Do Geometry and Calculus have to die?

In the book ‘This Idea Must Die: Scientific Theories That Are Blocking Progress’, there are two entries which caught my eye because they both belong to interests of mine: geometry and calculus. The two articles are provided below. [I believe it is “fair use” as a reprint of these two articles helps not only to promote the book but also to promote the two authors (by the Streisand effect in “applied communication theory”, the best strategy to fight these two articles would have been to be quiet about it).] The idea of a provocative title and the concept of attacking various ideas is effective. Provocations trigger thoughts. The editors of the book knew what they were doing. It is also commercially a successful book and I can only recommend it. Still, on a very general note, the history of science shows that it scientific ideas rarely die. Yes, they can fall out of fashion but often come back in some or the other way or are used in legacy situations. Newtonian mechanics for example has been replaced by relativity theory but the theory is still used in daily life computations like space explorations, possibly with some relativistic corrections. The theory has been refined but it has not died. It is the unscientific ideas and speculations not based on experiments which have died for good.

But here is the summary of the two articles under consideration:

Rovelli pledges to go beyond geometry in order to understand space time. Lih repeats the argument for more computer science than calculus.

My main objection is that both geometry and calculus mean much more what the field originally signified. Much of physics is pure geometry. The poster child is general relativity or gauge theory which both are geometry. The power of geometry is illustrated that the latest Nobel prize was given for geometric ideas in physics (neither string theory nor loop quantum gravity have yet won a Nobel prize!). Calculus on the other hand is used in various generalizations including discrete settings, the focus of this blog. But the use of calculus goes much wider: generating functions or Fourier theory help to analyze discrete structures. In computer science, the complexity of algorithms uses functions in time t which are analyzed using limits. Even very calculus and physics heavy topics like Stokes theorem have very practical applications. The integral theorems are frequently left out in multi variable courses one reason being that they are harder subjects an other that some don’t see the relevance beyond physics. Here is a very concrete example from a very applied problem. How do we compute the volume of a geometric object in space which is given as a triangular mesh. It turns out that the best known way uses the divergence theorem (a particular case of the general fundamental theorem of calculus in higher dimensions): we can just add up the fluxes over all triangles using a vector field with divergence 1. A computer scientist who does not know calculus would probably split up the body into small cubes and count cubes. The best programming skills could not beat the fundamental fact that with a mesh size of 1/n, integrating over the boundary has O(n^2) steps while integrating the solids has a O(n^3) complexity. With n=1000, the second method would need 1000 times more resources. This is one of many examples, where calculus is essential to fundamentally improve on the complexity of the task. The example also illustrates that the study of the complexity uses calculus (applied computer scientists everywhere from computer vision to cryptology use it to express the complexity of tasks uses the Landau O notation). The call for ‘more programming’ and ‘less math’ is not the first. It is often expressed by folks who have seen little about the relation of math with computer science. Similarly, one can have the impression that statistics is independent of math or calculus. Also this has maybe be amplified that in the last decades not only the computer science departments but also the statistics departments have split from the mathematics departments. But there is much more to statistics than “big data” and much more to programming than making a machine do some stuff. Especially when dealing with lots of data or when computational time constraints appear, calculus helps to cut through the mess and speed up processes. On the theoretical side, it is hard to understand for example the central limit theorem, the “fundamental theorem of statistics” without calculus and in particular Fourier theory, which enters when one deals with characteristic functions. The topic of “real analysis” is generally considered crucial for applied fields like finance or economics, but it is actually just a fancy version of calculus. In the US, one calls things calculus, which in other places are called Analysis. Many real analysis courses are hard to distinguish it from a calculus course. A major difference is usually just the rigor or language. The two subjects climb the same mountains. Its just that “calculus courses” take the more pedestial approach while “real analysis courses” pursue more difficult paths or assume slightly less regularity, requiring more measure theory, topology and proof skills. Why a “physicist” is asked to put “geometry” on the death row and a communication scientist” was asked to trash calculus is hard to understand. One should have asked maybe an algebraist to attack geometry and a computer scientist to attack calculus. But maybe the editors could not find authors closer to the subjects. Not that I believe the two authors are unqualified: calculus and geometry are subjects which are common knowledge as they are taught early in schools. While geometry produces the foundation of many parts in physics, I agree with Rovelli, if one thinks of geometry as narrowly as say differential geometry. It is likely that we will need new models of space to understand quantum gravity. Still it is also very likely that a successful theory will be a geometric theory of some kind, geometric in the sense of Klein’s Erlanger program which brings in an algebraic aspect, the aspect of symmetry and symmetry is always close to invariants (Noether’s theorem or particle physics illustrates this). I see less the need to diminish the importance of calculus, even if one considers calculus in the narrow way as taught in schools. Here is an example close to the science of communication (Lih is from a school of communication). The father of communication, Claude Shannon was a mathematician who understood well the role of calculus in communication. The Shannon entropy for example S(p) = -\sum p(i) \log(p(i)) can hardly be appreciated without knowing calculus: it involves a function whose properties can only be appreciated when using derivatives (it is concave). The entropy functional has extrema which are essential everywhere in combinatorics: it is only through entropy that we can truly appreciate basic distributions in statistics. Much of more applied communication theory is also empirical data analysis close to other like psychology (perception), education (learning), economics (value), which all heavily borrow from calculus.

