Turing's paper is a landmark, and includes even an equivalence proof of the power of the lambda calculus and his universal machine. In doing so, he came up with a fixed-point combinator, an applicative combinator, now called the Turing fixed point combinator [1]. Especially appropriate on a website run by the Y combinator.
Turing's paper should run as a TV ad "But wait, there's more..." Many applications of topology (he argues why the alphabet should be finite, rather than simply saying that it is finite, by using the Bolzano-Weierstrass theorem, essentially), combinatorics, logic, functional programming - when FP was hardly a decade old and largely unknown outside Princeton - and all from an undergraduate course project report!
[1] https://en.wikipedia.org/wiki/Fixed-point_combinator#Other_f...
halting problem before [Turing 1936], which was written up
in a hurry after [Church 1935].
The Y-combinator does not work for strongly-typed systems.
Instead recursion must be explicitly added as an additional
primitive to the lambda calculus.
See following for more information:
And "know that history as well as he does" approach is not suited for knowledge fields. The notion that you need to know as much as a creator to critic their works is childish and immature.
I might not know as much as him about the field, but I am sufficiently well versed about the topics discussed in this article to point out his biases and fallacies.
I am a fan of people like LeCun and Schmidhuber, but I don't worship them.
I have read many articles and listened to many talks by JS, and the kind of biases present in this article is nothing new.
Schmidhuber is not a "creator" in the sense that Pekinpah, or Coppola, are "creators" and his works are not works of art that can be experienced and criticised by anyone based on aesthetics alone (even though aesthetics also can take a lot of honing).
In any case my comment was that, to propose a better interpretation of the work described by Schmidhuber (not "to point out their biases and fallacies", as you say), you need to know that work at least as well as him. Otherwise, you find yourself criticising the work of someone who knows more than you know.
I don't know about you, but I've done that in the past and it made me look and feel like a fool. You may think you are "sufficiently well versed about the topics discussed in this article" but if you so dismiss the need for general knowledge, beyond the contents of one article, then there may easily be any amount of knowledge that you're missing and that makes you think you know all you need, simply because you don't know what you don't know.
Remember what Socrates, the wisest of men, said: "I know one thing, that I know nothing, Jon Snow".
I may have garbled that a bit there. But you get the picture. Arrogance cannot replace knowledge. Never assume that you know more than you need to know to prove wrong an expert in a field of knowledge as deep and with as long a history as AI, just because you have an internet connection.
Finally, nobody "worships" LeCun or Schmidhuber. What people admire is their knowledge and the hard work they put into acquiring that knowledge. And the reason they admire it is because anyone who's ever try to get to grips with a scientific subject understands how much hard work it takes to acquire knowledge.
"In 1935, Alonzo Church derived a corollary / extension of Gödel's result by showing that Hilbert & Ackermann's famous Entscheidungsproblem (decision problem) does not have a general solution.[CHU] To do this, he used his alternative universal coding language called Untyped Lambda Calculus, which forms the basis of the highly influential programming language LISP.
In 1936, Alan Turing introduced yet another universal model which has become perhaps the most well-known of them all (at least in computer science): the Turing Machine.[TUR] He rederived the above-mentioned result.[T20](Sec. IV) Of course, he cited both Gödel and Church in his 1936 paper[TUR] (whose corrections appeared in 1937). In the same year of 1936, Emil Post published yet another independent universal model of computing,[POS] also citing Gödel and Church. Today we know many such models. Nevertheless, according to Wang,[WA74-96] it was Turing's work (1936) that convinced Gödel of the universality of both his own approach (1931-34) and Church's (1935).
What exactly did Post[POS] and Turing[TUR] do in 1936 that hadn't been done earlier by Gödel[GOD][GOD34] (1931-34) and Church[CHU] (1935)? There is a seemingly minor difference whose significance emerged only later. Many of Gödel's instruction sequences were series of multiplications of number-coded storage contents by integers. Gödel did not care that the computational complexity of such multiplications tends to increase with storage size. Similarly, Church also ignored the spatio-temporal complexity of the basic instructions in his algorithms. Turing and Post, however, adopted a traditional, reductionist, minimalist, binary view of computing—just like Konrad Zuse (1936).[ZU36] Their machine models permitted only very simple elementary instructions with constant complexity, like the early binary machine model of Leibniz (1679).[L79][LA14][HO66] They did not exploit this back then—for example, in 1936, Turing used his (quite inefficient) model only to rephrase the results of Gödel and Church on the limits of computability. Later, however, the simplicity of these machines made them a convenient tool for theoretical studies of complexity. (I also happily used and generalized them for the case of never-ending computations.[ALL2])"
"The formal models of Gödel (1931-34), Church (1935), Turing (1936), and Post (1936) were theoretical pen & paper constructs that cannot directly serve as a foundation for practical computers. Remarkably, Konrad Zuse's patent application[ZU36-38][Z36][RO98] for the first practical general-purpose program-controlled computer also dates back to 1936. It describes general digital circuits (and predates Claude Shannon's 1937 thesis on digital circuit design[SHA37]). Then, in 1941, Zuse completed Z3, the world's first practical, working, programmable computer (based on the 1936 application). Ignoring the inevitable storage limitations of any physical computer, the physical hardware of Z3 was indeed universal in the "modern" sense of Gödel, Church, Turing, and Post—simple arithmetic tricks can compensate for Z3's lack of an explicit conditional jump instruction.[RO98]"
"According to the text"
I don't agree with the text, that is the point.
