thesis of the church

The Church-Turing Thesis Explained: A Deep Dive into the Foundations of Computation

• by history tools
• March 28, 2024

The Church-Turing thesis is a fundamental tenet of computer science that provides the very definition of what it means to be "computable." In essence, it claims that any function that is intuitively "computable" by an effective procedure can be computed by a Turing machine. While this may sound simple, the implications are profound, touching everything from the limits of logical proof to the nature of human cognition.

As a digital technology expert, I find the Church-Turing thesis endlessly fascinating, both for its elegance as an idea and its relevance to the technology we use every day. Far from an archaic piece of mathematical trivia, it remains the beating heart of theoretical computer science nearly a century after its inception. Let‘s dive in and explore the origins, meaning, and implications of this landmark idea in the history of human knowledge.

Origins of Computability Theory in the 1930s

Independently, a young British mathematician named Alan Turing was exploring similar questions. In 1936, Turing published a groundbreaking paper introducing what he called "a-machines," later known as Turing machines.[^3] A Turing machine is an abstract device that can perform any computation that can be done by a human following a finite set of explicit instructions. It consists of:

• An infinite tape divided into cells that can each hold a symbol from a finite alphabet
• A read/write head that can move left or right on the tape and read or write symbols
• A finite set of states the machine can be in, with transition rules that specify how the state and tape contents change based on the current state and symbol being read

Turing showed that any function computable by a Turing machine was also expressible in the lambda calculus, and vice versa. This established the equivalence of the two formalisms and led Church to propose what became known as "Church‘s thesis" or the "Church-Turing thesis":

"A function is effectively calculable if and only if it is computable by a Turing machine or expressible in the lambda calculus."

While Church and Turing could not prove that their formalisms captured all possible notions of computability, the thesis has stood the test of time remarkably well. In the 85 years since it was proposed, no one has found a convincing counterexample of a function that is intuitively computable but not Turing computable. The Church-Turing thesis has become a foundational axiom of computer science.

Computable and Uncomputable Functions

To get a concrete sense of what the Church-Turing thesis means, let‘s look at some examples of computable and uncomputable functions. A classic computable function is primality testing: given a natural number n, determine whether n is prime (i.e. evenly divisible only by 1 and itself). Here‘s a simple Python program that implements a primality test:

This function takes a number n as input, checks if it‘s evenly divisible by any number between 2 and its square root, and returns True if no such divisor is found (meaning n is prime) or False otherwise. It‘s easy to see that this function is computable by a Turing machine: we can specify a finite set of rules for updating the machine‘s state and tape contents to implement the same logic as the Python code. Primality testing is a computable problem.

Turing‘s proof is a clever use of diagonalization and self-reference. Suppose a halting solver Turing machine H existed. We could then construct a machine M that takes a program P as input and does the following:

• Run H on P and P itself as input
• If H says P halts on itself, go into an infinite loop; otherwise, halt immediately

Now, what happens if we run M on itself? If M halts on itself, then H would have determined that M does not halt on itself, in which case the second step would cause M to halt. But if M doesn‘t halt on itself, then H would have determined that M halts on itself, in which case the second step would cause M to go into an infinite loop! This contradiction means our original assumption that H exists must have been false. The halting problem is uncomputable.

This kind of self-referential paradox crops up often in computability theory. It‘s reminiscent of Gödel‘s incompleteness theorems, and in a sense, establishes a fundamental limit on what can be algorithmically decided. Not every well-defined mathematical question has a computable solution.

Variations and Extensions to the Church-Turing Thesis

While the Church-Turing thesis is widely accepted, there have been various philosophical and mathematical challenges to it over the years. Some researchers have proposed notions of "hypercomputation" that go beyond the limits of Turing machines, such as:

• Oracle machines: Turing machines equipped with a black box "oracle" that can magically solve the halting problem or other uncomputable tasks
• Analog computers: Machines that perform computation using continuous physical quantities like voltages or fluid pressures instead of discrete symbols
• Quantum computers: Devices that harness quantum superposition and entanglement to perform computations, potentially offering exponential speedups over classical computers for certain problems

Personally, I find the Church-Turing thesis compelling both as a mathematical foundation and an empirical claim. The fact that nearly a century of research has failed to produce a convincing counterexample suggests that Turing machines really do capture something fundamental about the nature of computation. At the same time, I‘m excited by theoretical and technological developments that probe the boundaries of the computable, and I try to keep an open mind about the potential for new computational models.

The Computational Lens on Mind and Universe

Beyond its central role in computer science, the Church-Turing thesis provides a powerful conceptual framework for viewing the world at large through a computational lens. The notion that any effective procedure can be realized by a Turing machine suggests a kind of universal computability to the cosmos. And if the universe itself is a computer, might the human mind simply be an embodied subprogram?

The strong form of this view is that human cognition is Turing-computable – that everything from perception to reasoning to consciousness can in principle be implemented by a sufficiently advanced AI system. If this is true, then the Church-Turing thesis places ultimate limits on the nature of intelligence. No matter how sophisticated our technology becomes, the space of possible minds will be constrained by the space of Turing-computable functions.

As a computer scientist, I lean towards a computational view of mind, but I also recognize the difficulty of reducing something as complex and subjective as human experience to the cut-and-dried formalisms of Turing machines. While I believe artificial general intelligence is possible in principle, I suspect the Church-Turing thesis alone is too crude a tool to fully delineate the space of possible minds. We likely need a richer theory of computation that can account for context, embodiment, and interaction with the environment.

This connects to perhaps the grandest application of the Church-Turing lens: viewing the physical universe itself as a computation. Digital physics, as championed by thinkers like Konrad Zuse, Edward Fredkin, and Stephen Wolfram, models the cosmos as a giant (quantum) computer, with the physics constrained by the Church-Turing thesis.[^12] In this view, spacetime is the hardware, particles are the software, and the speed of light is the clock rate.

While a compelling metaphor, digital physics remains highly speculative. We have no empirical evidence that the universe is discretized at the Planck scale or that physical dynamics are bounded by Turing computability. In fact, some have argued that a discrete, computable universe would violate locality and Lorentz invariance.[^13] For now, digital physics is more of a philosophical stance than a scientific theory.

The Church-Turing thesis is a profound and enduring idea that has shaped the foundations of computer science and our philosophical understanding of the nature of mind and cosmos. By precisely defining what it means for a function to be "computable," Church and Turing gave us a powerful mathematical framework for reasoning about the limits of algorithmic problem-solving.

While the thesis remains unproven in a formal sense, its remarkable resilience over nearly a century attests to its conceptual power. No one has yet found a convincing example of an intuitively computable function that is not Turing computable. The Church-Turing thesis has become a bedrock assumption of modern computability theory.

At the same time, the thesis raises deep questions about the nature of computation in the physical universe and human minds. Are there forms of hypercomputation that transcend the Church-Turing limit? Is the brain itself bounded by Turing computability? Might the universe be a vast digital computer constrained by the laws of Church and Turing? These are heady philosophical questions that have inspired much debate and speculation.

As our digital technologies continue to advance at a dizzying pace, it‘s worth reflecting on the Church-Turing foundations that make it all possible. The smartphones in our pockets and the supercomputers in the cloud are all in a sense instantiations of Turing‘s original vision – an astoundingly general model of mechanical computation. Every time you run a program, send an email, or do a web search, you‘re implicitly relying on the Church-Turing thesis. That is the mark of a truly deep idea.

Moving forward, I believe the Church-Turing thesis will remain a vital touchstone for anyone seeking to understand the nature of computation – in silicon, in carbon, and in the cosmos. While it may not be the final word on computability, it is a crucial piece of the puzzle, and one that will continue to inspire and inform our thinking about the algorithmic universe we inhabit. As a digital technology expert, I find that an endlessly exciting prospect.

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Supplement to The Church-Turing Thesis

The rise and fall of the entscheidungsproblem, a. stating the entscheidungsproblem, b.1 a “philosophers’ stone”, b.2 the consistency of mathematics, c. partial solutions, d. negative vibes, e. the church-turing result, f. aftermath.

Turing gave two formulations of what he called “the Hilbert Entscheidungsproblem ” (1936 [2004: 84, 87]):

[Is there a] general process [and a few paragraphs later, he emphasizes “general (mechanical) process”] for determining whether a given formula \(A\) of the functional calculus K is provable

[Is there a] machine [Turing machine] which, supplied with any one \(A\) of these formulae, will eventually say whether \(A\) is provable.

