Visit to Armenia

After leaving Tblisi we crossed the border to Armenia. Again we had to walk a certain distance across the border but in this case it was not so far and we did not have to show our passports so often. We spent our first night in Dilijan. This was a place of no special interest in itself but rather a starting point for excursions. After that we continued to Yerevan. I must admit that prior to visiting Armenia my knowledge of the country was essentially zero. My impression now is of a very interesting culture. Our local guide told us a parable. The Armenians are like a swallow building its nest. The swallow invests a great deal of time and effort building the nest, laying eggs, feeding its chicks. Then a predator comes and destroys everything, the nest and its contents. Then the swallow begins again, and again, as long as it takes. The Armenians repeatedly built up their civilisation and were then attacked by different enemies who destroyed everything. Like the swallow they always started again. He said that when he named the centuries in which different buildings we saw were constructed we would notice that only certain centuries occurred and others did not. The reason is that in those other centuries there were wars which prevented such things being built.

As in the case of Georgia one of the most striking first impressions when entering Armenia was the script. It plays a central role in the identity of the Armenians. It was invented by Mesrop Mashtots in 405 AD. There is a theory that he also invented the Georgian alphabet but this is controversial. In Yerevan we visited the National Library which has a huge collection of manuscripts, both Armenian and international. We were told about some ancient Greek manuscripts which were translated into Armenian and then destoyed in the original. They would have been lost but could then be translated back into Greek so that they were saved. In many Armenian monasteries, which were centres of learning through the ages, there are jars buried in the floor. The purpose of these was as follows. If the monastery was attacked by Barbarians the valuable contents of the library would be hidden in these jars in the hope they would not be discovered. The invaders often knew that manuscripts were very valuable to the Armenians. Sometimes they would destroy them as a kind of cruelty. Sometimes they would kidnap them. In other words they would steal them and demand a ransom. Typically the ransom would be payed, showing how much the Armenians valued their culture. Our trip included more visits to monasteries than I would really have liked. At the same time I must say that when I entered one of these for the first time I was impressed. It had a special archaic feeling I have not had anywhere else. The fact that the walls were so dark and colourless is partly due to the ravages of time but even in the Cathedral of Etchmiadzin which is the seat of the head of the Armenian church and newly renovated the decoration is somehow modest, which I see as a positive thing. The church in the first monastery we visited communicated a sense of solemnity which affected even me as a non-religious person. The acoustics there are fantastic. Our local guide illustrated this impressively by singing traditional Armenien religious music there. A striking symbol of the cultural richness of Armenia is the statue of Mesrop Mashtots in front of the National Library, raising his arm to point at the alphabet.

Statue of Mesrop Mashtots

Arriving in Yerevan I felt like I was arriving in the Soviet Union with lots of depressing Soviet architecture. On the other hand the pedestrian zone of the town is very attractive. Apart from the surroundings I was impressed by the large number of fashionably dressed young women walking around. It felt very cosmopolitan. I thought of a video which I have often watched of a young woman playing guitar and singing in the streets of Yerevan. The music is Latin American but the singer is Armenian. When I was there I was not able to identify the street – in the video the background is mostly quite blurred.

There was one point in the programme of our trip which was very important to me. This was a visit to the Genocide Memorial and Museum. In the museum I had a lot of striking impressions and learned a lot of interesting information. As I mentioned in a previous post, contrary to what one might think, the central example on the mind of the inventor of the concept of genocide was not the Holocaust but the case of Armenia. I mention in passing the statistic that there are about ten times the number of Armenians living outside the country as inside the country and that there is a big flow of money into the country coming from
expatriates. In the museum there are very graphic descriptions of the atrocities conducted during the genocide, with the perpetrators going out of their way to invent new ways of torturing their victims. The Turks even had a special philosophy behind this. I was reminded of accounts of the Hamas massacre of the 7th of October. One place which played a central role in these events was Van. I was in Van many years ago but at that time I was not paying attention to politics. Who was behind this genocide? It was started by Turkish nationalists, the ‘Young Turks’ but a lot of it was also carried out by Kurds, who attacked their Armenian neighbours. The border between Armenia and Turkey is closed to this day. Mount Ararat is very visible in Jerevan but it is not possible to approach it from the Armenian side. Some of those involved in the Armenian genocide were put on trial but these trials were stopped by Atatürk when he came to power. I always had a positive opinion of Atatürk and so this information disturbed me. When exiting the exhibition in the museum you can read the following telling quote from Adolf Hitler: ‘Wer redet heute noch von der Vernichtung der Armenier?’ [Who still talks today about the destruction of the Armenians?]

