Turing Machines, on the other hand, were instantly obviously mechanizable. It was clear that one could build a physical machine to run any Turing program without human input. By proving that they could simulate the other systems, Turing showed that those other systems could be mechanizable as well.

I don't understand why Schmidhuber continues to ignore this crucial point.

Seems to be a lot of uneasiness, of late, about the way credit is allocated in science. IMO, it's mistaken to point this at the top: nobel laureates, heroic icons like Einstein or Turing. These figures are supposed to be idolized and idealized. Yes, this is "untrue," technically. But, it serves many purposes. A nobel prize win elevates science by singling out scientists for hero status. Achilles elevated Greece by giving Greeks something to collectively aspire to or adulate.

If you're already deeply interested in computer science, of course the detailed narrative recognizing dozens of brilliant early computer scientists is richer. Of course!

Where poor credit allocation matters isn't historical hero narratives, it's at the working scientist level. The grants & positions level. Here, it's important to be accurate, fair, etc. Being inaccurate, unfair or corrupt at this level creates actual deficits.

I feel the era of great thinkers who single handledly performed disruptive breakthroughs in their field, the Galileos and Newtons, was over with the Einstein-era (and even Einstein also stood in the shoulders of giants).

No one works in isolation any more, and that is not a bad thing. You can subject any relevant figure to a similar analysis and come with the same results, it's absurd to try and come up with someone with such an overwhelming figure like Albert Einstein these days.

But if you need to choose a Founding Father of Computing Science for the general public, I'd say Alan Turing is the best candidate. Scholars will give due credit to Church, Zuse, von Neumann and all the others.

Just look at his previous blog post [3], in which he explains that the most cited neural networks all cite works by him. These papers cite dozens of papers, so a lot of other groups that are active in AI can claim the same thing...

[1]: For example, Turing published an independent proof of the Entscheidungsproblem, in the [TUR] article, just a month after Church, that the article forgets to highlight.

[2]: https://en.wikipedia.org/wiki/J%C3%BCrgen_Schmidhuber#Views

[3]: https://people.idsia.ch/~juergen/most-cited-neural-nets.html

However, his greatest impact came probably through his contribution to cracking the Enigma code, used by the German military during the Second World War. He worked with Gordon Welchman at Bletchley Park in the UK. The famous code-breaking Colossus machine, however, was designed by Tommy Flowers (not by Turing). The British cryptographers built on earlier foundational work by Polish mathematicians Marian Rejewski, Jerzy Rozycki and Henryk Zygalski who were the first to break the Enigma code (none of them were even mentioned in the movie). Some say this was decisive for defeating the Third Reich.

Yes, Turing worked on Enigma and the Bombe to automate breaking of the code. However, Colossus wasn't for Engima (it was for Tunny) and Turing didn't work on it. This paragraph seems confused about which is which.

Also, the fact that the Polish who did the original work weren't mentioned in the movie is just one of many things horribly wrong with that movie. It's so bad it shouldn't be considered canon.

But science fields do need marketing. All children have heroes they look up to. Putting focus on Turing's achievements is merely creating a pop star figure in the mainstream, which I think is a good thing: a smart dude works on a problem that saves World War 2 and now powers your phone and your TikTok app. Once you are actually interested in the field you can work out the nuances and the falsehoods in that claim.

Evaluating earlier work in some field throughout history always leads to a complex graph of achievements, but you cannot put that graph in the name of an annual prize. Do we change "Turing Award" to "Frege-Cantor-Godel-Church-Turing"?

Bletchley Park work was still classified in those days, so not much was known about that. Much to the annoyance of Tony Flowers, who built the Colossus code-breaking machine but couldn't mention it in post-war job-hunting. (Incidentally, Colossus was not a general-purpose computer. It was a key-tester, like a Bitcoin miner ASIC.)

