**Paul Schmutz Schaller for The Saker Blog**

**Introduction**

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.

* * *

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.

·at 1:48 pm EST/EDTThe ability to do good math cannot be faked and is essential to doing good physics and good engineering.Using the Lagrangian requires the ability to abstract, Maxwell’s equations demand do 3-dimensional capabilities, and of course QM requires the ability to suppress “common knowledge” and form knew schemas based on surprising experimental evidence.

Faux liberals (I call them fake b/c their actions frequently seek to deprive me of the ability to think freely) cannot do math and abhor science. Their small brains result in small ideas that they then try to impose on the rest of us.

·at 4:41 am EST/EDTJames Jones

An interesting article. I would just like to add that the word “mathematics” cannot be taken out of context from the word “history”, as both are interconnected. Watching the situation in the West one gets the impression that the West has entered a period of complacency, living in the past and being unable to grasp the changes occurring in the world, namely the rise of the East. Yes, Western governments are aware of this, but apparently not your average man in the street. It’s questionable how many politicians in the West understand what is going on. If you look at the US Congress, what do you see ? Infantile political theatrics, like Russiagate and that political show involving Judge Kavanough, where US politicians had nothing better to do than investigate what the Judge did when he was 15 years old.

Can the West break away from this complacency ? We shall see, but I am not much of an optimist in that respect.

·at 12:00 pm EST/EDTB.F.

Can the West break away from its complacency? Only if forced to do so, and there is no guarantee of success. Necessity may be the mother of invention, but to summarize Kunstler, Orlov, and others, the necessity we will face will be the mother of all mothers.

·at 1:25 am EST/EDT“Faux liberals (I call them fake b/c their actions frequently seek to deprive me of the ability to think freely) cannot do math and abhor science. Their small brains result in small ideas that they then try to impose on the rest of us.”

Is it just “faux liberals” who can’t do math and abhor science? Are social conservatives free of the impulse to “impose on the rest of us”? Are Christian creationists lover of science? Bio and nuclear weapons are products of people who can do math and love science. Have they done good for humanity?

·at 1:02 pm EST/EDTThe emphasis was on the hypocrisy. Faux liberals seek to deprive me of the freedom to think, and whether they are less or more immature than conservatives is another question.

At least with self recognized rigid thinkers (“God said, I believe it. That settles it.” – bumper sticker, or “If English was good enough for Jesus it’s good enough for me.” – Archie Bunker) are consistent.

·at 9:00 pm EST/EDT@B.F., @Begemot

Agree. I use a rule of thumb.

– Faux liberals are like 4 year olds. The 4 year old BELIEVE 2+2=7 because the big 6 year olds are saying it, as the 6 year old are scientists in life and must know it and the 4 years ref to popular rumours.

– Faux Conservatives are like the 6 year old. They discovered the 6 year old weren’t that experienced in life as they thought, so they invent and BELIEVE in “something completely new” 2+2=5 and ref to sceptical rumours.

-When you become 10 year old, you now KNOW 2+2=4. This is a very important stage, because here you don’t need UN, Nato, MSM, Yale University, EU, Government and your friends. Nobody can fool you anymore.

To Begemot.

True Christian Creationists are super interested in Math because God built our universe and life by Math and physics. God and Math are twins, the universe wouldn’t exist without both.

The bible is the oldest scriptures describing the water cycle around 1000 BC as another example.

First in 1675 a French Scientist started to find some right ends.

Even the ancient Egyptians and Leonardo da Vinci couldn’t figure out the water cycle.

We can only understand the universe and life by Gods Creation. People who don’t believe in our Creator are walking around in darkness and will constantly meet barriers in their searches.

However, look up for false religions clame they represent God and the sun circulate around a flat earth and like.

They do not belong to our Creator.

·at 2:20 pm EST/EDTExcellent piece, Paul Schmutz Schaller.

One thing that I would like to add is how the West’s neoliberal decadence “makes use” of maths, adding lots of formal mathematical “rigor” to parasitic, voodoo economics. Needless to say, this hogwash has been aggressively promoted by quite a few economic “Nobel” prizes courtesy of the Swedish Riksbank, starting with the charlatan Milton Friedman in 1976. This degradation of maths is accompanied by gender theory and assorted perversions in vogue.