Given Turing’s thesis, the two formulations are equivalent.

Church stated the Entscheidungsproblem more generally:

By the Entscheidungsproblem of a system of symbolic logic is here understood the problem to find an effective method by which, given any expression Q in the notation of the system, it can be determined whether or not Q is provable in the system. (Church 1936b: 41)

While Turing and Church both formulated the Entscheidungsproblem in terms of determining whether or not a formula is provable , Hilbert and Ackermann had formulated it in terms of universal validity (“ allgemeingültigkeit ”):

The Entscheidungsproblem is solved once we know a procedure that allows us to decide, by means of finitely many operations, whether a given logical expression is universally valid or, alternatively, satisfiable. (Hilbert & Ackermann 1928: 73)

A universally valid formula of the functional calculus (e.g., \((x)(Fx \lor \neg Fx)\) or, in modern notation, \(\forall x(Fx \lor \neg Fx)\)) must contain neither free variables nor individual constants, and is such that a true assertion results no matter which replacements (of suitable adicity) are made for the formula’s predicate symbols, and no matter which objects are allocated to its variables (Hilbert & Ackermann 1928: 72–73). In the case of a satisfiable formula (e.g., \((Ex)Fx\) or, in modern notation, \(\exists xFx\)) there must be at least one way of replacing its predicate symbols and of allocating objects to its variables so that a true assertion results. Universal validity and satisfiability are related as follows: \(A\) is universally valid if and only if \(\neg A\) is not satisfiable (1928: 73). In the above quotation, therefore, Hilbert and Ackermann are presenting two different but equivalent forms of the Entscheidungsproblem , one employing the concept of universal validity (or simply validity, as we would say today) and the other the closely related concept of satisfiability. The hunt was on for what they called “a determinate, finite procedure” (1928: 17) for deciding, of any given formula \(A\) of the functional calculus, whether or not \(A\) is universally valid, or, equivalently, for deciding whether or not \(\neg A\) is satisfiable.

Turing’s and Church’s provability formulation of the Entscheidungsproblem and the Hilbert-Ackermann formulation in terms of validity are in fact logically equivalent, as Church noted in 1936 (1936b: 41). This equivalence is a consequence of Gödel’s proof that (where \(A\) is any formula of the functional calculus) if \(A\) is universally valid then \(A\) is provable in the calculus. (The proof was presented originally in Gödel’s 1929 doctoral dissertation and then published as Gödel 1930.) Nevertheless, the provability version of the Entscheidungsproblem is arguably superior, since it asks a question about a specific axiom system, as do the allied problems of consistency and completeness. In modern treatments, the problems of consistency, completeness and decidability for an axiom system lie at the heart of proof theory, the area of modern logic founded by Hilbert.

In 1928, Hilbert and Ackermann were certainly aware of the provability formulation (1928: 86, and see below) but they gave the validity formulation the starring role. It was von Neumann who emphasized the provability formulation, calling the process of proof the “real heart of mathematics” (von Neumann 1927: 10).

B. Why the problem mattered

Why was it that the Entscheidungsproblem was regarded as “the main problem of mathematical logic” and “the most general problem of mathematics”? There were fundamentally two reasons.

In his turn-of-the-century Paris lecture, Hilbert explained the idea of axiomatizing a subject-matter and using provability from the axioms as the touchstone of truth:

When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science…. [N]o statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps. (Hilbert 1900: 264 [trans. 1902: 447])

He also famously said in his lecture that there is “no ignorabimus ” in mathematics—there is nothing that we shall not know (Hilbert 1900: 262). It was a phrase he would often use to express his “conviction of the solvability of every mathematical problem” (Hilbert 1900: 262 [trans. 1902: 445]). Thirty years later he reiterated the same standpoint, in a valedictory lecture in Königsberg:

[I]n my opinion, there is no such thing as an unsolvable problem. On the contrary, instead of foolish ignorabimus, our slogan is: We must know, We will know. (Hilbert 1930b: 963)

It was Hilbert’s great invention, proof theory, that would, he thought, supply the basis for answering all “meaningful questions”:

[O]n the basis of the proof theory I have outlined, every fundamental question can be answered in a way that is unambiguous and mathematically precise. (Hilbert 1930a: 8, 9)

However, proving a statement in an axiom system can be a tricky matter. Suppose the mathematician fails to discover a derivation of the statement from the axioms, what follows then? The failure could be because the statement is indeed not provable—but, on the other hand, there might be a way to prove it that escaped the mathematician’s attention. Success in finding a proof brings certainty, whereas failure leaves us uncertain. That is why a decision method is so desirable. The method enables the mathematician to determine whether or not the statement is provable. Thinking up proofs is a serendipitous activity that involves, as Behmann put it, “inventive insight”, “creative thought”, and searching “in every direction of the compass” (Behmann 1921: 2–3 [trans. 2015: 174]). Whereas a decision method is a purely mechanical process that is guaranteed to produce a definite answer.

Herbrand wrote of the “present great ambition, the solution of the Entscheidungsproblem ”, saying this “would provide a general method in mathematics” and enable us to “to decide with certainty whether a proposition is true in a given theory” (Herbrand 1930b: 254–255). In his 1921 lecture, Behmann had referred to this desired general method as a “philosophers’ stone” (“ Stein der Weisen ”), by means of which mathematicians “would be able to solve every problem posed ”—or even “delegate the work of proving mathematical theorems to mathematical laborers” (Behmann 1921: 3–4 [trans. 2015: 175], emphasis added). Turing’s mentor Newman, familiar of course with the work of the Hilbert group, also referred to the solution of the Entscheidungsproblem as a philosophers’ stone. Hilbert and Ackermann themselves said, in 1928:

Solving this general Entscheidungsproblem [for the second-order functional calculus] would … enable us to decide on the provability or unprovability of any mathematical proposition , at least in principle. (Hilbert & Ackermann 1928: 86, emphasis added)

Schönfinkel went even further. He was another member of the Göttingen group who praised “Leibniz’s bold conjectures” (Schönfinkel 192?: 2). (For an excellent biographical article on Schönfinkel, see Wolfram 2021.) In an early draft of what would become Bernays and Schönfinkel (1928), Schönfinkel wrote of “the great Entscheidungsproblem of mathematical logic” which, thanks to Hilbert, mathematicians now had “the courage and the boldness to dare to touch”. He described the Entscheidungsproblem as “the problem of ‘solving all problems’”—not just all mathematical problems but all problems. Because once there is a method for solving all mathematical problems:

thereafter everything else is indeed lightweight, once this “Gordian knot” is undone (since “the world is written in mathematical characters”). (Schönfinkel 192?: 1–2)

The second reason the Entscheidungsproblem was regarded as so important was its connection with the quest for proofs of the consistency of mathematical systems. A system is said to be inconsistent if there is some statement \(A\) such that both \(A\) and \(\neg A\) are provable from the axioms. The system is consistent if (and only if) there is no statement for which this sorry situation is the case.

By the early twentieth century, mathematics—until then a “paragon of reliability and truth”, Hilbert said—was in trouble (Hilbert 1926: 170, trans. in van Heijenoort 1967: 375). Its reputation had been “lost due to the paradoxes of set theory” (Hilbert 1922: 160). Hilbert explained:

[C]ontradictions appeared, sporadically at first, then ever more severely and ominously…. In particular, a contradiction discovered by Zermelo and Russell [“Russell’s paradox”] had, when it became known, a downright catastrophic effect in the world of mathematics. (Hilbert 1926: 169 [trans. 1967: 375])

Herbrand takes up the story:

One must take great care … Set theory has produced famous examples [of inconsistencies] … But there is nothing to show us that similar issues do not arise for the theories that seem to us the most finished, like arithmetic. Could it be that arithmetic is inconsistent? (Herbrand 1930b: 250–251)

“[P]roofs of consistency would be very useful to dispel lingering doubts”, wrote Herbrand (1930b: 251). Hilbert put it even more forcefully:

[T]he situation in which we presently find ourselves with respect to the paradoxes is in the long run intolerable. (Hilbert 1926: 170 [trans. 1967: 375])

Proving “the consistency of the arithmetic axioms” is “urgent”, he said—“a burning question” (Hilbert 1926: 179 [trans. 1967: 383]; Hilbert 1930a: 3). Hilbert’s overarching concern was to “re-establish mathematics’ old reputation for incontestable truth” (Hilbert 1922: 160), and he was “convinced that it must be possible to find a direct proof of the consistency of the arithmetical axioms” (Hilbert 1900: 265).

