During the first half of the 20th century, mathematicians found themselves with a conundrum. They celebrated the breakthroughs within pure mathematical thinking but lamented that its successes also enabled refinements to devastating weapons technology during the two world wars and beyond. Working through these questions in 1942, Lillian R. Lieber, a professor in the mathematics department at Long Island University, ultimately described mathematics as a tower: topped by pure (and good) abstraction, but connected to and often rooted in applications both good and bad.
Another way of describing the extremes of Lieber’s tower is the term dual use technology: tech with both civilian and military applications. Improvements to all-terrain vehicles, for example, change how such machines can be used for recreational purposes, as well as military ones. Research that experimentally explores ways to curb the virulence of infectious pathogens can provide insights for treatment and prevention of transmission but also illustrate how a pathogen can be modified to inflict more damage. Innovations, whether in machines or in molecules, can lead to outcomes that are both helpful and harmful. Mathematical innovations can yield the poetic beauty of a sequence of logical steps; yet the same innovation also can enable critical steps in the development of a weapon or define the optimal distribution of a biologic agent in order to maximize harm.
Originally, dual use technology would have applied only to a dichotomy between military and civilian uses. But in the wake of the 9/11 terrorist attacks in the US, the term evolved to refer to research like that of pathogens, yielding new insights that can both help and hinder, regardless of the military/civilian nature of application. Now, the question of the age might be whether there are dual use ideas. Can innovations in mathematics enable both beauty and harm? Can computational algorithms speed ranking of resumes, yet simultaneously amplify historical biases in hiring?
The dangers of dual use in mathematics and machine learning are explored in three texts, two recent and one from the mid-20th century: Lieber’s 1942 The Education of T. C. Mits, Alma Steingart’s 2023 Axiomatics, and Brian Christian’s 2020 The Alignment Problem. Taken together, these books show how breakthroughs in mathematical and computational thought are often believed to transcend potential misuse, but have darker shadows that always follow closely behind. Eighty years after some of the first critical thinking on modern dual use, these texts make one thing absolutely clear to boosters of AI and algorithms: rapid development does not eliminate the need to be watchful for misuse.
In the field of mathematics one moves from a set of assumed truths (axioms) to a set of derived truths (theorems), proved by the logical consequences of the axioms and their interactions. This is the logical basis of mathematics. And this foundation suggests that math is a science based on the discovery of truths that already exist but were simply previously unknown. In this sense, the goal of the field of mathematics simply is.
Or is it? Despite such high ideals, mathematical thought occurs within historical and cultural contexts; moreover, mathematical results and their applications can have cultural consequences. Contrary to protestations of some students, mathematics does indeed have implications in the “real world.”
The history of mathematical thought and thinking considers both the history of mathematics itself and the history of motivations driving mathematicians at different times. Alma Steingart provides a history of mathematical thinking over the twentieth century: a compelling review of the increased abstraction of mathematical thought as well as its embrace of deep exploration of alternative axiomatic systems.
Historically, Steingart notes, the axioms from which mathematical reasoning derived connected to observable, often physical systems. For example, the ancient Greek geometry of Euclid was based on a set of axioms connected to humans’ physical sense of two- and three-dimensional space.
However, in the late 19th and early 20th century, mathematicians began exploring the truths and theorems resulting from new sets of axioms, often without direct connection to physical objects and relationships. While Euclidean geometry is based on axioms such as “two parallel lines never intersect” and “nonparallel lines intersect at a single point,” the field of non-Eulidean geometry derives from a similar but fundamentally different set of axioms: “parallel lines do intersect.” Although these axioms have little to no basis in our observed reality, mathematical logic can define and prove theorems from them, creating, in a sense, a new type of geometry. This new geometry is self-consistent, and found applications in relativity theory in the early 20th century.
Mathematical thought occurs within historical and cultural contexts; moreover, mathematical results and their applications can have cultural consequences.