For the mathematically inclined, here is a little trigonometric problem: A square whose side = 3 length units shares one of its corners with a right-angled triangle — the square fitting exactly into the triangle. The diagonal which starts in their common corner ends at the triangle’s hypothenuse = 10 length units. Please calculate the triangle’s other two sides.

·at 4:26 pm EST/EDTI think being able to do math quiz’s is all well and good, but it doesn’t really add up to anything that affects peoples daily lives. To require people to say, do as they do in Italy, (especially when it comes to things like aquiring a drivers licence), they must pass a test on how an engine works. This type of thing could go a lot further toward saving money and recycling resources rather than filling up the dump and simplydoing the math to buy another one.

One person learns how an engine works, another person teaches them how to make it last virtually forever, and suddenly the cost of living starts dropping rather than the expodential rise you are currently trapped into. One problem solved leads to another problem solved and so on until living a comfortable life becomes almost seemingly free.

We are a long ways from heaven and solving a math problem on the fly is not going to get us there any sooner my friend.

·at 4:52 pm EST/EDTAs usual, the least relevant part gets the biggest attention, LOL. Personally, I would have preferred to get some feedback on the degradation of mathematics in the service of neoliberal political reaction — my bad.

Here is a more Western style mathematical problem:

Avogadro’s number = 6 x 10^23. If this is the number of infants Stalin devoured for breakfast per day, how much Novichok would it take to achieve the same result?

·at 5:49 pm EST/EDTI think its about as relevant as trying to determine how many grains of sand are in the world by counting a bucket of it, and applying the relevant formula(s). It don’t pay the bills at the end of the day.

·at 6:13 pm EST/EDTWithout Gauss, Carnot, at al laying the (mathematical and physical) foundation you wouldn’t know what the engine is freeing you from trouble to learn how it works.

·at 7:11 pm EST/EDTNot sure I get your point, but the realization that god (a woman naturally) was by the side of all these inventors at just the right designated time, has not gotten past me. But if anyone was able to fool all the people all the time,(for the most part), it was her.

·at 3:15 am EST/EDTAs a simple workman I solved it by looking up an ancient Egyptian / Babylonian list of practical building standards: For a square corner you must choose either a 3 4 5 triangle or a 6 8 10. The first is obviously too small to hold your 3 by 3 square, so choose the second.

Of course a “pure” mathematician would solve it by Euclid + Al’Jibra: 10 squared = 6 squared + 8 squared.

·at 12:51 pm EST/EDTI put a few minutes into it, and this is no elegant solution, but …

The solution is 3cos(alpha) +3sin(alpha) = 10sin(alpha)cos(alpha)

I guess the thing to do is to express cos(alpha) as sin(pi/2-alpha) and start diddling with

angle identities …. but lacking time, I used a numerical solution.

The sides of the triangle are 6.6368 and 3.3632, approximately.

·at 2:00 pm EST/EDTShort reply regarding the exercise: Start by drawing the square to scale (3 x 3), and then superimpose the triangle; make sure that there is a tangent point where one of the square’s corners touches the hypothenuse whose length = 10. Measure the sides of the triangle — their lengths then turn out

to be 6 and 8, but (approximately) 4.5 and 8.9, respectively.notIn the West, maths are being used for bogus science such as economics and for corroborating permanent slanders about historical ”genocides” (via nonsensical demographic extrapolations). When you know what you are expected to ”prove”, you don’t really have to be talented. Just assume what was the supposed subject of investigation, and wrap it up in some pseudo-scientific jargon, think ”highly likely” 😄

·at 3:17 pm EST/EDTI wrote that the sides were 6.6368 and 3.3632. This is wrong. (I was in a hurry.)

Those two numbers add up to 10 and are the segment lengths for the two minor hypotenuses. As Nussiminen stated, the legs of the right triangle are 8.920 and 4.520. Approximately.

·at 4:29 am EST/EDTI was already lost when I read the word ‘trigonometric’.

·at 2:33 pm EST/EDT“Mathematics in the West are declining. /…/ 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?”Absolutely not, and I also cannot imagine the idiotic self-worship, political propaganda, and all the other hallmarks of manifest brainrot.