A solution to the Entscheidungsproblem would furnish a route to establishing consistency: “By means of the decision method, issues of consistency would also be able to be resolved” (Hilbert & Ackermann 1928: 76). This is because the decision method could be used to establish whether or not \(A\) and \(\neg A\) are both provable—so gaining a definite answer to the question “Is the system consistent?”. Hilbert believed that, following the discovery of the decision method, arithmetic and analysis would be proved consistent, thereby “establish[ing] that mathematical propositions are indeed incontestable and absolute truths” (Hilbert 1922: 162). Mathematics’ reputation would be regained.

By the time Turing and Church engaged with the Entscheidungsproblem , a number of decision methods were known for parts of the functional calculus.

Besides the previously mentioned Hilbert-Bernays decision method for the propositional calculus (and Peirce’s much less well-known method), and also the Löwenheim method for the monadic fragment of the functional calculus (later improved by Behmann in his 1922 paper, where it was additionally proved that the monadic fragment of the second-order functional calculus is decidable), there were various decision methods that succeeded providing the functional calculus was restricted in one way or another. Bernays and Schönfinkel (1928) showed there is a decision method when formulae containing at most two individual variables are considered. At Cambridge, Ramsey devised a method that worked provided existential quantifiers were omitted from the calculus, and any universal quantifiers were stacked one after another at the very beginning of a formula (with no negation sign preceding any of them, and with their scope extending to the end of the formula). Such formulae are interesting, Ramsey pointed out, since they represent “general laws” (Ramsey 1930: 272; Langford 1926b: 115–116). Ackermann, Bernays, Schönfinkel, Gödel, Kálmar, and Herbrand devised methods for other fragments of the calculus, in which only certain patterns of quantifiers were permitted. Gödel’s 1933 paper on the Entscheidungsproblem gave a summary of some of the developments.

Despite all this attention to the decision problem, no method had been found for the full functional calculus. But according to Hilbert it was just a matter of time. He and Ackermann emphasized that “finding a general decision process is an as yet unsolved and difficult problem” (1928: 77). They exuded optimism, writing buoyantly of moving “closer to the solution of the Entscheidungsproblem ” (1928: 81).

Even in the 1920s, however, negative opinion on the solvability of the Entscheidungsproblem was building. Behmann had pointed out in his 1921 lecture that, once the “philosophers’ stone” was found, “the entirety of mathematics would indeed be transformed into one enormous triviality ” (Behmann 1921: 5 [trans. 2015: 175]). Some found this consequence highly unpalatable. In 1927, Hilbert’s student Weyl insisted that “such a philosopher’s stone has not been discovered and never will be discovered”, and a good job too, Weyl thought, since if it were, “Mathematics would thereby be trivialized” (Weyl 1927: 20 [trans. Weyl 1949: 24]). Then a year later, in 1928, Hardy announced to the Cambridge mathematicians, assembled at his Rouse Ball Lecture, that he expected no “system of rules which enable us to say whether any given formula [is] demonstrable or not” (Hardy 1929: 16). “[T]his is very fortunate”, he continued, since if there were such a system,

we should have a mechanical set of rules for the solution of all mathematical problems, and our activities as mathematicians would come to an end. (ibid.)

Hardy explained that his description of Hilbert’s ideas was “based upon that of v. Neumann, a pupil of Hilbert’s”, saying that he found von Neumann’s exposition “sharper and more sympathetic than Hilbert’s own” (Hardy 1929: 13–14). Hardy was referring to von Neumann’s “Zur Hilbertschen Beweistheorie” ( On Hilbert’s Proof Theory ), published in 1927 but completed by the middle of 1925. Von Neumann said:

So it seems that there is no way to find the general decision criterion for whether a given normal formula [i.e., a well-formed formula with no free variables] is provable. (von Neumann 1927: 11, emphasis added)

The day that undecidability lets up, mathematics in its current sense would cease to exist; into its place would step a perfectly mechanical rule, by means of which anyone could decide, of any given proposition, whether this can be proved or not. (von Neumann 1927: 12)

But how to prove that there is no general decision criterion? Von Neumann confessed he did not know:

At present, of course, we cannot demonstrate this. Moreover, no clue whatsoever exists how such an undecidability proof would have to be conducted. (von Neumann 1927: 11)

Nor did he find a proof. With a hint of resignation he said in 1931 that the Entscheidungsproblem was “far too difficult” (1931: 121). Four years later, in 1935, he visited Cambridge for a term (arriving in April, a few weeks after Newman had lectured on the Entscheidungsproblem ) and he struck up an acquaintance with a young mathematician who admired his work (Copeland & Fan 2023). The young man was of course Alan Turing, who within a few months would devise his groundbreaking proof that—just as von Neumann had hypothesized—“It is generally undecidable whether a given normal formula is provable or not” (von Neumann 1927: 12). Was there discussion, during the spring of 1935, between von Neumann and Turing (the latter already primed by Newman’s lecture) about the Entscheidungsproblem —which was, after all, the main problem of mathematical logic? Did von Neumann perhaps play a role in Turing’s decision to take on this fundamental problem and prove it unsolvable? These are tantalizing questions, and we may never know for sure.

Gödel’s famous incompleteness theorems of 1931 placed unexpected new obstacles in the way of Hilbert’s desired consistency proof for arithmetic (Gödel 1931). Suspicion also began to build that Gödel’s incompleteness results might further imply the unsolvability of the Entscheidungsproblem . Herbrand said cautiously:

[A]lthough at present the possibility of a solution to the Entscheidungsproblem seems unlikely, its impossibility has not yet been proved. (Herbrand 1931a: 56)

Carnap, who took a close interest in Hilbert’s Göttingen group and had worked with Behmann (Schiemer, Zach, & Reck 2017) wrote the following about the Hilbertian idea of a “definite criterion of validity” or “decision procedure for mathematics”, using which “we could so to speak calculate the truth or falsity of every given proposition”:

[B]ased on Gödel’s latest results, the search for a definite criterion of validity for the entire system of mathematics appears hopeless. (Carnap 1935: 163–4)

But, as Herbrand noted, nobody had proved the Entscheidungsproblem to be unsolvable. That was where Turing and Church entered the story. Newman later summed up matters as they had appeared at the time, before Church and Turing published their transformational papers in 1936:

A first blow was dealt [to the “Hilbert decision-programme”] by Gödel’s incompleteness theorem (1931), which made it clear that truth or falsehood of \(A\) could not be equated to provability of \(A\) or not-\(A\) in any finitely based logic, chosen once for all; but there still remained in principle the possibility of finding a mechanical process for deciding whether \(A\), or \(\neg A\), or neither, was formally provable in a given system. (Newman 1955: 256)

Turing summed up his show-stopping result in his usual terse way: he had shown, he said, that “the Hilbert Entscheidungsproblem can have no solution” (Turing 1936 [2004: 84]). Church, also not one to waste words, compressed his proof into a paper barely two pages long, and concluded pithily:

The general case of the Entscheidungsproblem of the engere Funktionenkalkül is unsolvable. (Church 1936b: 41)

In establishing this, Turing and Church showed moreover that there can be no decision method for the second-order functional calculus (where quantification is permitted over not just individuals but also over properties and relations), since the second-order calculus contains the first-order calculus as a fragment. The same applies to every other mathematical system containing the first-order calculus, such as arithmetic.

However, it is one of the great ironies of the history of logic that, all along, Gödel’s first incompleteness theorem of 1931 did in fact suffice to establish the undecidability of the functional calculus—although this was certainly not apparent at the time. More than three decades passed after the publication of Gödel’s paper before this corollary of his theorem was noted, by Davis (Davis 1965: 109; Kleene 1986: 136).

