nunosempere.github.io/projects/mathematicians-under-the-nazis.md

Notes of Sanford L. Segal's Mathematician's Under the Nazis.

INTRO

The history of mathematics as an academic discipline in a time of intense political and ideological pressure.

There were psychologists like E.R. Jaensch and mathematicians like Ludwig Bieberbach who saw Mathematics as a fruitful area in which Nazi ideology could be exemplified and play a significant role.

“There is no attempt in this book to fit the narrative to a procrustean bed of underlying interpretation” 

When evil is present it is tempting to see things only in starkly contrasted black and white

Deutsche Mathematik: ethnic styles in the doing and teaching of mathematics (while not denying the validity of any particular fact)

Indeed, its supporters argued that the universal validity of mathematical fact is precisely what makes mathematics most suitable for discerning such different styles.



CHAPTER 1: WHY MATHEMATICS

Relation between extra-scientific culture and pure science.

Image of science proceeding in a vacuum.

Truth is necessary but not sufficient for real mathematics.

“They argued that exactly the apparent culture-free nature of mathematical abstraction and mathematical causality makes mathematics the ideal testing ground for theories about racially determined differences in intellectual attitudes”

Rassenseelekunde: theory of the racial soul.

The mode of intellectual discovery → feeling and attitude towards the world 

Associate the political argument with various philosophical differences within mathematics

Put succinctly, a Nazi argument promoted by Bieberbach was that because Jews thought differently and were suited to do mathematics differently, they could not be proper instructors of non-Jews. Indeed, their presence in the classroom caused a perversion of instruction. Thus, an elaborate intellectual rationale for the dismissal of Jews was established, discussed and defended.

Biological reductionism

Irrationalism

Philip Lenard (Deutsche Physik, Nobel): it was important for students to avoid studying too much mathematics. Mathematics was the “most subordinate intellectual discipline” because, under Jewish influence, it had lost its “feeling for natural scientific research”

matiz: p. 10

The Nazi typology cannot be dismissed out of hand

Other researchers, say in America, were attempting to recognize Jewish students by their writing style (A. A. Roback, pro-Jewish bias)

// In an expert, absence of evidence may be evidence of absence



CHAPTER 2: THE CRISIS IN MATHEMATICS

Uncomfortable axiomatic procedures: anomalies, paradoxes, but seemingly essential for ordinary analysis

Banach-Tarski

ZFC does not assume that, for every property there is a set of things satisfying that property

Type theory: every term has a type and operations are restricted to terms of a certain type.

Mathematiker oder Jongleur mit Definitionen?



Hilbert: Wir mussen wissen – wir werden wissen: Debemos / necesitamos saber, sabremos.

Comte: The chemical composition of stars will never be known

Some other guy: Temperature is unmeasurable



For the Nazi mathematician, the issue was the ability of mathematics to relate to real objects.

For Ludwig Bieberbach, the distinction was between defining pi as the ratio of the perimeter of a circumference to its diameter or, f.ex. As twice the smallest positive 0 of the cosine.

Jews attempt to create, true Germans discover.

Undeutsch

Categorization of method according to racial and national boundaries.

What is mathematics and what should it be? The conflict is deep, but precisely because it is so fundamental and so removed from everyday practice, many working mathematicians could simply get on with the job.

Intellect has no necessary connection with the ability to reason



CHAPTER 3: THE GERMAN ACADEMIC CRISIS

There was no Nazi party line on mathematics or science, and mathematicians who espoused similar cultural politics might disagree on how that politics affected mathematics.

The German professor was generally a conservative establishment figure who, under Weimar, had largely lost his establishment status.

Part of the academic culture was the notion of the professor as a prestigious state servant who had been declassed by the collapse of the empire.

“German academic freedom as defined by the government in 1898 did not include the right to be a politically active Social Democrat; thirty-five years later it did not include the right to be a Jew”.

Kulturstaat: The state would support learning in a widely humanistic sense; in return, the educated would become the trained civil servants and defenders of the state

Manifesto of the 93

„Fort also mit dem landfremden und abstoßenden Schlagwort »demokratisch«!”

“Away with the repugnant motto “democratic”, foreign to our country!” - Thomas Mann

Salonkommunist – Salonbolschewist

Emil Gumbel.

The majority of German academics simply “went along”, neither supporting nor condemning the regime.

