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Notes and Advice 1
Problems and Answers



A couple of years ago, I was taking a Lufthansa flight to Germany. Because I am quite a big guy, I had a seat near the emergency exit where I could stretch my feet out. However, after a while the space was taken away by a stewardess, who during a longer-lasting turbulence occupied her seat facing me. I was reading an early 19th-century math book. She noticed the age of the book and asked me: "What are you reading?" I showed her. Her face displayed disgust and she replied: "Mein lieber Gott (my beloved God), I hasse Mathematik (I hate mathematics). I never could see the use of it." And the lady beside me agreed by nodding emphatically.

I answered: "You probably cannot equal the amount of aversion I felt against mathematics for a long time. But let me ask you a question: Have you ever watched the Rocky movies?" She shook her head disdainfully and replied: "Solch einen Mist schaue ich nicht an (I do not watch shit like that)."

I retaliated: "As flies can tell you, there might be something good even in shit. And as I can tell you, there is something good in Rocky I. Let me explain: The trainer of Rocky made him chase a chicken once. Rocky resisted because he did not see what that would do for his boxing. The trainer explained that he would acquire speed and fast reactions.Although chicken-chasing does not seem to have to do anything with anything, it can be very useful for boxing--and maybe even other activities, for which you need speed and good reactions.

Mathematics is like chicken-chasing. Although it does not seem to be useful for anything (at least not for the so-called general public), it is very useful in an indirect way. I would not have spent so much time and money on it, if it were not."

Let me begin the discussion with some advantages of mathematics, which I experienced myself, then go into a historical excursion, and finally conclude with another set of advantages. I want to stress though that you do not need to learn mathematics before you can succeed in my sessions. The training, which I received and which influenced my reasoning, will just seep over to you. It will mobilize your own reasoning skills.

The first question you may have is probably the following: "What has that got to do with the problems I have?" Let me try and give you a preliminary answer: "Psychiatrists. psychologists and counselors diagnose(d) their patients according to the DSM's various editions, which differ from each other, but whose common denominator is exhibited by their title: Diagnostic and Statistical Manual of Mental Disorders. Let me concentrate on the last two words now. Our mind is composed of thoughts and feelings. After many years, psychologists seem at least to agree that thoughts influence emotions--and emotions also thoughts. Now, if one part is disordered, the other one will be, too--and then we have a so-called vicious circle. If, however, you can insert sound reasoning into your thoughts, this will change the circle from vicious to beneficial. Got it?"

Let me give you now three pages of certificates of supervision, which concentrate on low-achieving math students:




There is a book by Professor Morris Kline (see BIBLIOGRAPHY) called Why Johnny Can't Add: The Failure of the New Math. But the above certificate shows that I could profit from modern mathematics. The goal is not to do away with the new mathematics, but to add something to it. Humankind has a tendency to erase things, not to add alternatives. And so it was within mathematics: An older mathematics was erased during the twentieth century and replaced by modern mathematics. This older mathematics was much closer to our everyday reasoning and, therefore, could shape and improve it. The result was the rise of the enlightenment, which created the USA. What I had essentially done in my writings for students was rediscovering this older way of reasoning myself. At this time, however, I still was a firm believer in the New Mathematics, which did not hinder me, however, to add a new option. I did not read any old mathematics books back then. I thought they were inferior to the new ones. I believed in linear progress--my way of proceeding was just a way to help students go from one stage to the next. (Some recent European university professors believe that a good grasp of certain subjects can only be achieved by teaching it in a historical order. See Hairer and Wanner's Analysis by Its History, BIBLIOGRAPHY.) But when I started reading old texts, I noticed that they were in many ways superior to modern texts; they just knew much more about certain subjects than we do today. And eminent mathematicians of the time wrote the school books, which made them much better than today's books:



I have scanned in the first pages of Lacroix's book in order to show you the style of reasoning (see LACROIX). There is no need for the modern, abstract, counterintuitive approach to achieve results in mathematics. Normal human reasoning is validated. One of the bad things of modern mathematics is that this everyday-reasoning was invalidated and frowned upon because it may lead to contradictions. But this is as silly as to say that you should not walk because there is the danger that you can fall.