·at 3:06 pm EST/EDTRe:”This power is not only related to the present level of mathematics, but also to the attitude with respect to mathematics.”

“attitude” is primary, and “level” is it’s derivative.

·at 4:47 pm EST/EDTI would have to vehemently disagree with this article’s thesis about mathematics and the USA.

America is by far the world leader in terms of math ability.

After all, you have to be a veritable mathematical genius to “cook the books” like Enron, Arthur Anderson, Global Crossing, the US Department of Labor (and its unemployment statistics), or the American war machine (and burying the fact that it has “lost” trillions of dollars) have been able to do.

This is America mathematical wizardry in action!

It’s pure genius.

·at 5:38 pm EST/EDTTo take this large tongue in cheek remark one step further, we still have the pentagon with its problems with its “books” and its “accounting.” It cannot account for $35 trillion (with a “T”); to be sure, accounting in large measure is adding and subtracting after the entries have been properly posted but that has proven to be much too difficult for the current civilian and military geniuses in the pentagon.

I would say Martyanov may be giving western societies are far too much credit (at least as interpreted by the author of this article) when their record is obviously abysmal:

“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;”

·at 4:47 pm EST/EDTThere is a distinction between pure and applied mathematics – both play their part in military supremacy, the latter being particularly relevant to strategy and tactics. The understanding of probability theory and its application, I would assert, should be mandatory for the officer class.

·at 4:48 pm EST/EDTThe ability to do good math cannot be faked and is essential to doing good physics and good engineering. Using the Lagrangian requires the ability to think in the abstract, Maxwell’s equations demand the ability to do 3-dimensional thinking, and of course QM requires the ability to suppress “common knowledge” and form new schemas based on surprising experimental evidence.

Faux liberals (I call them fake b/c their actions frequently seek to deprive me of the ability to think freely) cannot do math and abhor science. Their small brains result in small ideas that they then try to impose on the rest of us.

·at 5:06 pm EST/EDTIt is now being said the election fiasco in Iowa is the result of faulty math (that is after the app failed.) Iowa’s whole election system is about to be overturned because of people unable to do their math. But what kind of math could there be in counting votes other than addition? Are we to say that people in the U.S.A. cannot do simple addition? Pretty scary.

But then again a lot of people born, raised, and educated in the U.S.A. do not know where Canada is, and do not even know which State Nebraska is in. So adding numbers together might be too much to ask for.

Suddenly I’m scared of driving over bridges around here.

Imagine this place without immigrants.

·at 5:13 pm EST/EDTYes, excellent example of what occurs when nations stop aiding the development of their primary asset–their human capital. When the dominant political-economic method is to create money on top of money recently created on the basis of nothing–what’s known as a Ponzi Scheme–who needs engineers, or industrial capitalists, or anyone else elated to the making and maintenance of things. But who fixes the printing press when it breaks when the technicians have all become

Rentiers?·at 6:45 pm EST/EDTThanks for a most interesting article, great to see mathematics presented in a positive light

Your example of vodka bottles brings back memories of teaching my sons fractions. I asked them if they preferred half a pizza or an eighth of a pizza.

They have never forgotten since.

Sometimes the mathematics are done as an afterthought.

The ancient Greek Navy were able to outmaneuver and defeat the Roman ships because they could turn faster. The arc and stroke length the rowers could make with their oars was greater because they put a sheep-skin under their buttocks and sat on a highly waxed and polished board. This was considered a state secret.

·at 7:07 pm EST/EDTCan anyone imagine the construction of the Parthenon, of the Lighthouse of Alexandria, the Colossus of Rhodes, the Roman Pantheon, the Colosseum, Hagia Sophia, to name a few, without mathematics (applied mathematics at that, without Vitruvius and Archimedes)? Roman artillery, the ballistae, catapultae, carroballistae, scorpiones, without mathematics? Astronomy, navigation, computing of time, the Antikythera mechanism without mathematics? ”Let no one unversed in geometry enter here” Μηδείς άγεωμέτρητος είσίτω μον τήν στέγην” was the motto of Plato’s Academy, because ““God eternally geometrizes,” Άεί θεός γεωμετρεΐ”, because “You have ordered all things in measure and number and weight”/ἀλλὰ πάντα μέτρῳ καὶ ἀριθμῷ καὶ σταθμῷ διέταξας” as the Wise King Solomon said.