Naturally, this unnoticed implication of Gödel’s theorem does not diminish Turing’s and Church’s great achievement, which at the time broke new ground. However, in recent times their work is sometimes regarded as amounting to merely a smallish development of Gödel’s previously published results, on which Church’s and Turing’s work was supposedly based. This is particularly true of Turing. Turing, it is said, “merely reformulated Gödel’s work in an elegant way” (Schmidhuber 2012: 1638) and “recast” Gödel’s findings “in the guise of the Halting Problem” (Dawson 2006: 133). (For further discussion of these and similar views, see Copeland & Fan 2022.) In this connection, it is worth remembering the words of Kleene, who worked closely with Church and played a leading part in the development of computability theory in the 1930s. Kleene noted that

One sometimes encounters statements asserting that Gödel’s work laid the foundation for Church’s and Turing’s results

and commented:

Whether or not one judges that Church would have proceeded from his thesis to these [undecidability] results without his having been exposed to Gödel numbering, it seems clear that Turing in [“On Computable Numbers”] had his own train of thought, quite unalloyed by any input from Gödel. One is impressed by this in reading Turing [“On Computable Numbers”] in detail. (Kleene 1987: 491–2)

What became of the Entscheidungsproblem after the Church-Turing negative result?

In letters written in the wake of the result, Turing and Newman discussed the idea Newman had presented in his 1935 lecture, a machine that is able to decide mathematical problems. Turing wrote “I think you take a much more radically Hilbertian attitude about mathematics than I do”, responding to Newman’s statement that

If all this whole formal outfit is not about finding proofs which can be checked on a machine it’s difficult to know what it is about. (Turing c.1940 [2004: 215])

Turing noted that he saw no essential difference between a proof-checking machine and a proof-finding machine. He challenged Newman:

When you say “on a machine” do you have in mind that there is (or should be or could be, but has not been actually described anywhere) some fixed machine on which proofs are to be checked, and that the formal outfit is, as it were about this machine?

Turing called this an “extreme Hilbertian” position, and said:

If you take this attitude … there is little more to be said: we simply have to get used to the technique of this machine and resign ourselves to the fact that there are some problems to which we can never get the answer.

Turing rejected this attitude to mathematics, because, he said, “there is a fairly definite idea of a true formula which is quite different from the idea of a provable one”—mathematicians’ judgements about whether formulae are true can outrun the pronouncements of the Hilbertian machine. Turing continued in his letter:

If you think of various machines I don’t see your difficulty. One imagines different machines allowing different sets of proofs, and by choosing a suitable machine one can approximate “truth” by “provability” better than with a less suitable machine … (Turing c.1940 [2004: 215])

In Turing’s opinion, that was how the debate on the Entscheidungsproblem had panned out. The Hilbertians had wanted a single decision method for the whole of mathematics—a single computing machine or algorithm—whereas Turing showed there can be no such thing; but he emphasized that there are, nevertheless, different decision methods (machines) for different areas of mathematics (see further Copeland & Shagrir 2013). In place of the one great unsolvable decision problem, there are many lesser, but often solvable, decision problems.

In the long aftermath of the Church-Turing result, as those rough pioneering days gave way to modern computer science, Turing’s opinion became the mainstream view: Today, computer science makes use of a multiplicity of algorithms for deciding different parts of the functional calculus and other mathematical systems.

Copyright © 2023 by B. Jack Copeland < jack . copeland @ canterbury . ac . nz >

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Martin Luther and the 95 Theses

By: History.com Editors

Updated: June 6, 2019 | Original: October 29, 2009

Born in Eisleben, Germany, in 1483, Martin Luther went on to become one of Western history’s most significant figures. Luther spent his early years in relative anonymity as a monk and scholar. But in 1517 Luther penned a document attacking the Catholic Church’s corrupt practice of selling “indulgences” to absolve sin. His “95 Theses,” which propounded two central beliefs—that the Bible is the central religious authority and that humans may reach salvation only by their faith and not by their deeds—was to spark the Protestant Reformation. Although these ideas had been advanced before, Martin Luther codified them at a moment in history ripe for religious reformation. The Catholic Church was ever after divided, and the Protestantism that soon emerged was shaped by Luther’s ideas. His writings changed the course of religious and cultural history in the West.

Martin Luther (1483–1546) was born in Eisleben, Saxony (now Germany), part of the Holy Roman Empire, to parents Hans and Margaretta. Luther’s father was a prosperous businessman, and when Luther was young, his father moved the family of 10 to Mansfeld. At age five, Luther began his education at a local school where he learned reading, writing and Latin. At 13, Luther began to attend a school run by the Brethren of the Common Life in Magdeburg. The Brethren’s teachings focused on personal piety, and while there Luther developed an early interest in monastic life.

Did you know? Legend says Martin Luther was inspired to launch the Protestant Reformation while seated comfortably on the chamber pot. That cannot be confirmed, but in 2004 archeologists discovered Luther's lavatory, which was remarkably modern for its day, featuring a heated-floor system and a primitive drain.

Martin Luther Enters the Monastery

But Hans Luther had other plans for young Martin—he wanted him to become a lawyer—so he withdrew him from the school in Magdeburg and sent him to new school in Eisenach. Then, in 1501, Luther enrolled at the University of Erfurt, the premiere university in Germany at the time. There, he studied the typical curriculum of the day: arithmetic, astronomy, geometry and philosophy and he attained a Master’s degree from the school in 1505. In July of that year, Luther got caught in a violent thunderstorm, in which a bolt of lightning nearly struck him down. He considered the incident a sign from God and vowed to become a monk if he survived the storm. The storm subsided, Luther emerged unscathed and, true to his promise, Luther turned his back on his study of the law days later on July 17, 1505. Instead, he entered an Augustinian monastery.

Luther began to live the spartan and rigorous life of a monk but did not abandon his studies. Between 1507 and 1510, Luther studied at the University of Erfurt and at a university in Wittenberg. In 1510–1511, he took a break from his education to serve as a representative in Rome for the German Augustinian monasteries. In 1512, Luther received his doctorate and became a professor of biblical studies. Over the next five years Luther’s continuing theological studies would lead him to insights that would have implications for Christian thought for centuries to come.

Martin Luther Questions the Catholic Church

In early 16th-century Europe, some theologians and scholars were beginning to question the teachings of the Roman Catholic Church. It was also around this time that translations of original texts—namely, the Bible and the writings of the early church philosopher Augustine—became more widely available.

Augustine (340–430) had emphasized the primacy of the Bible rather than Church officials as the ultimate religious authority. He also believed that humans could not reach salvation by their own acts, but that only God could bestow salvation by his divine grace. In the Middle Ages the Catholic Church taught that salvation was possible through “good works,” or works of righteousness, that pleased God. Luther came to share Augustine’s two central beliefs, which would later form the basis of Protestantism.

Meanwhile, the Catholic Church’s practice of granting “indulgences” to provide absolution to sinners became increasingly corrupt. Indulgence-selling had been banned in Germany, but the practice continued unabated. In 1517, a friar named Johann Tetzel began to sell indulgences in Germany to raise funds to renovate St. Peter’s Basilica in Rome.

The 95 Theses

Committed to the idea that salvation could be reached through faith and by divine grace only, Luther vigorously objected to the corrupt practice of selling indulgences. Acting on this belief, he wrote the “Disputation on the Power and Efficacy of Indulgences,” also known as “The 95 Theses,” a list of questions and propositions for debate. Popular legend has it that on October 31, 1517 Luther defiantly nailed a copy of his 95 Theses to the door of the Wittenberg Castle church. The reality was probably not so dramatic; Luther more likely hung the document on the door of the church matter-of-factly to announce the ensuing academic discussion around it that he was organizing.

The 95 Theses, which would later become the foundation of the Protestant Reformation, were written in a remarkably humble and academic tone, questioning rather than accusing. The overall thrust of the document was nonetheless quite provocative. The first two of the theses contained Luther’s central idea, that God intended believers to seek repentance and that faith alone, and not deeds, would lead to salvation. The other 93 theses, a number of them directly criticizing the practice of indulgences, supported these first two.

In addition to his criticisms of indulgences, Luther also reflected popular sentiment about the “St. Peter’s scandal” in the 95 Theses:

Why does not the pope, whose wealth today is greater than the wealth of the richest Crassus, build the basilica of St. Peter with his own money rather than with the money of poor believers?

The 95 Theses were quickly distributed throughout Germany and then made their way to Rome. In 1518, Luther was summoned to Augsburg, a city in southern Germany, to defend his opinions before an imperial diet (assembly). A debate lasting three days between Luther and Cardinal Thomas Cajetan produced no agreement. Cajetan defended the church’s use of indulgences, but Luther refused to recant and returned to Wittenberg.