When I mentioned my trip to Azerbaijan, Georgia and Armenia to people, before and after it took place, they often reacted in a way that they felt a trip of this kind to be dangerous. I would just like to state that during the whole trip I never encountered anything which I felt to be threatening or dangerous. Of course we spent a lot of our time in the protected context of a group of tourists but Eva and I wandered around alone, off the beaten track, in various towns and we had no problems.

Visit to Georgia

After leaving Shaki we crossed the border into Georgia. For this we had to walk several hundred meters uphill with our suitcases but otherwise crossing the border was unspectacular. The land border between Azerbaijan and Georgia can only be crossed in one direction. That between Azerbaijan and Armenia is closed in both directions. This means that if you want to visit all three countries travelling by land and not entering any other country in between the order is fixed uniquely to be the one in which we visited them, namely Azerbaijan, Georgia, Armenia. When we arrived at the Georgian border one of the most prominent things was the national flag. In all three countries the national flag was often to be seen, not only on official buildings. In Baku there is an enormous flag next to the lake which is almost impossible to ignore. This is in stark contrast to the situation in many west European countries where flying the flag of your own country is frequently politically incorrect or even illegal. National pride has negative aspects: it no doubt contributes to the conflicts between the countries in the Caucasus. However it also has positive aspects: it can inspire the inhabitants of a country to positive achievements and Europeans should not forget that. Another prominent thing upon arrival in Georgia was the script, which looks quite exotic to the uninitiated. There are three versions of the script. The newest of these, Mkhedruli, the standard one today, goes back to the 10th century. For me many of the Georgian letters look like Latin letters. Thus learning the Georgian alphabet could largely reduce to learning a permutation without fixed points. This does not apply to all letters. The first one I picked up from road signs corresponds to the Latin $i$ and looks in first approximation like the mathematical symbol for an intersection. A difficulty in pronouncing Georgian is the occurence of clusters of consonants. An example is the river Mtkvari which flows through Tblisi or, slightly simpler, Tblisi itself. In comparison with Baku Tbilisi appeared to me quite classical. I heard that the town in Georgia with the most modern architecture is Batumi on the Black Sea.

We first travelled to Tblisi. During our arrival we drove along the river bank where there are houses that look that they could fall down a cliff any time. In fact they are very stable. At one point I saw a tree full of egrets but only briefly. A couple of days later I was able to return on foot and see that they (or at least the majority of them) were cattle egrets. We drove from Tblisi to Stepantsminda. This a main route for lorries travelling between Georgia and Russia. It passes through the town Vladikavkas on the other side of the border. There are lots of road works and traffic jams on that route. We passed a huge queue of waiting lorries wanting to travel the other direction. They are not allowed to drive during the day and are only allowed to continue their journey at a certain hour. This extreme measure is needed to ensure that the traffic on the road can move at all. Thus it seems that there is a lot of trade between the two countries although I heard that they have no diplomatic relations. The mountain landscapes were very impressive. I particularly liked the huge grass-covered mountains. From Stepantsminda there is (weather permitting) a fantastic view. In the middle there is the Gergeti Trinity Church and in the background the mountain Kasbek with an altitude of over 5000 meters. We had good weather and optimal views, both from Stepantsminda and from the church. According to one account Kasbek is the mountain where Prometheus was chained by the gods to have his liver eaten by an eagle. I did not see any eagles there but I did see some choughs around the church.

Kasbek with Gergeti Trinity Church

Visit to Azerbaijan

Eva and I recently visited Azerbaijan. The trip started with a flight from Frankfurt to Baku. Apparently the only European cities which have direct flights to Baku are Frankfurt and Warsaw. In saying this I am not counting Moscow as being in Europe, which is based more on emotional then geographical factors. There are several flights a day between Baku and Moscow and the relationship between Azerbaijan and Russia seems quite good. At the airport in Baku I was surprised that there was a flight coming from Tel Aviv. This is perhaps a reflection of the fact that although the great majority of the inhabitants of Azerbaijan are Muslims there is a strict separation of religion and state. There were less women to be seen wearing obviously Muslim dress in Baku than in Mainz. In Baku there is a large statue representing a woman throwing off her veil.