·at 7:15 pm EST/EDTWell, I would not go over the top with the geographic shift in mathematics education and achievement on the university level.

The top anglo world universities still shape a lot of what happens. People all around the world apply at the US top institutions, but not people from a lot of places go to China. To oversimplify: Havard draws from 7 billion best, top China uni draws from 0.5 billion best (because more than half would be happy to leave China for more research opportunities, money and prestige).

If you would look where do US undergrads apply for math grad school, most go for USA with some exception to also apply for UK (Cambridge, Oxford, others), maybe Paris, Zurich, University of Bonn, BMS (collaboration of Berlin universities), the Euro ones more for fun / change. To Asia, practically no one.

To sum it up US math, physics, I now less about other sciences still is running well. But it is only about 40% homegrown, a lot of internationals coming in as PhD candidates, Postdocs, professors.

This is pretty different from let’s say Germany, where about 70% of university professors in sciences are locals (at least in the education sense, but very often also ethnically). I even read about ETH Zurich something like: “Yes, we could recruit more foreign students. But there are enough talented Swiss people.” So that’s a very different ethos / mission from the one of an US university.

Also I would like to say that late Russian Empire science was already pretty successful, and exchange with central Europe very extensive, and the turmoil of founding Soviet Union, some repression and not perfect university system led too it being running okay I would say. My impression is schools were very good and respect for science was big, but the pure math did not grow much more than compared to the west. Very strong in applied math of course. The Russian often, still I think, like to have a certain concrete task as their focus.

In my impression, taking in account to economic power and population, the most successful math (as a science) nations are USA, France and Japan, Israel also. But how they achieve it is very different, with USA clearly the one that uses “mercenaries” the most.

·at 10:08 am EST/EDT“But not a lot of people go to China” (for higher mathematics studies). I have met som counterproof to that statement at CHinese universities. May I venture a possible explanation to the lemma coated here? Probably fewer of le crème de la crème of math studs in the world go to China 1) because those going to the 50 US of North A . have already learned the English language before going htere and not Chinese, and 2)Much of the top talent have been borne and raised in China already. HED (Hic erad damonstrandem).

·at 1:08 pm EST/EDT@Ou Si (區司)/Turkmänijïv

Hahaha… Very good points. I have to admit that teaching in China is fantastic, as the students are genuinely interested in their topics, and are brilliant. I suggest that MOTIVATION is a fundamental element in education (any topic). Students with the right motivation can certainly become

crème de la crème, as you say. Which motivation do the US students have to study and do well?·at 8:59 pm EST/EDTParaphrasing Einstein; you do not understand General Relativity(GR) unless you understand the mathematics behind it. We can get a general idea of the principles of GR by watching a science documentary on GR. Although, if we can derive the equations for GR then we not only understand it, we know the roots of the underlying concept.

·at 10:03 pm EST/EDTNice article, thank you very much. Being an experienced physicist, I would lake to comment some points:

1- Actually most of the things are calculable, except for the phenomena ruled by physical law that are unknown. Still in these cases, one can make predictions and compare to experiments and get a clue of how inaccurate current models and theory are, as well as get directions to develop or extend new theories formalizing the knowledge about the category of phenomena under scrutiny.

2- Calculation of physical phenomena involves a trade off many people ignore. The trade off is between the cost to calculate or simply go straight and make an experiment. Example: due the lack of enough computing power, during most of the last century, research and development in chemistry was driven by experimental efforts. Nowadays, with the increased computing power made available by new and faster computers, as well the capability to store and process large samples of data, O(10) or more petabytes, it is increasingly common to design and simulate a new material and predict its properties way ahead of hitting the laboratory to manufacture it. Sure enough, non-relativistic quantum mechanics provides a complete and accurate description of underlying physical phenomena.