As I've pointed out before, the big problem was memory. IBM had electronic addition and multiplication, with tubes, in test before WWII. But the memory situation was really bad. Two tubes per bit. Or electromagnetic relays. Or electromechanical counter wheels, IBM's mainstay in the tabulator era. To store N bits, you had to build at least N somethings. Babbage's Analytical Engine called for a memory of a ring of rows of counter wheels, a mechanism the size of a locomotive. Access time would have been seconds.

Much of early computing was about kludges to get some kind of memory. The "Manchester Baby" had a CRT-based memory, the Williams tube. That was the first computer to actually run a program stored in memory. Frederic C. Williams, Tom Kilburn, and Geoff Tootill, 1948.

After that, everybody got in on the act. Mercury tanks and long metal rods were built. Really slow, and the whole memory has to go past for each read, so twice the memory means twice the access time. Then there were magnetic drum machines. Magnetic tape. Finally, magnetic cores. At last, random access, but a million dollars a megabyte as late as 1970. Memory was a choice between really expensive and really slow well into the 1980s.

Turing was involved with one of the early machines, Pilot ACE/ACE. But he quit before it was finished.

If the problem is that some important early contributions to CS are being overlooked, the solution is to promote those contributions.

By framing this as Turing vs. others, the focus is squarely on Turning and his contributions. It puts itself in the position of having to beat down and minimize Turing’s contributions before raising up other contributions. Pretty much setting itself up to fail to convince very many.

Instead, e.g., present the narrative of those early contributions, showing how they provided the foundation Turing worked from.

(edit: I should add: I understand perfectly that the point of this article probably isn’t to actually convince anyone of anything, but is just a hot take meant to get people worked up for the page views. So mission accomplished, from that perspective, I guess.)

It passes the popular bar for new knowledge because it illustrates a conflict and resolves it, but it's in a cognitive style that seems to conflate debasing knowledge with discovering it. It is how people are educated these days, where it is sufficient to destabilize and neutralize target ideas instead of augmenting or improving them, so it's no surprise, but the article seemed like a good example to articulate what is irritating about this trend in educated thinking.

I think everyone with an interest in theoretical CS should work through Turing's 1936 paper at one point in their life. For me, the important part of that paper is how convincingly it argues that the proposed machine model is not just yet another model of computation, but a complete model of computation: that everything you could reasonably call computation is encompassed by these machines.

So there's a finality to Turing's definition of computation: these problems are unsolvable not just with this model, but with any reasonable machine model. It's very hard to make the same case for lambda calculus, which is (in my opinion) a large part of what made Turing's paper so groundbreaking.

- The core mechanisms of the machine for running the Enigma "quickly" was from the Polish - The machine wasn't even a generalized computer!

I just felt really misled! Perhaps the biggest thing is Turing probably ended up doing good amounts of contributions to the academic/theoretical side and the practical side, but it feels like we are missing opportunities to describe the confluence of so many people's ideas in this period of history to end up at the current "machines that read instructions and then act upon them in a generalized fashion, very quickly".

This article seems to be that, and it's super intersting

That might seem pedantic and it is, but you need to define exactly where the line is drawn and more so, give a good reason why. In fact its not even that simple, WE need to decide and agree on where the line is drawn and all of us agree why. Otherwise one mans pedantic is anothers important creditation.

Obviously that's not going to happen anytime soon, so for now, figureheads like Einstein and Turing do the job. And they do certainly deserve credit to some degree. That or we stop giving credit completely, which a). seems like a good way to destroy knowledge and b). isn't going to happen anytime soon.

Edit: As another commenter pointed out, if Einstein or the like were born somewhere else and lived around a different group of people, theres a chance he wouldn't become a figurehead, or he would make less or more or different discoveries. Therefore, theres a third option for creditation, in which everyone who has ever lived up until those discoveries has equal credit. If I were 60 or so years older, I'd be as much to credit for the turing machine as Turing himself. So would you. Of course, this is pretty much as good as no credit to anyone at all, but fixing it again requires a joint agreement on where the line is drawn

Goedel apparently believed that it was impossible to explicitly enumerate all the partial computable functions over the integers, until Turing proved him wrong. He reasoned as follows:

- You can't enumerate the total functions because Cantor's diagonal argument is constructive. Given a claimed enumeration of N^N, you can perform Cantor's diagonal construction to produce a computable element of N^N not in the enumeration. So N^N is computably uncountable.