The Rovelli text


Carlo Rovelli
Theoretical physicist, Professeur de classe exceptionelle, Université de la Méditerranée, Marseille

Carlo Rovelli

We will continue to use geometry as a useful branch of mathematics, but it’s time to abandon the longstanding idea of geometry as the description of physical space. The idea that geometry is the description of physical space is ingrained in us and might seem hard to get rid of, but getting rid of it is unavoidable and just a matter of time. Might as well get rid of it soon. Geometry developed at first as a description of the properties of parcels of agricultural land. In the hands of the ancient Greeks, it became a powerful tool for dealing with abstract triangles, lines, circles, and the like, and it was applied to describe paths of light and movements of celestial bodies with great efficacy. In the modern age, with Newton, it became the mathematics of physical space. This geometrization of physical space appeared to be further vindicated by Einstein, who described space (actually, spacetime) in terms of the curved geometry of Riemann. But in fact this was the beginning of the end. Einstein discovered that the Newtonian space described by geometry is in fact a field, like the electromagnetic field, and fields are nicely continuous and smooth only if measured at large scales. In reality, they’re quantum entities that are discrete and fluctuating. Therefore, the physical space in which we’re immersed is in reality a quantum-dynamical entity that has very little in common with what we call “geometry.” It’s a pullulating process of finite interacting quanta. We can still use expressions like “quantum geometry” to describe it, but the reality is that a quantum geometry is not much of a geometry anymore. Better to rid ourselves of the idea that our spatial intuition is always reliable. The world is far more complicated (and beautiful) than a “geometrical space” and things moving in it.

The Lih text


Andrew Lih
School of Communication, American University.

Andrew Lih

I do not propose that we should do away with the study of change or the area under the curve, or bury Isaac Newton and Gottfried Leibniz. However, for decades now, learning calculus has been the passing requirement for entry into modern fields of study by combining the rigorous requirements of science, technology, engineering, and math. Universities still require undergraduates to take anywhere from one to three semesters of calculus as a pure math discipline, typically featuring complex math concepts uncontextualized and removed from practical applications, and heavily emphasizing proofs and theorems. Calculus has thus become a hazing ritual for those interested in going into one of the most essential fields today: computer science. Calculus has very little relevance to the day-to-day work of coders, hackers, and entrepreneurs, yet poses a significant barrier to recruiting sorely needed candidates for today’s digital workforce. This problem is particularly urgent in the area of programming and coding. Undergraduate computer-science programs are starting to bounce back from a dearth of enrollment that plagued them in the early Internet era, but we could do a lot more to fill the ranks if we rid ourselves of the lingering view that computer science is an extension of mathematics—a view that dates from an era when computers were primarily crafted as the ultimate calculators. Calculus remains in many curricula more as a rite of passage than for any particular need. It’s one way of problem solving, and it contributes to our ability to absorb more complex concepts, but retaining it as an obstacle course that one must navigate in order to program and code is counterproductive. Leaving in this obtuse math requirement is lazy curricular thinking. It sticks with a model that weeds out people for no reason related to their ability to program. This leads us to the question, What makes for good programmers? The answer “showed “a statistically significant improvement in retention in engineering” when it reconfigured its approach to introduce math in later semesters. We need more of these experiments and further radical curricular thinking to get past the prerequisite model that has dominated the field for decades. How can so many people be interested in coding and programming yet not be served by our top institutions of higher learning? By treating computer science largely as a STEM discipline instead of as a whole new capability cutting across several fields, we haven’t evolved with the times. The sooner we move beyond STEM-oriented thinking, the better.