In a sense, each axiomatic system defines its own reality, waiting to be discovered through development and proofs of theorems. Such work is referred to as axiomatics, a type of mathematical research that deeply explores different systems defined by different core sets of axioms. Mathematicians responded eagerly to this challenge, and as detailed in Steingart, axiomatics quickly dominated mathematics research and graduate education through much of the twentieth century. The growth, expansion, and influence of axiomatics drove the development of thinking in mathematical research, both pure and applied, as well as the “mathematization” of other fields of study in the biological, physical, and social sciences.
By World War II, however, the field of mathematics was grappling with the excitement of discoveries (thanks, in large part, to axiomatics) and dismay about their misuse in wartime applications. This struggle is winningly undertaken by Lieber in her uniquely styled, pseudo-free verse description of modern mathematics in 1942, The Education of T. C. Mits (The Celebrated Man in the Street). Her target audience is a fictitious individual named T. C. Mits (acronym for “The Celebrated Man in the Street”), and she deftly presents a very readable and approachable description of axiomatic thinking and non-Euclidian geometry.
Lieber’s book is truly unique and a delight to read. It reveals the struggle firsthand in a broad review of complicated axiomatic mathematical concepts (e.g., non-Euclidean geometry). Her husband, also a mathematician, illustrated the text with cartoons in the style of mid-century modern art (itself subject to “mathematized” axiomatic thinking, as discussed in Steingart ). The illustrations capture the tension of the beauty of pure mathematical thinking with horror relating to its application to war.
The struggle between the uses of mathematical thinking for benevolent and malevolent purposes carries forward to the current development, expansion, hype, and practice of machine learning algorithms driving current research and applications of artificial intelligence (AI). Brian Christian captures an arc of history through the last quarter of the twentieth century and into the first quarter of the twenty-first. In many ways, the thrill of researchers pushing envelopes, enabled by new breakthroughs in thinking and computational technology, is quite similar to Steingart’s documentation of the rapid growth of and enthusiasm for axiomatics in the mathematics community some 50 years earlier.
Christian highlights both technological and algorithmic developments in AI, with their promise and peril. In so doing, he mirrors Lieber’s own excitement and dismay (but, admittedly, without the creative poetry/prose structure that makes Lieber’s book uniquely appealing). (In a twist of publication history, the same publishing company produced both books.)
Christian documents advances in computing, data availability, algorithmic structure, and, importantly, modes of thinking in the people defining the development of machine learning. The computer scientists (the people) reveled in the rapid progress and breakthroughs in the second half of the twentieth century, paralleling mathematicians’ excitement surrounding axiomatic thinking in the first half of the century.
In both settings, the pace of development raised optimism about new ways of thinking. Advances dominated attention; worries about negative applications were consciously or unconsciously pushed to the far back burner.
Such heady days eventually came to an end. In mathematics, World War II offered sobering reflection, and the text and illustrations in Lieber’s book—as well as the metaphor of the tower—serve as a snapshot of this moment. While axiomatic thinking continued to dominate mathematical theory, fully extending the concepts of core axioms to other fields (e.g., biology, art) proved elusive and generated pushback. The “mathematization” and “axiomatization” of these other fields slid further down Lieber’s tower.
And Lieber’s tower is still with us. Today, in machine learning, algorithms in image classification and text translation have rapidly improved. But such successes are now offset by examples of racial and gender bias in facial recognition and resume recommendations. Such concerning examples have fueled a growing realization that seemingly neutral AI algorithms based on historical data can identify and amplify past patterns of bias.
Such a mix of help and harm is analogous to the dual use technologies of yesteryear, those that Lieber noted may have lofty ideals but nevertheless remain connected to real-world applications that can include harm and violence. Even so, it is comforting that while both mathematics and machine learning have dual use tendencies, this does not mean innovations are equally likely to have beneficial or malevolent consequences. Positive applications typically outweigh negative applications, and the problem-solving nature of mathematics and machine learning often motivates a quest for positive solutions once a problem is identified. In short, this brief summary serves as a reminder that practitioners in these fields should use their genius for good, not evil.