metamathblog
Studying the motion of bodies has been reasonably straightforward for us, but not so, the dynamics of souls. Socrates says "know thy self", Descartes argues "I think, therefore I exist" and Upanishads flatly assert "tat tvam asi" (that thou art). In all this, the intriguing concept that bothers all of us is the invisible life force and intelligence which is within all living beings.
For the last two thousand years we have used mathematics to study the forces of nature, with great success. But it was only in the last century the concepts of computation and information was defined satisfactorily and led us to take a closer look at the working of computers and more generally the capabilities of our rational mind. With this, the old terms like knowledge, understanding, and meaning acquired a new deeper connotation for us. Soon it became clear that metamathematics, which is concerned with the axioms and methods of arguments used in proving theorems are as important as the study of mathematics itself, if not more. In its broad sense, it will not be unreasonable to say that metamathematics is only one step away from metaphysics.
This blog is about metamathematics.
Tuesday, April 10, 2007
Intuitive Set Theory: for Scientists and Engineers
Engineers know that they can land a man on the moon without using the Lebesgue integral and they will never encounter Skolem paradox in their nuclear reactor design. Intuitive Set Theory (IST) defined here, de-emphasizes concepts that are not required by scientists in their practical work.
Axiom of combinatorial Sets. A set as important as the powerset of Cantor is what I call the combinatorial set of \aleph_0, which is defined as the set of all subsets of \aleph_0 with cardinality \aleph_0.
Tuesday, April 10, 2007
Infinitesimal Graph: Visualizing the Unit Interval
Briefly stated, infinitesimal graph is what we get when we consider an infinitesimal in the unit interval as a dedekind edge separating rational numbers above and below a number. The dedekind cut itself we imagine to be on the right side of the edge and call it a dedekind node.
Infinitesimal Graph. In representing the unit interval as a graph, we consider only the recursive numbers, that is, the numbers that are programmable on a computer. Since the programs in a computer can be listed, it
Tuesday, April 10, 2007
The Book: Visualizing the Intellectual Universe
The Book is an invention of Paul Erdos, one of the most prolific mathematicians of the twentieth century. This book lists all the proofs of mathematics, written in the formal language of mathematical logic, according to length and in lexical order. With our knowledge of formal logic today, we know that a computer can be set up to start writing this book, and the computer will write as many pages as we desire, but we also know that the computer will never be able to complete the book.
Tuesday, April 10, 2007
White Hole and Black Whole: Visualizing the Physical Universe
Here, white hole is the terminology we use when we cram a set of points of cardinality 2^\aleph_0 into an interval of infinitesimal length and black whole is the terminology for the entire space beyond the finite space.
We will consider only the real line in its entirety, infinite on both sides, with the understanding that we can have a visualization of the three-dimensional space, if we have a clear mental picture of the one-dimensional line.
White hole. First of all, let us note that
Sunday, April 8, 2007
Mysticism and Logicism: Reality is Unrealizable
We want to show that the part of reality that can be understood from set theory can be represented by a specific point in a unit interval, and further, we want to claim that this point is illusive and beyond our grasp.
Since set theory forms the foundations of mathematics, it will not be unreasonable for us to assume that a substantial part of our knowledge of reality can be derived from set-theoretic concepts.
We need the following classification of formulas to proceed further. Originally
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happiness is when, what you think, what you say, and what you do are in harmony
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