Tony’s Take January 2023

This month’s topics:

Doing math with ChatGPT.

Paul T. von Hippel posted ChatGPT Is Not Ready to Teach Geometry (Yet) (last update January 4, 2023) on the Math Technology blog of Education Next. Von Hippel’s point is that even though ChatGPT gives fluent, coherent answers to mathematics questions, the program can give wrong information as glibly as the truth. He put it to the test with questions about the Pythagorean theorem: it did OK for a while, but then veered into nonsense. “[I]t had revealed something important: it didn’t have a coherent internal representation of the triangle that we were talking about.”

As I understand it, ChatGPT is based on the premise that knowing a huge amount of things that people have written about the world can substitute for knowing about the world itself. (In that way, it is the digital epitome of “book learning”). I encourage readers to follow von Hippel’s example and experiment with ChatGPT and math. It’s fun if you know the stuff, but it might be dangerous for a student.

Here’s an example. I asked: “Can you chart a great-circle route between Chicago and Beijing?” ChatGPT answered: “Yes, a great-circle route is the shortest path between two points on the surface of a sphere. To find the great-circle route between Chicago and Beijing, you can use a map projection that displays the surface of the Earth on a two-dimensional plane. One such projection is the Mercator projection, which is commonly used for navigation purposes. On a Mercator projection, a great-circle route between two points appears as a straight line. To chart a great-circle route between Chicago and Beijing on a Mercator projection, you would simply draw a straight line between the two cities on the map. … ” The last two sentences are completely wrong. (See for example this Feature Column for a detailed explanation).

How did this happen? One can imagine sentences in Chat’s corpus like “A great-circle route is the shortest path between two points on the surface of a sphere.” “A straight line is the shortest path between two points.” “The Mercator projection displays the surface of the Earth on a two-dimensional plane.” And one can imagine Chat stitching pieces together to get “On a Mercator projection, a great-circle route between two points appears as a straight line.” It’s engineered to sound plausible but it’s just not true, confirming von Hippel’s observation: ChatGPT has literally no idea what it is talking about.

Mathematics and reality.

Nature ran Davide Castelvecchi’s review of Alma Steingart’s Axiomatics: Mathematical Thought and High Modernism, on January 16, 2023. Steingart is a historian of science; part of her thesis in Axiomatics, as Castelvecchi explains it, is that the axiomatizing movement in mathematics was not special to math but was a central part of Modernism, a movement that “dominated the mid-twentieth century in the arts and social sciences, particularly in the United States.” Whether or not axiomatization (more than relativity theory, for example) had an impact on art, Steingart gives evidence, in Castelvecchi’s words, of scientists “who saw their liberation from merely explaining the natural world as analogous to how abstract expressionism freed painting from the shackles of reality.”

Castelvecchi is more interested in the impact of the axiomatic method on mathematics itself. As he explains, the mathematical tradition exemplified by Euclid had traction because its concepts were “rooted in physical reality.”

To illustrate early axiomatization, Castelvecchi shows a picture of this page from Oliver Byrne’s The First Six Books of The Elements of Euclid (London, 1847). Byrne emphasizes the physical-tactile basis of Euclid’s concepts to the point of substituting images for Euclid’s symbols in his edition of the Elements. See for example his proof (scroll down) that the sum of the angles in a triangle is two right angles, from Bill Casselman’s website where the whole book can be perused.


Starting around the turn of the 20th century, mathematicians turned to structures as a way of encapsulating known phenomena and engendering new problems. For example, even though every finite group is the permutation group of its own elements, freeing the concept of group from these and other concrete examples was an essential step in the development of modern mathematics.

In Castelvecchi’s presentation of Steingart’s narrative, this new love of abstraction caught on first in the United States. But those mathematicians, far from pursuing abstraction for its own sake, understood that the ability to recognize abstract patterns, to “reveal a hidden skeleton of conceptual relationships” was what mathematicians could contribute to the study of the real world. Unfortunately, the story continues, most academic mathematicians, even in the U.S., were seduced by the thrill of ever purer abstraction and left the important connections with the concrete and perceivable to applied mathematics departments. The tide only turned in the last part of the century: William Thurston is mentioned as “an enormously influential topologist who delighted in making his complex geometric constructions feel physically real.”

Castelvecchi notes that Steingart’s story omits the interaction between very abstract mathematics and theoretical physics, which has become a central area of activity in both fields. In fact, what seems to be missing in this discussion is the realization that mathematics is part of the structure of the universe, and that the distinction between pure and applied is essentially an illusion. This is not to say that in teaching mathematics one should jump into abstraction prematurely. The subject is difficult, and as Byrne understood, we need all the help that our senses and terrestrial experience can give us.

Photonic smoke-rings and the Hopf fibration.