Part of our flight was over the Black Sea and on the map there was a point where our subjective distance to Sebastopol was not very great. I know that in the last few years terrible things have been happening in the Ukraine, At the same time it all seemed to me very far away. Passing Sebastopol made it more real.

While we were in Baku we experienced some very strong winds. Perhaps it could be called the Chicago of the Orient. I also see a certain analogy between the Caspian Sea and Lake Michigan. The city of Baku appears modern and it has quite a few high-rise buildings. The most striking of these are the Flame Towers. They are already striking in the daytime due to their being in the form of flames. This is enhanced during the night due to the dynamic lighting. The use of flames as symbols has to do with the deposits of gas and oil near Baku. In earlier centuries there were flames coming out of the ground and this was important for the development of the Zoroastrian religion. We visited a fire temple. This is now a museum. There is still a flame but it is now fed with gas artificially. Here I heard the idea that Zoroastrianism can be thought of as a predecessor of Judaism, Christianity and Islam. Around Baku there are still oil fields on land and these remind me of pictures I have seen from the US. In fact Baku was the first place where mechanical oil drilling started, before it did so in the US. Another architectural highlight in Baku is the spectacular Heydar Aliyev Center. Outside the building there are curious sculptures of animals and we were immediately reminded of similar sculptures we had seen at Ground Zero in New York. In fact it turns out that both sets of statues are due to the same artists, Gillie and Marc.

In the West Mikhael Gorbachev is usually seen as a very positive figure. The picture of him in Azerbaijan is a very different one. In January 1990 he sent troops to Baku to crush anti-government protests. As a result more than a hundred protestors were killed. We saw their graves which display pictures of them. He later said that this was the biggest mistake of his career but this does little to reduce his guilt in the eyes of the Azeris.

We visited the cave paintings in Gobustan. Unfortunately the whole thing failed to capture my imagination. The place was overrun by tourists. It was our first encounter (but not the last during this trip) with Chinese tourists who struck us as very loud and inconsiderate. The contrast with the more familiar Japanese and Korean students, of which there were very few, could not have been greater.

From Baku we travelled to Shaki and on the way we stopped in Shamakhi to visit the Juma Mosque there. It was the first time in my life that I had been in a mosque.

The flame towers

Biography of Elon Musk by Walter Isaacson

In the modern world Elon Musk is a very influential person. He receives a lot of hate and envy for what he does and says. In the past my attitude to Musk was rather neutral. This had to do with the fact that I was not well informed about him. Now I set out to change that by reading the book quoted in the title of this post. The book is well written and pleasant to read. It contains a lot of interesting information. I was glad to see that it appeared to be unbiased: the author is neither trying to damn his subject nor to idealise him.

What are the most important influences of Musk? One consists of his achievements in new technologies. I was suprised to see how much the ability of the Western world to be active in space is dependent on technologies coming from his company SpaceX. I also learned how great his influence has been in the field of electric cars. These were themes that I had simply not been following. Another influence of Musk came through his direct political involvement in the present Trump administration. Since the book was published in 2023 it of course cannot say anything about that subject. One theme which particularly interested me was that of Twitter and X. The book provided me with new insights into that. Musk bought Twitter because he wanted to so something against the splits in public opinion. He spectacularly failed to achieve that aim. Instead we have seen an increasing split between the political atmosphere on X and that of the woke culture, many of whose representatives have left the platform. A lot of these things happened after the time the book was published but it is possible to learn there about the roots of certain developments. An aim he did achieve with X was to provide a forum for free speech. Among his contributions that is the one I appreciate most. He did produce some very damaging tweets and there is no doubt a lot of evil content on X. However I see X as making a central positive contribution to modern society. It helps to prevent powerful people from hiding their dirty secrets. Of course using X in a positive way requires critical thinking on the part of the user. Modern technologies are important but what is even more important is a society which is willing and able to produce and use such technologies.

