[HN Gopher] Opinions of Doron Zeilberger
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Opinions of Doron Zeilberger
Author : kaladin-jasnah
Score : 38 points
Date : 2024-01-19 15:55 UTC (7 hours ago)
(HTM) web link (sites.math.rutgers.edu)
(TXT) w3m dump (sites.math.rutgers.edu)
| nickdrozd wrote:
| Zeilberger is a hardline finitist. According to him, a lot of
| mathematics is meaningless or at least woefully misguided due to
| its reliance on infinite sets, which he believes to be nonsense.
| His opinions are far from mainstream, but they sure are
| entertaining. Here's a taste:
|
| > But just like Godel, [Turing] missed the point! Neither of them
| proved that "there exist true yet unprovable statements".
| Rubbish! Every meaningful statement is either provable (if it is
| true) or disprovable (if it is false). What they did
| (meta-)prove, by a Reductio Ad Absurdum clever argument, is that
| many statements that were believed to be meaningful, are really
| utterly devoid of meaning. As I have already preached in this
| sermon, the problem is the fictional (and pernicious!) infinity.
| In particular, Turing thesis's is utter nonsense, talking about
| "oracles", that lead to lots of beautiful, but fictional and
| irrelevant work by logicians and theoretical computer scientists.
| monoscient wrote:
| Interesting contrast with Godel's view:
|
| "Everyone believes that the Empire State Building is real,
| because it is possible for almost anyone to go and see it for
| himself. By the same token, anyone who takes the trouble to
| learn some mathematics can "see" the set of natural numbers for
| himself. So, Godel reasoned, it must be that the set of natural
| numbers has an independent existence, an existence as a certain
| abstract possibility of thought.
|
| I asked him how best to perceive pure abstract possibility. He
| said three things, i) First one must close off the other
| senses, for instance, by lying down in a quiet place. It is not
| enough, however, to perform this negative action, one must
| actively seek with the mind, ii) It is a mistake to let
| everyday reality condition possibility, and only to imagine the
| combinings and permutations of physical objects -- the mind is
| capable of directly perceiving infinite sets, iii) The ultimate
| goal of such thought, and of all philosophy, is the perception
| of the Absolute. Godel rounded off these comments with a remark
| on Plato: "When Plautus could fully perceive the Good, his
| philosophy ended.""
|
| [https://medium.com/@rudyrucker/memories-of-kurt-
| godel-84f66e...]
|
| Also far from mainstream, but in an opposite direction
| boothby wrote:
| Careful, now. Ultrafinitists exist within the mathematical
| community and sometimes we overstate our opinions for funsies.
| Doron is much more reasonable in person than one may conclude
| from this website.
|
| To me, ultrafinitism is an intellectual curiosity. Proofs which
| depend upon the axiom of choice, or a notion of infinity, do
| not always correspond to algorithms which terminate in finite
| time. Such results are "useless" to a computational
| mathematician. The goal of determining how much of mathematics
| can translate to an finite universe is, in my opinion, a noble
| one.
|
| A hill I would die on, for sport: "the logarithm of the biggest
| number I see reason to believe in is the combined size of my
| RAM and HDD."
|
| The key word being "believe." Mathematicians do not tend to
| operate on faith. Proofs invoking unrealistic axioms are
| perfectly acceptable, valid proofs. One needn't "believe"
| anything.
| cubefox wrote:
| There are three levels:
|
| - Infinitism, people who believe in the existence of
| completed infinite sets, and transfinite numbers more
| generally.
|
| - Finitism, people who believe that "infinite" means
| "arbitrarily large" or "unbounded", but nothing more.
|
| - Ultrafinitism, people who think every quantity has an upper
| bound due to mathematics not being separate from the physical
| world.
|
| Zeilberger identifies as an ultrafinitist. I personally think
| finitism is the most plausible one. It accepts an intuitive
| notion of potential infinity, but it avoids basically all the
| mathematical paradoxes of infinity and the philosophically
| extravagant mathematics of transfinite set theory.
