What unifies mathematics

A grove. Bamboo stalks to the left and right. They have been nicely lined up in parallel. They run together at a far distance and seem to be nearing a common point. Do they touch somewhere?

This is the image that the Harvard mathematician Barry Mazur has selected as the theme of the Paul Bernays Lectures that he will give on 11 and 12 September 2018 at ETH Zurich. “When you observe things from the ‘right’ perspective, significant patterns can emerge”, adds the American.

Barry Mazur is a versatile and creative mathematician: “He has covered an impressively broad spectrum of mathematics”, says Giovanni Sommaruga. “Based on his works, you could write a good part of the history of mathematics in the second half of the twentieth century.” Sommaruga organises the Paul Bernays lectures and is a member of the committee that invited Barry Mazur.

Like a city with flourishing quarters

Mazur started his research in a sub-area of topology and related topics. From there, he moved on to algebraic geometry and number theory. At the same time, he is interested in the historical and philosophical issues of his discipline. Evidence of this can be found in his books entitled Imagining Numbers (particularly the square root of minus fifteen) and Circles Disturbed. The Interplay of Mathematics and Narrative, that deals with the relationship between mathematics and narrative, as well as with the role of stories for mathematical knowledge.

In Zurich, Mazur will present his thoughts on the subject The unity and breadth of mathematics – from Diophantus to the present day and investigate the question of to what extent mathematics today represents a unified subject and how much or why the different sub-areas or theories can be unified.

“The diversity and distinctiveness of the mathematical sub-disciplines is enormous, and seems to even be increasing throughout the history of mathematics. So the matter of their unity remains a constant challenge for mathematics and its philosophy”, says Giovanni Sommaruga, an ETH lecturer for Philosophy of the Formal Sciences.

Mathematics could be captured in the notion of a city: its sub-areas would be flourishing quarters, each with their own architectures, languages and rules. Inquiring about their unity would mean asking whether the individual quarters would really create a coherent cityscape.

The common foundations of the subject are posed by the philosophy of mathematics. A classic work in this regard is the two-volume book Foundations of Mathematics, which the former ETH Professor of Higher Mathematics, Paul Bernays (1888–1977), published in the years 1934 and 1939. ”I see the ‘Foundations of Mathematics’ as one of the great unifying steps in the history of modern mathematics”, says Mazur, who has been motivated by this work, “because the book asks the general question: ‘What can be stated about the unifying forces that structure our subject?’”

Momentum of the progress of insights

Not least, there are fundamental insights that arise when you can create and link surprising new connections between seemingly unrelated sub-disciplines. “There are exceptionally profound ‘analogies’ that link very disparate areas of mathematics”, says Mazur, “Today, mathematicians are attempting to join great mathematical field – each with its own intuition – to grand syntheses which, as soon as they are connected through an analogy, have an intuitive power that cannot be attained by any of the individual areas on its own.”

Barry Mazur has experienced first-hand how a new area of research can arise when a connection between previously separate topics is established: his observations in the 1960s of analogies between prime numbers and knits led to the area of arithmetic topology when other mathematicians utilised his works in the 1990s.

One example cited by Mazur, and commonly referred to as the “Grand Unified Theory of Mathematics”, is the Langlands Programme, whose creator, Robert Langlands, received the Abel Prize this year. The programme examines the relationships between the various areas of mathematics such as algebra, geometry, number theory, analysis and quantum physics.

Langlands and the breath of history

Langlands himself established a link between number theory and harmonics analysis, which deals with vibrations. One example of this is elliptical curves, thanks to which the British mathematician Andrew Wiles succeeded in proving Fermat’s Last Theorem in 1994. The Theorem is considered a highlight in the history of mathematics of the twentieth century, after number theorists had toiled over it during the previous 350 years. Barry Mazur himself – in addition to others – had done much of the important preliminary work for the proof.

The proof confirmed Pierre de Fermat’s (1607-1665) intuition that the equation aⁿ + bⁿ = cⁿ does not have any positive whole-number solutions if n is greater than 2. The problem is traced back to antiquity and the question of whether there are any right-angled triangles whose sides all have lengths that are whole numbers. Diophantus of Alexandria (approx. 250 AD) was the first to make a connection between algebra and geometry, and showed that there are infinitely many triples of three whole numbers a, b, and c, thus satisfying the equation a² + b² = c².

“Number theory has very strong ties to its predecessors, which date back to the mathematics of Ancient Greece. Philosophy and the history of mathematics suggest profound paths that serve as an inspiration to pursue them further”, concludes Mazur.