In my video 3 weeks ago about natural groups (see video link), I had claimed that metric spaces are the champions among all topological structures and that groups are the champions among all algebraic structures with one operation. In order to back this up quantitatively, I initiated two “data analysis research tasks” to investigate this quantitatively. There was some initial back an forth discussion of course on how to set this up (similarly as one would do with a person) and which structures to take and what data to take and what methodology to use. After agreement, I let the assistant to write two papers. They are both authored by GPT 5.6 Sol. The two documents are linked here. I have also got the raw data files, just in case somebody should be interested. I myself only take credit however for the idea to run such a “championship on mathematical structures”, similarly as I’m proud about the idea of letting these two structures compete using the notion of “naturality”. Coming up with this definition emerged over years. (I still remember the exact spot in Arlington heights I had been running through, when it stuck that one should have “metric space forces unique group that is compatible” as the central definition. And also now, after 18 years since that idaea, I still like it, even so the results turned out to be rather soft, The reason is that the naturality problem appears to be very nice prototype question in the world of all of mathematical structures. And I personally like “soft mathematics”, mathematics that is humanly accessible and teachable to younger mathematicians without much experience or mathematicians coming from an other field. I personally also like to think in areas with like steep learning curves (“climbing” even) in a new territory, where one has experiences the danger to fall (and indeed can fall, as my project on Graphs, Groups and Geometry shows). I personally also avoid crowded research areas, where I (as an average mathematician) could not compete with giants in the field. I also like in general (like for holidays) places that are not so crowded. I hope this file will also soon be updated on the ArXiv. [8/25/26: ] Graphs, Groups and Geometry is updated]
The data analysis confirms what I suspected. Groups and Metric spaces are the champions!
https://www.quantumcalculus.org/wp-content/uploads/2026/08/metric.pdf “How Metric Is the Mathematics of Spaces?” with (from the abstract): “The source corpus is the English Wikipedia page List of theorems, as consulted on August 24, 2026. Rather than counting occurrences of words such as “metric” and “topological”, we isolate theorems substantially concerned with spaces and divide them into three classes: metric (M), nonessential general-topological (N), and essentially nonmetric (E). A preliminary conservative census of 339 space-related entries gives 299 in M, 34 in N, and 6 in E. Thus the corresponding proportions are approximately 88.2, 10.0, and 1.8 percentages“.
https://www.quantumcalculus.org/wp-content/uploads/2026/08/groups.pdf “Groups Among One-Operation Algebraic Structures” with (from the abstract of the paper) “The source corpus is the same frozen 1282-entry English Wikipedia List of theorems last edited July 20, 2026. We restrict to theorems whose content essentially involves a one-sorted algebraic structure with one binary operation. Rings, fields, lattices, Boolean algebras, vector spaces, modules, and other intrinsically multi-operation structures are excluded unless the theorem itself is genuinely about one distinguished binary operation. The relevant entries are partitioned into group-essential (G), non-group one-operation (N), and equivocal/mixed (E). A conservative first-pass classification gives |G|=70, |N|=3, |E|=3. Thus 76 of the 1282 Wikipedia entries enter the restricted corpus, and among them approximately 92.1 percent are group-essential. The three clear non-group cases are Moufang’s theorem for loops and the Hille-Yosida and Lumer-Phillips theorems for operator semigroups.”
The two data research project (done by the machine) have both a similar methodology and just use the list of Wikipedia theorems. (I have decided to focus on that because this is auditable as Wikipedia allows to make time capsuled versions of an entry). An other run, where groups had to compete with vector spaces showed that vector spaces in general win over groups. This is clear: rings and vector spaces and fields are very strong and heavily compete with group among all algebraic structure. But one can argue that they are in an other category as they involve more operations. Here was the outcome (from the abstract of the generated paper): “Rather than count occurrences of the word “group”, we identify theorem entries in which an algebraic structure is mathematically substantial and partition them into four classes: group-essential (G), non-group algebraic (N), equivocal or mixed algebraic (E), and excluded (X). A first-pass conservative census gives 85 entries in G, 175 in N, 79 in E, and 943 in X. Thus 339 of the 1282 entries are classified as substantially involving algebraic structure, and among these the proportions are approximately 25.1, 51.6, and 23.3 percentages”.
