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1 LIMITATIONS ON OUR UNDERSTANDING OF THE BEHAVIOR OF SIMPLIFIED PHYSICAL SYSTEMS by Harvey M. Friedman* Distinguished University Professor of Mathematics, Philosophy, and Computer Science Ohio State University October 30, 2008 Abstract . Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of discrete deterministic physical systems involving a few moving bodies (twelve point masses) in potentially infinite one dimensional space. There are two kinds of such limiting results that must be carefully distinguished. Results of the first kind state the nonexistence of any algorithm for determining whether any statement among a given set of statements is true or false. Results of the second kind are much deeper and present much greater challenges. They point to specific statements A, where we can neither prove nor refute A using accepted principles of mathematical reasoning. We give a brief survey of these limiting results. These include limiting results of the first kind: from number theory, group theory, and topology, in mathematics, and from idealized computing devices in theoretical computer science. We present a new limiting result of the first kind for simplified physical systems. We conjecture some related limiting results of the second kind, for simplified physical systems. TABLE OF CONTENTS 1. EXAMPLES OF ALGORITHMS. Arithmetic ops, gcd, primality, factoring, solvability of equations. 2. NO ALGORITHMS. Robust model of computation. Behavior of abstract machines, solvability of equations. 3. SIMPLIFIED PHYSICAL SYSTEMS. Linear Order Systems. No algorithm for determining boundedness.
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2 4. SIMULATION OF TURING MACHINES. Twelve body linear order system. 5. ZFC AND INCOMPLETENESS. Two kinds of undecidability. 6. CONJECTURES. 1. EXAMPLES OF ALGORITHMS. Arithmetic ops, gcd, primality, factoring, solvability of equations. We want to provide some significant context by discussing demonstrable successes before I discuss demonstrable limitations. Long before there were any appropriate models of computation, there were actual interesting algorithms. Schoolchildren are still taught the standard algorithms for adding, subtracting, and multiplying two integers given in base 10. They are also taught the standard division algorithm with remainder. Algorithms for these standard arithmetic operations have been revisited in a very powerful way because of the computer revolution. There is a real need for optimizing speed as much as possible, for numerous applications. These modern algorithms take full advantage of actual circuit designs, and exploit their capacity for parallelism. See, for example, [CLR90], [BP00]. It is instructive to consider one the most famous of all
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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