Luther the Heretic

On November 9, 1518 the pope condemned Luther’s writings as conflicting with the teachings of the Church. One year later a series of commissions were convened to examine Luther’s teachings. The first papal commission found them to be heretical, but the second merely stated that Luther’s writings were “scandalous and offensive to pious ears.” Finally, in July 1520 Pope Leo X issued a papal bull (public decree) that concluded that Luther’s propositions were heretical and gave Luther 120 days to recant in Rome. Luther refused to recant, and on January 3, 1521 Pope Leo excommunicated Martin Luther from the Catholic Church.

On April 17, 1521 Luther appeared before the Diet of Worms in Germany. Refusing again to recant, Luther concluded his testimony with the defiant statement: “Here I stand. God help me. I can do no other.” On May 25, the Holy Roman emperor Charles V signed an edict against Luther, ordering his writings to be burned. Luther hid in the town of Eisenach for the next year, where he began work on one of his major life projects, the translation of the New Testament into German, which took him 10 months to complete.

Martin Luther's Later Years

Luther returned to Wittenberg in 1521, where the reform movement initiated by his writings had grown beyond his influence. It was no longer a purely theological cause; it had become political. Other leaders stepped up to lead the reform, and concurrently, the rebellion known as the Peasants’ War was making its way across Germany.

Luther had previously written against the Church’s adherence to clerical celibacy, and in 1525 he married Katherine of Bora, a former nun. They had five children. At the end of his life, Luther turned strident in his views, and pronounced the pope the Antichrist, advocated for the expulsion of Jews from the empire and condoned polygamy based on the practice of the patriarchs in the Old Testament.

Luther died on February 18, 1546.

Significance of Martin Luther’s Work

Martin Luther is one of the most influential figures in Western history. His writings were responsible for fractionalizing the Catholic Church and sparking the Protestant Reformation. His central teachings, that the Bible is the central source of religious authority and that salvation is reached through faith and not deeds, shaped the core of Protestantism. Although Luther was critical of the Catholic Church, he distanced himself from the radical successors who took up his mantle. Luther is remembered as a controversial figure, not only because his writings led to significant religious reform and division, but also because in later life he took on radical positions on other questions, including his pronouncements against Jews, which some have said may have portended German anti-Semitism; others dismiss them as just one man’s vitriol that did not gain a following. Some of Luther’s most significant contributions to theological history, however, such as his insistence that as the sole source of religious authority the Bible be translated and made available to everyone, were truly revolutionary in his day.

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Church-Turing Thesis

The Church-Turing thesis (formerly commonly known simply as Church's thesis) says that any real-world computation can be translated into an equivalent computation involving a Turing machine . In Church's original formulation (Church 1935, 1936), the thesis says that real-world calculation can be done using the lambda calculus , which is equivalent to using general recursive functions .

The Church-Turing thesis encompasses more kinds of computations than those originally envisioned, such as those involving cellular automata , combinators , register machines , and substitution systems . It also applies to other kinds of computations found in theoretical computer science such as quantum computing and probabilistic computing.

There are conflicting points of view about the Church-Turing thesis. One says that it can be proven, and the other says that it serves as a definition for computation. There has never been a proof, but the evidence for its validity comes from the fact that every realistic model of computation, yet discovered, has been shown to be equivalent. If there were a device which could answer questions beyond those that a Turing machine can answer, then it would be called an oracle .

Some computational models are more efficient, in terms of computation time and memory, for different tasks. For example, it is suspected that quantum computers can perform many common tasks with lower time complexity , compared to modern computers, in the sense that for large enough versions of these problems, a quantum computer would solve the problem faster than an ordinary computer. In contrast, there exist questions, such as the halting problem , which an ordinary computer cannot answer, and according to the Church-Turing thesis, no other computational device can answer such a question.

The Church-Turing thesis has been extended to a proposition about the processes in the natural world by Stephen Wolfram in his principle of computational equivalence (Wolfram 2002), which also claims that there are only a small number of intermediate levels of computing power before a system is universal and that most natural systems are universal.

This entry contributed by Todd Rowland

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What is the Church-Turing Thesis?

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We aim to put some order to the multiple interpretations of the Church-Turing Thesis and to the different approaches taken to prove or disprove it.

[Answer:] Rosser and its inventor proved that its beta-reduction satisfies the diamond property, and Kleene (pron. clean-ee) proved that it was equivalent to his partial recursive functions. The previous result combined with a similar one with the Turing Machine, led to the Church-Turing thesis. [Question: “What is ...?”] —Quizbowl Tournament (2004)

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Peter Wegner and Dina Goldin asserted that the Church-Turing Thesis should be interpreted with respect to functions over natural numbers [ 17 ]. Yet, if one considers it in the more general form, they claim that he/she must admit it wrong due to the actuality of interactive computations. All this means is that the original thesis needs to be adapted to refer to the nature of non-function-computing uses of computation. See, for example, [ 19 ].

There are numerous other names for the various versions of the thesis in the literature. Even the terms “mathematical” and “physical” have different connotations for different authors.

In [ 13 , 33 ] it is shown that while strict containment of function sets is in general not enough to infer the presence of extra computational power, general recursion is indeed more powerful than primitive recursion, even under rigorous definitions of power comparison.

Sieg [ 49 ] in explaining Church [ 4 ]: “Calculability is explicated by that of derivability in a logic.”

Gandy’s idea of basing a formalization of computability on hereditarily finite sets, which are precisely the finite objects of set theory, was presaged by Moto-o Takahashi [ 53 ].

Another argument of Bringsjord, explicitly touted as “a case against Church’s thesis”, takes its cue from the perceived ability of humans to judge the literary quality of writings that they cannot for the life of them produce themselves [ 101 ].

Herbert Robbins, the statistician: “Nobody is going to run 100-meters in five seconds, no matter how much is invested in training and machines. The same can be said about using the brain. The human mind is no different now from what it was five thousand years ago. And when it comes to mathematics, you must realize that this is the human mind at an extreme limit of its capacity” [ 105 ].

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Acknowledgements

We are enormously grateful for comments and suggestions from Arnon Avron, Erwin Engeler, Oron Shagrir, Wilfried Sieg, and an anonymous reader.

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Boker, U., Dershowitz, N. (2022). What is the Church-Turing Thesis?. In: Ferreira, F., Kahle, R., Sommaruga, G. (eds) Axiomatic Thinking II. Springer, Cham. https://doi.org/10.1007/978-3-030-77799-9_9

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Church’s Thesis for Turing Machine

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In 1936, A method named as lambda-calculus was created by Alonzo Church in which the Church numerals are well defined, i.e. the encoding of natural numbers. Also in 1936, Turing machines (earlier called theoretical model for machines) was created by Alan Turing, that is used for manipulating the symbols of string with the help of tape.

Church Turing Thesis :

Turing machine is defined as an abstract representation of a computing device such as hardware in computers. Alan Turing proposed Logical Computing Machines (LCMs), i.e. Turing’s expressions for Turing Machines. This was done to define algorithms properly. So, Church made a mechanical method named as ‘M’ for manipulation of strings by using logic and mathematics. This method M must pass the following statements:

• Number of instructions in M must be finite.
• Output should be produced after performing finite number of steps.
• It should not be imaginary, i.e. can be made in real life.
• It should not require any complex understanding.

Using these statements Church proposed a hypothesis called

Church’s Turing thesis

that can be stated as: “The assumption that the intuitive notion of computable functions can be identified with partial recursive functions.”

Or in simple words we can say that “Every computation that can be carried out in the real world can be effectively performed by a Turing Machine.”

In 1930, this statement was first formulated by Alonzo Church and is usually referred to as Church’s thesis, or the Church-Turing thesis. However, this hypothesis cannot be proved. The recursive functions can be computable after taking following assumptions:

• Each and every function must be computable.
• Let ‘F’ be the computable function and after performing some elementary operations to ‘F’, it will transform a new function ‘G’ then this function ‘G’ automatically becomes the computable function.
• If any functions that follow above two assumptions must be states as computable function.

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The Mission of the Church

Other essays.

The mission of the church is the task given by God for the people of God to accomplish in the world.

After defining the terminology this essay will explore the nature of the church’s mission in light of the missio Dei and the apostolic pattern in the New Testament and the book of Acts in particular. It will evaluate contemporary broader ideas of mission and conclude with a re-emphasis on the gospel-centered focus of the New Testament pattern.

