I was struck today by the title of an article in Science News that read, Before his early death, Riemann freed geometry from Euclidean prejudices. The piece, by science writer Tom Siegfried, was no doubt inspired by the recent claim from award-winning mathematician Michael Atiyah that he has proved the long standing Riemann hypothesis, one of the most famous unsolved problems in mathematics for close to 160 years. But Siegfried’s article was more about Riemann’s extraordinary insights than it was about Atiyah’s claim (which I’ll get to before I’m done here). First, let me say this: by using the term ‘Euclidean prejudices,’ Siegfried is telling us that we could be mislead by prejudices even in mathematics. And what prejudices usually do is conceal the truth. The word actually appears in the lecture itself. This is from a translation of the lecture by William Kingdom Clifford:
Researches starting from general notions…can only be useful in preventing this work from being hampered by too narrow views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices.
Siegfried seems not so much interested in talking about mathematics itself as he is in illustrating the significance of a change of perspective within mathematics. Most young students of mathematics would never imagine that there could be more than one mathematical way to think, or even that within the discipline there is mathematical thinking that’s not just problem solving. Referring to Riemann’s famous lecture, given in 1854, that essentially redefined what we mean by geometry, Siegfried says this:
In that lecture, Riemann cut to the core of Euclidean geometry, pointing out that its foundation consisted of presuppositions about points, lines and space that lacked any logical basis. As those presuppositions are based on experience, and “within the limits of observation,” the probability of their correctness seems high. But it is necessary, Riemann asserted, to “inquire about the justice of their extension beyond the limits of observation, on the side both of the infinitely great and of the infinitely small. (emphasis added)
Physics gets us beyond the limits of observation with extraordinarily imaginative instruments, detectors of all sorts. But how is it that mathematics can get us beyond those limits on its own? How is it possible for Riemann to see more without getting outside of himself? I don’t think this is the usual way the question is posed, but I have become a bit preoccupied with understanding how it is that purely abstract formal structures, that we seem to build in our minds, with our intellect, can get us beyond what we are able to observe. Again from Siegfried:
Riemann’s insights stemmed from his belief that in math, it was important to grasp the ideas behind the calculations, not merely accept the rules and follow standard procedures. Euclidean geometry seemed sensible at distance scales commonly experienced, but could differ under conditions not yet investigated (which is just what Einstein eventually showed)…
…Riemann’s geometrical conceptions extended to the possible existence of dimensions of space beyond the three commonly noticed. By developing the math describing such multidimensional spaces, Riemann provided an essential tool for physicists exploring the possibility of extra dimensions today.
Riemann appears in a number of my posts. I’ve taken particular interest in the significance of his work in part because in his famous lecture on geometry he cited the philosopher John Friedrich Herbart as one of his influences. Herbart pioneered early studies of perception and learning and his work played an important role in 19th century debates about how the mind brings structure to sensation. In Jose Ferreiros’ book Labyrinth of Thought, he takes up Riemann’s introduction of the notion of manifold and says this:
Herbart thought that mathematics is, among the scientific disciplines, the closest to philosophy. Treated philosophically, i.e., conceptually, mathematics can become a part of philosophy.~According to Scholz, Riemann’s mathematics cannot be better characterized than as a “philosophical study of mathematics” in the Herbarian spirit, since he always searched for the elaboration of central concepts with which to reorganize and restructure the discipline and its different branches, as Herbart recommended [Scholz 1982a, 428; 1990a].
I think the way I first grappled with the depth of Riemann’s insights was to consider that he was somehow guided, by the cognitive processes, that govern perception, despite the fact that they operate outside our awareness. Some blend of experience, psychology, and rigor worked to establish the clarity of his view. I wrote a piece for Plus magazine on this very topic.
Herbart’s thinking foreshadows what studies in cognitive science now show us about how we perceive space and magnitude — it may be that Riemann’s mathematical insights reflect them.
More recently I’ve become focused on asking a related question, but maybe from a different angle, and that is what is actually happening when we explore mathematical territories? How is this internal investigation accomplished? What does the mind think it’s doing? These questions are relevant because it is clear that there is significantly more going on in mathematics than calculation and problem solving. Riemann’s groundbreaking observations make that clear. The questions I ask may sound like impossible questions to answer, but even just organizing an approach to them is likely to involve, at the very least, cognitive science and neuroscience, mathematics, and epistemology, which makes them clearly worthwhile.
About Atiya’s breakthrough, an NBC news article said this:
Atiyah is a wizard of a mathematician, but there’s a lot of skepticism among mathematicians that his wizardry has been sufficient to crack the Riemann Hypothesis,” John Allen Paulos, a professor of mathematics at Temple University in Philadelphia and the author of several popular books on mathematical topics, told NBC News MACH in an email.
This skepticism is present in almost every article I read, but Atiya remains confident and is promising to publish a full version of the proof.
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