It was fall of 1986. I was attending the Master’s Program in Mathematics at Sambalpur University. It is a small University in the western part of the state of Orissa, India. With a small group of students and teachers, we had a very friendly environment. The department was headed by Prof Swadhin Pattanayak – a well known mathematician and a great teacher. Besides, we were lucky to have some other very good teachers like Prof A Mishra and Prof B B Panda. There was a very stimulating learning environment. This environment has made this department is proud of producing many good mathematician of national and international reputation.
We had a professor, who was the one of the most brilliant of all faculty members in the university (I will call him Professor.). He had unfathomable knowledge in a very wide domain - starting from Cosmology, General Theory of Relativity to Bhagbad Geeta, Old and New Testament and Quran. He used to attend most of the seminars organized in the university and delighted the audience with his wit and wisdom. He commanded a lot of respect from the university community. He led a very simple life. His doors were always open to all. Anybody can come to him and discuss on any topic from mathematics to any academic topic including religion. We always felt he was the Almighty’s gift to the university and to us in particular.
(Gate of Sambalpur University, Jyoti Vihar, Burla, Orissa)
Professor had a great passion for teaching. He can extemporaneously teach many Masters level Mathematics courses. On many occasions we visit his residence with course-related questions and he always welcome us with a smiling face. His life was (and still is) dedicated to popularize mathematics among students and among masses. He used to dream of a time when people enjoy mathematics as much as literature - when commuters will be solving math puzzles instead of reading novels in the buses and trains.
He was a great teacher of Mathematical Analysis. In one of his classes, he very passionately proved a really complex theorem. His teaching always makes very abstruse concepts look obvious. After proving the theorem he smilingly and lightly commented – even God cannot disprove it now. His favorite student in the class was quite friendly to him and was aware of his agnostic attitude towards God. So he teasingly commented – “God is Almighty, he can do anything. He can disprove any theorem established logically".
Professor looked at him smilingly. “Is it? “, he asked, “If God is Almighty, can He create something which He can not break”?
The student could understand his Professors's intentions. He was pushing the student to a paradoxical situation. If he says yes, his next question would be - can He break it?
It is something like Russel’s paradox. If God can create something which he cannot break, then he is unable to break it, which implies he is not Almighty. If He can’t create it, then also there is something which the Almighty can’t do. Professor proved that there can not be someone with absolute power like that of Almighty. British philosopher and mathematician Bertrand Russel in his famous treatise on oundations of Mathematics, Pricipia Mathemtica, has described similar paradoxes and had warned mathematicians about overusing mathematical reasoning. The student had no answer to this paradoxical situation, so surrendered.
Should we try to prove God’s existence using mathematical logic? We know there are two ways to explore the unknown – one is experimentation and observation and the other is reasoning. Mathematics is the first to use reasoning as a tool to explore and expand knowledge, while many of the pure sciences were initially using observation and experimentation for the same. So it is easy to entice a student of Mathematics or Philosophy to use logic to try to prove the unprovables.
Let us see, how does mathematics use logic? It axiomatizes. It starts with a very small set of axioms. Axioms are self-evident facts, which can neither be proved nor be disproved. They don’t contradict our observations and every day experiences. Assuming that the axioms are true, mathematics explores other questions in that realm of knowledge using deductive and inductive logic. For example, the whole of Geometry starts with five basic axioms enunciated by Euclid. They are very simple and self-evident facts like “One and only one line can pass through any two points” or “only one perpendicular line can be drawn to a straight line from a point external to it.” Usually mathematicians try to minimize the number of axioms to give their theory a strong logical foundation. That makes mathematics – the king of all sciences and the queen of all arts.
In 1929, an Austrian Mathematician named Kurt Godel wrote a seminal paper. In order to effectively describe his theory, he had proposed 46 definitions at the beginning of the paper. The celebrated paper is well known in the mathematical community as Godel’s paper and it lead to birth of new mathematical concepts like ‘undecidability’ and ‘unprovability’, thus giving birth to metamathematics . These concept were later exploited in theoretical computer science to explore computability and computational complexity. Godel’s celebrated paper proved that given any axiom system, one can not prove nor disprove all the propositions (or theorems) arising in that domain. Some propositions would just remain as undecidable.
The greatest contribution of Godels’ paper is that, it established the limitation of mathematical logic. We can not expect mathematical logic to prove all theorems arising in a branch of mathematics that develops from a given axiom system. Some propositions will remain improvable or undecidable. In the light of this paper, logic looks feebler than we use to believe. There is a limit to its strength.
So can we use logic to answer the 'ultimate' questions like existence of God?
Einstein using his theory of relativity has proved that this universe is finite! He has also established the shape of the universe – it is spherical. He has also calculated its diameter - it is some billion light years. So what is there beyond that big sphere that is universe?? Einstein replies – never bother about that, because you can never get an answer to that question. His theory of relativity assumes that nothing in the universe with a positive mass can move faster than light. Under that condition we can never be able to explore beyond the limit of this universe. That is another example, where we can see the limit of mathematical logic and science.
These two examples unequivocally prove that mathematical and scientific means are not even good enough to answer all our mundane questions. So we should not try to answer the the 'ultimate' questions using these means.
Probably, Professor was looking for this kind of answer to his paradoxical question that day.