Anti Semitism

Jews were represented in German universities far out of proportion to their numbers in the population

Promotion for Jews was the exception

Heinrich von Treitschke: Jews as the eternal foreigners within the people.

Elitist, genteel anti-Semitism: condemn the rabble rousing anti-Semitic agitator Herman Anwardt while understanding and condoning his position.

Refugee scientists from Germany were not always welcome in the USA.

Harvard began to restrict Jewish admissions in 1926, and Jewish quotas at Harvard, Princeton and Yale persisted until well after WWII.

The dismissal of the Jews meant more academic posts available for those not so tainted

Stennes-Putsch: by crushing the more violent elements of the S.A. (Stennes) and embracing the legality principle (z.B. forbidding street fights with the communists), the Nazis gained respectability in the eyes of conservative businessmen: Posible investigación histórica.

Hitler’s speech of Jan 27, 1932 to Düsseldorf industrialists.

“Nothing gives more credence to the correctness of our idea than the triumph of National Socialism at the University”, Adolf Hitler, July 8, 1930

Thus, it was natural that the parliamentary Weimar Republic was viewed as a corrupt and deceitful intrusion into the naturally Germanic order of things.

Jews, as outsiders, took posts away from real Germans

The mathematician, whose work basically involves self-generated mental constructions.

During the Nazi regime, mathematicians not only exhibited the same range of behaviours as their colleagues, but shared the same fundamental attitudes and reactions.

Phenomenon of initial enthusiasm followed by serious second thoughts, when it was “too” late



CHAPTER FOUR: THREE MATHEMATICAL CASE STUDIES

Politicization of Mathematics: Struggle among individuals, each purporting that they represented the true aims of the Nazi state in their actions

Once adherence to the state was established and its pariahs were eliminated, the system, with its overlapping bureaucracies, not only allowed conflict but was even said to encourage it. Mathematics was no exception.

For Doetsch, a law of nature was a description of a causal connection, not an explanation thereof 

“That the two men became somewhat close may perhaps be indicated by a letter written by Kamke to Süss on February 24, 1943, which ends “Mit freundlichem Grüss – Ihr” instead of the mandated “Heil Hitler”

niggard: avaro

Such symbolic appointments were presumably quite important. But absent these, once the regimes basic demand – elimination of all Jews and political undesirables from academic posts – was satisfied, no one seems to have cared much about academic mathematics, including even its role for military purposes.

A decree existed by which inadequate academic performance could be compensated for by political service.

What Hasse did not understand was that national socialism meant the politicization of all life according to its Weltanschauung. This effectively was his irremediable and unredeemable flaw in Weber’s eyes.

Infighting between shades of Nazi, each striving to prove itself the true participant in standard bearing for the “national will”.

In the Nazi bureaucracy, if an initial approach somewhere had failed, there were always other ways, other completing agencies to which one could turn.

Given the positions to fill in Göttingen, especially the institute directorship, which was arguably the most important mathematical position in the entire country, the political demands of the Nazi regime could be met in two ways: seek the best politically acceptable mathematicians or seek the best mathematically acceptable true Nazi believer. Weber seems to have oscillated between both antipodal options, leaning towards the second and finally adopting it.

“the completely judaized mathematics institute at Göttingen had been waiting for over two semesters for National Socialist leadership”.

Questions of academic value only rarely played a significant role in academic placement under the Nazis.

“Sometime after that September, Mohr apparently told a female friend he thought Hitler was a megalomaniac and the German situation hopeless. He was led to this by the overthrow of Mussolini and the illegal listening to English radio broadcasts. She denounced him. On may 17, 1944, the Gestapo arrested him”. He was sentenced to death.

“English public opinion seems to be badly prejudiced against Germany. The papers pick up every bit of bad news they can get hold of about Germany, but they hardly think it worthwhile to write about good things brought about by National Socialism. One would not expect enthusiasm, since the whole movement has a certain and very outspoken tendency against the allies because of the Versailles Treaty. But one would expect a certain detachment and aloofness, otherwise so strong in the English. It is not their business after all.”

The only condition was that all activity, all struggle, as carried on in the name of ideology.



CHAPTER 5: ACADEMIC MATHEMATICAL LIFE.