- Classical set theory says that a superset of an uncountable set is also uncountable.

The problem is that while a superset of an uncountable set is uncountable, a superset of a computably uncountable set may instead be computably countable. The partial functions over the integers show that this is indeed the case.

Cantor's construction is blocked because a partial function can output a degenerate value called non-termination. A computable function must map the non-termination value to itself. Since it isn't mapped to something different, Cantor's construction can't produce something outside an enumeration.

It's almost like Schmidhuber hates Turing and wants the world to worship Godel.

It's personal for him.

He also wrote a tweet and maybe another post where he ranked up all countries of Europe and says that Europeans collectively got the most medals in Olympics and hence they are superior.

He suffers from this weird and serious case of European Supremacy.

I know about his actual works (not what he claims to have done).

And based on that I respect him.

He also has got a cult of people following him and who will gang up on any dissenters, i.e. people not thinking of him as a walking god.

Schmidhuber has actual contributions, I know.

Now he moved to a country with extremely bad record of human rights, and has policies and implementation that put the Taliban to shame.

Now he is promoting the institution left and right any opportunity he gets.

He is very hard to take seriously.

History seems to like a single origin story.

What Turing did in 1937 might not have been an advance over what Godel and Church did. But if you want to make the case for building a general purpose computer out of mechanical relays or vacuum tubes in 1946, you need a semantic advance. Turing machines did that.

Nobody would read Godel and say "let's build ENIAC." Tommy Flowers did read Turing and say that. THAT is the difference.

It's like Einstein versus Lorenz. Same mathematics.Different semantic interpretation.

It was written by Charles Petzold, who also wrote the immensity popular book “CODE”.

https://www.amazon.com/Annotated-Turing-Through-Historic-Com...

On Computable Numbers [1] is perhaps one of the top 3 papers in the history of Mathematics. One of the most remarkable thing about Turning Machine is its simplicity.

Then again, in 1950, Can Machine Think[2] is perhaps the top 3 papers in the history of Philosophy. And then again, one of the most remarkable thing about Turing Test and the Imitation Game is its simplicity.

The impact of these two papers in the academia, industry and in our lives is huge.

Alan Turning is easily one of the top 3 Mathematicians and Philosophers of all time.

[1]. https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

[2]. https://academic.oup.com/mind/article/LIX/236/433/986238

The Turing machine is a key conceptual model for understanding the basics of computation. The Turing Test is a great model for thinking about what being intelligent means. Hardly a week goes by without the term Turing Complete appearing somewhere in a HN comment. The fact that he also played an important role in the design and construction of actual practical computing machines, and did so to fight nazis seals the deal.

Of course there's more to it, there's plenty of credit to go around, but Turing is the perfect entry point for people to appreciate and learn more about all the work that went into the founding of computer science. It elevates the profile of the whole field.

If there was no narrative, no idols, no celebrities, would people be less motivated to do science? Why do we need to lie to ourselves so?

> If you're already deeply interested in computer science, of course the detailed narrative recognizing dozens of brilliant early computer scientists is richer. Of course!

Personally I'm mostly uninterested in who did what, but maybe that's just me. It seems obvious to me that nearly every scientific discovery could have been done equally well by millions of people, it's just a matter of who had the resources to be educated, who decided to research the problem, who managed to snipe the answer first, and who had the right connections to get it acknowledged. They're still great achievements, for sure, but they're not the markers of exceptional genius we want to think they are, not for Turing or Einstein, but not for anyone at all, really.

