MKLab - HISTORY

W. T. Tutte—The Graph Theorist

Thu, 29 Oct 2026 01:09:23 +0200

W. T. Tutte — The Graph Theorist Whose Code-Busting Algorithms Powered the D-Day Invasion

William Thomas Tutte was one of the twentieth century’s most remarkable mathematicians, whose contributions remained largely unknown for decades because of wartime secrecy. Initially a chemistry student at Cambridge University, Tutte developed a deep interest in mathematics, particularly graph theory. In collaboration with three fellow students, he investigated how squares of different sizes could be arranged to form rectangles, connecting the problem to electrical circuits and Kirchhoff’s laws. Their research, published in 1940, helped establish important connections between graph theory, mathematics and engineering.

During the Second World War, Tutte worked at Bletchley Park, Britain’s secret codebreaking centre, where he achieved an extraordinary breakthrough by deducing the internal structure of the German Tunny cipher machine without ever having examined the machine itself. Tunny protected highly sensitive communications between Hitler’s headquarters and German military commanders. Tutte’s mathematical insights, combined with the work of Alan Turing and other codebreakers, enabled the Allies to decipher large volumes of encrypted messages. The resulting intelligence revealed German military plans and contributed significantly to Allied operations, including the preparations for the D-Day invasion and the campaign across Europe.

After the war, Tutte continued to make fundamental contributions to graph theory and combinatorics, becoming one of the field’s leading figures. His wartime achievements demonstrated how abstract mathematical reasoning can solve practical problems with enormous historical consequences. The article highlights the often-overlooked role of mathematicians in scientific and technological progress, showing that deep theoretical insights can become powerful tools when combined with computation, engineering and collaboration. Tutte’s story is also a reminder that some of the most influential contributions to history have been made by people whose names remained hidden for many years.

Source: Link to the original article in The Mathematical Intelligencer.

Postcard from Albert Einstein to David Hilbert

Fri, 23 Oct 2026 18:03:22 +0300

The postcard sent by Albert Einstein to David Hilbert on 18 November 1915 highlights the scientific competition between the two mathematicians and physicists as they worked to formulate the field equations of general relativity. Einstein acknowledged that Hilbert had also discovered the correct mathematical foundations for the theory, while emphasizing that the main challenge lay in the physical interpretation of the equations and their relationship to established theories of gravity.

In particular, Einstein noted that Hilbert’s work had not yet clarified the connection between his equations and Newton’s theory of gravitation. The postcard illustrates a crucial moment in the development of general relativity, demonstrating how mathematical formulation and physical interpretation were both essential to establishing a revolutionary scientific theory.

POSTCARD

Antisemitism in Soviet mathematics

Tue, 29 Sep 2026 00:45:52 +0300

Antisemitism in Soviet mathematics refers to the discrimination experienced by Jewish mathematicians and students in the Soviet Union, particularly from the late 1940s through the 1980s. According to the Wikipedia article and its cited sources, discrimination affected admission to universities, postgraduate education, academic employment, dissertation defenses, publication, international travel, and participation in scientific conferences. The problem became especially visible in mathematics because several prestigious Soviet mathematical institutions had highly selective admissions systems.

A major example was the Faculty of Mechanics and Mathematics at Moscow State University (MSU). Jewish applicants reportedly encountered unusually difficult or ambiguous entrance examinations. In 1978, for example, 21 graduates from one Moscow mathematical school applied: all 14 Russian students were admitted, while only one of seven Jewish students was admitted. That Jewish student had won the International Mathematical Olympiad and had repeatedly won All-Union Olympiads. According to testimonies cited by the article, some Jewish applicants were given exceptionally difficult problems, sometimes drawn from advanced mathematical olympiads, or questions that could not reasonably be solved within the examination time.

The discrimination was part of a broader Soviet history of restrictions on Jewish education and professional advancement. The article traces earlier restrictions to the Russian Empire, where Jewish university quotas existed, while noting that Soviet antisemitism became particularly prominent in the late 1930s and especially the late 1940s and early 1950s. After the 1967 Six-Day War, Soviet anti-Zionist propaganda intensified and, according to the cited sources, sometimes became intertwined with anti-Jewish prejudice. Jewish mathematicians also participated in the broader Soviet dissident movement; for example, 99 mathematicians signed a 1968 letter protesting the forced psychiatric hospitalization of Alexander Esenin-Volpin, after which some signatories experienced professional repression.