3- It is almost never practical to calculate a detailed microscopic picture of a phenomena. Let me explain better this point, because it is crucial. Imagine a perfectly spherical piece of stone, falling from a certain height. One can engage in a calculation to predict the movement such body and if it would brake or not when hitting the ground. Deploying a microscopical description, from atom level towards, would implies to model the quantum interactions among the atoms, easily finishing with million time of Avogadro’s numbers of coupled differential equations, a similar number parameters and so on. I am unsure if there is, or will ever exists, a distributed computer system able to store the problem. The crucial point about this approach is actually more basic: would be one able to formulate such problem? To my current knowledge, the answer is not. Mathematics is not there yet. To continue our exercise, lets suppose we can do both: formulate and store the problem, and somehow solve it. At this point one would have so much information, each atom’s position and speed, to store and make sense of, before finally getting the answer to the initial problem. So, if you reflect a bit more, knowledge of atom’s positions and speeds are not required. Detailed description of the stone is microscopical properties are not required neither. Instead of engaging in a such hairy approach, one could make a pair of simple 10 minutes long experiments to measure properties like density, resistance, geometry of the stone and given it is a rigid body only, cares about the movement of it’s center. The answer for the initial problem would arrives as a probability to the stone get broken when hinting the ground. Probabilities quantifies our ignorance about underlying, not controlled phenomena. To conclude this point, I would say that the example about the falling glass is not a good one. It is calculable. Maybe not simple, maybe irrelevant as information, but sure calculable.

3- As explained in previous points, performing strict calculations to predict physical phenomena is almost never feasible. The few cases that would be feasible, are not practical. So, one needs to deploy models. Mathematical models. Models abstract away irrelevant or intractable features and allowing to focus in the core ones. Good models provides outcomes followed by a error estimation ( probabilities again, see!) and often have scale parameters that can be tuned to provide more accurate outcomes, at expenses of computing time. More complex the physical phenomena more complex the modelling required to describe it.

4- The connection with mathematics is clear. Mathematics provides the language to formulate the problems and the methods attack the problems. Training in mathematics also provides the capability form a abstract view of the phenomena, which allows to visualize the structures behind apparent chaotic phenomena.

5- The decline of quality of education in math and physics in the west, which in my humble opinion never reached the URSS levels by the way, is very connected to the capability of the hegemon to brain wash westerns with the most random and stupid bullshit. Example: I have a lot of friends who believed some though guy with a manpad downed the plane over Ukraine in 2015. The plane was flying at 10Km of altitude. Only people without an idea of the trajectory, energy, speed and weight a rocket needs to have to reach such altitudes can believe in a such thing. I could give many other examples…

6- Finally, I think Chinese are investing heavily in acquire and update their skills in math, physics and engineering, but the effort is very western framed. Oppositely to former URSS and currently Russia, Chinese does not seem focused in formulate and solve problems using in frameworks developed in house. This makes them particularly vulnerable in my opinion.

I am very bias. I am a Latin American, but educated my self mostly in reading soviet books from MIR editor house.

·at 12:47 pm EST/EDTThanks [Wasp] Lisbeth Salander for the very interesting and insightful comment.

Could you elaborate more on your last point (point 6.)? What do you mean by “frameworks developed in house”? Do you mean their system of education/funding/selecting the students and researchers, or do you mean developing their own math theory/symbols/methodology?

And why does this make them “particularly vulnerable”? Is there a risk of being sabotage or what.

I am quite curiously of what you are trying to say in this point.

·at 1:28 am EST/EDTI have performed mathematical research for 45 years, 26 of those at Los Alamos National Laboratory. When I came to Los Alamos I was very excited; it had alot of applications of national importance and had a strong historical legacy of mathematical significance.

However, despite being a foundational contributor to the field of machine learning with a good publication history of solving important problems, it became very clear that Los Alamos don’t need no stinking mathematics. Indeed, I left in 2012, and a year later the very last working mathematician left Los Alamos National Lab. There are people who are quite talented mathematically still there, but they really don’t work in mathematics -the culture has no use for it.

The current culture at our National Laboratories can be summed up by G K. Chesterton:

“It isn’t that they can’t see the solution. It is that they can’t see the problem.”

To assist someone by using mathematics, one needs a problem to solve. However, the current culture, there are no problems, there are only, as Dewey described, “problematic situations”.