Of late? You should read up on Newton/Leibniz hysterics over who invented calculus. The arguments over who invented the first light bulb, car, etc. Whether greek knowledge ( the foundation of western civilization ) originated in the near east or even in india. Heck, people still argue about who "discovered" america first. There is national, religious, ethnic, racial, gender, sexuality pride tied to "priority". It's not just in science/math, it's applies to everything.

> These figures are supposed to be idolized and idealized

Why? They weren't particularly good people. Neither were saints.

> Achilles elevated Greece by giving Greeks something to collectively aspire to or adulate.

Are you talking about science/knowledge or politics? But you are right on the point. It's what this is all about at the end of the day. Politics.

Without politics, the discovery/knowledge would be what is important. Because of politics, the people become the focal point.

But one soon realizes you probably won't get heroic credit even if you do contribute something heroic, neutralizing that encouragement.

Therefore, you'd better do it for the love of the work itself or for how it helps others.

There's no limit to what you can accomplish if you don't mind who gets the credit.

This is always been the case---medieval and renaissance thinkers would publish anagrams of their key findings because they didn't want to give someone else the advantage of knowing the finding but also wanted to prove that they thought of the idea. IIRC, Isaac Newton did not publish any of his findings until someone else threatened to publish their independent results. And he's known as the creator of calculus because the British Royal Society got into an academic slap-fight with the French.

Move Newton, Faraday, Maxwell and Einstein 10kms away from where they were born, surround them by a different set of chimps and the story doesnt end the same way.

A good book from Niall Ferguson - the Sqaure and the Tower - makes the case tradionally Historians have studied individuals instead of groups because its easier to collect data on one chimp versus the entire troupe.

https://en.wikipedia.org/wiki/Von_Neumann_architecture

von Neumann simply described the work of Eckert and Mauchly on the ENIAC in a written report. And his name was on the report which made people think that he came up with the idea, which was false. It also resulted in a patent dispute -- it's interesting to imagine what would have happened if the concept had been patented. The book goes into detail on this.

The wikipedia article also talks about Turing machines as a precedent that store data and code in the same place. But ironically I'd say that probably gives him too much credit! I think a large share should go to the people who designed a working machine, because it's easy to say come up with the idea of an airplane; much harder to make it work :) And to me it seems unlikely that the Turing Machine, which was an idea created to prove mathematical facts, was a major inspiration for the design of the ENIAC.

Finally, even though the author of this web page has his own credit dispute, I appreciate this elaboration on the credit assigned to Turing.

People have had that perched-on-giants feeling for some time:

This concept has been traced to the 12th century, attributed to Bernard of Chartres. Its most familiar expression in English is by Isaac Newton in 1675: "If I have seen further it is by standing on the shoulders of Giants."

https://en.wikipedia.org/wiki/Standing_on_the_shoulders_of_g...

Once you start digging you realize that nothing is as simple. For example for physics, "Physics for Poets" by Robert H. March is an eye opener.

I agree with this. It's certainly the case that I wish more people knew of Alonso Church and Kurt Gödel, but you have to realize in a "PR" sense that it's simply not going to be feasible to teach the general public about their contributions.

And Turing's contributions were genuinely ground-breaking, there's a reason that computer science is lousy with concepts named after or by him (Turing machines, Turing-completeness, even the word "computing" was arguably coined in "On Computable Numbers"). He also thought deeply and hard about the philosophical implications to computing in a way that others didn't (the "Turing test" being the obvious example).

In addition: when a mathematically inclined person describes any kind of mathematical concept to laymen, the first question is always "Yeah, but what is that actually useful for?", asked with a certain amount of disdain. With Turing, the answer is powerful: "How about defeating the Nazis and laying the foundation for modern society?". That case is harder to make for Church or Gödel: they obviously didn't work for the GCSE, and "lambda calculus" as a concept is a much more abstract thing than Turing machines, which laymen can readily understand (i.e. it's "just" a computer).

Add to that the fact that Turing's story is not just about computing, or code-breaking, it's also the story of the suffering that society inflicted on gay men. The fact that he was shamed into suicide is just all the more reason to celebrate him now.