The consequences extended beyond individual careers and contributed to the emigration of Jewish mathematicians from the USSR. The article also discusses accusations against influential mathematical figures such as Ivan Vinogradov and Lev Pontryagin, although Pontryagin himself denied accusations of antisemitic policy. Importantly, the Wikipedia page currently carries editorial warnings that it contains argumentative or personal-reflection-style material and needs improvement, so its claims should be understood in conjunction with the primary and scholarly sources cited in the article, rather than treated as an unquestionable historical account.

Key takeaways

Jewish mathematicians faced institutional discrimination in parts of Soviet higher education and mathematics.
Moscow State University's Mechanics and Mathematics Faculty became one of the most prominent examples.
Some entrance examinations allegedly used disproportionately difficult or inappropriate mathematical problems against Jewish applicants.
The discrimination extended beyond admission to employment, publications, academic advancement, dissertations, and international travel.
The phenomenon was connected to the broader history of Soviet antisemitism and restrictions on Jewish education.
Political repression and ethnic discrimination sometimes overlapped, particularly during the dissident period of the 1960s–1980s.
Discrimination was one factor associated with the emigration of Jewish mathematicians from the Soviet Union.
Claims concerning particular mathematicians and institutions remain historically contested, so the cited scholarly and primary sources are important for evaluating individual cases.

===>ARTICLE<===

A piece from the mathematics of Ted Kaczynski

Sun, 27 Sep 2026 01:56:13 +0300

This paper is particularly interesting in relation to Ted Kaczynski, because it is not a paper about his life or political ideas; it is a modern mathematical paper that takes several problems and conjectures from his 1967 mathematics PhD thesis and later mathematical correspondence and proves new results about them.

The paper is Parker Kuklinski, “An uncountable union of line segments with null two-dimensional measure” (2023). Its central problem is geometrical/measure-theoretic: how can one connect two sets of points by an enormous collection of line segments while making the resulting planar set have surprisingly small—or even zero—area?

1. The basic mathematical problem
Imagine two horizontal line segments:
$X_0={(x,0):0\leq x\leq1}$
and
$X_1={(x,1):0\leq x\leq1}$.
Now connect points on $X_0$ to points on $X_1$ with straight line segments.
You might intuitively expect that if you connect every point of the bottom interval to a corresponding point of the top interval, the resulting collection of lines should occupy a substantial amount of the square.
Kaczynski studied precisely this kind of object, which the paper calls an unrestricted trapezoid. The surprising possibility is that an uncountable collection of line segments can have zero two-dimensional Lebesgue measure.
In other words, you can have something that contains an enormous number of curves and nevertheless has
$\mu_2(T)=0$.
That counterintuitive phenomenon is the central theme.

2. Kaczynski's original contribution
This is where the paper becomes particularly interesting.
Kaczynski's 1967 PhD thesis, “Boundary Functions,” contained a theorem concerning unions of line segments that the new paper builds upon. The authors explicitly cite his thesis as reference [1].
Kaczynski considered a trapezoid
$T=\bigcup \ell$
where each line segment $\ell$ connects a point of $X_0$ with a point of $X_1$.
For the special case where the line segments are disjoint, he proved

$\mu_2(T)=\frac{\mu_1(T_0)+\mu_1(T_1)}{2}$

This is a remarkably clean relationship between the two-dimensional measure of the interior construction and the one-dimensional measures of its boundary sets.
He then used this machinery to construct a collection of non-intersecting paths whose union has measure zero, even though the paths correspond to every point of an interval.
The paper explains that this construction also gives an example of a measurable function whose boundary function is non-measurable.
That is an important distinction:

Quote:Kaczynski's mathematical work here was not about technology or politics. It was genuine research in real analysis, measure theory and complex analysis.

3. The really strange phenomenon: zero boundary measure but huge interior measure
Kaczynski also considered what happens if the line segments are allowed to intersect.
The result becomes dramatically different.
Kuklinski reproduces a result from Kaczynski showing that there is an unrestricted trapezoid with
$\mu_1(T_0)=\mu_1(T_1)=0$
but
$\mu_2(T)=\infty$.
So the two boundary sets can have zero one-dimensional measure, while the union of the connecting segments has infinite two-dimensional measure.
This demonstrates that once intersections are permitted, the intuitive relationship between the boundary and interior breaks down completely.
And this leads directly to the question that Kuklinski investigates:
Can the opposite happen?
Can the boundary sets have large measure while the union of the connecting segments has zero area?

