Computers still can't do beautiful mathematics

Article Abstract:

More and more mathematical researchers are using computers to prove theorems, but some researchers object to this. According to Daniel Kleitman, a mathematician at the Massachusetts Institute of Technology (MIT), mathematicians read mathematical papers not only to learn results, but also to understand how proofs are formulated and how they work. If a mathematician is forced to depend on a computer and on assumptions made by a programmer, in a sense, a programmer's work becomes part of a proof, and mathematicians are usually not qualified to check such a program's reliability. Computer-aided proofs have been a cause for concern since 1977, when Kenneth Appel and Wolfgang Haken published a computer-generated solution to the 'four-color problem,' which involved coloring of adjacent countries on maps. The repetitive and inelegant way that the computer solved the problem was not satisfying to traditional mathematicians.

Technology application, Criticism and interpretation, Mathematical logic, Graph theory, Theorem Proving, Automated theorem proving

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Math problem, long baffling, slowly yields; the traveling salesman problem still isn't solved, but computers can now get most answers

Article Abstract:

The traveling salesman problem is a complex mathematics problem that computers are beginning to find solutions for, albeit imperfect solutions. The problem asks for the shortest route between a number of cities or points, and has a wide variety of applications in telecommunications, computer chip manufacture and factory work order management. Computers have not found the solution to the problem, which still evades mathematicians, but they are providing reasonable solutions by doing brute number crunching. Although it is impossible to enumerate all possible tours for even a small number of cities, for example 100, reasonably close answers can be approximated. The solution of a 2,392-city problem is the greatest achievement to date, but a 3,038-city problem is being attempted in early 1991.

Tests, problems and exercises, Computers, Digital computers, Problem solving, Mathematics, Optimization theory, Traveling-salesman problem, Mathematical programming

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In a frenzy, math enters age of electronic mail

Article Abstract:

A mathematical advance transforms from a promising concept to a completed result in a matter of weeks through an informal worldwide competition conducted by electronic mail. Noam Nisan suggests that it might be possible to use a probabilistic proving system to calculate the most efficient route for the Travelling Salesman problem, a problem where a salesman must travel through a group of cities in the shortest possible route while visiting each city once. The solution suggested by Nisan found enthusiastic response from researchers over an electronic mail system. Many suggest that networks will increase collaboration on difficult problems but others note the scientists in the third world and Eastern Europe, who are not connected to networks, would be left out.

Research, Computer networks, Electronic mail systems, E-mail, Email, Mathematical research, Networks, Conferences, Nisan, Noam

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