Tony’s Take April 2024

This month’s topics:

Galileo was wrong about mathematics?

In 1623 Galileo published Il Saggiatore (“The Assayer”), a salvo in his polemic war against the Jesuit astronomer Orazio Grassi, an apologist for a mixed geo/helio-centric solar system model. In Il Saggiatore, he compares the universe to a huge book which one can’t understand without first understanding the language and the letters in which it is written: “[That book] is written in the language of mathematics, and the letters are triangles, circles and other geometric figures.” The prize-winning British science writer Philip Ball disagrees. “Galileo was, to put it bluntly, wrong: maths is not the language of nature, but a tool with which we are able to make quantitative predictions about some aspects of nature,” Ball wrote in an April 24 opinion piece in ChemistryWorld.

Ball, who writes mostly about the life sciences, is actually taking Galileo out of context. Galileo was talking about astronomy and physics (“the universe”) without addressing medicine or the life sciences of his day, as the word “nature” might suggest. On the other hand, leaving Galileo out of the picture, Ball has a point. As he puts it, “There is a profound difference between using the maths and understanding the science.” That is, we need pictures and stories (“causal narrative”) for the equations to make sense; too often, Ball says, “maths can even be something to hide behind: if you can crank the handle and get results, you can disguise (for a while) the fact that you don’t quite grasp the underlying science.”

But why does science have to make sense? Ball starts his piece, in fact, with a 1930 quotation from Werner Heisenberg, the discoverer of quantum mechanics, who wrote: “It is not surprising that our language should be incapable of describing the processes occurring within the atoms, for it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms.” More generally, why should our minds be able to comprehend the fundamental structure of the universe or, for that matter, of the human brain itself? Luckily, as Heisenberg also remarked, we have mathematics at our disposal, which can lead us safely beyond what we can comfortably “understand.” There may be parallel universes where the speed of light is different, but there can’t be one in which 47 is not a prime. We should be grateful that we have this reliable guide, and not complain when it sees better than we can.

Updates on the shape of the universe.

If our three-dimensional universe is not infinite, then what does it look like? Presumably, it doesn’t stop at a wall, so it goes on being 3-dimensional everywhere; it’s what we call a 3-dimensional manifold. But just as a 2-dimensional finite surface with no boundary can have different topologies—it can be a sphere, a torus, a 2-holed torus, etc.—there are many possible topologies in dimension 3.

Some 3-dimensional topologies have been excluded as possibilities for our universe because they don’t match with current observations. For example, by current consensus the universe is geometrically flat, so that in any region we can examine the overall geometry is essentially like 3-dimensional Euclidean space. This rules out the simplest model for a finite topology, which would be the 3-dimensional sphere since the sphere requires overall positive curvature. “Promise of Future Searches for Cosmic Topology”, published in Physical Review Letters on April 26, 2024, reviews the remaining options. One possibility is the 3-dimensional torus, the space obtained by identifying opposite sides of a cube. This image of what that space would look like inside was used in a Physics Magazine commentary on the PRL article.

A lattice of cubes marked by frames that extends forever.
Looking around inside the 3-torus (except the frames would not be visible). Every object in the cube appears infinitely many times as the identification forces lines of sight to circle around in each of the three dimensions. Credit: Jeff Weeks, from the Curved Spaces flight simulator for multi-connected universes, used under a GNU General Public License.

The authors of the PRL article (the 15 members of the international “COMPACT collaboration”) explain that the place to look for information on the topology of the universe is in the map of the cosmic microwave background (CMB). If topology is the explanation for anomalous features of this map, then it must contain detectable topological information. The most obvious sign of a topology like the 3-torus would be recurring images of features like galaxies, or, more generally, what the authors describe as “pairs of circles with matched temperature (and polarization) … visible in different parts of the sky,” but there is no sign of these.

A map of cosmic background radiation from when the universe was around 380,000 years old. The 2-dimensional sphere of directions looking out from Earth is mapped onto an ellipse. Credit: NASA / WMAP Science Team.

