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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, wh

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 Gödel, [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 3y ago
Interesting contrast with Gödel'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, Gödel 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. Gödel 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-84f66e3b91a7 https://medium.com/@rudyrucker/memories-of-kurt-godel-84f66e...] Also far from mainstream, but in an opposite direction
boothby 3y ago
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 3y ago
  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 3y ago
    > 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.
    - 3y ago
tome 3y ago
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 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 3y ago
  > 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/Linear_bounded_automaton https://en.wikipedia.org/wiki/Finite-state_machine https://en.wikipedia.org/wiki/Finite-state_machine https://en.wikipedia.org/wiki/Template:Formal_languages_and_grammars https://en.wikipedia.org/wiki/Template:Formal_languages_and_...
pron 3y ago
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/S0002-9947-1965-0170805-7/S0002-9947-1965-0170805-7.pdf https://www.ams.org/journals/tran/1965-117-00/S0002-9947-196...
- _0ffh 3y ago
  > 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 3y ago
    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 3y ago
      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 3y ago
        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_mathematics) https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...

monoscient 3y ago

Interesting contrast with Gödel'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, Gödel 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. Gödel 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-84f66e3b91a7 https://medium.com/@rudyrucker/memories-of-kurt-godel-84f66e...] Also far from mainstream, but in an opposite direction

boothby 3y ago

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 3y ago
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 3y ago
  > 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.
  - 3y ago

tome 3y ago

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 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 3y ago
> 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/Linear_bounded_automaton https://en.wikipedia.org/wiki/Finite-state_machine https://en.wikipedia.org/wiki/Finite-state_machine https://en.wikipedia.org/wiki/Template:Formal_languages_and_grammars https://en.wikipedia.org/wiki/Template:Formal_languages_and_...

pron 3y ago

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/S0002-9947-1965-0170805-7/S0002-9947-1965-0170805-7.pdf https://www.ams.org/journals/tran/1965-117-00/S0002-9947-196...

_0ffh 3y ago
> 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 3y ago
  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 3y ago
    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 3y ago
      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_mathematics) https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...