I agree with the basic point of the article, but I have no issue with giving Alan Turing this title. He earned it.

So, Founding Fathers of computing science becomes mixed - starting from those low brow thinkers we call journalists - with the idea of Founding Father of computing. And this is not only unfair, but technically wrong.

Originality is easily defined as who did sth first.

This might not be the same as influence of some work. It might be that someone else does a lot of groundbreaking work which actually makes sth work (e.g. Goodfellow et al for GAN). You can say the GAN paper had more influence than Schmidhubers Adversarial Curiosity Principle.

Also, of course some newer authors might not know of all the old work. So it might be that people get the same ideas. So when Goodfellow got the idea for GAN, he might not have known about Schmidhubers Adversarial Curiosity.

The problem is, sometimes people did know about the other original work but intentionally do not cite them. You can not really know. People of course will tell you they did not know. But this can be fixed by just adding the citation. It looks bad of course when there are signs that they should have known, so it was really intentionally.

There is also a lot of arguing when sth is the same idea, or when sth is a different novel idea. This can be ambiguous. But for most cases which are discussed by Schmidhuber, when you look at the core idea, this is actually not so much the case. Also, this is also not so much a problem. There is less argumentation about whether sth is at least related. So this still should be cited then.

The question is then, which work should one cite. I would say all the relevant references. Which is definitely the original work, but then also other influential work. Many people just do the latter. And this is one of the criticism by Schmidhuber, that people do not give enough credit (or no credit) to the original work.

It seemed more like he felt he was unfairly being uncredited. Which is probably why he wrote this - he now cares deeply about giving credit to the right people.

You_again wants his work and that of others properly recognised. For example, his article, titled Critique of Paper by "Deep Learning Conspiracy" (Nature 521 p 436) [1] that is referenced by your link to wikipedia, cites a couple dozen pioneers of deep learning, quite apart from Schmidhuber hismelf. Quoting from it:

>> 2. LBH discuss the importance and problems of gradient descent-based learning through backpropagation (BP), and cite their own papers on BP, plus a few others, but fail to mention BP's inventors. BP's continuous form was derived in the early 1960s (Bryson, 1961; Kelley, 1960; Bryson and Ho, 1969). Dreyfus (1962) published the elegant derivation of BP based on the chain rule only. BP's modern efficient version for discrete sparse networks (including FORTRAN code) was published by Linnainmaa (1970). Dreyfus (1973) used BP to change weights of controllers in proportion to such gradients. By 1980, automatic differentiation could derive BP for any differentiable graph (Speelpenning, 1980). Werbos (1982) published the first application of BP to NNs, extending thoughts in his 1974 thesis (cited by LBH), which did not have Linnainmaa's (1970) modern, efficient form of BP. BP for NNs on computers 10,000 times faster per Dollar than those of the 1960s can yield useful internal representations, as shown by Rumelhart et al. (1986), who also did not cite BP's inventors.

That is not "wanting to be the center of attention". It is very much asking for proper attribution of research results. Failing to do so is a scientific scandal, especially when the work cited has contributed towards a Turing award.

__________

[1] https://people.idsia.ch/~juergen/deep-learning-conspiracy.ht...

  > Polish mathematicians Marian Rejewski, Jerzy Rozycki
  > and Henryk Zygalski who were the first to break the Enigma
  > code

The person who toured us around was telling us a brief story of how the Enigma was broken, starting with Bletchley Park. Me or maybe my friend asked who was the first to break Enigma, and he immediately answered that it was Turing, then noticed our puzzled face expressions, and added something along 'ah.. yeah.. and Polish did some minor work too' :). Just an anecdote.

Since when are movies supposed to be accurate historical references? They are made to be entertaining, so facts get kicked out of the door from Day 1.

Since it wasn't linked, Bombe was based on the Polish Bomba machine:

https://en.wikipedia.org/wiki/Bomba_(cryptography)

After reading that I sat here for a minute and racked my brain as to who my childhood 'hero' might be. I can't remember a single person.