This happened shortly before the end of the 19th century. Formal logic was discovered by the German Frege. Informal logic was replaced by formal logic--at least officially. And since there was an official neglect of informal logic (today's "critical thinking"), mathematics ceased to be enlightening with the result of two nasty World Wars. The destruction of informal reasoning on a mass basis lies at the heart of the "mass madness" behind these two wars, the second being much madder than the first one--on the German side, where the destruction of reasoning had taken place first.

The older mathematics was a fortification of the human mind, which was achieved through years of public education. Baron von Vega, an Austrian artillery colonel and mathematician reports that even lower ranks of his army read mathematics books between battles. (By the way, von Vega came from very poor circumstances, could not even speak German as a child--but made it in the "highly authoritarian and suppressive" Austrian monarchy.) Now, if you pour self-doubt into the souls of impressive children through years of public education, you should not wonder if they look for certainty somewhere else. Together with the erosion of any reasoning standards, you then might believe in the bullshit Hitler was spewing forth.


The advent of Frege's formal logic and Hilbert's formal approach to mathematics at the end of the 19th century is antedated by Martin Ohm's formal approach to school mathematics at the beginning of the 19th century. Ohm was a school teacher and later a professor and very influential. His brother was the well-known physicist Georg Simon Ohm.


The above book, written for teachers and 10-14 year-olds, is meant as introduction to his elementary mathematics, which was condensed into the following book and which had at least five editions:


In his first book he says:

"Because my books are more scientific and logical than others, they also are more abstract." He then continues and admonishes the reader to be patient and not to use preconceived expectations because he has to learn something quite new; however, that his newness would pay off because it would lead to a love of mathematics. In order to realize his program, he instructs teachers to tell students to forget and doubt everything they have learnt before in their mathematical training. He then anticipates the arguments of the New Math movement of many years later, that mathematics is not a conglomeration of statements, which the student just has to learn by heart, but that it is logically interconnected system of statements.

This is, of course, a straw-man argument: You attack the worst kind of practice--which is still around in spite of all the New Math--in order to advance your agenda. Morris Kline actually says that now it is the New Math that is stuffed down students throats! Formal systems are actually something really fun and one should not treat them unthinkingly "in the old way." But, certainly, neither Ohm nor most teachers of today can really sell them to students. Just compare the practice to an excellent account of formal systems in the following book:


One should be really highly cautious with crusades, which burn down alternatives: Euler and Gauss did not need Ohm to do mathematics, of which Ohm did only little himself. And then formal reasoning is under attack from within mathematics even today after its "complete" victory:


One may now object that Chaitin is highly controversial and not mainstream. But there was a famous mainstream mathematician, who had the following assessment of one of Leonard Euler's (1707-1783) works:



Let us now look once more at how Ohm would teachers like to proceed:

Paragraph 11: Beginning of instruction. Development of the notion of the sum.

TEACHER: Here on this table, we imagine a heap of pennies, we do not know how many; and another stack, of which we do not know either how many pennies it contains. So, here we then have two numbers, of which we do not know anything yet. But since we want to talk about them anyway, we designate the first one by p and the other one by q.  .... What do we think of if we use q?

STUDENT: The number of pennies in the second heap.

TEACHER: So, you think that q designates pennies?

STUDENT: No, but a number; the number that indicates how many pennies are in the second stack.

TEACHER: Good. But when we push both heaps together, how many pennies are then in the bigger heap?

STUDENT: That we cannot know.

TEACHER: I think we can. Are there not as many pennies in the new heap as were in the two initial ones taken together?

STUDENT: Yes, sure.

TEACHER: Well; more we do not want to know.......We want to designate the number of pennies in the third heap by a. What is a? Is it a number?


TEACHER: No! It is a mere sign, which shall designate a number, as much as p and q do. When we say the number a, what do we mean by this?

STUDENT: The number, which has been designated by a.

