The Central Limit Theorem Rediscovered Alone

King's College Fellowship and the Probabilistic Bridge to Mathematical Foundations

Experimental physicists measuring the transit of a star or the charge of an electron constantly encounter the same bell-shaped curve. Whenever an observer combines dozens of small, independent measurement errors—thermal expansion in a brass micrometer, air turbulence in a telescope tube, optical parallax at the eyepiece—the total error clusters around Carl Friedrich Gauss's normal probability distribution. During the autumn term of 1933 at Cambridge, King's College undergraduate Alan Turing attended a lecture series on the methodology of science delivered by Plumian Professor Arthur Stanley Eddington. When Eddington sketched why observational errors obey the Gaussian curve, Turing noticed a major gap. The mathematical justification rested on circular heuristic assumptions rather than a rigorous proof.

Existing British textbooks proved the Central Limit Theorem only for simplified cases. Those proofs assumed that every individual random shock followed the exact same probability distribution or possessed bounded higher moments. Turing wanted to find the exact minimal condition for a far broader case. Suppose one sums completely different, independent random variables $X_1, X_2, \dots, X_n$ with zero means and finite variances $\sigma_1^2, \sigma_2^2, \dots, \sigma_n^2$. When does that sum converge to a smooth Gaussian bell curve as $n$ grows large?

Turing preferred to work problems out from scratch in his rooms at King's College rather than comb through continental journals in the university library. He attacked the convergence problem alone across the winter of 1933 and spring of 1934. First, he defined the cumulative variance $s_n^2 = \sum_{k=1}^{n} \sigma_k^2$ and the normalized sum $Z_n = s_n^{-1} \sum_{k=1}^{n} X_k$. Next, he analyzed how adding one more small random step smooths the cumulative distribution function through convolution. He proved that the normalized sum $Z_n$ converges to the standard Gaussian integral $(2\pi)^{-1/2} \int e^{-t^2/2} dt$ if and only if no single random variable dominates the total variance. That requirement is captured by the asymptotic tail condition:

Here $F_k(x)$ is the probability distribution of the $k$-th variable and $\varepsilon$ is any positive threshold. Only after completing the manuscript did Turing learn that the Finnish mathematician Jarl Waldemar Lindeberg had published the exact same theorem twelve years earlier. Lindeberg's paper had appeared in the German journal Mathematische Zeitschrift in 1922, and William Feller added the necessity proof in 1935.

Even though the main result was no longer new to continental probability theory, Turing had bypassed characteristic functions in favor of a direct quasi-physical smoothing operator. Encouraged by group theorist Philip Hall, Turing submitted the typescript titled On the Gaussian Error Function for the King's College fellowship competition in late 1934. Referee Abram Besicovitch noted that discovering Lindeberg's theorem unaided showed remarkable analytical power. In March 1935 King's College elected Turing a Fellow at twenty-two, providing him with rooms, dining rights, and an annual stipend of three hundred pounds.

That spring term of 1935, the newly elected Fellow enrolled in a Part III lecture course on the Foundations of Mathematics taught by Cambridge topologist Max Newman. Newman walked his students through David Hilbert's formalist program, which had aimed to place all of mathematics on a complete, consistent, and mechanically decidable footing. Newman then proved Kurt Gödel's 1931 incompleteness theorems. Those theorems showed that any consistent formal system rich enough to express arithmetic contains true statements that cannot be proved inside the system.

One final question from Hilbert's 1928 Bologna address remained open: the Entscheidungsproblem, or Decision Problem. Could there exist a definite mechanical procedure that takes any logical formula written in first-order predicate calculus and decides in finite steps whether that formula is provable? Within weeks of hearing Max Newman ask whether a definite mechanical method could decide the truth of any mathematical assertion, Turing set aside continuous probability distributions to construct an idealized clerk working on an endless strip of paper.