The fact that Musk is so extremely rich is a reason for many people to envy and hate him. Many people on the left suggest that Musk is only interested in making money and he just exploits the people who work for him rather than making a positive contribution to the technologies developed by his companies. The book shows how wrong this is. Musk is very good at making money but that is not his central concern. He pursues certain key technological goals and the money is just a means to help him do so. In his pursuit of these goals he is uncompromising. This determines the way he treats himself and others. He requires absolute commitment from those who work for him but he also spends huge amounts of time on the factory floor, looking for possibilities to improve his products. He often sets more or less impossible deadlines and targets which are sometimes achieved and sometimes not. He is often authoritarian. If someone does not do what he wants or fails to achieve a goal he has set them he will immediately fire them. I would not like to be an engineer working for Musk but I would be of no use to him anyway. I am not a person who can efficiently carry out technical tasks under pressure of time. Musk is often opinionated in his work but he does allow himself to be
persuaded to change his mind by evidence. The difficult task of his associates is to know how to feed him with this kind of information.

On a personal level Musk is completely different from me. He loves computer games. He is good at computer programming. He likes Windows. He thrives on conflict and often tries to generate conflicts. If I look for common ground I think of ‘The Hitchhiker’s Guide to the Galaxy’ and ‘Monty Python’. I think he should be praised for his influence on the Ukraine war by providing access to Starlink. On the other hand he later introduced more restrictions on that than I would have wished. I am happy that in some cases he has acted as a bastion against wokeness. After reading the book my attitude to Elon Musk is still somewhat neutral but the tendency is positive rather than negative. I certainly came away with a much more positive impression than I did when I read a biography of Bill Gates years ago.

Suicidal Empathy by Gad Saad

In a previous post I wrote about the book ‘The Parasitic Mind’ by Gad Saad and I mentioned that I saw my reading of that book as a preparation for reading his new book ‘Suicidal Empathy’. I have now finished reading the latter. I found it stylistically more pleasant than the previous book. Either his style has got milder or I have just got used to it. The most significant thing about the book is its contents. I think that it is a very important book and that it should be read by as many people as possible. One aspect of it is that the author presents many examples which are so extreme as to be breathtaking. Incredible extremes of wokeness. The main theme of the book is how Western society is being destroyed by excessive amounts of empathy applied to the wrong targets. The examples are backed up by sources where it is possible to learn details about the cases presented. I made use of this in a couple of cases. For instance I got more information about the case of the young German politician who initially lied to the police about the nationality of the men who had raped her so as to not risk arousing racist sentiments. Another important element of the book is its discussion of many themes from psychology, many of which were not familiar to me.

The first chapter is concerned with the notion of empathy in general and an initial presentation of its pathological variants. The second is concerned with things that people are not allowed to say in our society and even better (from the point of view of those promoting this ideology) not allowed to know. The central idea of third chapter is how people in our society make the fateful assumption that members of other cultures share certain of our basic principles. A key element of the fourth chapter is the way in which criminals are not held responsible for their acts, instead being portrayed as victims of circumstances. Subjects of the fifth chapter include COVID, climate change and trans activism. There I am less in agreement with the author than in other parts of the book, particularly concerning COVID. Chapter six treats topics related to DEI, or as Saad calls it, with deliberate misspelling, the DIE cult. It includes a quote which for me was one of the most shocking in the book, an oath taken by graduates of the University of Minnesota Medical School in 2026. This may be summarized by Saad’s title for the section: reject the hippocratic oath, choose the woke oath – be empathetic. Suddenly being well educated about medicine and doing everything to promote the health of the individual patient are qualities of a doctor which are of minor importance. In the seventh chapter there is a discussion of the way governments take the resources of those who are productive and distribute them to those who are not. The Canadian health service is taken as a central example.

When the central ills of our society have been presented the obvious question is what we can do to change this situation. The last chapter of the book makes some suggestions. When I ask myself what I can do I am disappointed by my lack of answers. I do not see this book as one which is to be read and then given a more or less honourable place on a shelf. I expect to be able to learn a lot more things by following the lead of the footnotes in this book. My recommendation is: buy this book, read it, recommend it whenever you get the chance. Its relevance is not confined to North America – it is relevant for all Western countries. As far as I am aware neither this book nor any other by Gad Saad has been translated into German. This is a pity. Presumably editors are afraid of touching an author with such a questionable reputation. By contrast, some of the books of Douglas Murray have been published in German. They appeared in the FinanzBuch Verlag and belong to a series run bei Tichys Einblick. Could that be a German home for Saad’s books?