Hannah Arendt

Dictatorial regimes: Merely insisted on enforced obedience to the dictator or monarch

Totalitarianism: Revolt of the masses that ended with all individuals being portions of the state machinery, in which each individual’s connections to the others was mediated by the state.

Thus the life of the mathematician was replete with intrusions of non-academic nature intent on making each individual a subservient member of the state, without connections to others except as mediated by the state. Academics had always been civil servants; such intrusions were easy.

~Under the Nazis, academic life became a progression of increasing restrictions and continual evaluations. No respect for academic rank: it was enforced from the bottom

Gradually all able bodied men who were not over-age were drawn into the armed forces.

The conservative Catholic atmosphere in which the university of Munster dwelled kept it from excesses prior to Hitler’s ascension to power.

“One often learned of one’s own dismissal from the newspapers”

he had to go back to Germany because he could go back to Germany, to make room for those who could not do so.

Cleansed of Jews, the remaining faculty needed only worry about the students who, inspired by the “new order”, might consider them “calcified”.

One consequence for mathematics was that the number of female students, originally discouraged by the Nazi government, increased.

Decline in effective secondary school instruction ⇒ decline in student quality

Bohn book burning on May 10.

Incidentally, anyone physically unable to raise his or her right arm was instru8cted to use the left if possible.

Three protected exceptions: WW1 veterans who had served at the front / Those who had been in civil service since 1/09/1914 continuously / Those who had lost a father or son in WW1.

Führerprinzip.

Rektor, and thus, in the Nazi interpretation, effectively Führer of the University.

“What a benefit, to find a pure German in such a hopelessly judaized discipline”

The Nazi believed in a sort of scientific reductionism, but for them the basic science was biology.

July 1933: All civil servants were ordered while on the job or inside their place of work to greet one another with the Hitler salute.

All servants of the state had to take a civil service oath. Under the Nazis, this was transformed into a personal oath to Adolf Hitler.

Dozentenschaft: Zumo de docente

All organizations were replaced by National Socialist ones.

Assistent – Ordinarien – Privatdozent – Extraordinarius.

People find their Nazi heart to climb in their Beruf.

No discussion, written or oral, was permitted about the reform of the Reich Government.

Inner emigration: Retreat into one’s work, holding all opinions to oneself.

The Nazi state was not satisfied with just having easily manipulable, atomized and terrorized individuals; it needed to weld them into a unified whole whose only bonds to each other were those binding them to the state.

An individual cannot answer to his Volk that mathematics is done because of the yearning for knowledge.

Pure mathematics (in the narrowest sense) has meaning only as a means of education to a formal character building that is consciously employed for service tot the entire people.

A characteristically German spiritual attitude to the doing of maths.

In March 36, all activity with League of Nations’ organizations was to cease.

“Is it really German to do something for its own sake?”

“Documentation cannot provide the “feeling” of a period by itself; while the personal narrative is subject to the strictures of subjectivity and the plague of an idiosyncratic point of view, it also provides the nuance of emotive detail that can only be inferred from documents. Thus, the two kinds of sources, when available, need to be used as complimentary to one another, providing symbiotically a clearer picture than either alone would.”

As Barbara Marshall has shown, in those years of late Weimar, Münster’s Catholic conservatism prevented it from becoming the hotbed of anti-Semitism and Nazi leaning nationalism that the town of Göttingen was.

While the onset of the war does not seem to have been greeted enthusiastically, the early German victories quickly changed this view.

There were deficits in instruction created by the many extracientific demands on the students.

Universities are depleted

“save qui peut”

Hitler, Mein Kampf: “A man of little scientific education but physically healthy, with a good, firm character, imbued with the joy of determination and willpower is more valuable for the national community than a clever weakling.”

[Thomsen] defends the fact that it takes perhaps decades for theoretical mathematics to find a practical utility by citing the length of time it took racial theory to develop, and “what is right for racial theory, for the benefit of our state, is also right for theoretical natural science and mathematics”.

“We need today a student youth that has ever so little fear of a murderous weapon in wartime as of the monster of a five-fold integral with singularities in the integrand”. [Sadly not taken seriously by the Nazi bureaucracy]

Secondary and elementary mathematics were corrupted: militarized, racialized and politicized.

To teach National Socialistially.

What does a bad achievement by an upper level student in Latin and Mathematics trouble him if he is loyal to his SA leader and comrades.