4. The problem Kaczynski left behind
Kaczynski later returned to this problem, decades after his PhD.
The paper says that more than 30 years later, he reformulated the problem using increasingly thin parallelograms instead of infinitely thin line segments and proposed three conjectures.
Define
$\mu_2^*(T)=\frac{\mu_1^*(T_0)+\mu_1^*(T_1)}{2}$
where $T_n^\sigma$ is constructed from $n$ thin parallelograms according to a permutation $\sigma$.
Kaczynski proposed:
Conjecture 1
$\lim_{n\to\infty}\alpha(n)=0$.
That means that with increasingly fine constructions, the minimum possible area should approach zero.
The fascinating part is that Kaczynski claimed to have proved this. Kuklinski supplies a different proof.
Conjecture 2
Kaczynski suggested that the decay should essentially behave like
$\frac{1}{\log n}$
up to constants.
More specifically, the paper records his proposed form
$c_-\frac{1}{\log n}<\alpha(n) and notes that Kaczynski claimed a weaker upper bound than this.
Conjecture 3
Kaczynski conjectured that
$\alpha(n)$
is monotonically decreasing.
This remained open in the formulation discussed by the paper.

5. What Kuklinski actually proves
This is the main mathematical contribution of the 2023 paper.
The author connects Kaczynski's trapezoid problem to the geometry of the Sierpiński gasket.

The connection is quite elegant.
Kuklinski constructs a function based on ternary expansions. Essentially, digits $1$ and $2$ are exchanged in a base-3 representation:
$1\leftrightarrow2$.
This produces a highly discontinuous/fractal correspondence $f:[0,1]\to[0,1]$.
The associated collection
$S={((1-t)x+tf(x),t):x,t\in[0,1]}$
can nevertheless have
$\mu_2(S)=0$.
That is the key result.
So the paper constructs an uncountable union of line segments connecting two full intervals, yet the entire planar set has zero area.
That's the result behind the paper's title.

6. Why the Sierpiński gasket appears
This is probably the most beautiful part mathematically.
The ternary construction naturally generates Cantor-set-like slices.
Those slices can be related to projections of the Sierpiński gasket.
The paper then uses a quantity called the Favard length/distance, essentially an average of the lengths of projections of a planar set:
$\operatorname{Fav}(E)=\frac{1}{\pi}\int_0^\pi
\mu(\operatorname{proj}_\theta(E)),d\theta$.
The authors use known estimates concerning projections of finite approximations to the Sierpiński gasket.
The important chain of ideas is:
Kaczynski's trapezoids
$\downarrow$
ternary/fractal construction
$\downarrow$
Cantor-type sets
$\downarrow$
Sierpiński gasket
$\downarrow$
projection lengths
$\downarrow$
Favard length
$\downarrow$
area of the trapezoid tends to zero.
That's the modern mathematical machinery used to attack Kaczynski's old problem.

7. The main new bound
The paper obtains
$\mu_2(T_{3^n}^{\sigma})\leq Cn^{-p}$
for some constants $C,p>0$.
Then, for general $n$, it obtains the weaker but still significant result
$\alpha(n)<\frac{C}{(\log n)^p}$.
Consequently,
$\boxed{\lim_{n\to\infty}\alpha(n)=0}$.
Thus Kaczynski's first conjecture is established by the paper's argument, although through a different route from Kaczynski's claimed proof.
Importantly, the paper does not prove the full sharp $1/\log n$ conjecture recorded from Kaczynski. Instead it establishes a polynomial-in-$1/\log n$ upper bound.

8. The connection to Kaczynski's PhD work
There's a deeper historical point here.
Kaczynski's mathematical career had two quite different stages.
1960s:
He was working in complex analysis, geometric measure theory and boundary behavior of functions. His 1967 thesis was concerned with boundary functions and unusual behavior of functions near the boundary of domains.
Later decades:
He returned to some of the geometrical questions from his earlier work and reformulated them in terms of finite collections of parallelograms, producing the conjectures discussed in this paper.
Kuklinski's paper effectively creates a bridge:
Kaczynski 1967 → Kaczynski's later conjectures → modern fractal geometry → new proof.
That is probably the most interesting aspect of the paper if you're specifically studying Kaczynski.

9. One particularly striking detail
Kaczynski's mathematical work is sometimes overshadowed by his later crimes and manifesto. This paper gives a concrete example of why the distinction between “Kaczynski the mathematician” and “Kaczynski the terrorist” matters.
The paper's bibliography treats his 1967 thesis simply as a mathematical source:

Quote:Theodore John Kaczynski, Boundary Functions, PhD thesis, University of Michigan, 1967.

And the author is able to take a problem originating in that thesis and connect it to modern fractal geometry and harmonic/measure-theoretic ideas.
So, mathematically speaking, Kaczynski's relevant legacy here isn't the manifesto. It is a set of unusual questions about measure, boundary behavior and geometric constructions that survived long enough to become useful starting points for later research.