The mission of the church is the task given by God for the people of God to accomplish in the world. In simplest terms, the mission of the church is the Great Commission—what Philip Ryken calls “a clear, unambiguous statement of [the church’s] mission to the world.” 1 Our task as the gathered body of Christ is to make disciples, by bearing witness to Jesus Christ the Son in the power of the Holy Spirit to glory of God the Father. 2

Defining Our Terms

In talking about the mission of the church, we are not trying to enumerate all the good things Christians can or should do to love their neighbors and to be salt and light in the world. The issue at hand relates to the church as church. What collectively as an organized institution must we be about as God’s people if we are to faithfully accomplish his purposes for us in the world?

If the word “church” is important, so is the word “mission.” While “mission” does not appear in most English Bibles, it is still a biblical word. Eckhard Schnabel— who, with almost 2000 pages on Early Christian Mission and another 500-page work on Paul the Missionary , is probably the world’s leading expert on mission in the New Testament—makes this point forcefully:

The argument that the word mission does not occur in the New Testament is incorrect. The Latin verb mittere corresponds to the Greek verb apostellein , which occurs 136 times in the New Testament (97 times in the Gospels, used both for Jesus having been “sent” by God and for the Twelve being “sent” by Jesus). 3

The apostles, in the broadest sense of the term, were those who had been sent out. This sent-outness is also the first thing we should note relative to the term missionary . It is, after all, the first thing Jesus notes about his mission, that he was sent to proclaim a message of good news to the poor (Luke 4:18). Being “on mission” or engaging in mission work suggests intentionality and movement. 4 Mission, at the very least, involves being sent from one place to somewhere else.

Every Christian—if we are going to be obedient to the Great Commission—must be involved in missions, but not every Christian is a missionary. While it is certainly true that we should all be ready to give an answer for the hope that we have (1Pet. 3:15), and we should all adorn the gospel with our good works (Titus 2:1), and we should all do our part to make Christ known (1Thess. 1:8; 2Thess. 3:1), we should reserve the term “missionary” for those who are intentionally sent out from one place to another. Strictly speaking, the church is not sent out but sends out workers from her midst. Our fundamental identity as church ( ekklesia ) is not as those who are sent into the world with a mission, but as those who are called out from darkness into his marvelous light (1Pet. 2:9). 5

Jesus’s Mission and Ours

Before the sixteenth century, “mission” was primarily a word used in connection with the Trinity. The “sending” theologians talked about was the sending of the Son by the Father, and the sending of the Holy Spirit by the Father and the Son. This is a crucial point. We will not rightly understand the mission of the church without the conviction that “the sending of Jesus by the Father is still the essential mission .” 6

And what was the nature of Jesus’s ministry? Jesus ministered to bodies as well as souls, but within this holistic ministry, he made preaching his priority. Preaching is why he came out in public ministry and why he moved from town to town (Mark 1:38-39). The purpose of his Spirit-anointed ministry was to proclaim good news to the poor (Luke 4:18-19). He came to call sinners to repentance and faith (Mark 1:15; 2:17). Although Jesus frequently attended to the physical needs of those around him, there is not a single example of Jesus going into a town with the purpose of healing or casting out demons. The Son of Man never ventured out on a healing or exorcism tour. His stated purpose was to seek and to save the lost (Luke 19:10).

Of course, Jesus’ mission must not be reduced to verbal proclamation. Unique to his identity as the divine Messiah, Jesus’s mission was vicariously to die for the sins of his people (Matt. 1:21; Mark 10:45). Concomitant with this purpose, Jesus’s public ministry aimed at the eternal life that could come to the sinner only through faith in Christ (John 3:16-17; 14:6; 20:21). We see this in Mark’s Gospel, for example, where the entire narrative builds toward the centurion’s confession in Mark 15:37 where, in fulfillment of the book’s opening sentence (Mark 1:1), the Roman soldier confesses, “Truly this man was the Son of God!” Leading people to this Spirit-given conviction is the purpose of Mark’s gospel and of Jesus’s ministry. The Messiah ministered to bodies as well as souls and made preaching his priority so that those with ears to hear might see his true identity and follow him in faith.

It’s no wonder, then, that all four Gospels (plus Acts) include some version of the Great Commission (Matt. 28:16-20; Mark 13:10; 14:9; Luke 24:44-49; John 20:21; Acts 1:8). The mission given to the bumbling band of disciples was not one of cultural transformation—though that would often come as a result of their message—but a mission of gospel proclamation. To be sure, God’s cosmic mission is bigger than the Great Commission, but it is telling that while the church is not commanded to participate with God in the renewal of all things—which would, presumably, include not only re-creation but also fiery judgment—we are often told to bear witness to the one will do all these things. In short, while the disciples were never told to be avatars of Christ, it is everywhere stated, either explicitly or implicitly, that they were to be ambassadors for Christ (2 Cor. 5:20).

A Mission Too Small?

No Christian disagrees with the importance of Jesus’ final instructions to the disciples, but many missiological scholars and practitioners have disagreed with the central or controlling importance of the Great Commission. John Stott, for example, in arguing for social action as an equal partner of evangelism suggested that “we give the Great Commission too prominent a place in our Christian thinking.” 7 Similarly, Lesslie Newbigin concluded that the “Christian mission is thus to act out in the whole life of the whole world the confession that Jesus is Lord.” 8 The mission of the church, in other words, cannot be reduced to our traditional understanding of missions.

In the past fifty years, we have seen, to quote the title of one seminal book, “paradigm shifts in theology of mission.” 9 At the heart of this shift has been a much more expansive view of the mission of the church, one that recasts the identity of the church as missional communities “called and sent to represent the reign of God” or as “communities of common people doing uncommon deeds.” 10 No longer is the role of the church defined mainly as an ambassador or a witness. Instead, we are collaborators with God in the missio Dei (mission of God), co-operators in the redemption and renewal of all things. As Christopher Wright puts it, “Fundamentally, our mission (if it is biblically informed and validated) means our committed participation as God’s people, at God’s invitation and command, in God’s own mission within the history of God’s world for the redemption of God’s creation.” 11 The church’s task in the world is to partner with God as he establishes shalom and brings his reign and rule to bear on the peoples and places of the earth.

The Mission of the Church in Acts

As attractive as this newer model may seem, there are a number of problems with the missio Dei paradigm for the mission of the church. It undervalues the Great Commission, underemphasizes what is central in the mission of the Son, and overextends our role in God’s cosmic mission on earth.

Besides all this, the new model has a hard time accounting for the pattern of mission in the earliest days of the church. Acts is the inspired history of the mission of the church. This second volume from Luke describes what those commissioned at the end of the first volume were sent out to do (Luke 24:47-48). If Luke’s Gospel was the book of everything Jesus began to do and teach (1:1), then Acts must be the record of all that Jesus continues to do and teach.

We could look at almost any chapter in Acts to gain insight into the mission of the church, but Acts 14 is especially instructive, verses 21-23 in particular. At the beginning of Acts 13, the church at Antioch, prompted by the Holy Spirit, set apart Paul and Barnabas “for the work to which I have called them” (v. 2). This isn’t the first time the gospel is going to be preached to unbelievers in Acts, neither is it the first gospel work Paul and Barnabas will do. But it is the first time we see a church intentionally sending out Christian workers with a mission to another location.

Paul and Barnabas traveled to Cyprus, then to Pisidian Antioch, then to Iconium, then to Lystra, then to Derbe, and from there back through Lystra, Iconium, and Pisidian Antioch, and then to Perga, and back to Antioch in Syria. The final section in Acts 14 is not only a good summary of Paul’s missionary work, it is the sort of information Paul would have shared with the church in Antioch when he returned (v. 27). These verses are like the PowerPoint presentation Paul and Barnabas shared with their sending church. “This is how we saw God at work. Here’s where we went and what we did.” In other words, if any verses are going to give us a succinct description of what mission was about in the early church, it’s verses like these at the end of Acts 14.

Acts 14:21-23 presents us with the three-legged stool of the church’s mission. Through the missionary work of the Apostle Paul, the early church aimed for:

• New converts: “when they had preached the gospel to that city and had made many disciples” (v. 21)
• New communities: “And when they had appointed elders for them in every church” (v. 23)
• Nurtured churches: “strengthening the souls of the disciples, encouraging them to continue in the faith” (v. 22).

