Coordinator: Dr. G. Kaur
Coordinator: Prof. S. Chandrasekharan, Ms. D. Dutta & Mr. Durgaprasad K.
Coordinator: Dr. Aniket Sule
Coordinator: Dr. N. D. Deshmukh
Jointly organized by Homi Bhabha Centre For Science Education, Mumbai & NSC, Mumbai; Exploratory, Pune; Vigyan Ashram, Pabal & funded by RGSTC, Mumbai
Venue: G2 & Integrated Lab
Coordinator: Prof. Jyotsna Vijapurkar
Dr. Sriram Nambiar Guest Professor, Department of Humanities and Social Sciences, IIT Madras
How do we, finite beings, know anything about every number? That we indeed do, no matter how inexplicable, is irrefutable.
The axioms of a field are known immediately, that is, without mediation by a proof. A theorem of arithmetic may be known by a proof from these axioms. However, propositions about the natural numbers were known millennia before the axioms for them were discovered. What, then, is the relation of knowledge of these ancient theorems to the knowledge of the axioms? We examine ancient proofs of arithmetic (Pythagorean pebble patterns) and discover that the central concepts involved in the axioms do participate, though tacitly, in these proofs.
We also present a definition of a universal property of the numbers, that is, a property which belongs to every number. An example of such a universal property is the property of being exceeded by a prime. Many, if not most, theorems of arithmetic may be stated in terms of the numbers having some such universal property.
Surprisingly, it turns out that this definition characterizes the numbers well enough to serve as a sufficient axiom for the natural numbers. It appears that the two interrelated concepts involved in this axiom–a successor function and the universal property–are tacit participants in many standard proofs about the natural numbers.
Dr. Sriram Nambiar is currently a Guest Professor in the Department of Humanities and Social Sciences, IIT Madras. He has a Ph.D in philosophy from the State University of New York, Buffalo. He has held faculty positions in several institutes including Chennai Mathematical Institute, IIT Madras and St. John's College in Annapolis, Maryland. He has also worked in industry (for example, in TCS) on training programs for software developers.
His areas of interest include History and Philosophy of Mathematics, History and Philosophy of Logic, and Mathematical Logic. He is currently involved in the design of software for interactive teaching of logic, and in investigating the underlying logic of science as well as the origins of mathematical logic.