Paul Schmutz Schaller for The Saker Blog


In some sense, this article is a comment on some aspects of Andrei Martyanov’s two books „Losing Military Supremacy: the Myopia of American Strategic Planning“ (2018) and „The (Real) Revolution in Military Affairs“ (2019). I am not aiming at discussing these excellent books; if you do not intend to read them, you may at least look at the book reviews of The Saker, see here and here.

My interest here is the content of these books which is related to mathematics. In this regard, here is a try to condense some of Martyanov’s writing into simple theses:

1) The power of a nation very much depends on its military strength.

2) Military strength is a complex thing, depending on the strength of the whole society: its economy, its culture, its scientific level, its historical experiences.

3) Mathematics are crucial for the military strength, not only for the production of sophisticated weapons and military technology, but also for strategic planning and for warfare.

4) Western societies in general and the USA in particular suffer from a lack of education in mathematics and this is an important reason for their decline.

5) Without understanding mathematics in relation to military, one cannot understand military strength, nor geopolitical developments.

While I agree with the first four theses, the last one is not convincing. More generally speaking, my overall ideas about mathematics are somewhat different from those presented in these books. Here are two examples.

Martyanov underlines the efforts aiming at calculating warfare (he criticizes some of them as too unrealistic). It is certainly true that there is a strong tendency in (Western?) societies to consider nearly everything as calculable; I do not like at all this tendency. My favorite „counterexample“ is the following. No machine is able to predict or calculate what will happen when you let fall a glass. By the way, once, when I was talking about this topic, somebody told me that he has seen that the glass jumped up (like a ball), intactly. Quite surprising! Ok, you could now try some thousand times whether you can repeat this phenomenon; but in general, as everybody knows, letting fall a glass just produces a dirty mess.

The second example concerns my conviction that mathematics have general cultural values, independently of direct applications; these values are not mentioned in Martyanov’s books. In my eyes, Euclid’s proof that there are infinitely many primes, is an eternal jewel of human culture. Of course, you might say that for all practical purposes (big primes are used for codes), the knowledge of, say, 100 trillions different primes is by far sufficient – and this is still nothing compared to infinity. So why is it of interest to know that there are infinitely many? Yes, indeed, why? This „why“ is one of these really big questions, humans have always tried to find an answer for. I would say that, in the end, it is a spiritual question.

Mathematics and rationality are certainly of huge importance, but also have limits. As Martyanov writes: „While we may endlessly discuss the already deployed or future combat technologies, in this deadly mix of machines and people, people remain what, in the end, decides the outcome of the battle, and indeed, of the war. […] people with all their knowledge, skills, will, morale, culture and patriotism“ (Martyanov 2019, pg. 192). In my words: The decisions that really matter, are taken (or should be taken) by our hearts, not by our heads – even if the heads may be very helpful. Or, as I like to say, humanity has survived not because of (Cartesian) rational thinking, but in spite of it. In this sense, I understand this article not as a criticism on Martyanov’s books, but as a small supplement.

Mathematics and Power in Ancient Greece, in (modern) West, in USSR – and an Excuse

Mathematics in Ancient Greece attained a high level. Euclid’s „Elements“ as well as „The Works of Archimedes“ easily reach my list of the best ten mathematical books ever written. And, in a slightly more general context, Aristotle stands comparison with modern philosophers of science without any problem. These examples are by far not isolated. Moreover, quite a lot of the work has been destroyed and is not anymore available.

One often hears that mathematics in Ancient Greece were a purely Intellectual matter without important influence over the society. This idea lacks logic. For example, mathematics played a crucial role in the philosophical thinking, especially in the work of Plato and Aristotle, and their work was of course authoritative. Moreover, Greek culture was (part of) the basis of two empires, that of Alexander the Great as well as the Roman Empire. For both, Greek teachers were significant. Aristotle was a teacher of Alexander. Evidently, Aristotle instructed Alexander in mathematics and Alexander could apply this training in his strategical planning. This being said, I, of course, do not intend to take position for the wars of Alexander.

Another illogical proposal pretends that Roman mathematics were of no interest, may-be apart from Roman numerals. This would be a big exception from the rule that all powerful cultures produced valuable mathematics. It is surprising that this idea about mediocre Roman mathematics is not more questioned; unfortunately, I have not the knowledge in order to give counterexamples.

Nobody would deny that natural sciences in general and mathematics in particular were of huge importance in the emerging of the Western empire(s). There is no real technological progress without mathematics. At least in the beginning, mathematics also played a central role in philosophy, similarly as in Ancient Greece. Descartes, Leibniz, or British empiricism are examples.

I consider the 19th century as the peak of Western mathematics. There were already great mathematicians before, notably Newton and Euler, but the development in the 19th century was more dynamical and creative. Important figures were Gauss, Maxwell, Riemann, and Poincaré. The 20th century was of lower level, qualitatively speaking (of course, there were, on the other hand, much more mathematicians than ever before in history). The last big push was probably quantum theory (or quantum mechanics), starting around 1920.

During this period, United Kingdom, France, and Germany were definitely the leading countries in mathematics. Around 1930, Germany was foremost. When Hitler took over, this was changing, due to a broad exodus, mainly to the USA. After WWII, the USA became clearly the major power in Western mathematics. Certainly, the exodus of European mathematicians contributed to this position of the USA, but it is not correct to see this as the decisive factor. Mathematics in the USA were already strong before.