Key takeaways

Kaczynski's 1967 PhD thesis is the starting point for the mathematical problem.
He studied boundary functions, measure and unusual geometric constructions.
He proved a striking theorem relating the measure of certain trapezoidal unions to their boundary measures.
He later formulated the problem using thin parallelograms.
He proposed three conjectures concerning the minimum possible area $\alpha(n)$.
Kuklinski proves that $\alpha(n)\to0$, establishing the substance of Kaczynski's first conjecture through a different argument.
The proof uses Cantor-type sets, ternary expansions, the Sierpiński gasket and Favard-length estimates.
The resulting construction gives an uncountable union of line segments with zero two-dimensional measure.
The paper therefore represents a genuine example of a modern mathematical development growing out of one of Kaczynski's old research problems.

One correction to a common impression is important: this paper does not show that Kaczynski was secretly a major modern mathematician whose later work was ignored. It shows something more precise and interesting: he posed some genuinely nontrivial mathematical questions, and at least one of those questions has subsequently been connected to modern fractal geometry and solved in a new way.

ARTICLE [PDF]

The Mathematical Center of the Universe

Thu, 24 Sep 2026 19:15:01 +0300

The article “The Mathematical Center of the Universe: 1800–1933” by Jørgen Veisdal traces the extraordinary rise of the University of Göttingen as a global center of mathematics and mathematical physics. It begins with Carl Friedrich Gauss, who established a tradition that attracted figures such as Bernhard Riemann, Richard Dedekind and Peter Gustav Lejeune Dirichlet.

Their work transformed algebra, number theory, geometry, analysis and the foundations of mathematics. Later, figures including Felix Klein, David Hilbert, Hermann Minkowski and Edmund Landau reinforced Göttingen's reputation, with Hilbert in particular attracting a remarkable generation of doctoral students and developing influential ideas in algebra, geometry, logic, physics and the foundations of mathematics.

The article then describes Göttingen's extraordinary connection with the birth of modern quantum mechanics during the 1920s. Max Born became a central figure, working with and mentoring scientists including Werner Heisenberg, Wolfgang Pauli, Pascual Jordan, Paul Dirac and J. Robert Oppenheimer, while John von Neumann studied mathematics under Hilbert. Göttingen therefore became a meeting point where mathematics and physics interacted at an exceptional level.

This intellectual environment was catastrophically disrupted in 1933, when the Nazi regime's antisemitic policies forced many Jewish and politically targeted academics to leave Germany. The resulting exodus transferred enormous intellectual talent to Britain, the United States and elsewhere, and the article presents 1933 as the decisive end of Göttingen's remarkable period as the world's leading mathematical center.

ARTICLE [PDF]

Figures from the History of Probability and Statistics

Wed, 16 Sep 2026 19:39:30 +0300

The University of Southampton’s “Figures from the History of Probability and Statistics”, compiled by John Aldrich, is a chronological guide to the people and ideas that shaped probability and statistics. It begins around 1650, when probability emerged from mathematical problems involving games of chance and statistics developed through the systematic analysis of mortality and population data.

The site highlights figures such as Pascal, Fermat, Huygens, Graunt, Jakob Bernoulli, de Moivre, and Daniel Bernoulli, showing how their work gradually connected probability with empirical data, demography, insurance, and decision-making. Graunt’s analysis of mortality records, for example, helped establish population statistics and the life table, while Bernoulli’s work introduced the law of large numbers and important ideas about expected utility.

The collection then follows the development of the subjects through the eighteenth, nineteenth, and twentieth centuries, covering major figures including Bayes, Laplace, Gauss, Quetelet, Galton, Pearson, Fisher, Neyman, Kolmogorov, Markov, and Savage. Rather than presenting isolated biographies, Aldrich emphasizes how ideas evolved and influenced one another: probability moved from games of chance toward a general mathematical theory of uncertainty, while statistics increasingly became a method for extracting information from real-world data. The website also provides links to biographies, original texts, historical books, and other resources, making it essentially a historical map of probability and statistics from the seventeenth century into the modern era.

ARTICLE

Fourier, one man, several lives

Tue, 15 Sep 2026 19:56:56 +0300

The article “Fourier, One Man, Several Lives” by Bernard Maurey presents Joseph Fourier not simply as the mathematician associated with Fourier series, but as a remarkable figure whose life unfolded amid the upheavals of the French Revolution and Napoleonic era. Born in 1768, Fourier received a strong education from the Benedictines,

initially considered entering the military, and later became a teacher. His involvement in revolutionary politics brought serious danger: he was investigated, imprisoned, and narrowly avoided more severe punishment. After the Revolution, he studied and taught at institutions including the École Normale and École Polytechnique, where he came into contact with leading mathematicians such as Monge, Laplace, and Lagrange.