Let me give an example. We had someone from high up in the government, who came and gave a talk on the urgent need for cybersecurity at our government institutions. He said something like “ we need a way to detect network intrusions” and listed some difficulties of doing so. After an hour of discussion, I raised my hand and said “I can do that for you, right now, with minimal cost”

How so he asked; “ simply declare all packets intrusions” They all laughed and said I would get too many false alarms. My response was “then simply declare all packets good”. This time the laughter was a little nervous. I said that if you don’t specify how you are going to compare my gizmo with Susan’s here, you can never make real progress in such a quest. Needless to say, I didn’t win any friends that day, but it is true nevertheless. Without a problem formulation, the overhead slide metric will rule, and it is the one with the prettiest slides that wins the competition for funds.

As John Dewey said in

Dewey, J., Logic-Theory of Inquiry, pg. 108,Henry Holt and Company (1938).

1) “It is a familiar and significant saying that a problem well put is half solved” and

2) “Without a problem, there is blind groping in the dark.”

Imo, we do alot of “blind groping in the dark” at our National Laboratories.

I have read both Andrei Martyanov’s books, and I identified very strongly with what his assertions are. Im not sure its his quote, but I associate it with him; “the Russians make weapons to kill and Americans make weapons for profit”. Essentially he is saying that we have a vast “spiritual gap” between the US and Russia with regards to activities like this. The culture at our national labs seeks not to serve the laboratory or the nation, but seeks a career and a stable funding stream.

I have a friend of mine who does avionics for US fighters and fighter bombers. He told me he was having nightmares about his planes falling out of the sky. I asked him to explain. He tells me that the project is on a rush, so he rented a car and drove across a large western state and when he gets to the job he asks “where is the guy who knows about such and such” and the answer was that it was his regular friday off. He said “this is an emergency” and they said there was nothing they could do about it. I asked him

;” do you think the Chinese have such problems?” The Russians?The Iranians?

To me the depth of mathematics is not as important as the foundation of rational analysis and the commitment to actually solve the problem. What I hear from Martyanov is not so much complete theories of mathematics in warfare, but an ethos of mathematical rigor as it relates to rational thought.

Let me quote Lewis Fry Richardson:

************************

To have to translate one’s verbal statements into mathematical formulae compels one to carefully to scrutinise the ideas expressed therein. Next, the possession of formulae makes it easier to deduce the consequences. In this way, absurd implications which might have passed unnoticed in a verbal statement are brought clearly into view and stimulate one to amend the formula. An additional advantage of a mathematical mode of expression is its brevity, which greatly diminishes the labour of memorising the idea expressed. If the statement of an individual become the subject of a controversy, this definiteness and brevity will lead to a speeding up of discussions over disputable points, so that obscurities can be cleared away, errors refuted, and truth found and expressed more quickly than could have been done had a more cumbrous method of discussion been pursued. Mathematical expressions, however, have their special tendencies to pervert thought. The definiteness may be spurious, existing in the equations but not in the phenomena to be described. And the brevity may be due to the omission of the more important things, simply because they cannot be mathematised. Against these faults we must constantly be on our guard. … Mathematics has been used in this book both inductively, to summarise facts, and deductively, to trace the consequences of hypotheses. By mistaking the intention, it is possible to complain that the inductions do not follow from the previous hypotheses, and that the deductions go beyond the known facts. Certainly that is so, but it is no cause for complaint, rather for rejoicing. Those who say ‘you can prove anything by statistics’ should instead say ‘unfortunately, we do not understand statistical methods sufficiently well to enable us to distinguish arguments that are genuine from those that are false.’ To anyone who believes that words are the proper medium of expression, I recommend the following exercise: translate into word ax^2 + bx + c = 0, remembering that x, a, b, and c are not words; proceed in words, not using any algebraic symbols, to find and prove the solution of the equation.

***************************

I have a friend at Los Alamos Laboratory who was in charge of interviewing post doc applicants for a postdoc in material science. He told me that out of the 50 or so applicants, they were distributed 1/3 as

1) Chinese

2) Indian

3) Iranian

yes, you read that right, Iranian at our National Security Lab.

