User:Ray Calvin Baker/PicardsCourtMartial

 This is the "PicardsCourtMartial.txt" file, created WED 2011 AUG 10 11:26 PM, revised SUN 2011 NOV 27 06:49 PM.

The Court-Martial of Captain Jean-Luc Picard

In one of the Star Trek: Generations episodes, Captain Picard made a statement consistent with the following text, taken from a twentieth century computer science textbook.

Statement presented in evidence by the defense:

3.1.1 History of Computability

The idea of having an algorithm, or recipe for performing some task, has existed for thousands of years. For many years, people also held the following belief: if any problem could be precisely stated, then with enough effort a solution could eventually be found (or else a proof that no solution exists could eventually be provided). In other words, it was believed that there was no problem so intrinsically difficult that, in principle, it could never be solved.

One of the great supporters of this belief was the famous mathematician David Hilbert (1862 - 1943). He once said, "Every definite mathematical problem must necessarily be suseptible of an exact settlement either in the form of an actual answer to the question asked, or by the proof of the impossiblity of its solution and therewith the necessary failure of all attempts ... one of the things that attracts us most when we apply ourselves to a mathematical problem is precisely that within us we always hear the call: here is the problem, search for the solution; you can find it by pure thought, for in mathematics there is nothing which cannot be known."

Hilbert's aim was to devise a formal mathematical system in which all problems can be precisely formulated in terms of statements which are either true or false. His idea was to find an algorith which, given any statement in the formal system, would determine whether or not that statement was true. If Hilbert had achieved this objective, then any problem which was well-defined could have been solved simply by executing the algorithm. Deciding the truth of a given statement in the formal system was called the Entscheidungsproblem by Hilbert, who considered it to be a fundamental open problem in mathematics.

(Source: _Computer_Science_A_Modern_Introduction_, by Les Goldschlager and Andrew Lister, Prentice Hall, 1982, ISBN 9-13-165704-8, page 73)

As a student of mathematical logic, I was shocked by Captain Picard's statement! I was also shocked that his senior offices seemed to take no notice of the statement. How would this scene have played out, had the senior officers regarded Picard's statement with the alarm that a twentieth-century student felt? Could it be that Gene Roddenberry (or his associates) had missed one of the most important developments in the history of mathematics? I believe Picard's careless (and completely false!) statement should have been perceived as sufficient reason for his officers to relieve him from duty immediately!

Would the following have been brought forth by a prosecutor at Picard's court-martial hearing?

Statement presented in evidence by the prosecution:

Unfortunately for Hilbert's objective, the 1930s brought a wave of research which showed that the Entscheidungsproblem is not computable. That is, no algorithm of the type for which Hilbert longed, exists. A cynic might say that mathematicians could heave a sigh of relief, for if such an algorithm did exist, they would be out of a job just as soon as the algorithm was found! In fact, however, mathematicians were stunned by the remarkable discovery.

The first news of the discovery came in 1931 when Kurt Godel published his now famous incompleteness theorem. Among other things, this showed that there is no algorithm whose input can be any statement about the integers and whose output tells whether or not that statement is true. Closely following on Godel's heels, further mathematicians such as Alonzo Church, Stephen Kleene, Emil Post, Alan Turing and many others, found more problems which had no algorithmic solution. Perhaps the most remarkable feature of those early results on problems which cannot be solved by computers is that they were obtained in the 1930s, before the earliest [electronic] computers had been built!

(Source: as above, pages 73, 74)

How do you think this episode should have continued? Should Picard have been reprimanded for conduct unbecoming a Star Fleet Captain? Should his officers have been commended for relieving the Captain of duty, an unpleasant and distasteful task, but correct and in accord with essential Star Fleet Regulations?

I don't know, because this episode was never written. But I think it should have been, because of the importance of the issues in modern mathematics.

The end. 