It's amusing to me how much of intellectual work deals in a currency of status. Getting/giving credit for things appears to be the Prime Directive, at least among the observers. We've now graduated to not only stressing who is responsible but what demographic groups they are a part of.

Now, it could be that the real deal groundbreaking folks don't give a damn. Tip o' the hat to those people.

The vast majority of people working anywhere near mathematics, physical sciences or electrical engineering (the 3 founding pillars of CS) in the 1920s and 1930s probably worked on problems related to WW2 during WW2. You can equally state that motivating claim for a lot of other people.

I think Turing gets the Media Treatment^TM because there's a lot of Tragic Hero Energy in his story:

<A gay man in an era that summarily rejected him [and we tell this story in an era that is extremely oversensitive and hyper-reactive to this particular sort of injustice]; a smart, shy pupil whose closest childhood friend (and suspected lover) died early of a now-extinct illness; a mathematician who dreamed of how numbers and lookup tables could hold a conversation, saw them used and counter-used to destroy cities and murder millions, then was finally rewarded with prison and humiliation by the people he fought for.>

Turing himself off course deserves all praise and glory and the righteous anger for how he was treated in his last years, but I think our era's affinity for him is just the old mechanism of people digging the past for battles that reflect the moral values they're currently (fighting for|winning|losing), see also the misguided claim that Ada Lovelace is the first "computer programmer", usually followed by a looong screed about Women In Tech.

We just like a good story to exaggerate and make it reflect our current moral memes, and the idea of a Man|Woman Ahead Of Their Times is a catch by this standard.

This is a misunderstanding of the Turing machine model. The Turing machine is not designed to be a realistically implementable physical machine, and indeed there are no details in Turing's paper on how such a physical machine could be achieved.

Instead, the Turing machine model is designed to be a mechanistic model of what a mathematician does (in rote day-to-day tasks at least). The tape is a representation of the notebook where the mathematician writes down notes. The read head is a representation of the mathematician reading from the notebook.

It's fascinating to read the paper because of this, since it spends quite a few paragraphs showing that this simplification doesn't lose anything of the mathematician's work. It spends some time noting that even though paper is two-dimensional, it can be losslessly compressed on unidimensional tape. It spends time noting that writing/reading one symbol at a time is a good enough approximation for human writing/reading. It spends time noting that the next step doesn't need to depend on more than the current symbol + internal state, as human attention is also focused on a limited number of symbols.

This is actually why Turing's paper is so convincing on the argument of universal computation - not because the machine is realizable, but because it's hard to invent anything that a mathematician does while calculating that is not captured by the Turing machine model.

I very much recommend reading the first part of the paper [0] to see this argument (the second part, where it is proven that this abstract machine can in fact compute all known numbers, is more technical and flew over my own head).

[0] PDF https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

Not proving, conjecturing. It's not proven until this day: https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis

That's really important in an industry saturated with hype, elitism and general nonsense. Anything I can do in Rust, you can do in Assembly, so I've got some explaining to do (I can probably succeed in this example, others maybe not).

If Turing actually failed to deliver the goods on "completeness", I'd really like to resolve that.

Of course Turing machine is more easy to understand because... it's a theoritical machine, after all. On the other side, lambda calculus was weirder, I didn't get it until learning Haskell :D

I think register machines are more intuitive than Turing machines - they are much closer to how real world computers work.

Among so much that’s just plain wrong, I really dislike the insidious idea that Turing’s horrible punishment at the hands of the state was wrong because he was a unique genius and war hero. No, it was wrong because he was a human being and being gay should not be a crime!

That line of thought makes it harder to argue that no, Turing may have been a genius but wasn’t unique, he was just a significant player in a rich field. That doesn’t make him any less interesting.

https://en.wikipedia.org/wiki/The_Imitation_Game#Historical_...

