GOD DOES NOT EXIST

This is the third post of the series 'Happy Days Gone by'. It talks about the second notch of the key to unlock the door to the glorious future - Make your subject interesting. 

Dr. Banwari Lal Sharma in his traditional dress - Dhoti- Kurta in an international conference at the Allahabad University. He was one of the finest teachers in the University. He modernised the syllabus of the mathematics department of the University - making it toughest, and finest in the country.

Happy Days Gone By

Introduction।। Students' Enthusiasm at Allahabad - Unparalleled।। God Does Not Exist।। Christ Birth In Stable - Courtesy Statistician।। The Unforgettable Teacher - prof. RD Pathak।। Don't Worry, It Will Cool Down।। Can Parliament Abolish Constitution।। Fall from Grace।। We Shall Overcome।।

I took admission in the graduate classes of the Science Faculty of the Allahabad University in late 1960s with Physics, Mathematics and Statistics. At that time, the best students joined the Allahabad University. In our batch, we had at least 10 merit holders of High School and Intermediate including Intermediate topper Deepak Dhar, the renowned physicist. They always chose Mathematics as a subject. 

Nonetheless, the result of Mathematics used to be the poorest. 

The reason was not far - mathematics syllabus of our University was the most modern, toughest and perhaps the best in the country. 

Deepak Dhar receiving Pdam Bhushan in 2023. Instead of Statistics, he had Chemistry. He was inmate of Sir Sunder Lal Hostel. 

The credit for modernising syllabus for Mathematics goes to Dr. Banwari Lal Sharma. He had done his research in France with Henri Cartan, the Algebraic Topologist. In the early 1960s, Prof Sharma modernised Mathematics syllabus. 

It was common knowledge that one paper of our Mathematics required more effort than all the papers in MA. 

The toughest paper in our graduate curriculum was of Modern Algebra, perhaps because it is abstract. It was necessary to make it interesting. This is how it was done.

There Is No Universal Set

Set theory is a part of Modern Algebra. Set is a well defined collection of objects. One day our teacher asked us if we believed in GOD. Our answer was in affirmative. 
“Is entire Universe within him”, he put another question. We again answered in affirmative. 
“Then he must be the set of all sets or a universal set” said the teacher. “Yes” we shouted. 

“Is set of all sets or Universal set possible?”, queried the teacher, “Why not”,  our firm reply. 
“Let us assume that a universal set or set of all sets exists.” teacher started speaking. “Let’s consider two mutually exclusive members of such a set ‘A’ & ‘B’. ‘A’ is a set of those sets that are not members of themselves and set ‘B’ is set of those sets that are members of themselves. There is no problem, with regard to position of set ‘B’. But where would set ‘A’ lie?”

A bombshell - it took us sometime for us to realise its effect.

Set ‘A’ can not be within itself as by definition it has only those sets that are not within themselves. Then set ‘A’ should lie in set ‘B’. This is again not possible as it has those sets that contain themselves. We realised a universal set, or a set of all sets, is not possible. 

The teacher smiled and said, “God doesn’t exist - QED.” He also He informed us that: 

• It is known as Cantor’s paradox and its another version is ‘Russel’s paradox’ - who shaves the Barber; 
• This was the second of the twenty three questions posed by David Hilbert in his inaugural speech during 1900 International Congress of Mathematicians (ICM) – the solutions of which, were to take Mathematics and Science to the next level; 
• ICM is the occasion to award the Field's medal, the Nobel prize in Mathematics as well as the interesting anecdote, why there is no Nobel prize in Mathematics.

We also dreamt of winning Field's medal. 

World of Infinities

The other day, while teaching one to one onto mapping, teacher asked us if we knew how many points are there in 1” line. ‘Infinite’ was our answer.
Another question, “How many points are there in 5” line.” Again ‘Infinite’ was our answer again.
“Are points on both lines equal?” Our answers differed – some said 5” line will have more; some said equal. Then we were asked to prove our answers. We were still guessing what it had to do with mapping. 

The proof that both lines contained same number of points not only answered its connection with one to one onto mapping  but led us to Cantor’s world of infinities: the smallest Aleph null (set of natural numbers); bigger Aleph one (set of real numbers); even bigger Aleph two (set of all geometrical curves) then to Continuum Hypothesis that there is no infinity between Aleph null and Aleph one.  

This made dull and abstract subject of Modern Algebra interesting. 

Point-2: Our teachers created interest in the  subject - Do it in your subject as well.

1988 Conference in the Allahabad University on Algebraic & Differential Topology
Dr. Banwari Lal Sharma is sitting on the chair and is second from right

#AllahabadUniversity #ModernAlgebra #Mathematics #BanwariLalSharma #GoodTeacher