| A_D_E_P_T wrote:
| > _philosophically extravagant mathematics of transfinite
| set theory_
|
| And poorly-founded, I would add. Cantor's diagonal proof
| is, quite literally, question begging. Sure, you can
| construct a number not in the infinite set of natural
| numbers via that diagonalization method, but only if you're
| _at_ infinity, or if you make very liberal use of "..." --
| otherwise, you cannot.
|
| And Skolem was quick to discredit Cantor's extravagances,
| on grounds that tend to support finitism generally.
| epgui wrote:
| I don't think I agree, but this is difficult to
| communicate in words.
|
| Cantor's diagonalization argument doesn't rely on the
| contents of an infinite list of numbers, or on numbers
| themselves. You don't need to know anything about what
| "..." represents and you don't need to be "at" anywhere.
| There is no need to compute anything with concrete
| numbers.
|
| The only thing the argument relies upon is whether the
| fundamental structure of numbers allows for defining a
| bijection. The proof/computation is an abstract relation,
| not a concrete one with actual number objects.
| flir wrote:
| > Zeilberger identifies as an ultrafinitist.
|
| So, um, does that mean there's a "biggest number"? One you
| can't +1?
| epgui wrote:
| Within certain parameters, that's right.
|
| And that probably makes as much sense (if not more) as
| the alternative for all I know. There is a way to think
| of mathematics as corresponding to computation in a way
| where anything that cannot be computed cannot be proven.
| If you cannot "construct" or "compute" infinity, then you
| cannot say that it exists.
|
| I think what I am describing corresponds to the
| "constructivist" view of mathematics, but tbh I don't
| know enough about this to say whether the framework
| allows for infinities.
| mensetmanusman wrote:
| Axioms are faiths.
| boothby wrote:
| No, axioms are hypotheses. In general, a mathematical proof
| has the form "If [conditions] are met, then [result] is
| guaranteed." This requires _no_ faith whatsoever that
| [conditions] will ever be met.
|
| But those numbers that require the totality of my
| computer's N bits of storage to represent? I can, without
| reservation, believe that every integer between zero and
| 2^N-1 exists. But incrementing that biggest number will be
| challenging. I hear you scoff; surely there are bigger
| numbers, you can just buy more hard drive. And I can! But
| there are only finitely many hard drives available for
| purchase, many more than I can afford, so I'd rather save
| my money and believe in less.
| epgui wrote:
| > No, axioms are hypotheses
|
| IMO they're neither "faith" nor "hypotheses", although
| "whether a set of axioms is self-consistent" is probably
| a hypothesis.
|
| Axioms are just building blocks. They don't even need to
| be "true" in any real sense. They're just arbitrary
| determinations or rules that we make up, and then we
| figure out what the consequences of these choices are.
|
| In practice we often try to come up with axioms that have
| some practical value in the real world, but this is not
| necessarily or always how we go about doing maths.
| tome wrote:
| I quite sympathetic to finitism, even ultrafinitism, but I find
| Zeilberger's contention to be hard to reconcile with the
| observation that there is a < 8,000 state (one-tape, two-
| symbol) Turing machine whose behaviour cannot be proved in ZFC:
| https://scottaaronson.blog/?p=2725
|
| Maybe Zeilberger would respond that that machine might need to
| use unbounded amounts of memory. If you fix a bound on memory,
| i.e. take infinity out of the question, maybe you can't
| construct such machines.
| nsajko wrote:
| > If you fix a bound on memory, i.e. take infinity out of the
| question, maybe you can't construct such machines.
|
| A Turing machine with a tape of only constant length is
| equivalent to a finite-state automaton. If the length of the
| tape is a _linear_ function of the length of the input, on
| the other hand, the machine is a linear bounded automaton.
