PAI Private Arithmetics Institute       

Natural Numbers, Recursion, Geometry, Symmetry, Axioms, Logic, Metalanguage and more 


It's a long way to Kazan

Personal reflections of the author:

In October 1960 I started studying physics at LMU Ludwig-Maximilians-Universität München. Aiming for  its theoretical branch mathematics, logics and philosophy of science became other major fields of interest. The welcome for the freshmen in their first semester was entertained by a certain Dr. Bauer, who would also later give the lecture '1353 Higher Mathematics IA' , supposedly Privatdozent (private  lecturer) Dr. Bauer, whom, however I cannot identify in the university calendars of that period.

Dr. Bauer would provide us with the basics of being a student in the department of natural sciences (where strangely enough the  institute of mathematics was part of). After he was finished with the mandatory lectures for the first semester students of physics and mathematics the other present faculty members were asked to give a short survey on their lectures that could be relevant to us. At the very end it was Professor Perron's turn. He was introduced by his titles Geheimrat (privy counsellor) Professor Oskar Perron. Born in 1880 and emeritus since 1951 he was an impressive person with a white beard and  wearing a black frock coat. The doyen of München mathematicians told us that he would lecture on non-Euclidean elementary geometry. This sounded strange and -  interesting. No previous knowledge besides high school mathematics would be necessary for that class. Franz Geiger, a graduate from my high school and a math freak too, we exchanged looks: that's something for us. And so we attended Perron's weekly lecture 1369, Monday and Thursday 17:00 -18:00 , together with about 10 more students, to have a new look at geometry, that had been put forward by the Russian mathematician Nikolai Ivanovich Lobachevsky in the early 19th century.

It was there, where I was introduced to the axiomatic method on a scientific level. Strangely enough, Perron was no friend of new math, meaning that he did neither like set theory nor did he think that it was necessary or any good. He left his legacy at the final presentation of a paper at the age of 93 with a finishing statement: "ceterum censeo novam quam vocant mathematicam esse delendam, classicam autem mathematicam, ut Gaussianam, Dirichletianam, Hilbertianam etc. esse amandam et numquam delendam". By the way, some years later I arrived at mathematics without set theory myself, however, not as Perron who actually believed in some higher mathematical intuition ("we all know a priori what points or numbers are, let's do some precise reasoning about them") but rather on a logical basis with 100 % precise formal languages. It seems absurd that the person who introduced me to the axiomatic method of mathematics contradicted himself from the very beginning. On page 11 of his book he even gives a 'definition' of a 'point': "Ein Punkt ist genau das, was der intelligente, aber harmlose, unverbildete Leser sich darunter vorstellt" - "A point is exactly that what an intelligent, but harmless, unpretentious reader imagines it to be". Compare this to the first sentence in Euclid's Elements "A point is that which has no parts, or which has no magnitude"  - no progress in 2.300 years!

After two weeks of Perron's lecture  I was perfectly familiar with the axiomatic method. The  unwelcome side effect, however, was that I put the newly acquired scrutiny to the other lectures, e.g. Dr. Bauer's lecture '1353 Higher Mathematics IA' , which he claimed to put forward on a set-theoretical and purely axiomatic basis. At first I was very much impressed, but even then I realized that it is problematic to characterize the real numbers as an Archimedes-ordered field plus the so-called 'interval-nesting' axiom, which (like all other 'transcendency' methods) necessitates second-order logic. Dr. Bauer did not really answer my other question, how all of a sudden natural numbers were introduced in addition to real numbers: out of nowhere there were series of real numbers that used natural numbers as so-called indices. What had this indices to do with the real numbers 'zero' and 'one' of the ordered field ? In hindsight I regret that I did sort of accept some explanation of Dr. Bauer, authority won over a novice's concerns. But it kept hunting me for the rest of my life until the time when I thought that I had found at least the correct language to express the problems correctly.  

It was of great influence in my education to attend lectures and seminar of Professor Wolfgang  Stegmüller who then held the chair of philosophy of science at LMU Ludwig-Maximilians-Universität München. At that time Stegmüller appeared to be close to the Vienna Circle (Wiener Kreis) of Logical Empirism.

Many years later, in September of the year of 2017 I paid a visit to the city of Kazan (capital of the Republic of Tatarstan in Russia) while on a cruising tour on the Volga river. Sitting in the sightseeing bus I suddenly remembered about Lobachevsky having taught at the University of Kazan. I got off the bus immediately and together with my wife, I walked over to his monument that is located right across the main building of Kazan Federal University. I paid my tribute to the critical and creative original thinker, who did not get full credit for a long time, although still doing much better than the other inventor, the Hungarian Janos Bolyai. At least Gauss understood the importance of Lobachevsky's work and arranged for Lobachevsky's membership at the renowned 'Akademie der Wissenschaften zu Göttingen'. Poor Bolyai never really appeared to full size on the public scientific stage.

After returning home from Kazan I pulled out older notes and put them into proper form, presenting my ideas about planar geometry with  the method that I had developed during the last decades.