Related papers: On John Mather's Work
Not only did Turing help found one of the most exciting areas of modern science (computer science), but it may be that his contribution to our understanding of our physical reality is greater than we had hitherto supposed. Here I explore…
Mathematical maturity is a key concept for the professional life of a mathematician. This paper is not only a brief discussion of the importance of mathematical maturity but also presents some unusual ways we can use the concept to help our…
Modern mathematics is known for its rigorous proofs and tight analysis. Math is the paradigm of objectivity for most. We identify the source of that objectivity as our knowledge of the physical world given through our senses. We show in…
This paper is an investigation into Cantor works about representing a function with trigonometric series, and his proofs about its uniqueness. These works are important, because they cause invention of point-set topology, and foundation of…
The 75th anniversary of Turing's seminal paper and his centennial year anniversary occur in 2011 and 2012, respectively. It is natural to review and assess Turing's contributions in diverse fields in the light of new developments that his…
Prof. K.G Ramanathan was a legendary Indian Mathematician, working in Number Theory and a prolific Institution builder. Apart from this, he was an excellent teacher and influenced several brilliant students. In this article, we overview his…
This article reflects on the life and mathematical contributions of Pierre Cartier, a distinguished figure in 20th- and 21st-century mathematics. As a key member of the Bourbaki collective, Cartier played a pivotal role in the formalization…
The received Hilbert-style axiomatic foundations of mathematics has been designed by Hilbert and his followers as a tool for meta-theoretical research. Foundations of mathematics of this type fail to satisfactory perform more basic and more…
We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement's truth value remains the same. This view of mathematics as…
A sketch of some of the fundamental notions related to the nature of knowledge is offered, with special focus on the role of mathematics and my own opinions. No single idea exposed here is entirely original; indeed, this topic has been…
In his celebrated paper "Generic projections", John Mather has shown that almost all linear projections from a submanifold of a vector space into a subspace are transverse with respect to a given modular submanifold. In this paper, an…
The purpose of this project is to outline various philosophies on the metaphysics of mathematics that have been prominent since the time of Cantor, highlighting some biographical aspects that have influenced these ideas as well. The main…
In this paper, the notion of generic transversality and its characterization are given. The characterization is also a further improvement of the basic transversality result and its strengthening which was given by John Mather.
The foundations of mathematics have long been considered settled by the Zermelo-Fraenkel-Choice axioms. But set theory abounds in models with different truths and even classical questions such as the measurability of projective sets can…
This is the first part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. We prove two main results concerning systems that are regular singular at…
David Mumford made groundbreaking contributions in many fields, including the pure mathematics of algebraic geometry and the applied mathematics of machine learning and artificial intelligence. His work in both fields influenced my career…
Classical mathematics (involving such notions as infinitely small/large and continuity) is usually treated as fundamental while finite mathematics is treated as inferior which is used only in special applications. We first argue that the…
In his celebrated paper "Generic projections", John Mather has given a striking transversality theorem and its applications on generic projections. On the other hand, in this paper, two transversality theorems on generic linearly perturbed…
There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been…
We consider a definition of mathematics as the art of thinking in terms of formalized systems, and the science of relations, structures and algorithms. We also touch upon the relation of mathematics to other sciences, in particular through…