If the apostles are meant to be the church’s model for mission, then we should expect our missionaries to be engaged in these activities and pray for them to that end. The goal of mission work is to win new converts, establish these young disciples in the faith, and incorporate them into a local church. 12

Schnabel’s definition of missionary work sounds the same note:

• “Missionaries communicate the news of Jesus the Messiah and Savior to people who have not heard or accepted this news.”
• “Missionaries communicate a new way of life that replaces, at least partially, the social norms and the behavioral patterns of the society in which the new believers have been converted.”
• “Missionaries integrate the new believers into a new community.” 13

Evangelism, discipleship, church planting—that’s what the church in Antioch sent Paul and Barnabas to do, and these should be the goals of all mission work. Missionaries may aim at one of these components more than the other two, but all three should be present in the church’s overall mission strategy.

Keeping the Main Thing the Main Thing

As is true with almost every Christian doctrine, there are ditches on either side of the road when trying to define the mission of the church. On the one hand, we want to avoid the danger of making our mission too small. Some well-meaning Christians act like conversion is the only thing that counts. They put all their efforts into getting to the field as quickly as possible, speaking to as many people as possible, and then leaving as soon as possible. Mission becomes synonymous with first-time gospel proclamation. Clearly, Paul did not practice blitzkrieg evangelism, nor was he motivated by an impatient hankering for numbers to report back home.

On the other hand, we want to avoid the danger of making our mission too broad. Some well-meaning Christians act like everything counts as mission. They put all their efforts into improving job skills, digging wells, setting up medical centers, establishing great schools, and working for better crop yields—all of which can be wonderful expressions of Christian love, but bear little resemblance to what we see Paul and Barnabas sent out to do on their mission in Acts.

Without denigrating the good work Christians do as salt and light in the world, we must conclude from Acts 14—and from the New Testament more broadly, that the church’s mission is more specific than common people doing uncommon deeds. As Schnabel argues, those demanding a “‘revolution’ in our understanding of mission—away from the traditional missionary focus on winning people to faith in Jesus Christ, concentrating rather on a ‘holistic’ understanding of Jesus’ claims” do so without strong supporting evidence. 14 We see over and over in Paul’s missionary journeys, and again in his letters, that the central work to which he has been called was the verbal proclamation of Jesus Christ as Savior and Lord (Rom. 10:14-17; 15:18; 1Cor. 15:1-2, 11; Col. 1:28). Paul saw his identity as an apostle, as a sent-out one, in terms of being set apart for the gospel of God (Rom. 1:1). That’s why in Acts 14:27 the singular summary of his just-completed mission work is that God had opened a door of faith to the Gentiles. His goal as a missionary was the conversion of Jews and pagans, the transformation of their hearts and minds, and the incorporation of these new believers into a mature, duly constituted church. What Paul aimed to accomplish as a missionary in the first century is an apt description of the mission of the church for every century.

Further Reading

• Kevin DeYoung and Greg Gilbert, What Is the Mission of the Church? Making Sense of Social Justice, Shalom, and the Great Commission (Wheaton, IL: Crossway, 2011).
• Andreas J. Köstenberger and Peter T. O’Brien, Salvation to the Ends of the Earth: A Biblical Theology of Mission (Downers Grove, IL: InterVarsity, 2001).
• Eckhard J. Schnabel, Early Christian Mission , 2 vols. (Downers Grove, IL: InterVarsity, 2004).
• Eckhard J. Schnabel, Paul the Missionary: Realities, Strategy and Method (Downers Grove, IL: IVP Academic, 2008)
• Jason Sexton, Jonathan Leeman, Christopher J.H. Wright, John R. Franke, and Peter J. Leithart, Four Views on the Church’s Mission (Grand Rapids, MI: Zondervan, 2017).
• Denny Spitters and Matthew Ellison, When Everything is Missions (Orlando, FL: BottomLine Media, 2017).

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The Messages of the Restoration: A Study of the Doctrinal Development of the Missionary Message of The Church of Jesus Christ of Latter-day Saints , Sheldon Nixon

A Historical Analysis of How Preach My Gospel Came to Be , Benjamin H. White

An Analysis of the Teaching Aids Provided for Sunday School Teachers in The Church of Jesus Christ of Latter-day Saints , Kevin Douglas Whitehead

Theses/Dissertations from 2009 2009

Corporeal Man: A Latter-day Saint Perspective , Todd S. Davis

Doctrinal and Historical Analysis of Young Women's Education in The Church of Jesus Christ of Latter-day Saints , Andrea Erickson

Theses/Dissertations from 2008 2008

The Church of Jesus Christ of Latter-day Saints Enters Albania, 1992-1999 , Nathan D. Pali

Theses/Dissertations from 2007 2007

Joseph Smith's View of His Own Calling , Tucker John Boyle

Joseph F.Merrill: Latter-day Saint Commissioner of Education, 1928-1933 , Casey Paul Griffiths

A Comparative Study of Muhammad and Joseph Smith in the Prophetic Pattern , Todd J. Harris

The Influence of the First World War on The Church of Jesus Christ of Latter-day Saints , James I. Mangum

The Church of Jesus Christ of Latter-day Saints in National Periodicals: 1991-2000 , Casey William Olson

James E. Talmage and the Nature of the Godhead: The Gradual Unfolding of Latter-day Saint Theology , Brian William Ricks

Theses/Dissertations from 2006 2006

A Study of the History of the Office of High Priest , John D. Lawson

Personal Scripture Study of Prospective Missionaries , Eric Lyon Wing

Theses/Dissertations from 2005 2005

Cornelius P. Lott and his Contribution to the Temporal Salvation of the Latter-day Saint Pioneers Through the Care of Livestock , Gary S. Ford

The Etoile Du Deseret: Portrait of the French Mission, 1851-1852 , Douglas James Geilman

An Analysis of the Newspaper Coverage of Latter-Day Saint Temples Announced or Built Within the United States from October 1997 Through December 2004 , Kevan L. Gurr

The Church of Jesus Christ of Latter-Day Saints in National Periodicals, 1982-1990 , Matthew E. Morrison

Theses/Dissertations from 2003 2003

A History of "Especially For Youth" - 1976-1986 , John Bytheway

Latter-Day Saints in Popular National Periodicals 1970-1981 , Adam H. Nielson

Theses/Dissertations from 2002 2002

A History of the Concepts of Zion and New Jerusalem in America From Early Colonialism to 1835 With A Comparison to the Teachings of the Prophet Joseph Smith , Ryan S. Gardner

Corporeal Resurrection: The Pure Doctrine Restored Through the Prophet Joseph Smith , J. Peter Hansen

A History of the Latter-Day Saints in the Columbia Basin of Central Washington 1850-1972 , Rick B. Jorgensen

A Study of the Hill Cumorah: A Significant Latter-Day Saint Landmark in Western New York , Cameron J. Packer

Theses/Dissertations from 1980 1980

The World and Joseph Smith , Lane D. Ward

Theses/Dissertations from 1976 1976

The Southern States Mission and the Administration of Ben E. Rich, 1898-1908: Including a Statistical Study of Church Growth in the Southeastern United States During the Twentieth Century , Ted S. Anderson

A Study of Historical Evidences Related to LDS Church as Reflected in Volumes XIV Through XXVI of the Journal of Discourses , Terry J. Aubrey

Effects of Human Relations Training on the Social, Emotional, and Moral Development of Students, with Emphasis on Human Relations Training Based Upon Religious Principles , Stephen R. Covey

Missionary Activities and Church Organizations in Pennsylvania, 1830-1840 , V. Alan Curtis

Functional Problems and Informational Needs of Latter-Day Saint Chaplains Serving in the United States Armed Forces , N. Vernon Griffeth

A History of the Growth and Development of the Primary Association of the LDS Church From 1878 to 1928 , Conrad A. Harward

The Teacher Training Program Administered by the Sunday School of The Church of Jesus Christ of Latter-Day Saints , Steven A. Hedquist

Origin and Development of the San Juan Mission in Southeastern Utah in its Work with Indian People (Principally Since 1940) , Lyle S. Heinz

A History of Female Missionary Activity in The Church of Jesus Christ of Latter-Day Saints, 1830-1898 , Calvin S. Kunz

History and Functions of the Aaronic Priesthood and the Offices of Priest, Teacher, and Deacon in The Church of Jesus Christ of Latter-Day Saints, 1829 to 1844 , Robert L. Marrott

Formal Reporting Systems of The Church of Jesus Christ of Latter-Day Saints, 1830-1975 , Dennis H. Smith

The History of the Emery Stake Academy , Paul Robert Tabone

A History of the Young Men's Mutual Improvement Association 1939 to 1974 , John Kent Williams