It reminds me of voting systems, but maybe that’s just because of the election yesterday. If you want to give singular nontransferrable credit, the things you say are important because giving someone credit takes it away from someone else. Division and fighting become the right answers. But if you spread the credit around, saying Leibniz and Newton both get calculus credit (and probably not just those two!), then discussions of which one should get the title of The One And Only Calculus Hero just seems absurd.

> The computable countability of the partial maps from N to N.

Can anybody give an example?

does this have any to do with rationals? or is it more related to limits and calculus?

Yes, the set of functions computable with mu recursion, or with lambda calculus, is the same as the set of functions computable with a Turing machine. (Turing in his original paper showed that the set is the same for his system and Church's, and the proofs for mu recursion and many other systems are well-known.)

> I'll totally agree that from a teacher's perspective, the Turing machine is the better explanation.

When I learned this, the instructor did not use Turing machines. They used mu recursion. That has the advantage that you don't first define a machine and then derive the set of functions, instead you go straight to the functions. But I agree that Sipser (which I understand to be the most popular text today) defines a computable function as one that is computed by Turing machine, and to my mind that has the advantage of honestly suggesting to students what they are about.

The Rekursiv advantage is its ability to do recursion on the bare-metal. It's described as an object-oriented architecture. Its memory was a persistent store of objects, with each object having its size, type, and position known in memory.

[1] https://en.wikipedia.org/wiki/Rekursiv

That theory continues to bear fruit as the history of programming languages is almost entirely about bringing new and "better" abstractions to problems and then translating them to "dumber, more practical" machines. We have programming languages today modeled directly off the elegance of (though now sometimes still a few steps removed from being direct implementations of) the lambda calculus and mu-recursive functions, and the amazing thing is that they work great even given how "dumb" and "inelegant" our machines can be in practice.

See also https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis

This is already proven to be the case. Mu-recursion (which IIRC is not Godel's general recursive functions, despite what Wikipedia says; Kleene was the originator of the mu operator. Godel's general recursive functions are defined in a separate, but yet again equivalent way that directly extends primitive recursion), Turing machines, and the lambda calculus are all proven to be exactly equivalent to each other. The fact that these three independent approaches to computability are all equivalent is why we have strong informal justification that the Church-Turing Thesis (a non-mathematical statement) holds.

Separately, there's a sentiment that I've seen come up several times on HN that somehow the lambda calculus, mu-recursion, or general recursion is more "mathematical" and Turing machines are less "mathematical."

I want to push back on that. The mathematical field of computability is based almost entirely off of Turing machines because there are many classes of mathematical and logical problems that are easy to state with Turing machines and extremely awkward/almost impossible to state with the lambda calculus and mu-recursion (this is consistent with my previous statement that the three are all equivalent in power because computability theory often deals with non-computable functions, in particular trying to specify exactly how non-computable something is). The notion of oracles, which then leads to a rich theory of things like Turing jumps and the arithmetical hierarchy, is trivial to state with Turing machines and very unwieldy to state in these other formalisms.

Likewise the lambda calculus and mu-recursion (but not general recursion) provide a very poor foundation to do complexity theory in CS. Unlike Turing machines, where it is fairly easy to discern what is a constant time operator, the story is much more complicated for the lambda calculus, where to the best of my knowledge, analyzing complexity in the formalism of the lambda calculus, instead of translating it to Turing machines, is still an open problem.

There is indeed a mathematical elegance to Turing machines that makes it so that most of the mathematics of computability is studied with Turing machines rather than the lambda calculus.

The lambda calculus on the other hand is invaluable when studying programming language theory, but we should not mistake PLT to be representative of the wider field of mathematics or theoretical CS.

EDIT: I should perhaps make clear that if I put on my mathematical hat, mu-recursive functions seem like the most familiar formalism, because they align with a common way families of things are defined in mathematics (specify individual members and then generate the rest through some relationship). However, I would contend that for the majority of mathematicians outside of computability theory, the lambda calculus and Turing machines seem equally "strange."