The central scientific theme is Fourier's work on heat propagation, which ultimately led to his monumental Théorie analytique de la chaleur. The article explains how Fourier developed mathematical methods for representing complicated functions through trigonometric series, now known as Fourier series.

These ideas were initially controversial because Fourier's arguments did not always conform to the rigorous standards mathematicians would later demand, yet they proved extraordinarily powerful and became fundamental to harmonic analysis, differential equations, physics, and engineering. Maurey therefore portrays Fourier as having “several lives”: revolutionary, teacher, administrator, politician, and—above all—a mathematician whose ideas transformed the way mathematics describes physical phenomena.

ARTICLE [PDF]

An introduction to the mathematics in the Alhambra of Granada

Tue, 15 Sep 2026 18:45:52 +0300

The article “An introduction to the mathematics in the Alhambra of Granada” by Miguel Ortega explores the mathematical ideas underlying the extraordinary geometric decoration of the Nasrid Palaces. The author explains how medieval artisans used Euclidean geometry and the four basic isometries of the plane—translations, rotations, reflections, and glide reflections—to construct elaborate patterns from relatively simple geometric lattices of squares, triangles, and rectangles. Examples include the famous pajarita (bow-tie), the “bone” pattern, and the black-and-white “airplane” tiling.

The article then introduces rosettes, patterns with finite rotational or reflectional symmetry, and explains that mathematically they can have either cyclic or dihedral symmetry. Particularly remarkable are the many order-9 rosettes found in the Alhambra, since a regular nonagon cannot be constructed exactly with the classical ruler-and-compass methods.

The second major theme is plane crystallographic groups, which provide a mathematical framework for understanding repeating decorative patterns. Ortega introduces the crystallographic restriction theorem, showing that rotational symmetries in a plane crystallographic group can only have orders 2, 3, 4, or 6, and uses Conway-style notation to describe the different symmetry combinations. He then identifies examples corresponding to all 17 plane crystallographic groups among the Alhambra's decorations, including patterns based on translations, reflections, glide reflections, and rotations.

The central message is that the Nasrid artists, despite working centuries before modern mathematical terminology and technology, achieved an astonishing variety and precision of geometric constructions. Their artistic achievements provide a remarkable visual demonstration of concepts from group theory, geometry, symmetry, tilings, and crystallography.

ARTICLE [PDF]

An interview with Jean-Pierre Serre

Tue, 15 Sep 2026 18:24:17 +0300

The article is an extensive 2026 interview with Jean-Pierre Serre, one of the most influential mathematicians of the 20th and 21st centuries, conducted by Javier Fresán shortly before Serre’s 100th birthday. Serre reflects on how he discovered mathematics as a teenager, his early involvement with Bourbaki, and the mathematicians who shaped him, including Henri Cartan, André Weil, Claude Chevalley, and Alexander Grothendieck.

He discusses his fascination with algebra, number theory, algebraic groups, Galois representations, and modular forms, while offering candid assessments of his contemporaries. A recurring theme is his preference for concrete, elegant mathematics and his belief that mathematics should be approached with curiosity rather than excessive specialization.

The interview also provides fascinating accounts of Serre's major mathematical discoveries. He describes his work on homotopy groups of spheres, including the insight that emerged during a train journey and led to his thesis, as well as later work on finite groups, representations, modular forms, and Galois representations. Serre emphasizes that his mind "likes to play" rather than concentrate on a single famous problem, explaining why he never seriously pursued the Riemann hypothesis.

He also discusses mistakes and periods of difficulty, showing a remarkably experimental attitude toward research: formulate a plausible idea, calculate, test it, and move on when necessary. Overall, the interview portrays Serre not simply as a monumental mathematician, but as a remarkably curious, independent, and playful thinker whose work has connected topology, algebra, geometry, and number theory.

ARTICLE [PDF]

Alan Turing in America

Wed, 09 Sep 2026 23:21:50 +0300

Alan Turing in America
Author: Jørgen Veisdal

The article traces Alan Turing’s two important visits to the United States, first as a young mathematician studying at Princeton from 1936 to 1938 and later during the Second World War as part of Britain’s cryptographic cooperation with the United States. Turing arrived at Princeton shortly after completing his revolutionary paper On Computable Numbers, with an Application to the Entscheidungsproblem. Working under Alonzo Church, he developed ideas that helped establish the mathematical foundations of computation. His concept of the Turing machine provided a remarkably concrete definition of an algorithm: instead of starting with abstract formal systems, Turing analysed how a human performs a calculation step by step and showed that such procedures could be represented mechanically. At Princeton he interacted with an extraordinary mathematical environment that included figures such as John von Neumann, Hermann Weyl and Albert Einstein, although, despite his interest in Kurt Gödel, the two apparently never met. Turing completed his PhD in 1938 with Systems of Logic Based on Ordinals, introducing ideas such as oracle machines and extending the study of what can and cannot be computed.