To show that “no matched circles” does not rule out an interesting topology for the universe, the authors examine several flat models and conditions under which they would not be expected to show such circles. They remind us that there are 17 non-equivalent flat possibilities $E_1, \dots, E_{17}$ and concentrate on the first three: $E_1$, the 3-torus, where the faces to be identified can be parallelograms, or rhombi; $E_2$ (one of the rhombic faces has been rotated by $\pi$ before the identification) and $E_3$ (a square face has been rotated by $\pi/2$ before the identification). In these examples they work out a range of parameters (e.g. relative sizes of faces) for which the topology would not manifest matched circles, and so which are still viable candidates as models of a finite universe.

Three of the flat models. Each is shown as a 3-dimensional solid; when the opposite faces (yellow, blue, and transparent) are identified, it defines a 3-dimensional manifold. The lengths $a,b,c$ are the parameters in question. For E1, a parallelepiped, the identifications give a 3-dimensional torus, the same model illustrated by Weeks above. To define E2, one of the rectangular faces is rotated clockwise by $180^{\circ}$ before the identification. The image shows a rectangular cross-section halfway along. In E3, one of the faces is a square of side $a$; it is rotated $90^{\circ}$ before identification. Image by Tony Phillips.

Topology and schizophrenia.

An application of graph theory to the diagnosis of mental disease is reported in the Schizophrenia Bulletin, April 26, 2024. In the article, Topological Perturbations in the Functional Connectome Support the Deficit/Non-deficit Distinction in Antipsychotic Medication-Naïve First Episode Psychosis Patients, by Matheus Teles, Jose Omar Maximo, Adrienne Carol Lahti (all at the University of Alabama, Birmingham) and Nina Vanessa Kraguljac (Ohio State), the authors seek physiological support for the understanding that deficit syndrome (DS, also called Deficit Schizophrenia) is actually a distinct disease, and not just a peculiarly severe manifestation of schizophrenia. Deficit Syndrome, as its name implies, is marked by enduring negative symptoms, e.g., restricted affect, poverty of speech, diminished sense of purpose.

The team worked with 61 patients, 18 DS and the others diagnosed with non-deficit schizophrenia (NDS), comparing them with 72 healthy controls (HC). Their hypothesis was that DS is special in that it involves disruptions in the connectivity of the workings of the brain. To test it, they defined 246 regions of interest in the brain, and considered these regions as the nodes of a graph. They used functional magnetic resonance imaging to infer the strength of connections between the regions by correlating their activity, and based on these strengths, assigned weights to edges between the corresponding nodes. This yielded a weighted graph for each subject. The graphs were evaluated with graph-theoretic criteria including:

  • Global efficiency. For an unweighted graph, global efficiency is defined as the average of the reciprocal of the length of the shortest path between any two different nodes (so longer paths contribute less). For a connected graph $G$ with $N$ nodes, with $d_{ij}$ the length of the shortest path between node $i$ and node $j$, the formula for the global efficiency $E(G)$ is $$E(G)=\frac{1}{N(N-1)}\sum_{i\neq j}\frac{1}{d_{ij}}.$$
  • Global shortest path length here refers to the average length of the shortest path between any two nodes.
  • Local efficiency at a node $i$ is the global efficiency $E(G_i)$ of the graph $G_i$ made up of node $i$, all the nodes directly connected to $i$, and all the edges between them. Global local efficiency of $G$ is the average of $E(G_i)$ taken over all the nodes of $G$.

    Global efficiency, global shortest path length, and global local efficiency for 3 graphs. Row 1: The graph is 5 nodes connected in a line. The global efficiency is 0.64, global shortest path length is 2, and global local efficiency is 0.9. Row 2: The nodes are connected in a cross shape, with one at the center that is connected to four surrounding nodes. Compared to the line, this graph has higher global efficiency (0.7) and global local efficiency (1.6), and smaller global shortest path length (0.94). Row 3: The cross shaped graph, except now the outer nodes are also connected to form a square surrounding the cross. This increases global efficiency (0.9) and lowers the global shortest path length (1.2), but the global local efficiency is lower than for the cross shape (0.913).
    Examples of the three measurements for three small, unweighted graphs. Image by Tony Phillips.

The findings show that the global efficiency, global shortest path length, and global local efficiency differed—in a statistically significant way—between the deficit syndrome groups and the others. (In their conclusion, the authors remark that the topological metrics showed no statistically significant difference between the controls and the NDS (schizophrenia without deficit syndrome) subjects.) Put together, this is evidence that deficit syndrome really is a distinct disease.

—Tony Phillips, Stony Brook University