Theses/Dissertations from 1975 1975

An Analysis of the Problems in Obtaining and Maintaining Released Time for Seminary in the Central Idaho Seminary District , Arthur A. Bailey

The Settlement and Development of Wayne County, Utah, to 1900 , Aldus DeVon Chappell

The History of the Church of Jesus Christ of Latter-Day Saints in South America, 1945-1960 , Joel Alva Flake

Stephen Markham: Man of Valour , Mervin LeRoy Gifford

A History of Kirtland Camp: Its initial Purpose and Notable Accomplishments , Gordon Orville Hill

An Analysis of the Doctrinal Teachings of President George Albert Smith , Robert K. McIntosh

History of Four Mormon Landmarks in Western New York: The Joseph Smith Farm, Hill Cumorah, the Martin Harris Farm, and the Peter Whitmer, Sr., Farm , Rand H. Packer

Conceptual Patterns of Repetition in the Doctrine and Covenants and their Implications , Richard Cottam Shipp

Reasons For Non-Enrollment and Low Attendance in LDS Early Morning Seminary at Minneapolis-St. Paul , Wayne P. Smith

A Comparison of Mission Programs Used in the Three Language Training Missions of The Church of Jesus Christ of Latter-Day Saints , Rawn Arthur Wallgren

Joseph Smith the Colonizer , Brent L. Winward

Theses/Dissertations from 1974 1974

A Documentary History of the Lord's Way of Watching Over the Church by the Priesthood Through the Ages , Rex A. Anderson

A Study to Determine the Possible Influence of Public School Curriculum Development On Course Outlines Used by the Released-Time Seminary Program of The Church of Jesus Christ of Latter-Day Saints , Merrill Dean Briggs

The Development of The Church of Jesus Christ of Latter-Day Saints in Hawaii , Richard C. Harvey

Proselyting Techniques of Mormon Missionaries , Jay E. Jensen

History of Mormon Exhibits in World Expositions , Gerald Joseph Peterson

The Life and Contributions of Zebedee Coltrin , Calvin Robert Stephens

A History of the Nauvoo Legion in Illinois , John Sweeney Jr.

Kelsey, Texas: The Founding and Development of a Latter-Day Saint Gathering Place in Texas , James Clyde Vandygriff

Theses/Dissertations from 1973 1973

A Study to Determine the Understanding of the Nature and Mission of Jesus Christ by Third Year Seminary Graduates of The Church of Jesus Christ of Latter-Day Saints , Terry R. Baker

Prophetic Authority in the Teachings of Modern Prophets , Clifford Gary Bennett

A Study of the Opinions of LDS Athletes Concerning Excellence in Gospel Living Contributing to Excellence in Sports , Robert L. Cummings

A History of the Involvement of The Church of Jesus Christ of Latter-Day Saints in the Tanning Industry in Utah From 1847 to 1973 , Paul Edwards Damron

History of the Swedish Mission of The Church of Jesus Christ of Latter-Day Saints 1905-1973 , Carl Erik Johansson

The Divine Nature of God: A Study of What has Been Said and Taught About the Divine Nature of God by Ancient and Modern Apostles and Prophets , Lester Young Moody

The Box Elder Stake Academy in its Historical Setting , Byron L. Parkinson

Mormonism in National Periodicals, 1961-1970 , Dale P. Pelo

The Correlation Program of The Church of Jesus Christ of Latter-Day Saints During the Twentieth Century , Jerry Rose

Zion's Camp , Wilburn D. Talbot

Theses/Dissertations from 1972 1972

The Lord's Definition of Woman's Role as He has Revealed it to His Prophets of the Latter Day , Mildred Chandler Austin

A Study of A Teaching Method Called Seminary Bowl , Max G. Hirschi

An Evaluation of Instructional Television in the Brigham Young University College of Religious Instruction , James Frank Killian

A Study to Determine Reasons Why LDS Students were not Enrolled in LDS Seminary in the Southern Alberta Seminary District During 1970-71 , Robert Owen McClung

A Comparative Study of the Book of Mormon Secret Combinations and the American Mafia Organization , Ray G. Morley

The office of Associate President of The Church of Jesus Christ of Latter-Day Saints , Robert Glen Mouritsen

A History of the Schools and Educational Programs of The Church of Jesus Christ of Latter-Day Saints in Ohio and Missouri, 1831-1839 , Orlen Curtis Peterson

A Study of Evidences Related to LDS Church History as Reflected in Volumes I Through XIII of the Journal of Discourses , Paul C. Richards

Theses/Dissertations from 1971 1971

A Study of Rock Music to Determine its Declared Position Relative to Unchastity, the Use of Drugs and the Departure from Traditional Concepts of Family and Religion , E. Lynn Balmforth

An Analysis of the Role of Temples in the Establishment of Zion , C. Max Caldwell

A Comparative Study of the Teaching Methods of the LDS and Non-LDS Religious Educational Movements Among the Indians in Southeastern Utah Since 1943 , James A. Carver

A Study to Determine Duplication, Gaps, Emphasis, and Location of Lesson Concepts in the 1967-68 Religious Education Lesson Manuals for High School Age LDS Youth , Lowry K. Flake

Causes of the Mormon Boycott Against Gentile Merchants in 1866 and 1868 , Peter Neil Garff

George Reynolds: The Early Years , Grant R. Hardy

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Ep 1: The ACNAtoo Story—The Thesis Wall of Silence

• Christianity

Welcome to the Wall of Silence. In this first episode we introduce the sexual abuse case of lay pastor Mark Rivera. From there we begin to look outward to see how church leadership responded to it, from the local diocese to the national denomination, the Anglican Church in North America (ACNA). As "the thesis," this episode is the explanation for why this podcast exists and whose story we are telling: the advocacy group ACNAtoo and all victims, whistleblowers, and advocates against church abuse in the ACNA. If you want to support the mission of the Wall of Silence podcast please consider becoming a Patreon member at: https://www.patreon.com/WallofSilencePodcast Membership includes extra monthly interviews and discussions. Also featured: Megan Tucker, Lisa Weaver Swartz (https://lisaweaverswartz.com/), Stephen Backhouse (https://www.stephenbackhouse.com/, https://www.tenttheology.com/), and Audrey Luhmann. Here are the links referenced in this episode: https://twitter.com/ladyjessicahaze/status/1408916453848346629 https://www.instagram.com/mouthful.of.stars/ https://religionnews.com/2023/03/06/mark-rivera-a-former-anglican-lay-pastor-sentenced-to-15-years-in-prison/ https://twitter.com/luhmannaudrey/status/1734612110666244545?s=46&t=gMog_HbpmJgiGAYg7W8m4g https://nymag.com/intelligencer/article/anglican-church-of-north-america-sexual-abuse-scandal.html https://twitter.com/luhmannaudrey/status/1734612110666244545?s=46&t=gMog_HbpmJgiGAYg7W8m4g https://threadreaderapp.com/thread/1412957114620809221.html https://www.acnatoo.org/acnatoo-blog/umd-abuse-mishandling-summary https://www.acnatoo.org/acnatoo-news/ruchs-ecclesiastical-trial Show transcript link: https://www.acnatoo.org/episode-1

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• Chris Marchand

DEACON RICHFIELD KWAME ATINYO HAS BEEN AWARDED THE BEST GRADUATE THESIS

Deacon Richfield Kwame Atinyo has been awarded the Best Graduate Thesis in the Department of Human Resource Management at the University of Ghana. During the University of Ghana Vice-Chancellor’s Ceremony in Honour of Academic Award Winners for the 2022/2023 Academic year on Thursday 25th April, 2024, he received a cash prize, a certificate, and a book from Kimathi and Partners Corporate Attorneys.

In addition to his academic achievements, Deacon Richfield Atinyo is a devout member of The Apostolic Church-Ghana, serving as a Deacon in the Nii Boiman Central within the Adabraka Area. He is committed to spiritual growth and service within the church community. Throughout his academic and professional journey, he has held various leadership roles and community initiatives.

During his undergraduate studies, he served as an Executive of APOSA Legon chapter.

Deacon Richfield Atinyo embodies a rare blend of academic excellence, leadership acumen, and a heart for service, making him a valuable asset to any academic institution, religious and secular organizations for human resource enhancement.

The Leadership of the Apostolic Church-Ghana congratulates him and wishes him all the best.

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