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Comment by myWindoonn | Hacker News Reader

I don't want to be a professional Hewitt debunker, but his stance has been ripped to shreds; at this point, he is wrong about Direct Logic refuting Gödel and Turing, and his ActorScript can't be implemented on real machines.

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I don't know anything about Hewitt or his stance, but I don't feel like this adds a whole lot to the conversation. I mean I can tell that you're sure that he's wrong, but why should the strength of your belief convince me?

It would be helpful if you added some links or some explanation, or just anything beyond what you've written there.

It’s similar to claiming someone made a perpetuum mobile — extraordinary claims going against basically every mathematician/computer scientist since Gödel require extraordinary evidence. And the result is quasi always that the one claiming, looked over some small but crucial detail.

Veritasium, et. al. overlooked the crucial aspect that

orders on propositions play in maintaining the consistency

of foundations.

Neglecting to include the order of a proposition in in its

Gödel number makes it possible to mistakenly conclude that

the proposition I'mUnprovable exists as a fixed point of a

recursive definition.

See the following for details:

https://papers.ssrn.com/abstract=3603021

Isn’t the order of a proposition included in its Gödel number?

Each proposition is assigned to an increasing prime power, and the increasing list of primes has total order, such that swapping propositions yields a distinct Gödel number.

[Gödel 1931] encoded the characters of a proposition in its

Gödel number but did not include the order of the proposition.

Could you provide two statements with equal Gödel numbers but distinct proposition orders, and a detail of how you compute the Gödel numbers?

If two propositions have the same characters, then they have the same Gödel numbers.

Could you provide two statements with equal Gödel numbers and the same characters in different orders, and a detail of how you compute the Gödel numbers?

If two propositions have the same characters in the same order,

then they have the same Gödel numbers.

Unfortunately, Gödel number of a proposition leaves out the order of the proposition :-(

If two statements have the same characters in the same order, how are they not the same statement? And why would that make Gödel’s statement invalid in Principia Mathematica?

There might be two propositions of different orders that have the

same characters in the same order.

I think what ProfHewitt means here (based on other writing I've found, as he is frustratingly low on details in these conversations) is "order" in the sense of "first order logic", "second order logic"; not in the sense of "first proposition, second proposition" etc.

His claim is that the proposition "This proposition is not provable", formalized as "P equivalent to P is not provable" is not well formed in a typed logic, as "P"'s type is first-order, while "P is not provable"'s type is second-order. Therefore, his claim is that the proposition is simply not well-typed and therefore not interesting. Godel's proofs were discussing an untyped logic, but according to Hewitt that is not an accurate representation of mathematics.

I don't think anyone in the space agrees with him, though, as far as I could tell from some cursory reading.

[Gödel 1931] was for a system for the foundations of

mathematics with orders on propositions.

The [Gödel 1931] proposition I'mUnprovable is certainly of

historical interest and may perhaps be of philosophical

interest.

However, including the proposition I'mUnprovable in

foundations makes the foundations inconsistent.

I know of several computer verified proofs of the incompleteness theorem. See entry 6 of https://www.cs.ru.nl/~freek/100/ for a small catalogue. How does this relate to your objection? Are there bugs in the verifiers? Or did people verify the wrong theorem statement?

Proofs forgot to include order of proposition in its Gödel number.

So you say that the computer verified proofs of the incompleteness theorem of verifying the wrong theorem? Because there can't be bugs in a computer verified proof.

(Of course there can theoretically be bugs in a computer verified proof. If there are bugs in the verifier. But if 4 independent proof assistants verify a proof of a theorem, this is extremely unlikely.)

A proof is only as good as definitions and axioms used in the proof.

All of the computer-verified proofs of existence of [Gödel

1931] proposition I'mUnprovable left out the order of

proposition in its Gödel number :-(

Suppose that Turing is wrong. Then Turing gave an algorithm which should defeat Rice's Theorem. Therefore, I may simply ask, to Hewitt: Could you please write up a short program in a popular Turing-complete programming language which proves or disproves Goldbach's Conjecture? It should be trivial!

Of course, on actual computers, Turing's result applies, Rice's Theorem manifests, and such a program will likely run indefinitely, until we get bored and give up.

The point that Hewitt has made in another thread is typical misdirection. See, it doesn't matter which order of logic we're talking about; in general, we have results for higher order logic, all the way up to Lawvere's fixed-point theorem for any Cartesian closed category. To fight this, Hewitt has to disclaim the entirety of 20th-century mathematics. (At least he's internally consistent -- he does deny modern maths!)

Sad fact is that, ever since 1976 (http://www.laputan.org/pub/papers/aim-353.pdf) folks have recognized that Hewitt's claims about the foundations of computation are not just wrong, but flamebait. But he never had the humility to learn from his mistakes.

What's wrong with ActorScript? Does it have some kind of infinite-size features?

More interesting facts:

His Twitter account has been relatively recently deleted

https://mobile.twitter.com/drcarlhewitt/status/3972794314824...

And his PhD advisor was Seymour Papert.

Twitter account is @ProfCarlHewitt and blog is

https://professorhewitt.blogspot.com/

See following for a simple ActorScript program that cannot be

implemented using a Nondeterministic Turing Machine:

https://papers.ssrn.com/abstract=3603021

"...his ActorScript can't be implemented on real machines."

I think I understand why, but could you provide some more details?

Loosely, because ActorScript requires some sort of genuine non-determinism where an actor explores multiple states simultaneously, we need some sort of magic non-deterministic computer. A quantum computer isn't enough; simulating an ActorScript program is NP-complete.

Maybe there can be physical machines which run ActorScript, but I would expect them to come with new laws of physics, and also I think that there are probably more comfortable programming languages. We already have https://en.wikipedia.org/wiki/Constraint_Handling_Rules for example.

Thanks!

Maybe he should try writing it in Elixir instead of ActorScript?

Hewitt argues that erlang/elixir/beam isn't the actor model because of how Erlang processes receive/process messages

more here: https://www.erlang-solutions.com/blog/lets-talkconcurrency-p... and here: https://codesync.global/media/almost-actors-comparing-pony-l...

Of course, Erlang can be accurately modeled using the theory of Actors.

See discussion in Related Work section of the following article:

https://papers.ssrn.com/abstract=3603021

Prof. Hewitt himself!

myWindoonn: Do you have anything of actual substance to

contribute to the the discussion?

We found http://www.laputan.org/pub/papers/aim-353.pdf the other day. How's it feel to know that folks have been calling you out on your incorrectness since the 1970s?

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