CHAPTER SIX: MATHEMATICAL INSTITUTIONS. (How mathematical journals were affected by the political pressures of Nazi Germany and how they responded)

The three directors of the three most important mathematical journals were intent on upholding mathematics and protecting it from chicanery and political inference

Distinction between a fact and the “style” that led to it

Law of August 17, 1938: Jews were only allowed to have distinctly Jewish first names, and those who did not had to append the name Israel if male and Sara if female.

The Lachman paper incident shows how difficult it was in the Nazi atmosphere for decent people to uphold various professional attitudes they thought appropriate, not just because of the pressure of the state and its ideology, but also because of the mephitic atmosphere brought ordinary decent actions into conflict with one another.

“It itself, I have nothing against German citizenship, however, at this moment, since Germany has occupied my homeland, I would not gladly give up my previous neutrality and throw myself in a certain measure publicly on the German side”

Bieberbach maintained that only through the recognition of the differences between different peoples could proper cooperation take place, and that mathematical style was racially determined (bluterfüllt)

Many people at first saw Hitler’s ascension to power as a means of reestablishing German honour and respect at a time when they felt Germany was still unfairly disdained and still unfairly placed under a burden of guilt, both financial and moral, for WW1.

Their rhetoric is not mere lip service to the reigning ideology. Many people believed the things that wre said. The early Nazi government had reawakened in many a German national pride that was felt to have been demeaned since the end of the war. Genuine feeling

Teleology must step into the place of pure logic.

There was much appeal to educating intuitive understanding of three dimensional space and the motion of objects in it.

November 15, 1938: The ministry of education issued the so called academy decree (Akademieerlass), which among other things decreed the Führerprinzip for scientific academics.

Führerprinzip: Cada Führer es instituido por un Führer superior y tiene poder y responsabilidad absoluta sobre sus subordinados. Pero no puede haber una cadena infinita de Führers: Adolfo Hitler es el Führer en última instancia superior.

Hitler’s “new Germany” at its outset held forth to many the promise of a new and justifiable respect for Germany and Germans.

Let it be mentioned that with the scientific emigrants into enemy foreign countries we have delivered to the opposition a not-insignificant potential gain.

Americans were much better at coordinating the sciences with the war effort.

“[…] This was possible only if the ideological fundamentals of the regime were not transgressed. It would have been in vain and only have brought suspicion on one to defend Jews, but it was possible to save a distinguished “Aryan” mathematician who was “Jewish-related” from being sent to labour camp because of the importance of his scientific activity. However, Süss was only enabled to behave in this way because he had established a reputation of being, in Nazi jargon, politisch zuverlässig”.

Though many Germans knew that Germany was irrevocably collapsing, the bureaucracy behaved bureaucratically, simply carrying on in the face of increased difficulty – the “alas, your letter took five weeks to reach me” sort of thing.

Defeatism was a capital offence.

The total collapse and ensuing occupation were not part of the German vision.

~One would have thought that in a militarising and then warring Germany, appropriate applied mathematics, at least, might have nurtured. But this did not happen until late in the war, with the intervention of Süss.

An academic-military-industrial-political complex was forged during WW2 in GB and the USA. No such complex was built in Germany. This failure provably stemmed from the deep suspicion of academe present in National Socialism, and its emphasis on emotion and intuition as opposed to intellect and abstraction. Añadir a causas de la derrota alemana en WW2.

One of the two leading aeronautical experts, Theodor von Karman, whose mother was a Hungarian Jew, was forced into emigration and became the chief scientific advisor to the US Army Air Force.

Mathematics in the Concentration Camps.

Mathematical Slave Labour

By the end of July 1944, not only had there been military computations completed, but also a set of mathematical tables, and translations of Russian mathematical papers.

The numerical solutions of parial differential equations connected to ballistics investigations under an industrial contract, as well as the computation of the emphemerides of Mars and Jupiter.

March 25, Heinrich Himmler wrote to a subordinate: “Among the Jews whom we have now received from Hungary as well as also among out concentration camp prisoners without doubt are a whole lot of physicists, chemists and other scientists”.

A scientific research establishment in which the disciplinary knowledge of this people could be applied to stressful and time consuming computation of formulas.

“Fischer claims in his memories that he realized political opponents and prisoners could not be expected to have the requisite inner drive for scientific activity on behalf of the Third Reich”.