|
| https://en.wikipedia.org/wiki/Linear_bounded_automaton
|
| https://en.wikipedia.org/wiki/Finite-state_machine
|
| https://en.wikipedia.org/wiki/Template:Formal_languages_and_.
| ..
| pron wrote:
| I think that even a finitist would admit that it was Turing's
| work that paved the path to computational complexity theory
| [1], which could be presented as the finitist view of
| computability, and is not just plainly meaningful from a
| finitist's perspective, and not just eminently practical
| (perhaps indirectly), but perhaps also the very foundation of
| ascribing meaning to finitism.
|
| So if "nonsense" breeds meaning, is it really nonsensical?
|
| [1]: The birth of complexity theory is attributed to Hartmanis
| and Stearns's 1965 _" On the Computational Complexity of
| Algorithms_, which very directly builds on Turing's work:
| https://www.ams.org/journals/tran/1965-117-00/S0002-9947-196...
| _0ffh wrote:
| > So if "nonsense" breeds meaning, is it really nonsensical?
|
| How bad would the analogy be to a calculation with imaginary
| numbers that still yields a real result?
| senderista wrote:
| Very bad. Ordered pairs of real numbers are neither more
| nor less real than real numbers themselves.
|
| Would you say that the rationals (equivalence classes of
| ordered pairs of integers) are "less real" than the
| integers?
| _0ffh wrote:
| The point of the analogy was that a calculation might
| involve mathematical concepts that from some point of
| views may seem iffy on the face of them, but still arrive
| at a result that does not involve any of the intermediate
| perceived iffyness.
| pron wrote:
| Indeed, some people misrepresent Hilbert's formalism [1]
| to say that mathematical statements have no intrinsic
| meaning but are rather just a "game of symbols". What he
| really said was that we could split mathematical
| statements into "real" ones, dealing with the finitary,
| and "ideal" statements, with infinitary terms. He said
| that the manipulation of _ideal_ statements could be
| considered a possibly meaningless "game of symbols", but
| such a game can be used to help prove finitary, "real"
| statements that _are_ meaningful. As long as you cannot
| arrive at a contradiction involving finitary statements,
| infinitary ones are useful. In that view, infinitary
| statements could be considered meaningless (in the sense
| that they don 't themselves describe anything real) yet
| not nonsensical.
|
| [1]: https://en.wikipedia.org/wiki/Formalism_(philosophy_
| of_mathe...
| pron wrote:
| * s/finitis[t|m]/ultrafinitis[t|m]/g
| zero-sharp wrote:
| I'm pretty sure that making arguments under idealistic assumption
| often allows us to make progress. Yea certain methods can seem
| problematic and I definitely have worried about this before. But
| I'm pretty sure there's evidence that certain, useful, theories
| just wouldn't exist unless we used infinity in some capacity.
| BeetleB wrote:
| Doron's opinions are always fun to read.
|
| People refer to him as the official mathematics troll (in a nice
| way).
| Vecr wrote:
| See Godel, Escher, Bach [1979, pp. 452-456]) (that's pages 452 to
| 456 in the original) for more about problems caused by infinite
| precision, infinite numbers, and infinite "tape size" (for Turing
| machines). I and other people would also add "infinite time" as
| well.
| bigbillheck wrote:
| I don't usually agree with Zeilberger, but his heart's in the
| right place and I do, in fact, have to hand it to him.
| jasperry wrote:
| It's very interesting that he is strongly for the use of
| computation in math, but strongly against using computer logic
| systems to formalize proofs--see opinion #184. I'd love to hear
| other people's take on this.
|
| I saw how pro-computational he is from the one time I heard him
| talk. One of his most repeated phrases was, "you write a little
| computer program...", meaning as a way to enumerate some set of
| objects or calculate the size of something.