Hartman’s theorem on conjugacy of saddles in two dimensions

In a previous post I discussed the topic of conjugacy of vector fields on , concentrating on the case where the vector field and the conjugation are smooth, i.e. . In what follows I will continue to consider smooth vector fields but allow the conjugations to have only finite differentiability. In the last post I showed that in the case all vector fields are smoothly conjugate near a non-degenerate zero. In particular in this case each vector field is conjugate to its linearization at a steady state. Consider now the case . The theorem of Hartman mentioned in the title of this post concerns a hyperbolic saddle in two dimensions. (The reference is ‘On local homeomorphisms of Euclidean space’, Bol. Soc. Mat. Mex. 5 (1960) 220–241.) In other words we consider the case that the linearization of a vector field at a steady state has one positive and one negative eigenvalue. The theorem says that the vector field is locally -conjugate to its linearization. This is a very special case as we can see by varying the assumptions. For instance, a general two-dimensional vector field is not -conjugate to its linearization. An example is the vector field which generates the following system of ODE: , . Its general solution is given by and . If it were -conjugate to its linearization then the leading order terms in the solution in the limit would be exponential but this is not the case. In three dimensions a vector field can even fail to be -conjugate to its linearization. An example of this is given in Hartman’s paper.

The proof of the theorem in the paper is actually not formulated in terms of ODE but instead uses a related ‘integrated’ version of the problem where the central object is a diffeomorphism with fixed point. The relation between these problems is explained in Hartman’s book ‘Ordinary Differential Equations’. If we start with a system of ODE we can consider its time one flow, which is a diffeomorphism. Suppose now that we can find a mapping which gives a conjugacy of the time one flows. Then it can be shown that it also gives a conjugacy of the entire flow and thus (under a differentiability assumption) of the vector field. Conversely a conjugacy of the vector field will give a conjugacy of the flow.

Here I want to look into Hartman’s theorem and its proof in some more detail. Consider a local diffeomorphism in two dimensions which fixes the origin. Suppose that it is hyperbolic with an eigenvalue of the linearization which is less than one and an eigenvalue which is greater than one. Using the stable manifold theorem for maps it can be assumed that the the stable and unstable manifolds of the fixed point coincide with the coordinate axes. Moreover, using the (integrated version of the) one-dimensional result already mentioned it can be assumed that the diffeomorphism is equal to its linearization on the axes. The mapping will be written in the following form. , . By what has already been said we can assume that and . This is the statement about the stable and unstable manifolds. The mapping used to do the transformation is of the form , and its inverse is of the form , . The condition for a conjugation implies a functional equation for the function . This gives rise to an iteration and the interation can be shown to converge. What is special about the case of the two-dimensional saddle? The argument with the stable and unstable manifolds works in any dimension. The fact that the vector field can be transformed to its linearization on these invariant manifolds uses the fact that the manifolds are one-dimensional. Othewise resonances could cause problems. In the paper the convergence proof is carried out in general dimensions under an additional assumption on the eigenvalues, (8.1) of the paper. These allow estimates to be obtained for the matrices making up the linearization. In the case of a two-dimensional saddle these matrices are scalars and the additional estimates are unnecessary.

There is a case where the non-existence of a conjugation of higher differentiability for a two-dimensional saddle can be shown by relatively elementary means and I want to show this here. Exercise 3.3.1 of the book ‘Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields’ of Guckenheimer and Holmes asks to show that for a certain system there exists no smooth conjugation. Here I will do this exercise by showing that there is no -conjugation I have no idea if this level of differentiability is optimal in this example. The example is given by the equations , . The general solution is given by , . Suppose that there existed a conjugation of this vector field to its linearization at the origin defined by functions and . Then these functions satisfy certain functional equations. The equation for is . We are looking for a solution which is zero at the origin and whose linearization there is the identity. As a test take the first derivatives of these equations and evaluate the result at the origin. It turns out that the resulting equations are satisfied identically. Now do the corresponding procedure with second derivatives. The resulting equations imply that all the second derivatives of and vanish at the origin but no more than that. In looking at the corresponding procedure with the third derivatives the nonlinearity suggests that it could make sense to look at the derivative . In fact it can be concluded by doing so that , contradicting the assumptions. These calculations are justified if the transformation is assumed to be and this gives the desired result.