Turing returned to America in 1942, this time as a cryptanalyst connected with the secret work at Bletchley Park. His task involved exchanging information with American cryptographers while also protecting sensitive British intelligence about the breaking of the German Enigma system. He was often frustrated by American cryptographic methods, but the trip produced one especially important intellectual encounter: his discussions with Claude Shannon at Bell Labs. The two talked extensively about information, computers, the human brain and the possibility of machines capable of thought—questions that would later become central to computer science and artificial intelligence. Despite these productive exchanges, Turing developed a strong personal dislike of America and later declined invitations to return. The article therefore presents his American experiences as a mixture of immense scientific opportunity, wartime secrecy, cultural friction and conversations that helped shape the emerging age of computing.

Key takeaways:

Turing's Princeton years were crucial to the development of computability theory and theoretical computer science.
His Turing-machine model gave a fundamental mathematical description of what it means for a problem to be algorithmically computable.
His PhD work introduced oracle machines, extending the study of computation beyond ordinary Turing machines.
His wartime conversations with Claude Shannon anticipated later debates about computers, intelligence and artificial minds.

ARTICLE / ARTICLE [PDF]

Father of Cybernetics, Norbert Wiener

Wed, 09 Sep 2026 23:17:25 +0300

The Absent-Minded Father of Cybernetics, Norbert Wiener
Author: Jørgen Veisdal
Published: March 7, 2020
Publication:Cantor’s Paradise

Norbert Wiener (1894–1964) was one of the most remarkable mathematicians of the twentieth century. A child prodigy, he finished high school at the age of 11, graduated from Tufts University at 14, and earned a PhD in mathematical logic from Harvard before turning 19. He later studied with major figures such as Bertrand Russell, G. H. Hardy, and David Hilbert before joining MIT in 1919. Wiener made important contributions to probability, harmonic analysis, stochastic processes, and signal theory. His rigorous treatment of Brownian motion led to what is now known as the Wiener process, while other major results associated with his name include the Wiener–Khinchin theorem, Wiener’s Tauberian theorem, and the Paley–Wiener theorems.

During World War II, Wiener worked on the difficult problem of predicting the movement of enemy aircraft. Together with Julian Bigelow, he viewed the pilot, aircraft, observer, and weapon as interconnected parts of a probabilistic feedback system. This research contributed to the development of methods for extracting useful signals from noisy data, including the Wiener filter. These ideas eventually led Wiener to develop the field of cybernetics, which he described as the study of control and communication in animals and machines.

His influential 1948 book Cybernetics: Or Control and Communication in the Animal and the Machine introduced feedback as a general principle that could connect mathematics, engineering, biology, and computation. The concept of systems continuously receiving information, comparing their current state with a desired state, and adjusting their behavior became fundamental to control theory, robotics, neuroscience, computing, and later artificial intelligence.
The article also highlights the contrast between Wiener’s extraordinary intellect and his famously absent-minded personality. Numerous stories describe him forgetting colleagues, losing track of where he had parked his car, or even being uncertain whether he had already eaten. Despite these eccentricities, Wiener was known as a generous and thoughtful mathematician who supported younger researchers and colleagues.

Wiener’s lasting importance lies in the way he helped transform how scientists think about information, communication, prediction, and feedback. His work crossed the traditional boundaries between pure mathematics and engineering and provided many of the conceptual foundations on which modern computing, automation, robotics, and artificial intelligence were later built.

Key Takeaways

Norbert Wiener was a mathematical child prodigy who completed his PhD before the age of 19.
He made major contributions to probability theory, harmonic analysis, stochastic processes, and signal processing.
The Wiener process became one of the fundamental mathematical models of Brownian motion.
His wartime research on predicting aircraft movements helped lead to the development of the Wiener filter and modern signal-processing techniques.
Wiener founded the field of cybernetics, the study of control and communication in machines and living organisms.
He identified feedback as a fundamental principle common to biological, mechanical, and computational systems.
His ideas strongly influenced control theory, robotics, neuroscience, computer science, and artificial intelligence.
Despite his intellectual brilliance, Wiener became famous for his extreme absent-mindedness and eccentric personality.
He can be regarded as one of the major intellectual ancestors of modern thinking about AI, automation, and information systems.