In the 19th century, mathematics in Russia were developing rather fast, but did not catch the level of Western Europe. However, this changed in the Soviet period, between 1930 and 1990, say. Soviet mathematicians reached world class standard. After the dead of Hilbert (1943), there were no mathematician in the West who had the status of Kolmogorov and Gelfand in the USSR, I would say. Once again, this demonstrates the close relationship between economical development, power of a nation, and the level of mathematics.

The collapse of the USSR was also a catastrophe for mathematics. There was a big exodus of Soviet mathematicians to the West, again mainly to the USA. I cannot judge to what extent Russia has recovered from this weakening. Martyanov seems to be quite optimistic about this point. In any case, as Martyanov demonstrates, the capacity of the development and production of superior arms remained more than intact.

My description, focusing on Ancient Greece, (modern) West, and Russia/USSR, is very Europe-centric. I apologize for this shortcoming, which is due to my lack of competence. As I said, I think that every important culture/nation has produced own mathematics. This also holds – for example – for former nations in Africa or in the part of the world which now is called Latin America.

I just give a few examples of ancient Asian mathematics. I am impressed that Babylonians already applied a technique called „completing the square“, a technique which is still very much in use today. In the Chinese book „The Nine Chapters on the Mathematical Art“, the so-called Pythagoras’ Theorem was stated and correctly proved (in an elegant way) before Pythagoras. Needless to say that the theorem is known, in China, under a different name. The number „zero“ was not employed by Greek or Roman mathematicians. Apparently, it was invented in India, less than 2000 years ago. Of course, the word „nothing“ was used in mathematics long before. But creating a proper symbol for „zero“ was a major progress. Al-Khwarizmi, a Muslim mathematician who worked in Baghdad around 1200 years ago, has written the first book on algebra. The words „algebra“ and „algorithm“ were derived from this book. Al-Khwarizmi also heavily contributed to the replacement of the Roman numerals by the Indian-Arabic numerals which are much more suitable for calculations.

The Shift of Leading Mathematics to Asia

Mathematics in the West are declining. This is may-be not yet visible when regarding some top Western universities; they still are quite attractive for mathematicians from other countries due to their prestige and their money. The decline concerns above all the Western societies as a whole. They have become quite hostile towards mathematics. Increasingly, mathematics are just seen as a necessary evil. Typically, in the film „Salt“ (USA, 2010), starring Angelina Jolie, the latter says: „I hate math.“ Could you imagine this in a Chinese, Iranian, Indian, or Russian film?

The negative image of mathematics is strengthened by the often utterly elitist behavior of math teachers. I cannot resist to cite the Soviet mathematician Gelfand: „People think they don’t understand math, but it’s all about how you explain it to them. If you ask [someone] what number is larger, 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.“ It is very rare that first class mathematicians are so clearly engaged in the art of well explaining maths.

Every year, the International Mathematical Olympiad (IMO) is organized, with (up to) 6 students (under 20 years, not inscribed in a university) per country. Taking the cumulated results of the past ten years, one finds the following countries on the top (in this order): China, USA, South Korea, Russia, North Korea (has not participated every time), Thailand, Singapore, Vietnam, Japan, Iran. You can see the shift to Asia. Western Europe countries are quite behind and the teams of the USA give the strong impression to essentially consisting of students of Asian origin (when you look at the photos and read the names). Note also that in the last five years, the results of Syria were about as good as those of Switzerland and better than those of countries like Austria, Belgium, or Denmark. This is very amazing against the background of the difficult situation in Syria and shows the great resources of this country.

The film „X+Y“ (UK, 2014) describes a British student, taking part in the IMO. I think that the film is surprisingly realistic with respect to this student and clearly shows his problems with the fact that in his society, maths are very unpopular. On the other hand, during a common preparation of his team with a Chinese team from Taiwan, he becomes friends with a Chinese student; she has not at all experienced the same problems.

International Congresses of Mathematicians are hold, in principle, every four years, since 1897. There was one in Moscow (1966) and one in Warsaw (1983), all other were held in Western countries (including Japan), during 100 years. This is changing in the 21st century. In 2002, the host city was Beijing, in 2010 Hyderabad, in 2014 Seoul, in 2018 Rio de Janeiro, and in 2022, the congress will be held in Saint Petersburg.

These congresses also attribute prizes, the most prestigious being the Fields medal, which is restricted to mathematicians under 40 years (usually, each congress attributes four Fields medals). Until now, all winners were working in Western universities, at the time when they got the prize, except four who worked in USSR/Russia.

Let me make a comment concerning these kind of prizes. I think that the most important criterium for a winner is always a political one; this does of course not mean that the winners are not excellent. Nevertheless, one has to have an important and influential support group of mathematicians. To this day, it is just impossible that somebody working in Beijing or in Tehran can win a Fields medal. USSR/Russia was the unique exception. This may change in the nearer future, but we are not yet there.

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Coming back to the question of the power of a nation, we have to refine. This power is not only related to the present level of mathematics, but also to the attitude with respect to mathematics. In this regard, two items are obvious for everybody. First: The Western empire(s) are fading away. Second: Asia in general and East Asia in particular have already a huge advantage over Western countries.


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