Conjugacy of one-dimensional vector fields near a zero

A vector field on the real line can be written in the form and in this sense it is equivalent to a function. From a geometrical point of view it is different from a function because it has a different transformation behaviour under diffeomorphisms. If a function is transformed using a diffeomorphism to get a function then . This is a pull-back operation, transforming a function on the range to one the domain. By contrast the transformation of a vector field goes in the opposite direction with the vector field written above being transformed to a vector field represented by a function with . If is the inverse of then this is equivalent to . Suppose now that has a zero at zero, i.e. . The zero is called non-degenerate if . This is an invariant statement since , where the last equality holds at a zero of . If we have two vector fields of this kind does there exist a local diffeomorphism defined in a neighbourhood of with which transforms one to the other, up to a multiplicative constant? This would follow if we could show that any vector field of this kind can be transformed to the vector field . The latter statement is not invariant but more convenient that the former for proving it analytically. It seems at first sight that this is a very simple question and I am surprised that although I spent many years working with differential geometry I cannot recall having asked myself this question before. It is the question of conjugacy of vector fields satisfying the given conditions.

The answer to this question of conjugacy is given in Proposition 2.9 of the book ‘Qualitative Theory of Plane Differential Systems’ by Dumortier, Llibre and Artes. This proposition concerns a differential equation of the form where and is a smooth function with . Solutions of this equation are precisely the integral curves of the vector field with the coordinate function and this is the general smooth vector field with a non-degenerate zero at zero. The statement of the proposition is that there exists a smooth function with for a smooth function with such that in terms of the equation becomes . This is the most general diffeomorphism whose derivative at is the identity. The condition on the derivative is of little importance since any linear transformation leaves the vector field invariant. The proof given in the book is as follows. If we transform the equation and impose the condition that the transformed equation is then we get the equation . This corresponds to the equation involving written above. This correspondence arises from the fact that the right hand side of a system of ordinary differential equations can naturally be thought of as defining a vector field. Let . Since is smooth and the function is also smooth. It follows that must be a solution of the equation . This has a unique solution with . This theorem is the special case of Sternberg’s theorem in one dimension, where there exist no resonances.

Definitions and politics

In mathematics definitions play a central role. Before starting a mathematical discussion it is necessary to fix necessary definitions. When mathematicians discuss other subjects they also often want to know the definitions of the terms being used and this is often difficult. It can already be difficult in physics and of course it is much more difficult in areas such as law or politics. A definition involves explaining something in terms of something else which is already known and already defined. In many areas this becomes difficult because so many terms used are poorly defined. The extreme case is that where terms are not defined at all but at just phrases connected to an emotion or a value judgement. In law or politics it is often hard for people to agree on a definition since different individuals may have concrete advantages or disadvantages depending on what definition is chosen. There is an increasing tendency for people to change definitions for their own benefit or use undefined terms as a weapon to discredit certain people or even to mark them as targets of violence.

A first example is redefining the meaning of the words ‘man’ and ‘woman’ so that they correspond to something else than they have done in the whole history of the languages they belong to. Since I live in Germany the example which is most familiar to me is the so-called ‘Selbstbestimmungsgesetz’. This is a German law which allows individuals to choose their own sex as it pertains to legal matters. This choice can also be changed as often as once per year. Changing the meaning of these central words immediately changes the meaning of many laws although the text of those laws remain identical. It offers many opportunities to a man who is prepared to make a corresponding declaration. Criminals who have done so (including sex offenders) have the opportunity to spend their prison time in a womens’ prison. It is possible to legally enter spaces reserved for women such as toilets and changing rooms. It is possible to compete in women’s sports. A second example is that political opinions (and people expressing those opinions) which do not belong to a narrow left-wing corridor are labelled ‘extreme right wing’.

A third example, and the one which motivated me to write this post, is the definition of ‘genocide’. What is the definition of this term, which was introduced by Raphael Lemkin in the early 1940’s? He was thinking primarily of the case of Armenia and Turkey. Later the Holocaust became the central example in people’s minds. Conflicts of interest such as those I have already mentioned have prevented the development of a clear definition which is widely accepted. What are the central elements of the concept? It has to do with the murder of a large number of people who are chosen on the basis of the fact that they belong to a certain group and not because of their actions. In a war between a country X and a country Y it may happen that large numbers of citizens of Y are killed by citizens of X because those people are fighting for Y. This does not fit the definition since the people were killed as a result of their own activity (fighting). The definition is also not fulfilled if large numbers of citizens of Y are killed just because they happened to be at the wrong place at the wrong time and there was no specific intention of the citizens of X to kill those people. The one aspect of ‘genocide’ which everybody agrees on is that it is very very bad. Thus it is an excellent label to use when trying to discredit people or mark them for attack. This is exactly how it is used by pro-Palestian activists as a way of attacking Israel. This is particularly absurd given that the modern state of Israel came into existence as a reaction to the Holocaust which is the clearest example of a genocide.