ARTICLE / ARTICLE [PDF]

The Mathematics of Ted Kaczynski

Wed, 09 Sep 2026 22:54:21 +0300

The Mathematics of Ted Kaczynski
Author: Jørgen Veisdal
Published: March 16, 2020
Publication:Cantor’s Paradise

The article examines Ted Kaczynski’s largely forgotten career as a professional mathematician before he became known for terrorism. It stresses that the discussion is not intended to glorify him, but rather to assess realistically the frequently repeated claims about his mathematical genius. Kaczynski studied mathematics at Harvard before completing his M.Sc. and Ph.D. at the University of Michigan. His doctoral dissertation, Boundary Functions (1967), was supervised by Allen Shields and won the university’s Sumner Myers Prize for the best mathematics thesis. Shields reportedly regarded it as the best dissertation he had supervised. Kaczynski subsequently became an assistant professor at UC Berkeley in 1967, but unexpectedly resigned in 1969.

Most of Kaczynski’s research belonged to real and complex analysis, particularly geometric function theory and the study of boundary behaviour of continuous and harmonic functions. Roughly speaking, his work investigated what happens to a function $f(z)$ as $z$ approaches a boundary point along different curves or arcs. A central concept was the set of curvilinear convergence: the collection of boundary points at which a function approaches a definite limiting value along at least one suitable curve. His dissertation established general structural results about such sets and produced new proofs concerning boundary functions. Between 1965 and 1969 he published five papers arising from this research in respected journals including the Transactions of the American Mathematical Society and the Proceedings of the American Mathematical Society. He also published a short group-theoretic proof of Wedderburn’s theorem, which states that every finite division ring is commutative, and contributed an algebra problem to the American Mathematical Monthly.

The article ultimately presents a more restrained assessment of Kaczynski’s mathematical importance. His work was technically sophisticated and demonstrated exceptional ability, but it concentrated on a very narrow area with limited influence on the subsequent development of mathematics. Mathematicians quoted in the article describe the research as first-rate while noting that relatively few specialists were interested in the subject and that the field largely disappeared as an active research direction. Thus, the article distinguishes between mathematical talent and lasting mathematical impact: Kaczynski appears to have possessed considerable technical ability and research potential, but his surviving mathematical work cannot reasonably be placed alongside that of historically transformative mathematicians.

Key takeaways

Kaczynski was a serious research mathematician, not simply someone with a strong interest in mathematics.
His main research area was real and complex analysis, especially boundary behaviour and geometric function theory.
His Ph.D. dissertation, Boundary Functions, was regarded very highly and received the Sumner Myers Prize.
He published several papers in respected mathematical journals during the 1960s.
His work showed considerable technical ability and research talent.
However, his research focused on a narrow area and had relatively limited long-term influence on mathematics.
The article therefore distinguishes between exceptional mathematical talent and major historical mathematical impact.

ARTICLE / ARTICLE [PDF]

Richard Stearns (1936-2026)

Tue, 08 Sep 2026 00:24:38 +0300

Authors: Eugene H. Spafford and Simson L. Garfinkel
Published: September 3, 2026, Communications of the ACM
Area: Theoretical Computer Science — Computational Complexity Theory

The article commemorates Richard E. Stearns (1936–2026), one of the founders of modern computational complexity theory. His most influential contribution came through his 1965 paper with Juris Hartmanis, On the Computational Complexity of Algorithms. That work introduced a rigorous way of classifying computational problems according to the resources—especially time—required to solve them, developing notions such as $DTIME(T(n))$. It also established early time-hierarchy results, showing in a precise mathematical sense that giving a machine more computational time can allow it to solve strictly more problems. This conceptual framework helped transform questions about whether algorithms merely existed into questions about how efficiently computation could actually be performed. Stearns and Hartmanis received the 1993 ACM A.M. Turing Award for this foundational work.

Stearns' influence extended well beyond complexity theory. His research covered automata and formal-language theory, compiler construction, algorithm analysis, databases, and game theory. Among his important results were work on deterministic pushdown automata and, with Philip Lewis, research that helped introduce LL parsing, which became important in compiler design. He spent 17 years at General Electric Research Laboratory before joining the University at Albany, where he spent more than two decades and served as department chair. Even late in his career, he continued publishing research, illustrating an unusually long scientific life. Stearns died on August 29, 2026, at age 90, leaving behind concepts that are now part of the basic language of theoretical computer science.