If this is not the NEW MATH in spirit, then I don't what is--and that more than one hundred years before it came to the USA. Part of the mathematical community of Ohm's era resisted at the time. But they lost. More about this can be found in the following book:


On page 7, Bekemeier writes:

"General principles of Ohm's--I do not mean technical details ones--have become so much part of the mathematical practice at the end of the 19th century (as they still are) that it is difficult to imagine the difficulties, which Ohm had to fight when he formulated and applied these radical principles at the beginning of the 19th century."

The problem with revolutions, which kill previous practices, is that certain solutions become impossible. Another way to express it is that eradication of alternatives kills imagination. There are mathematical analogies: If you ask a first- or second grader the question "10 and how much is 3?," you will get baffled silence because they do not know about negative numbers. The eradication of informal mathematics makes the solution to solve the current mathematical crisis very hard.

If one compares contemporary mathematical texts from England and France with Martin Ohm's books, then he is unique in his approach. So, the question is whether his ideals were not somehow suggested by societal circumstances. Ohm mainly lived and taught in Prussia. The German philosopher Schelling says the following of the town the Prussian regents lived in: "Potsdam is nothing but a huge guardhouse, a prison from which there is no way out. All bridges are guarded so as to make it a virtual island. It was here that soldiers kept under constant control were trained to become the heroes of the Seven Years' War. What cruelties were the price of that glory! How much torture, despair, and disaster have those silent stone houses witnessed!" And the German poet Lessing called Prussia a "prison state."

What about Prussian universities of the time? Emil Ludwig describes the situation thus: "The enthusiasm of some university students for the Polish and Greek fight for liberation...was hastily quelled by professors, lest the movement grow to a desire for German liberties. When Berlin University once celebrated old Goethe's birthday, a royal receipt came down, warning the students against too much zeal. Members of fraternities which had dared to follow the French custom of erecting 'trees of liberty' were denied the necessary licenses to establish themselves as doctors or lawyers; thirty-nine of them were condemned to death, though the penalties were later commuted to long prison terms. But of the universities not one had lifted a finger in their defense." It is very clear that a mathematics that could fortify the human spirit and mind (like Lacroix's works) had to be eradicated and replaced by a new "law and order" mathematics, which invalidated human informal reasoning.

There was a German mathematician of this period, who stood for the development of the human spirit and who has a connection to


Brooklyn Bridge, without whom this monument would not have been built.

Let me introduce this mathematician to you. It is Dr. Epahraim Salomon Unger, a man of Jewish faith, who--when studying the Talmud (especially the Mishnah)--discovered some mathematical discussions there and became interested in mathematics. He first taught mathematics at the University of Erfurt and then privately to officers of the French garrison and members of the topographic institute. In addition, he prepared officers of the regiment in Erfurt for acceptance into the military academy. In 1820, he founded a private institute for mathematical instruction, which attracted many students, even from abroad. In 1844, the school was converted into a state institution. Because Unger was Jewish, he was removed as the head of this institution. One of his students, was a certain Johann August Roebling, who had flunked his courses in Latin and Religion at the high school and had to leave it. Unger accepted him into his instution a few years after its beginning. Thus prepared. Roebling could continue his studies as an engineer and architect in Berlin. After emigrating to the United States in 1831, he did not find a job in his profession, but a fellow student from Unger's institute helped him find one in canal operations. Roebling began working on the planning of Brookyn Bridge in 1865. His son continued his work after the father's death. After the death of his son in 1870, the son's wife took over and finished the bridge in 1883.

Roebling1.jpgJohn August Roebling


The Roebling family was an unhappy one: Hardworking, competent, and extremely cruel to themselves. The father had his foot crushed and refused to see a doctor, dying from tetanus two weeks later. He was very cruel to his son. There certainly is a connection to their Prussian upbringing.

Unger's books are excellent examples of informal mathematics. I have an almost complete collection of his books, which are very rare. Here is the title page of the first four volumes of his books on "Mathematical Analysis:"


There is a slip of paper inside several of my books


The stamps on the title page of Unger's books show that it was part of three different military libraries. The slip says that the previous owner of the book has bought it in the fall of 1944 at Buber's in Potsdam, a shop selling old books and artwork. It was a few months later, and the Russians had moved into Berlin and Potsdam. Adolf Hitler and Eva Braun had committed suicide. Later on, the Russians established a communist dictatorship in East Germany, which called itself the German Democratic Republic and which ruled in the old Prussian spirit of slavish obedience and prison state. The soldiers wore the old German uniforms, goose-stepping as before--with the modification of having Russian helmets and AK-47 assault rifles--shooting their own citizens who wanted to cross the heavily fortified border into the free Federal Republic of Germany. Unger's portrait at his former school had even survived Hitler (although Unger was Jewish). Now it was destroyed by the German Democratic Republic because it was part of the "bourgeois past." And then this prison state collapsed, too.

