651, as well as inequalities In54312, and In34567, we prove Conjecture C84231." Please! Mathematics papers are hard enough to read the way they are written now, and this new policy would make reading them so much harder. It is true that few people read papers any more, and eventually, only computers will write and read papers, and for them using numbered-theorems would be preferred, so let's postpone this reform for another fifty years, when humans will no longer care how unreadable a mathematics paper is"

190 citations sounds impressive but considering they mostly come from possibly the two fields that gathers citations the quickest - biology and AI - this shouldn't be too surprising? Shalosh B. Ekhad, OTOH, is the real deal - https://sites.math.rutgers.edu/~zeilberg/ekhad.html https://en.wikipedia.org/wiki/Doron_Zeilberger

Sadly there is something here, but it is not that: https://sites.math.rutgers.edu/~zeilberg/akherim/PAM.pdf As an aside, from time to time I try to cut the filename out of these kinds of urls and I'm delightedly nostalgic whenever I can browse the server, as it was so common some years ago; these were the kind of "secret powers" that attracted a lot of kids to learning about computers...

>unprovable statements Nonsensical is better in modern discussions than "unprovable". It shows that statements that were previously (and currently by some) thought to be mathematical are actually nonsense just like "x tastes like chicken" but more devious. Asking if a Turing machine with M memory halts in less than N steps is sane and provable. http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enquiry.pdf

use google next time https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/UltraF.ht…

There are people like the late Edward Nelson who represent radical formalism: https://web.math.princeton.edu/~nelson/papers/rome.pdf and question the consistency of the commonly used axiomatic systems like peano arithmetic: https://www.lesswrong.com/posts/gsvQSpeDHKXxjXwuM/edward-nelson-claims… (Although his inconsistency proof of PA turned out to be flawed.) One should also mention Wittgensteins remarks: https://plato.stanford.edu/entries/wittgenstein-mathematics/ And in general thoughts going into the direction of ultrafinitism are quite provocative. See for example D. Zeilbergs opinions: https://sites.math.rutgers.edu/~zeilberg/OPINIONS.html

This reminds of Doron Zeilberger's statement that "real" analysis is just a degenerate case of discrete analysis [0]. [0]: https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf

Here's one with tenure: https://sites.math.rutgers.edu/~zeilberg/Opinion160.html

Maybe there isn’t a math replication crisis, but there is a kind of math fragmentation crisis. Sub-fields of mathematics have gotten so deep that specialists can’t communicate with one another [0]. I’ve heard this expressed by other mathematicians as having only 2 or 3 other people in the world that they can discuss their work with. The level of solitary study necessary to make progress is another sign of this phenomenon. Andrew Wiles famously spent six years alone in his attic to come up with his proofs for Fermat’s Last Theorem. [0] https://sites.math.rutgers.edu/~zeilberg/Opinion104.html

For example, the course "Experimental Mathematics" given by Doron Zeilberger. Interesting stuff. Link, for anybody else who's curious: https://sites.math.rutgers.edu/~zeilberg/teaching.html

> ATT 3B2 There are a number of papers in pure mathematics authored by Doron Zeilberger and Shalosh B Ekhad. (See https://sites.math.rutgers.edu/~zeilberg/ekhad/papers.html for a list. I don't think it's exhaustive) Shalosh B Ekhad is Zeilberger's computer, which once upon a time was an AT&T 3B1. In Hebrew, 3 is "shalosh" and 1 is "ekhad". (The 3B1 isn't actually much like the 3B2, despite the very similar names.)

The (actual) article has a fairly detailed literature review in the introduction, and makes it pretty clear that the main idea was sort-of known already if you squint - but it looks like nobody had put the whole theory together elegantly and advertised it properly. The fact that they couldn't find some natural slices of the hyper-Catalan numbers on OEIS supports that. The proof they give that the hyper-Catalan series solves the Lagrange inversion problem is very good from a pedagogical point of view - I don't think I'll ever be able to forget it now that I've seen it. The only thing this paper is missing is a direct, self-contained combinatorial proof of the factorial-ratio formula they gave for the hyper-Catalan numbers - digging though the chain of equivalences proved in the references eventually got too annoying for me and I had to sit down and find a proof myself (there is a simple variation of the usual argument for counting Dyck paths [1] that does the trick). Another thing to note is that the power series solution isn't just "a power series" - it's a hypergeometric series. There are lots of computation techniques that apply to hypergeometric series which don't apply to power series in general (see [2]). [1] https://jlmartin.ku.edu/courses/math724-F13/count-dyck.pdf (for instance) [2] https://sites.math.rutgers.edu/~zeilberg/AeqB.pdf

His argument against infinities is interesting. He's making an argument for finite constructive mathematics. It's interesting to see that coming around again. Several decades ago, when I was working on early program proof of correctness, I spent a lot of time with the original Boyer-Moore theorem prover. This is a constructive theory of mathematics, similar to the Piano axioms. It really is possible to have useful mathematics without undecidablity. But you have to give up infinity. Boyer-Moore theory has recursion, but all recursions must have an upper bound. There must be some nonnegative integer that gets smaller with each recursion. Set theory is possible in the Boyer-Moore system. Boyer and Moore usually added the traditional axioms of set theory. But I tried doing without that to get to a McCarthy-like theory of arrays without set theory first. Arrays had a concrete representation as an ordered sequence of (index, value) tuples. The sequence has to be ordered so that array equality works. So all the functions for operating on arrays are rather clunky. Here's the sequence of machine proofs for that.[3] Back then, mathematicians hated such approaches. Infinity is a useful trick for handling the edge cases within the main case. Formulas become simpler. You can do things on blackboards with chalk more easily. Constructive math is boring and kind of ugly, because there's a lot of case analysis. But it's sound. You can avoid undecidability and Russel's class of all…

I am reminded of two curious statements which are not normally linked: (0) There could be a Siegel zero [0], like an "indentation" in the Riemann zeta function. The prime numbers need to all collaborate to "shift" the zeta function a little in order to "buckle" the zeta function at the Siegel zero, so there's only really enough room in the primes to accomodate one Siegel zero at most. (1) Ultrafinitism [1] comes in many flavors, including a flavor where there is a single largest prime number [2]. In this ultrafinite physical world, the largest prime determines how large any object can be before folding back in on itself. Since this would be a true statement about natural numbers, it would also be a sort of conspiracy amongst the primes, with only one instance. Situations like this article remind me of both simultaneously: If there's any sort of conspiracy amongst the primes to forge a Siegel zero, it would have to be the same conspiracy which produces an ultrafinite prime candidate; the primes don't have bandwidth for two conspiracies. [0] https://en.wikipedia.org/wiki/Siegel_zero [1] https://en.wikipedia.org/wiki/Ultrafinitism [2] https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf