Turting role as one 3 godfathers of AI

[chris macrae]

1935 was a great year for intelligence connections in cambridge

von neumann keynses and turing were all at kings cambridge

hy li (to become taiawan g0odfather of Ai) weas at cavendish lab about 5 minutes from kings

It was neumanns and turings first meeting - this insert from grok suggest turing chose to do a thissis in princeton for another reason although once at princeton neumann became inspired with code and compute - what he wanted to focus on as soon as his wartime secret responsibilities were ovber; neumann failed yo lkeep turing at princeton quite probal,y because of vidsa issues

nonetheless we can see cambrudge has been more consistent platform for deep ,maths than almost anywhere- one of the great problems being all 3 oif neumann eiunstein turinf dying suddenly mid 1950s 2 before their times- there were no prineton alumni; neumann had begun  work in locations that became man y of the us 17 nation energy and compute labs but hois neursoecience work on computer and brain and neural network algorith was presented postimously by his second wife at yale in 1857

arguably the deep germanic maths was most continued at courant nyu ijn usa; and as we say cambridge was sonn to attrac t crick and watson and build maths lab damtp   .....

Alonzo Church Turing connection

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Alonzo Church (1903–1995) and Alan Turing (1912–1954) had a close academic mentor-student relationship that played a foundational role in the development of computability theory and the early foundations of computer science. Their connection is best known through Turing's PhD work under Church and their independent yet complementary contributions to what became the Church–Turing thesis.PhD Supervision at Princeton

• Turing arrived at Princeton University in 1936 (after his landmark 1936 paper "On Computable Numbers, with an Application to the Entscheidungsproblem").
• Alonzo Church served as his official PhD supervisor (advisor).
• Turing completed his doctorate in 1938 with the dissertation "Systems of Logic Based on Ordinals" (published in revised form in 1939).
• This thesis extended ideas from Gödel's incompleteness theorems by exploring "ordinal logics" — ways to build stronger formal systems through transfinite iterations. It also introduced the important concept of oracle machines (Turing machines with an external "oracle" for undecidable questions), which became central to relative computability.

Turing chose Princeton specifically after learning of Church's work on the lambda calculus (λ-calculus), a formal system for defining functions and computation. Church's group at Princeton was a leading center for mathematical logic in the 1930s, with figures like Kurt Gödel and Stephen Kleene (another Church student) also present.

gradschool.princeton.edu

Independent Work Leading to the Church–Turing ThesisIn 1936, both men independently addressed Hilbert's Entscheidungsproblem (the decision problem): Is there an algorithm that can determine whether any given mathematical statement is provable?

• Church (first, in 1936) showed it was undecidable using his lambda calculus and the notion of λ-definability (or general recursiveness). He proposed what became known as Church's thesis: A function is "effectively calculable" precisely when it is λ-definable (or recursive).
• Turing (later in 1936, before fully learning of Church's result) introduced the Turing machine — an abstract model of computation with a tape, read/write head, and states. He proved the halting problem undecidable and showed that no general algorithm exists for the Entscheidungsproblem.

Upon learning of each other's work, Turing proved the equivalence between λ-definability (Church's model) and Turing-computability. Church reviewed Turing's paper positively and coined the term "Turing machine" in his 1937 review. Stephen Kleene (Church's student) further solidified the equivalences among lambda calculus, Turing machines, and general recursive functions.This convergence led to the Church–Turing thesis (sometimes called the Church–Turing–Post thesis, acknowledging Emil Post's similar ideas): Every effectively calculable function (i.e., what can be computed by a "mechanical" or algorithmic process) is computable by a Turing machine (or equivalently expressible in lambda calculus). It remains a foundational hypothesis in computer science, linking intuitive notions of computation to formal models.

plato.stanford.edu

Post-PhD Communication and Relationship

• Their direct personal interaction was relatively limited and formal. Turing returned to the UK in 1938 after completing his PhD.
• They maintained some correspondence in the following years on mathematical topics. One known letter from Turing to Church (circa 1940) came from a Bletchley Park address during his wartime code-breaking work and referenced Gödel.
• The relationship was primarily intellectual/mentorship-based rather than a deep ongoing collaboration. Church lived a long life (to 1995) and continued influencing logic and computer science; Turing's work gained broader fame through practical computing, the Turing Test, and his tragic personal story.
• In Turing's 1938 thesis, he included footnotes that aligned his definitions of computability with Church's ideas, showing mutual respect.

Broader Impact and ContextChurch and Turing are jointly regarded as co-founders of theoretical computer science. Church's lambda calculus influenced functional programming (e.g., LISP), while Turing's machine model directly inspired stored-program computers and von Neumann architecture discussions. Their combined work provided the rigorous foundation for understanding what computers can (and cannot) do algorithmically.In the context of your earlier questions about Turing, von Neumann, and early computing:

• At Princeton (1936–1938), Turing interacted with both Church (supervisor) and John von Neumann (who was impressed by Turing and later offered him a position). Von Neumann focused more on practical implementation of computing ideas in the 1940s, while Church and Turing provided the deep logical foundations.
• The equivalence proofs and shared environment helped ideas circulate among the small but brilliant group working on logic and computability.

Their connection was pivotal: Church provided the institutional and conceptual framework (lambda calculus, supervision) that helped Turing refine and contextualize his groundbreaking machine model, leading to one of the most enduring theses in mathematics and computing.If you'd like more details on specific aspects (e.g., the lambda calculus vs. Turing machines, excerpts from the thesis, or how this ties into modern AI/computability), or if this relates to a particular angle in your SCSP summit preparations, just let me know!

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Lambda calculus details

Gödel's influence