It was established that then, if certain relief were granted, e.g. permission to work on ordinary clothing, the investigations were immediately more reliable.

Mathematical slave labour.

Originally the declared purpose of the camps was to arrest people for purposes of reeducation or for prophylactic security reasons. The first concentration camp opened, Dachau, was publicly celebrated. Conditions were never particularly good, however, and soon terror took over, as did the realization that the camps were an excellent source of slave labour. As slaves, the inmates could be worked to death and in general treated as human material, as for example, for the various medical experiments carried out in the camps. Thus the original purposes of indefinite preventive “detention” and “reeducation” became quickly subsumed by the better known and more permanent ones of terror and economic / scientific rationality.

The checking of computational work was not difficult. Just give the same task to two separate groups & then compare.

Mathematical intuitionism.

Importance of geometric intuition in mathematics, and mathematics should be more than just the surety of proof. 

CHAPTER SEVEN: LUDWIG BIEBERBACH AND “DEUTSCHE MATHEMATIK”

Talented mathematician & leading proponent among mathematicians of Nazi ideology

“A man’s reputation has a very considerable coefficient of inertia”

Since the end of WW1, largely through the influence of the French, the Germans had been barred from international congresses.

After the ban was lifted, Germans at first chose not to attend: The first congress was sponsored by the Union Mathematique Internationale, an organization from which Germans had been excluded / The congress was to take place in Ledrousse, Austro-hungarian befor 1914, Italian after 1918.

“Suppose we had to fight a war to rearm Germany, unite with Austria, liberate the Saar and the German part of Czechoslovakia. Such a war would have cost us half a million young men. But everybody would have admired our victorious leader. Now, Hitler has sacrificed half a million Jews and has achieved great things for Germany. I hope some day you will be recompensed but I am still grateful to Hitler.”

There could and did develop a conservative right wing opposition to Hitler, which, e.g. prevented the Habilitation of a “deserving National Socialist” because his Habilitationsschrift was inadequate.

Geist als Wiedersacher der Seele: Intellect as adversary of the Soul.

There can be no self sufficient mathematical kingdom independent of human activity and intuition, therefore also none independent of the styles in which human racial membership expresses itself.

Generally I am of the opinion that the whole dispute over the foundations of mathematics is a dispute of contrary psychological types, therefore in the first place a dispute between races. The rise of intuitionism seems to me a corroboration of this interpretation.

An activity of an idiosyncratically German nature needs no further justification.

A new Journal: Deutsche Mathematik.

Thus mathematics is a mirror of the races, and proves the presence of racial qualities in the intellectual domain with mathematical, thus incontrovertible certainty

An international version: There are different incompatible racial styles, all of which produce mathematical truth.

A racial struggle did not imply a valuation of races. Each style was valid for its own people. “Thus German science relinquishes Jacobi and leaves it to Judaism to see in him one of its greatest sons”.

Philip Lenard: Trash mathematics!

[To rewrite history through taxonomy]

Copernicus was a German, not a Pole! 

As soon as, however, the question arises whether the theorem concerned might be important, interesting or highly relevant, then one will hear the most various judgments, Opinion about it largely depends on the [Janeschian ≈ racial] type of the judge.

“I like mathematics as something organic”: una de las tipologías

For both the Heidelberg and Könisberg groups, the concept of force was particularly Germanic, and its elimination materialistic, whether French or Jewish. For both, religious grounding was Germanic. For both, Germanic attitudes were unitary and non-Germanic attitudes divisive, e.g. the cartesian mind body/dualism

“Only a people that, the danger known, allows the number of progeny to increase with the racial value of the parents can turn aside this danger of the erradication of fitness”

Students who wish to overthrow the status quo ante are a commonplace in universities at almost every time and place. In the Nazi context, however, the political establishment was on their side, and had provided a revolutionary ideology that had transformed the state/

The brilliant young mathematician and dedicated Nazi Oswald Teichmüller.

Nürnberg law: forbade sexual relations between Jews and non-Jews.

Even when dealing with intrinsically non-ideological material, such as book reviews, Weltanschauung could find its way in.

World Ice theory

Attack on acausality in physics and on those who want to introduce such acausal notions in biology (oh the heresy)

Science is not objective, and natural scientists should better understand this.