Normal forms of dynamical systems

Here I consider a system of ODE or equivalently a vector field on an open subset of Euclidean space. The question is in what way it can be simplified by a coordinate transformation. This leads to the theory of normal forms. My main source for what follows is the book ‘Qualitative Theory of Planar Differential Systems’ by Dumortier, Llibre and Artes. I will mainly concentrate on the two-dimensional case, as that book does. The aim is to simplify the vector field in the neighbourhood of a steady state, which is placed at the origin. Then we can write it in the form where is a matrix and vanishes faster than linearly at the origin. To avoid any subtleties with differentiability let us suppose that is smooth (). We want to apply a diffeomorphism to bring the vector field into the form where is in some sense as simple as possible. Let be the set of vector fields which are homogeneous of degree . The adjoint action of a linear vector field defines a linear mapping from to itself. We will be interested in this mapping in the case of the vector field defined by the matrix . In each degree choose a complement to the image of the linear map defined by the adjoint mapping. Then it turns out that there is a transformation which makes the sum of elements of the up to a certain order and an error term of higher order. It then follows using Borel’s theorem that there is a transformation which puts the whole -jet of at the origin into this form. The book also discusses a concrete choice for the complement .

We now restrict to the two-dimensional case. The analysis depends on the eigenvalues and eigenvectors of the matrix . Consider first the case that both eigenvalues have non-zero real part with the same sign. We can suppose without loss of generality that the real parts of the eigenvalues are positive (repelling case). Suppose now that is diagonalizable with real eigenvalues . It turns out that it is possible to reduce the system to the case where and . The normal form theorem shows that this can be done up to an error of infinite order. Sternberg’s theorem shows that it can be done exactly. Here can be chosen to be zero unless with . The case where is needed is the resonant case. Next suppose that is not diagonalizable. Then it turns out that there are no resonances and all non-linear terms can be removed. In other words the vector field can be transformed to its linearization by a diffeomorphism. The same is true in the case of complex eigenvalues with positive real parts.

In the case of a hyperbolic saddle the book starts by discussing the existence of stable and unstable manifolds, first in the analytic case and then in the smooth case. This can be used to rewrite the equation in a form where these manifolds have been mapped onto the coordinate axes. Only after that is the normal form theorem applied. In this case there can be resonances whenever the quotient of the eigenvalues is rational. There is a cautionary note that, in contrast to the cases previously considered, the theory of the smooth case cannot be straightforwardly be extended to the analytic case. This is related to the famous phenomenon of ‘small divisors’ in celestial mechanics. Returning to the smooth case note that when the quotient of the eigenvalues is irrational the system can be linearized. This completes the discussion of the hyperbolic case. I intend to return to the non-hyperbolic case in a later post.

A conflict of visions by Thomas Sowell

These days people often talk about how society is divided when it somes to various political issues. What is remarkable is a certain correlation. People who are on the same side in the case of one political issue often find themselves on the same side in the case of an apparently unrelated issue. This is a phenomenon I have thought about a lot and also discussed with other people. Recently I talked about it with Samuel Blotiu and he recommended me a book which is concerned with this subject and offerred to lend it to me. It is the book which is the title of this post. I was not immediately enthusiastic about reading the book but I was a little intrigued and I accepted his offer. I knew the name of Sowell only through some quotes I had seen on X but I usually liked them. When I started reading the book it immediately made a positive impression on me. The first sentence is ‘One of the curious things about political opinions is how often the same people line up on opposite sides of different issues’. Thus we meet exactly the theme I have just been talking about. I find Sowell’s style very pleasant. I also found myself confronted with unfamiliar and interesting ideas from the very beginning. When I started reading the book I stopped in a frame of mind which was similar to that when you stop eating chocolate. It tastes so good but you realise it makes sense to pause before eating more of it. I mentioned a correlation above and the aim of the book is to give some kind of causal explanation of it, even if that explanation cannot be complete. Here the notion of a ‘vision’ plays a key role and this was a new category for me. The short definition is ‘A vision is our sense of how the world works’. The author concentrates on two types of vision, the ‘constrained’ and the ‘unconstrained’. He emphasizes that it is not the purpose of the book to argue for one of these alternatives but instead to describe the differences between them. A first rough definition of this distinction is as follows. The constrained vision is that political action is limited by basic (negative) aspects of human nature. The unconstrained vision is that people can be made more altruistic by educating them, and that to an extent which is more or less unlimited.