I asked him “didnt you have any white applicants?”

He said, oh yeah, one guy was from the Netherlands.

Also, during the last 10 years the number of chinese authors on high quality mathematical publications has skyrocketed.

From what I understand, the days where they sent their best here to get good graduate educations is also coming to an end; they get good graduate educations at

home. Now, after they get their graduate education at home they come here ( like working at Los Alamos) to earn a good living for awhile.

As a funny irony, I say to those obsessed with the brown invasion on our southern border, that that is NOTHING. You have a lot more to worry about than people from south of the border. You are looking at the wrong invasion, imo.

I have, as the Saker suggested, bought several copies of Martyanov’s books and given them to my friends. We owe him a great deal of gratitude, although I am still recovering from the cognitive dissonance obtained but reading his thoughts on American exceptionalism and George Patton, gol.

Mustafa

·at 5:29 am EST/EDTThank you very much, this is very instructive! It is about what I had in mind of the present situation in the USA. And you are right to speak of rigor, I should have used this term.

·at 11:49 am EST/EDTPaul, you are welcome. Please forgive me, I forgot to thank you for your excellent article.

There is much more to say actually. A few tidbits:

1) after 50 years leading the world in numerical methods to solve field equations on the computer, no (0!) work has been done here to establish the accuracy of long runs for these field equations, equations such that for example, they use in the global climate models. So, in reality, after a long or medium run on these simulations they really have no idea what they have.

2) I was at a meeting where somebody asked the question “So how do we know how well we solved that equation?” The answer:

It doesnt matter, we know its not the right equation in the first place.

3) I worked for some time to develop a rigorous method for stockpile certification, and when I presented my preliminary results I was asked: why should we fund you to do this work? My naive answer was; because I have a track record of success at such things, and I have already laid the groundwork for it. Nothing became of that effort.

4) Two of my Los Alamos colleagues and I cracked the fundamental problem of machine learning; the 2-class Statistical Classification Problem. This took over 7 publications, in addition to a very popular book, and provided us with a strong international reputation. The results present a clear picture; we understand now how to make machines learn ( mea culpa mea culpa). Currently, machine learning is all the rage at Los Alamos, but despite me living 30 miles from there, no Los Alamos employee has contacted me to speak with me about what we learned. They are not interested.

·at 2:55 pm EST/EDTMustafa Kemal, perhaps you should move to another country where your knowledge would be appreciated… Rusia? China? Iran?

·at 12:16 pm EST/EDTMC, I am getting old, but still maintain an enthusiastic employer at a very good US university, so I am still appreciated. But I feel like a dinosaur.

However, I do recommend to my children and grandchildren that they learn Russian -not because they will come here, but because i think there is a future there. If we destroy ourselves before we do Iran, Iran, having preserved its culture, also may have a good future.

·at 2:54 am EST/EDT… well, the problem of the so called West might indeed a mathematical one, considering the fact that the ruling Talmudic Crime Syndicates rely on fanciful and freewheeling Kabbalistic ‘by way of deception mathematics’ …

: )

·at 4:08 am EST/EDTAnd what happens to humans when hundreds of millions of Life forms (animals, fish, insects and plants) are subtracted from one planet?

·at 7:39 am EST/EDTThanks for the challange, Nussiminen.

30 and 10/3 as the other two sides of that right triangle?

·at 3:31 pm EST/EDTNusssiminen, my proposed solution, leg 30 units > hyp 10 units. I’m learning more than what I expected!

·at 4:31 pm EST/EDTSolution to exercise:

If the two sides are denoted x and y, the following equalities apply (correct regardless of which side is denoted x / y):

Pythagoras’ theorem: x^2 + y^2 = 100 (Eq. 1)

Uniform triangles: (x – 3) / 3 = x / y (Eq. 2)

Eq. 2 can be re-cast into:

xy = 3x + 3y ==> (xy)^2 = 9 (x^2 + y^2) + 18 xy ==> (xy)^2 – 18xy – 900 = 0, by insertion of Eq. 1.