And, moreover, I don't need to. It is already "out there".
And what's up with the wall of quote? I read the article in full.
On top of what I already said, I add that crediting Lambda Calculus as only the basis of a high-level programming language does not do it justice.
He is giving Turing and Post credit in one paragraph, and taking it away in another. As if like he cannot decide.
More importantly, Turing's approach is completely reduced to adopting "a traditional, reductionist, minimalist, binary view of computing—just like Konrad Zuse". It is not giving even close to the credit deserved. Many of the child parents to my main address to that.
Machine [Turing 1936] were fundamental achievements. However,
they are inadequate models of computation because there are
digital computations they cannot implement.
See the following for an example:
Also, isn't that your own paper?
depend crucially on indeterminacy in order to gain
performance.
See the following
It's a concession.
a many-core computer has inherent indeterminacy.
Indeterminacy slows things down. It's a concession.
Enforced determinacy slows down processing.
Do you think the above statement is wrong? If so, why?
See
Which paragraph is repeated?
Unfortunately, the above article is woefully incomplete and
inaccurate because it is highly censored by Wikipedia.
More comprehensive, up-to-date, and accurate information can be found here:
> “Monster” is a term introduced in [Lakatos 1976] for a mathematical construct that introduces inconsistencies and paradoxes. Euclid, Richard Dedekind, Gottlob Frege, Bertrand Russell, Kurt Gödel, Ludwig Wittgenstein, Alonzo Church, Alan Turing, and Stanisław Jaśkowski all had issues with mathematical monsters as discussed in this article. Monsters can lurk long undiscovered. For example, that “theorems are provably computational enumerable” [Euclid approximately 300 BC] is a monster was only discovered after millennia when [Church 1934] used it to identify fundamental inconsistency in the foundations of mathematics that is resolved in this article.
> Euclid, Richard Dedekind, Gottlob Frege, Bertrand Russell, Kurt Gödel, Ludwig Wittgenstein, Alonzo Church, Alan Turing, and Stanisław Jaśkowski all had issues with mathematical monsters as discussed in this article. This article explains how the theories Actors and Ordinals recraft classical foundations without limiting mathematical power.
> Euclid, Richard Dedekind, Gottlob Frege, Bertrand Russell, Kurt Gödel, Ludwig Wittgenstein, Alonzo Church, Alan Turing, and Stanisław Jaśkowski all had issues with mathematical monsters as discussed in this article. Computer Science brings new concerns and considerations to foundations beyond earlier work requiring new machinery. This article explains how the theories Actors and Ordinals recraft classical foundations without limiting mathematical power.
> “Monster” is a term introduced in [Lakatos 1976] for a mathematical construct that introduces inconsistencies and paradoxes. Since the very beginning, monsters have been endemic in foundations. They can lurk long undiscovered. For example, that "theorems are provably computational enumerable" [Euclid approximately 300 BC] is a monster was only discovered after millennia when [Church 1934] used it to identify fundamental
For typos in the actual PDF, here are a few :
- teams of Wrens (female Naval Officers) operated large-scale simulations in [sic!] that discovered ways to defeat U-boat attacks that were crippling Britain.
- The reason that in practice that [sic!] an Actor can be hundreds of times faster is that in order to carry out a concurrent computation, the parallel [...]
- close request when received, send [...] [not a typo, just strange phrasing?]
- if it is not recorded as that the engine is [...]
- Actor event induction (cf. [Turing 1949]) can used to prove [...]
- Suppose to obtain a contradiction that there is [...]
Thank you very much for finding these typos!!!!
They should all now be fixed at SSRN :-)
Can anyone find more typos or make other suggestions for improvement???
Thanks! Carl
An automatic theorem prover is a kind of AI. Gödel showed its limitations.
There have been entire conferences dedicated to Gödel and AI.
Which of those limitations apply only to AI and not to human intelligence and mathematical reasoning?
Is it somehow surprising that AI can't prove mathematical contradictions to be true?
He sometimes overdoes these. Even when something is not related to Deep Learning and modern AI. He tries to push very narrow and one-sided views without much room for nuance.
For example- "Turing used his (quite inefficient) model only to rephrase the results of Gödel and Church on the limits of computability."
He is trying to say that Turing did nothing new, his work is a mere extension of Godel's (!), and he only accounted for practical computational reality. That is not true.
He goes on further by saying, "Later, however, the simplicity of these machines made them a convenient tool for theoretical studies of complexity." Like, Turing and Church's work was only good for that. I digress. I believe that Turing's work is more seminal and paved the way for modern general computers.
Another bit here says "The formal models of Gödel (1931-34), Church (1935), Turing (1936), and Post (1936) were theoretical pen & paper constructs that cannot directly serve as a foundation for practical computers."
It is putting the works of Turing, Post, Church, and Godel all together. Like they are the same. I see the logic as they are all impractical, but they are different levels of impractical. Putting them all under one bracket like that makes no sense.
I respect Schmidhuber a lot, but some of his behaviour is borderline childish which are more pronounced in his claims about AI history. You should be skeptical about what he says (and what anyone says).
Despite all this, I did not know many things written in this article. I learned a lot and had fun reading it.
I first read about Zuse in Isaacson's Innovators book. Did not know that he used a high-level language for a chess program. Fascinating. He is very underrated.