I am sure that any real mathematician will frown upon hearing this ill-posed question and would say that the word ‘ranking’ is not at all appropriate in this case. And they would probably add that both of them were mathematical geniuses who were extremely important for the development of a discipline, and it is impossible to compare their inputs. However, due to his long life and exceptional influence on many branches of modern mathematics, Gauss is often regarded as the most prominent mathematician of all times.

have I decided to comment on rankings? Because in media and around, one can often encounter definitions such as ‘the top-ranked scientist’ of the university, country, worldwide, or ‘belongs to one or two percent of highly ranked scientists in the world’. However, it is vital to understand on what basis these statements are made and what is meant by ranking. They are based solely on bibliometric characteristics of the published work of the authors, and mostly on the h-index (h=n, if there are, at least, n papers by the author that have each been cited at least n times). This can be a valuable characteristic describing the outputs of an author, but one should be very cautious to compare scientists on these grounds (better not to do it at all). In my opinion, one should also avoid phrases such as ‘the best, the second best’ in the university, country, continent, world. On the other hand – for university rankings – the number of researchers in top groups in terms of citation is a valuable, proper characteristic, among others. But in this case, it is a statistical one, since we are speaking here about individuals.

I will mostly refer to mathematical sciences, where bibliometric characteristics (h-index) can fail dramatically as the sole characteristic of a researcher’s international standing.

It should be well understood that only acknowledgement by the regional or worldwide professional communities, organisations, associations, etc. (e.g., the International Mathematical Union (IMU), European Mathematical Society, national mathematical societies) through reviews, competitions, and significant awards can provide sufficiently objective assessment of the outputs of researchers of relatively high standing, whereas the h-index can serve only as an additional characteristic. We, in South Africa, for example, have a unique system in the world of evaluating scientists through the NRF rating process, which is based on the reviews of the applicants’ outputs by peers worldwide. The latter is the main basis for evaluation and rating, where bibliometric characteristics play only a supportive role.

The highest prize in mathematics is believed to be the Fields Medal, which is awarded to two to four mathematicians once every four years at the IMU Congress. Many people call it the ‘Nobel Prize of Mathematics’. The latest recipients in 2022 were Profs Maryna Viazovska (Ukraine), James Maynard (UK), June Huh (USA), and Hugo Duminil-Copin (France). Surprisingly for the advocates of h-ranking, their h-indices are rather low for mathematicians of such calibre – in the range of 21-33. This means that their h-ranking in world mathematics is in the range of #2.500--#7.800. But these are the most talented, relatively young, outstanding mathematicians in the world!

Another example is Yale Professor Gregory Margulis, who is one of the very few mathematicians (I believe there have been six of them in history) to have received both the Fields Medal and the Abel Prize. The latter is also considered the main world award in mathematics and does not have an age restriction. He is, without any doubt, one of the most influential contemporary mathematicians. His h-index is 34 and his world ranking in mathematics according to bibliometric characteristics is #2000! Is there anything more ridiculous or non-relevant? I believe, not.

Finally, Prof Andrew Wiles (of course, the Fields laureate) – the groundbreaking, contemporary mathematician who 30 years ago solved Fermat’s Last Theorem, which humanity could not solve for 300 years. His h-index is just 17! No comment.

To conclude: I believe that the h-index-based ranking of scientists is a valuable statistical instrument for worldwide ranking of universities. However, for individuals, it should not be used for comparisons, as it can be misleading and unethical. My reasoning refers mostly to mathematical sciences, where the drawbacks of the h-index-based ranking can be seen more clearly.

Maxim Finkelstein

• Distinguished Professor, UFS Department of Mathematical Statistics and Actuarial Science and A1-rated researcher by the National Research Foundation of South Africa.