After WW2, Bieberbach apparently mantained to an Allied interrogator (who actually happened to know some mathematics) the validity of his views on ethnic personality types and mathematics – he was a true believer to this extent. […] On 1981, the ninety-five year old Bieberbach mantained that his distinction between a Jewish style and an Aryan style was valid but implied no comparative valuation of the styles.



CHAPTER 8: GERMANS AND JEWS.

Not only was the Nazi ideolog centered on nonrational processes as ideal, not only was it involved in a biologic reductionism of all learning to a fundament of racial genetics, but the general lay opinion of mathematics at that time was none too favorable.

The initial Nazi view of academia was a mixture of contempt and suspicion. 

The speed of the nazification of the universities improved that view.

Most mathematicians simply wanted to be left alone with their mathematics.

The only reason professional resistance could be at all useful was that the regime did not care what happened in the academic world, provided that Jews and active troublemakers were expelled from their positions.

Someone like Erich Hecke could mantain his distinguished position despite his apparent refusal to say Heil Hitler.

“The sole Roman contribution to mathematics was teh negative one of murdering Archimedes”.

Process of denazification.

The argument made to me by Christoph Mass that Blaschke’s real concern was with what the Nazi regime could do for him rather than what he could do for them neglects the fact that a certain quid pro quo was involved. Obtaining what Blaschke wanted in prestige and position from the Nazis involved aiding them.

The nickname “Mussolinetto” [Musolinito]

Nazi political correctness

“Tietz was somewhat afraid of Zachenhaus, who always wore the insignia of a Nazi organization, but Hecke calmed him by saying that Zachenhaus was their trusted agent who behaved in a politically correct manner so as to protect them”

Teichmüller: The family was poor, but the son, with perhaps the sense of moving up in social class as a student, always listed his father’s occupation as factory owner rather than employee.

Party activity seems to have provided friends for Teichmüller.

The letter, the letter.

Jews unsuitable to teach German youth.

The Nazi education ministry dismissed Jews under the April 7, 1933 law because they were Jews, not because of any intellectual rationale about Jews being unfit teachers for Germans.

“You are a reactionary burgeois and don’t comprehend the Führer’s ideas”

Britain guaranteed the Polish borders on August 25, 1939, something it had refused to do fourteen years earlier at Locarno: Trabajo de historia.

Nothing necessarily connects a person’s brilliance in one area with a particular insight into politics.

Heidegger.

Ernst Witt: As a student in Göttingen, he had the naiveté to appear at Emmy Noether’s private seminar in her home in his SA Uniform // [someone] complained that he, despite being a party member, was “politically colorless” and had yet to understand that mathematics and natural science were also racially connected; he mistakenly thought them international. // He didn’t pay overly much attention to other’s opinions and so, for example, wore braids in his hair for a while [en la Alemania nazi]

Richard Courant, jewish mathematician: At the time of the planned Nazi boycott of Jewish stores on April 1, 1933, Courant thought the whole trouble was Einstain’s fault, and a recent speech of Hitler had made “a quite positive impression on him”

Landau: His father Leopold was both a nationalist patriot and someone politically engaged in Jewish issues – not a contradiction in the first decade of the century.

There is a story that reveals both Landau’s cynicism and his understimation of the Nazis: Once, on a visit in 1932, Fritzt Rathonau told Landau that he didn’t know whether the Nazis were going to win, but if they did, he had heard they planned to construct a concentration camp for Jews on the Lüneburg heath. Landau’s response is supposed to have been “In that case I should immediately reserve for myself a room with a balcony with southern exposure”.

Teichmüller recognized Landau’s genius and would have gladly studied advanced material with him (and Landau would have gladly taught him).

The yound Hausdorf was pirmarily interested in literature and philosophy and moved in those circles.

Ernst Pechl is a good example of the young mathematician who, in order to begin a career, was orced into at least a nominal political stance in which he apparently did not believe.

Hans Peterson had the courage to marry the woman he loved, despite her “tainted” status, to hide it succesfully, and to convincingly go through the motions of being an ardent nazi even when he was not.

It is hard not to respect Kähler as principled while abhorring his principles.

Süss’s efforts were only succesfull because the meaningful political ideological powers found mathematics unimportant.

It was easy for many to see the Nazi movement, for all its “plebeian excess”, aimed primarily at reestablishing Germany’s place among the nations.

The notion that training as a mathematician permits better analytical and logical thinking is a common belief.