How reality ceased to define mathematics – and began to invent new ones

Euclid items Provided the foundation for mathematics.Credit: Tim Ockenden/PA/Alamy

Axioms: Mathematical Thought and High Modernity Alma Steingart University. Chicago Press (2022)

The work of mathematicians centuries or even millennia ago speaks to their living peers in ways that practitioners of other disciplines must find mystifying. Euclid’s proof that the list of primes never ends is as elegant and clear now as it was about 300B.C.E.when it appeared in his book items.

However, mathematics has undergone tremendous changes, especially during the twentieth century, when it moved deeper into the realm of abstraction. These disorders even involved redefining definition itself, as Alma Steingart explains in V axioms.

Steingart, a historian of science, sees this revolution as central to the modernist movements that dominated the mid-twentieth century in the arts and social sciences, particularly in the United States. Mathematicians’ push for abstraction was reflected—and often directly triggered—by parallel trends in economics, sociology, psychology, and political science. Steingart quotes some scholars who saw their liberation from merely explaining the natural world as analogous to how Abstract Expressionism liberated painting from the shackles of reality.

Mathematics encapsulated in Euclid items He started from foundational truths called axioms and built statements through logical deductions. For Euclid, axioms such as “any straight segment can be extended into an infinite line” were axioms because they were rooted in physical reality.

For centuries, Western culture has been respected items as a model of intellectual rigor. In the 17th century, the political theorist Thomas Hobbes referred to the book as a “structural model for a stable society,” and the philosopher Baruch Spinoza adopted its structure of axioms and deductions in his Treatise on Ethics. Meanwhile, the Catholic Church viewed Euclid’s work as dogma, a way to anchor itself amidst what it saw as the chaos brought on by the Protestant Reformation (and also to protect it from the revolutionary threat of the “extremely small” numbers introduced into calculus, as Prince Alexander explains in his book interesting for 2014 infinitesimal).

Divergent worlds

Modern mathematics turns Euclid’s principle of axioms derived from physical reality on its head. Mathematicians may follow their creative impulses to come up with any set of axioms they desire—for example, inventing non-Euclidean geometric shapes in which parallel lines converge or diverge.

In this concept, creating a mathematical theory is analogous to establishing the rules of a game like chess, in which the names, shapes, and roles of pieces are purely a matter of imitation. Just as a person can start with the rules of chess and a certain board configuration and come up with a prediction of “checkmate in five moves,” a mathematician can start from axioms and go through a series of logical steps to prove a theoretical fact, without worrying about which reality it represents.

Diagrams illustrating the mathematics of circles from the first six books of Euclid's Elements.

Euclid’s geometry was rooted in reality – but modern mathematics is less constrained.Credit: SSPL/Getty

For a mathematical theory builder, abstraction is not a destination, but a journey. As Steingart said, “abstract” is not an adjective but a verb: “abstract.” In the 1930s, due to the influence of the German mathematician Emmy Noether, mathematicians began to construct axiomatic systems that were increasingly abstract and general. This revealed familiar objects such as numbers, card changes, and geometric symmetries to be special cases of the same concept.

The tendency towards abstraction and generalization is often associated with a school of mathematics that flourished in France after World War II. But, as Steingart explains, it took hold in the 1930s in the United States and came to define the country’s mid-century sports culture. Steingart captures the trend with a story Foundations of algebraic topology, a 1952 book by American mathematicians Samuel Eilenberg and Norman Steenrod. It has dealt with various computational techniques for distinguishing between geometric shapes, but the authors present the topic backwards, claiming that students should first become familiar with high-tech algebraic tools and later learn how they fit shapes, or why the tools exist in the first place.

Central thesis in axioms is that American mathematicians saw abstraction and generalization not as antithetical to the real world, but as crucial to how mathematics—or rather mathematical thinking—was applied to practical problems. It does not mean “mathematicalizing” the problem of measurement and arithmetic, but rather revealing the hidden skeleton of conceptual relations: formulating the basic idea in abstract mathematical language.

Often, this meant not applying existing mathematical concepts, but inventing new ones. A poignant example is the book of 1944 Game theory and economic behavior, which established modern game theory and included not-so-fun ramifications for nuclear defense strategies. Recovering an impressive web of connections, Steingart shows how the authors, American mathematician John von Neumann and German economist Oskar Morgenstern, not only helped define the relationship between science and politics during the Cold War, but also provided a template for further information. Attempts at mathematics in the social sciences.

taken from the appendix

Ironically, even when some mathematicians argued that abstract reasoning was the key to applying mathematics to other disciplines—and suggested that even the most abstract mathematics was worthy of public funding—most mathematicians in academia seemed individually uninterested in getting involved. For decades, many advances in applied mathematics ended up not in universities, but in think tanks and industrial laboratories, or in newly created departments devoted to fields such as computer science or statistics.

Ultimately, much of the abstraction fell out with mathematicians. Steingart charts how the latter parts of the twentieth century saw a shift in tide: she chose the example of William Thurston, a hugely influential topologist who delighted in making his complex geometric constructs seem physically real.

One of the major developments that it does not touch on is the regenerative cross-pollination with theoretical physics in the late 20th century. This includes the application of topology to innovations such as “topological” materials – which could form the basis of super-powerful quantum computers – and the development of string theory, which may not have given physicists their long-awaited theory of everything, but has inspired several doctoral dissertations in mathematics. And there is some anecdotal evidence that in the past decade or so, the barriers between pure mathematics and applied mathematics have begun to lower: it is not uncommon now to see researchers from the most abstract areas of the field ‘get their hands dirty’ with applications. Like data analysis.

Meanwhile, Eilenberg and Steinrod’s approach to pedagogy has become something of a cautionary tale, with their techniques, though still widely used, affectionately dubbed ‘abstract nonsense’. But in other turns of fate often seen in the history of mathematics and physics, abstract absurd techniques are now seen by some physicists as a promising approach to devising a quantum theory of gravity—perhaps offering another path to the very real, albeit abstract, goal. From the theory of everything.

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