Mon, Feb 11, 2019
David Hilbert, a German mathematician challenged colleagues with a list of 23 unsolved problems.
The Russian mathematician Yuri Matiyasevich proved that problem 10 was unsolvable. This problem was whether a diophantine equation is solvable over the integers
We must know. We shall know.
History teaches the continuity of the development of science. Every age has its own problems so we must let these unsettled questions, as Professor Lewis argues. Why make this distinction between ages and problems? Aren’t present problems a natural continuity of past problems, forever evolving? Perhaps this is best explained by the unsettled questions that we have seen in David Hilbert’s problems.
A good mathematical problem depends on ease and comprehension, being able to be explained to the first person one meets on the street. Difficult mathematics always serves as the foundation for greater progress, never mocking us for our efforts. Has there ever been a case where other disciplines superceded math in creating a new age of discovery? Perhaps it is this time of ours that we live in when we start seeing the infancy, the burgeoning of machine learning and artificial intelligence, this rampant proliferation of technologies teaching themselves to an extent that human beings are unable to catch up. What machines lack in creative mathematical thinking, they make up for in brute force.
Hilbert’s spirit is endearing because it reminds us that in the possibility of certain equations never coming to fruition, we must continue to strive on, living for the truth.
Alan Turing had the goal in mind to solve Hilbert’s problems, a call to question.
For context, Godel used a diagonalization argument previously used by Cantor and showed the impossibility of Hilbert’s equation for negative integers.
Turing instead had to not only prove his theorems, but also needed to convince his computations were enough. He proceeds to define it as such:
A computable number, for example, may very well just be a representable as a set of non-negative values in [0,1] spread out over time. This configuration results in a “circular machine” that is known to print finitely many figures, which includes certain states.
This solution derived by Turing in many ways captured the imagination of mathematicians to begin formalizing “machines” in the mathematical repertoire. Thus, Alan Turing’s contributions were remarkable in this sense, bringing the very best minds of his day to recognize a new area of focus, changing the paradigm of thought.