It looks as if authoritarian hierarchical systems in society and mathematics are not working. However, we have to say that they look impressive--much more so than their democratic and "multicultural" counterparts.

The following Prussian professor, who gave formal mathematics its definite form at the end of the 19th and beginning of the 20th century, is really impressive:


Unger's work compares to Hilbert's as a fisherboat does to a battleship. But two swiftboats  (Goedel and Turing) torpedoed the battleship several decades later. However it is still drifting around dominating the seas. My opinion is: They should all stay afloat and cease hostilities and leave their fate to the passage of time. 

The Americans, who had more to do and study mathematics in Germany during the nineteenth century, came to Germany after the battleship had sunk the fisherboats. Page 48 of the above book reads:

"Klein's reputation drew students to Goettingen from all over the world, but particularly from the United States. The Bulletin of the newly founded American Mathematical Society regularly listed the courses of lectures to be given in Goettingen, and at one time the Americans at the University were sufficient in number and wealth to have their own letterhead: The American Colony of Goettingen."

Hilbert was in Goettingen at that time, too, and impressed the American students even more than Klein. That is how formal reasoning came to the USA and destroyed the informal French tradition there (compare Lacroix's book above and the following two below):



Just let us look at the last book: It contains much more material than is taught today--and this on far fewer pages: Plane Geometry (150 pages), Spatial and Spherical Geometry (100 pages), Trigonometry (70 pages), and Spherical Trigonometry (60 pages).

This puts all current methods to measure mathematical knowlegde and achievement in perspective. If we compare nationally and internationally today, the situation looks roughly like this:


No amount of national or international rank ordering (indicated in the yellow square) will give a true picture. The wiggles within the square are just  variations in an overall downhill trend. This assessment is reinforced by examining French books written for university students, such as Serret's books on calculus, which were published in 1868 for beginning calculus students and then even translated into German. There is no way that modern undergraduate--or even graduate mathematics students--will be able to follow them. The book was then rewritten in the new spirit by German professors until it had almost nothing to do with the French original.

Now, the procedure in schools is to make math more and more "easy" for students: They do not need to calculate any more. They do not need to understand the proofs. They just need to be able to solve certain types of problems. And students are disgusted because the pretty clever individuals, who they are, are forced to do stupid things. So, the solution might be to actually present them with the material that has been omitted to make mathematics easier for them. And they are so proud and happy to suddently understand what they have seen so far as a strange and unpopular activity! You fortify their minds instead of crapping into them.

Let me fulfill my promise now and give you a list of advantages of good mathematics:

  • Informal reasoning (another word for "critical thinking") will transfer over to everyday life, in the beginning a little--and increasingly more--if the benefits are experienced.
  • Mathematics is a gold standard of sound reasoning.
  • Sometimes, direct examples from it can expose a claim's absurdity. My history teacher once said: "Hitler does not seem to have understood mathematics when he said YOU ARE NOTHING, YOUR PEOPLE IS EVERYTHING. I learnt in mathematics that 60 million times zero is still zero."
  • It is learnt that many tiny well-reasoned steps can lead to an impressive outcome. Grandiose claims become suspicious.
  • Good solid reasoning and problem-solving skills will help us improve everything we are working or even playing at.
  • Sound reasoning can create predictions of how events will be going years from now. One person, whom I greatly admire (Emil Ludwig), wrote in 1940--five years before World War II--ended: "We can assume that, at the end of the war, Stalin will still be in power, Mussolini only if he remains neutral, and Hitler in no case."
  • Above all, we learn that humans are great--very different from what people want us to believe who want to exploit others for their own ends.