The Hopf fibration is a historically and conceptually central ingredient of modern topology. Heinz Hopf discovered around 1930 that the 3-dimensional sphere $S^3$ (we can think of it as regular 3-dimensional space plus a point at infinity) can be completely filled up with circles: any two are linked, and each one corresponds to a point on the 2-dimensional (ordinary) sphere $S^2$ in such a way that nearby circles match with nearby points. The term Hopf fibration represents the whole picture, packaged in the function $h: S^3\rightarrow S^2$ that sends each circle to the corresponding point.

In an article in Advanced Photonics (January 10, 2023) a U.K.-China team led by Yijie Shen (Southampton), Zhihan Zhu (Harbin) and Anatoly Zayata (King’s College, London) explain how they have used lasers to generate stable physical instantiations of the Hopf fibration (they call them hopfions) and how they can even get them to propagate through space.

(a) A sphere with color varying continuously from point to point. Several points around the equator are marked. (b) A torus with several colored loops on its surface. Each loop threads through the hole. Loops are evenly spaced around the surface. (c) Colored loops tracing out a complicated set of nested tori.
This illustration from the Open Access article Advanced Photonics5 015001 can be used as a pictorial definition of the Hopf fibration. Published under CC by 4.0 license. Points on the sphere (a) Are color-coded: hue is determined by longitude $\alpha$, measured in radians east and west from a central meridian; saturation by latitude $\beta$, measured down from the north pole; see the inset. To each point corresponds a circle in 3-space; the circles lie on a set of nested tori, one for each latitude: (b) The circles corresponding to the highlighted points around the equator. (c) All the tori together. The core black circle corresponds to the south pole, the unique point at latitude $\pi$; corresponding to the unique point at latitude 0 (the north pole) is a vertical white line. When a point at infinity is added to close up 3-space into a 3-dimensional sphere (this creates what mathematicians call the Hopf fibration) that line becomes a circle like all the others.

Where is the physics? Splitting a laser beam and recombining the two halves after manipulation, the team created an optical field where the polarization varies from point to point. Polarization is a physical phenomenon associated with waves, electromagnetic waves for example, that oscillate transversely to their direction of propagation.

Electromagnetic rays propagating along the $z$-axis, with three kinds of polarization: (a) linear, (b), (c) circular. The electric and magnetic fields oscillate in the transverse, $x,y$ directions. Only the electric field is shown (it traditionally defines the polarization).

Each of (a), (b) and (c) above shows three rays, represented by their electric fields, one drawn in blue, one in red and one in black (they all have the same wavelength, so if they were in the visible range they would all have the same actual color). In each case, the black field is the sum of the other two. The blue field oscillates back and forth along the $x$-axis: it is linearly polarized; likewise for the red field. When (a) blue and red fields are exactly in phase, their sum (black) is also linearly polarized. When (b) the red field has phase $\frac{1}{4}$-wavelength ahead of the blue, the sum rotates clockwise (looking down the $z$-axis). This is circular polarization. When (c) the red field is $\frac{1}{4}$-wavelength behind, the rotation is counter-clockwise. (The angle of the linear polarization in (a) can be controlled by varying the proportions of red and blue in the mix. Similarly for (b) and (c), when red and blue are not equal in magnitude the sum will manifest elliptical polarization; this also happens for phase differences different from $\frac{1}{4}$-wavelength.)

The polarization of a light beam can be represented by a point on the sphere: the south pole corresponds to clockwise circular polarization, the north pole to clockwise. Points along the equator represent linear polarization at the angle of their longitude. Intermediate points represent the gradation from a circle through more and more eccentric ellipses to a line, eccentricity varying with the latitude from 0 (circular, north pole) to infinite (linear, equator) and back to 0 (circular, south pole), with color corresponding to the orientation of their major axis.

When the points in the beam field are colored by polarization, using the coloring of the sphere described above, the lines of constant polarization (isospin lines) form the circles of the Hopf fibration. This is the simplest hopfion.

(d) Shows a collection of vectors associated to points 3D space, with directions varying continuously in space. (e) A square with a rainbow around the edge, a black spot in the center, and another rainbow in the center of that. (f) Two rainbows following hyperbolic curves, with two black spots near the foci.
Advanced Photonics5 015001, CC by 4.0 license. (d) The distribution of polarizations in the experiment. The coloring of the vectors tangent to the isospin lines follows the same convention as above. (e) and (f) show cross-sections along the (dashed black outline) planes they are linked to; again, the color scale corresponds to the polarization direction. The grey arrows are part of a different interpretation of the experiment.

The team reports that they were able to make their hopfion move through space, by varying a phase angle in the preparation of the beam. In the newsletter of the International Society for Optics and Photonics, the report of this achievement had the headline “Light shaped as a smoke ring behaves like a particle.”

Advanced Photonics5 015001, CC by 4.0 license. “Propagation of photonic hopfions in free space.” Five images, corresponding to the cross-section (f) above, showing how the whole structure moves as the phase $\varphi$ is varied.