So we get the

of the sides to be:productxy = 9 + sqrt(981) (Eq. 3)

Eq. 1 and Eq. 3 give us:

(x+y)^2 = x^2 + y^2 + 2xy = 118 + 2 sqrt(981) = A

(x-y)^2 = x^2 + y^2 – 2xy = 82 – 2 sqrt(981) = B

So if we stipulate that x > y, we finally obtain the results:

x = (sqrt(A) + sqrt(B))/2 = 8.92

y = (sqrt(A) – sqrt(B))/2 = 4.52

·at 10:50 pm EST/EDTThere have been three big waves of U.S. “import” (ruthless exploitation) of foreign labour, which have kept that made-up country floating.

In its feudal and early post-feudal period, those were the black slaves from Africa.

With world change to industrial mode of production, the U.S. “import” switched to poor white blue-collar labourers, to work under the whip in mines and factories – lured

en massefrom all impoverished corners of Europe, hoping for a “better life” – and now cruelly exploited (white slaves replaced the black ones, you might well say; once arrived, there was practically no way back for most of them).Starting from just around the WWII, and into the late XX century, with increasing demands of hi-tech industry, the “import

du jour” became the highly educated Europeans – engineers, mathematicians, physicists and other scientists. First there was “the war effort”, then “the cold war effort”… If one looks at the details, one comes to the conclusion – that intellectual slave labour had replaced the above two. These days, however, the U.S. has ceased to be a big magnet for this category, for Europeans in particular, for a number of very good reasons. The U.S. is indeed on the wane.·at 12:56 pm EST/EDTI haven’t read anything as refreshing to the soul as this article!

I particularly enjoyed the part about the ‘resources’ in places like Syria and Iraq, where algebra was thriving a millenia ago whilst western Europe was in moral and intellectual decay.

I live in Nairobi, Kenya…

·at 5:50 pm EST/EDTMaths were from the origin connected to philosophy. Pythagoras was viewed as a sage in Greece, and the philosophical academy bore the sign “Let no one enter unless he is a geometer.”

So it was really a matter of leveraging both what you call “heart” and “head”. Gödel’s theorems, by torpedoing Hilbert’s project of automating all theorem proving, showed that automated proof, by sticking to a syntactic level, the only level that automata operate at, could not fake understanding past a fairly basic level of sophistication. They showed there cannot be any purely syntactic specification of the axioms of elementary arithmetic.

And semantics is properly human. No machine understands the meaning of the symbols it manipulates. It is not even aware they are symbols: it just applies rules, like the Chinese-agnostic operator that, in Searle’s Chinese room example, read from a book of rules so as to automatically conduct a Chinese conversation without knowing its meaning. The brain, being a machine, doesn’t understand meaning either. Maths are an activity of the soul, akin to music, the brain only providing computing power.

Maths are not a natural science. Natural sciences have no access to proven truth. Only maths do. All truths of natural sciences, from the existence of atoms to the existence of energy quanta, are probable truths. Extremely precise measurements have always converged in keeping with the law; so it would be unreasonable to question the law. But it has not been proved from first principles.

Theoretically, some philosophical truths could be as well-founded as mathematical truths. This was Descartes’ ambition when writing his Metaphysical Meditations, namely to bring geometric rigour to metaphysical thinking. Since the goal was to establish whether we live in total illusion (“the matrix”) or can trust mathematical statements to be objectively true, his proof is of some interest; so it is one of those things I promise myself to dig into some day.

Re the shift to the East, I have worked in a US high-tech corp that had considerable contempt for theory and saw a young Chinese recruit that had placed well in a Chinese maths olympiad be pushed out of the company by an arrogant and ignorant boss. And I’ve seen a lot of disdain for maths in US software engineers. Maybe things have changed with the recent AI push; but I tend to think that societies in which proven truth is a value will have an edge over value-less societies.

Sciences in the West are ideological. Neodarwinism, inflation theory and strong AI are baseless and sometimes counterfactual speculations leveled at theism, and mainly Christianity. They are the West’s lysenkoism; and an indication of serious spiritual malfunction.

China would do well to retain the attitude so well encapsulated by a few years ago by a Chinese paleontologist: “In the USA, you can criticize the president but you cannot criticize Darwin; in China, we cannot criticize the president, but we can criticize Darwin.”