The constrained vision emphasizes the value of tradition and ideas which have evolved within society. The unconstrained vision emphasizes the value of explicit reasoning. This might suggest that I, as a lover and proponent of rationality, should prefer the unconstrained vision. However that conclusion is too hasty. An important fact (one which has been emphasized by Hayek, someone who Sowell often quotes) is that when we make a decision we are often lacking much of the information necessary to make it in a purely rational way. Thus it makes sense to make use of other resources. There is a wisdom in society which has evolved. An example is the knowledge embodied in the development of the free market. If this kind of argument appears implausible it suffices to look at Darwinian evolution. In that context many things have evolved which are very beautiful and many which make use of remarkable mechanisms to attain goals. All this comes from the combination of mutation and selection. In a social context progress can come from the combination of chance developments and competition. The idea that those who know better should control the behaviour of others is limited by the fact that in many contexts those who know better still do not know enough in order to have a decisive advantage when making a decision. This does not conflict with the idea that in certain limited contexts experts can be expected to make better decisions. I would prefer to have my broken arm treated by a surgeon than by a carpenter. I now list a few other points from the first part of the book, which is a general discussion of the topic. Advocacy journalism is anathema to those with constrained vision since it is seen a a misuse of an entrusted role. Sincerity is so central to the unconstrained vision that it is not readily conceded to adversaries. The unconstrained vision tends to promote the interests of the young against those of the old.

The second part of the book is entitled ‘Applications’. The first chapter is concerned with the notion of equality. It starts with a discussion where I felt myself on very familiar ground. This discussion concerns the distinction between equality of process (which belongs to the constrained vision) and equality of result (which belongs to the unconstrained vision). The first of these was always what I regarded as the type of equality that I wanted to see in society. If I put this in mathematical terms, the first concept of equality is a property of the dynamics of the process while the second is an exercise is control theory. The chapter goes on to the idea that in the unconstrained vision the state of poorer people is seen as being influenced negatively by the activities of the rich. In the constrained vision the state of poorer people is influenced positively by processes in society to which some of the rich contribute decisively. The first variant is what I see being propagated by many German political parties, particularly those on the left. The second reminds me of statements of Rainer Zitelmann who has recently written a book related to this topic with the title ‘Zero
Sum Mindset’. (I have not read the book.) The second variant is also related to another theme which Zitelmann likes to talk about, namely envy as one of the worst diseases of society. Sowell explains the different attitudes of the two visions towards war and crime. In the constrained vision these negative phenomena lie in the nature of human beings and need no special explanation. Society should combat them directly, for instance by punishment of criminals. In the unconstrained vision these phenomena are caused by negative features of society and responsibility for crimes is shifted from the criminal to the society in which he was brought up and lives. In the constrained vision the rules embodied in laws have evolved from long experience and the judicial system should limit itself to implementing these rules. Attempting to implement perfect justice is not wise because it is a goal which is impossible to reach. Sowell strongly criticises the notion of social justice and quotes Hayek saying that it is a concept which ‘does not belong to the category of error but to that of nonsense’. In the book it is repeatedly emphasized that the difference between the constrained and unconstrained visions is not one of moral values but one of the belief in different types of causation of social and political phenomena. An example which occurs to me spontaneously is given by the following two statements. Hamas attacked Israel on the 7th of October because many Palestinians are driven by an evil ideology (constrained). Hamas attacked Israel on the 7th of October because of the way Israel had treated Gaza in previous years (unconstrained).

This is one of the most rewarding non-fiction books I have ever read. I already notice that when confronted with certain political events there is a resonance with ideas in the book. I expect that in the future I will have a better understanding of the actions of certain people, not in the sense of finding these actions more justified but in the sense of seeing how they fit into a wider pattern.