| _0ffh wrote:
| I mean, he's the guy famous for listing his computer as co-
| author on some of his papers, so I guess being pro-
| computational figures. =)
| dang wrote:
| Related:
|
| _The Mathematical Opinions of Dr. Doron Zeilberger_ -
| https://news.ycombinator.com/item?id=485899 - Feb 2009 (8
| comments)
| yding wrote:
| Great to see. Not only a wonderful mathematician but also a
| wonderful human being.
| ykonstant wrote:
| I love reading Doron's opinions, whether I agree with them or
| not.
| paulpauper wrote:
| "Opinion 185: David Jackson and Bruce Richmond Should Retract
| Their Erroneous Attempted Proof of the Four-Color-Theorem
| [Written Dec. 30, 2022]"
|
| _Of course the arxiv is full of claimed proofs of Collatz and
| other major open problems, but there is a special section for
| them called GM (General Mathematics)._
|
| _It really damages the credibility of the arxiv if a non-GM
| department, in this case CO (combinatorics) has a false proof of
| a long-standing open problem. I am talking about the recent
| Erroneous computer-less proof of the Four Color Theorem posted by
| distinguished and accomplished combinatorialists David Jackson
| and Bruce Richmond, who hail from the Mecca of combinatorics,
| University of Waterloo. Their proof is definitely flawed in its
| current form, and very unlikely to be fixable. (See here)._
|
| And before that:
|
| "appendix:Why ArXiv Moderators (including Victor Reiner)
| Seriously Erred in Rejecting a One Page Gem Entitled "Five More
| Proofs of the Cosine Addition Formula (Inspired by Mark Levi's
| Perpetuum Mobile Proof)"
|
| He wanted arxiv to reject or reclassify the flawed four-color
| proof, but mad that arxiv rejected his paper. In the first
| instance, the pre-print processes worked as it was intended to.
| The paper was uploaded , critiqued, and then retracted. it would
| seem like he wants to be the arbiter of which papers get
| rejected/reclassified on arXiv.
| throw_pm23 wrote:
| I think he is generally consistent here: he wants papers with
| flawed proofs to be retracted, and he does not want papers to
| be rejected for not being sufficiently significant (if they are
| otherwise correct).
| adrian_b wrote:
| 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"
| bigbillheck wrote:
| Check the date.
| gurchik wrote:
| I might not have finished my Computer Science degree a few years
| ago if it wasn't due to Dr Z (as we called him at the time).
|
| I was required to take Linear Algebra to satisfy the requirements
| for the B.S. in Computer Science, but I simply could not get
| interested in the course. I had attempted to take this course 3
| times previously, but could not succeed. Twice, I withdrew from
| the course before the first exam. The third time, I missed that
| window and ended up failing the class. I just couldn't be
| motivated to study this boring subject. A flaw of mine, as I've
| discovered. But in my 4th attempt, it was taught by Dr. Z, and it
| was completely different. His passion for mathematics (and more
| importantly, teaching) was infectious. This was completely
| different than my experience with some other professors in the
| Math and CS departments. I never got a feeling that Dr. Z was
| "wasting" his time on this 200-level course when he could be
| doing much more important research. I easily got an A, and with
| that, satisfied all the math and CS requirements for my degree.
| Years later, I remember a lot more about that class than, say,
| Calculus.
| pessimizer wrote:
| 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
| Animats wrote:
| 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 classes paradox.
| You don't actually need infinities for much of mathematics.
|
| Today there's more acceptance of machine proofs and case
| analysis. Everybody has computers, after all. I'm not a
| mathematician; I just used formal math as a tool to improve
| programs. This is a problem for the theorists.
|
| (A few years ago, I converted the original Boyer-Moore code to
| run under Gnu Common LISP, so it can be run today.[2] Proving the
| basics of number theory now takes under a second. It took about
| 45 minutes in 1981.)
|
| [1] https://sites.math.rutgers.edu/~zeilberg/Opinion160.html
|
| [2] https://github.com/John-Nagle/nqthm
|
| [3] https://github.com/John-
| Nagle/pasv/blob/master/src/work/temp...
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