Key takeaways

Stearns was a father of computational complexity theory, helping formalize the study of computational resources.
The 1965 Hartmanis–Stearns paper provided foundations for complexity classes such as $DTIME(T(n))$ and the time hierarchy theorem.
His contributions also reached formal languages, parsing and compilers, databases, algorithms, and game theory.
His work helped establish one of computer science's central questions: not simply “Can a problem be solved?”, but “How much computational time or other resources are fundamentally required to solve it?”

ARTICLE

Hooke–Newton inverse square law controversy

Mon, 07 Sep 2026 22:23:21 +0300

Summary

The Hooke–Newton inverse-square law controversy concerns who deserves credit for recognizing that gravitational attraction decreases with the square of distance, approximately $F\propto 1/r^2$. Robert Hooke had developed important ideas about planetary motion during the 1660s and 1670s, arguing that celestial bodies attract one another and that planetary orbits could be understood as a combination of inertial motion along a tangent and attraction toward a central body. In correspondence with Isaac Newton in 1679–1680, Hooke explicitly proposed that attraction varied inversely with the square of the distance. However, Hooke did not provide a rigorous mathematical derivation, and some of his associated conclusions were incorrect.

The dispute erupted in 1686, shortly before the publication of Newton's Principia, when Hooke claimed that Newton had taken the inverse-square idea from him. Newton strongly rejected this claim, pointing out that inverse-square ideas had already appeared in earlier work by scientists such as Ismaël Bullialdus, and that Newton himself had investigated related inverse-square relations during the 1660s. More importantly, Newton transformed the hypothesis into a comprehensive mathematical theory. He demonstrated that an inverse-square central force produces the observed properties of planetary orbits and proved that a spherically symmetric body attracts an external object as though its mass were concentrated at its centre. Thus Newton established the universal gravitational law in the form
$F=G\frac{m_1m_2}{r^2}$.

Modern historians therefore generally distinguish between suggesting the idea and proving the theory. Hooke deserves significant credit for independently proposing the inverse-square relationship and for encouraging Newton to reconsider planetary dynamics in terms of centripetal attraction, but the inverse-square concept itself predates both men. Newton's decisive contribution was to connect the idea with mathematics, Keplerian orbital motion and observational evidence, turning a plausible hypothesis into a predictive theory of universal gravitation. Newton did acknowledge Hooke, Christopher Wren and Edmond Halley in the Principia, although debate continues over exactly how much Newton's thinking was stimulated by Hooke's correspondence.

Key takeaways

Hooke: clearly proposed $1/r^2$ gravitational attraction by 1680, but did not mathematically prove its consequences.
Newton: supplied the rigorous mathematical framework connecting inverse-square gravity with planetary orbits.
Neither man invented the inverse-square idea from nothing; related proposals existed earlier.
The controversy illustrates the crucial distinction between having an insightful hypothesis and demonstrating that it forms a quantitative scientific theory.

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Leibniz–Newton calculus controversy

Mon, 07 Sep 2026 22:19:09 +0300

Summary

The Leibniz–Newton calculus controversy was a major 17th- and early-18th-century dispute over who deserved priority for the invention of calculus. Isaac Newton developed his method of fluxions as early as 1665–1666, using ideas involving quantities changing with time, but he did not publish a full account for many years. Gottfried Wilhelm Leibniz began developing his own calculus around 1674 and published his first paper in 1684, well before Newton published his fluxional methods. Modern historians generally conclude that Newton and Leibniz developed calculus independently, although Newton reached many of the central ideas earlier.

The dispute became serious after 1699, when supporters of Newton accused Leibniz of having obtained his ideas from Newton's unpublished manuscripts and correspondence. The controversy intensified in 1712, when the Royal Society published the Commercium Epistolicum, a report supporting Newton's priority claim. The report's neutrality was questionable because Newton, then president of the Royal Society, was deeply involved in preparing it. Leibniz had indeed seen some of Newton's mathematical work, but his surviving notebooks show a distinct path toward calculus, providing strong evidence that the essential structure of his system was independently developed.

The two approaches also differed conceptually and notationally. Newton described changing quantities through fluxions, commonly represented by notation such as $\dot{x}$, whereas Leibniz introduced the differential notation $\frac{dy}{dx}$ and the integral sign $\int$. Leibniz's notation proved more flexible and became the foundation of the notation used in modern calculus. The controversy therefore ended without a simple winner: Newton appears to have developed calculus first, while Leibniz published first and independently created the notation and formalism that became dominant.

Key takeaways

Newton developed calculus earlier, beginning around 1665–1666.
Leibniz published calculus first, in 1684.
The modern historical consensus is that both discovered calculus independently.
Leibniz's notation, especially $\frac{dy}{dx}$ and $\int$, became the standard language of differential and integral calculus.

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