In our Discrete Mathematics University course, there was a discussion on the Knapsack problem (as it is called).
The problem goes like this:
A U.S. shuttle is to be sent to a space station in orbit around the earth, and 700 kilograms of its payload are allotted to experiments designed by scientists. Researchers from around the country apply for the inclusion of their experiments. They must specify the weight of the equipment they want taken into orbit. A panel of reviewers then decides which proposals are reasonable. These proposals are then rated from 1 (the lowest score) to 10 (the highest) on their potential importance to science… It is decided to choose experiments so that the total of all their ratings is as large as possible (Otto, Spence, Eynden, & Dossey, 2006).
After this outline, we’re asked to examine algorithmic variations that would allow us to postulate the most efficient experiments out of the 4096 possible variations that come about from the 12 possible experiments.
Isn’t it interesting how a mathematical question can become an existential question? While theoretically, one could evaluate the knapsack equation from a logical perspective, and get the ‘biggest bang for the buck’, one also has to wonder (if this were a real scenario) who assigned the rating values for these experiments, and what type of objective/subjective approach did they take?
For example: what if we had two experiments, one that would give us more information about cancer and one that gave us better insights to obesity. Most people might be inclined to include the research on cancer, as the rate of death directly attributed to cancer in the world is typically thought to be much higher than those attributed to obesity. However, what if the probable outcome of the research on cancer might move us a few years ahead in our research, but the research on obesity has a probable end goal of realizing the end of obesity within just a few years. What about all of the secondary causes of death that are indirectly linked to obesity. How does one decide the rating mathematically?
This seems to show that even while our capabilities of solving complex algorithmic variations using state machines can increase the efficiency of mathematical computation; the answer to Alan Turing’s fundamental question of whether or not a computer can ever make ‘human’ decisions seems to lie outside of the realm of algorithmic efficiencies!
References
Otto, A. D., Spence, L. E., Eynden, C. V., & Dossey, J. A. (2006). Discrete Matmatics – Fifth Edition. Boston: Greg Tobin.