Wyatt Brakeboer | Fulcrum Staff
HAVE YOU EVER been stuck on a problem, unable to come up with a solution? Ever wish there was some way to draw connections to make your problem simpler? Category theory is a branch of mathematics that examines different mathematical concepts with the intention of finding connections between them. It is a way of simplifying a problem by linking it to another type of easily solvable problem. This branch of mathematics is especially useful in fields like algebraic topology and geometry.
The researcher
Pieter Hofstra was hired by the University of Ottawa in 2007 as an assistant professor in the department of mathematics and statistics. Hofstra researches both logic and category theory in an attempt to develop a categorical approach to computability theory. The research that Hofstra does is not completed in a fancy lab, but instead in his office using good old-fashioned pen-and-paper technology. This time-consuming field of research cannot be sped up by modern technology, as computer technology is not capable of the logical deductions that the research requires.
The research
Hofstra’s research does not have set boundaries or limitations, as is the case with most research. He finds new problems through either working on a general problem or starting with a vague intuition and continuing from there. Once a specific problem is selected, Hofstra works on finding commonalities between that problem and one that simplifies the problem. This area of study involves a significant amount of thinking and does not always yield the expected results. Category theory can extend in many different directions due to its diverse nature in problem-solving. One of the current problems that category theory is trying to solve is how to link game theory and logic in a more in-depth manner. Currently, the connection only exists for basic, two-player, win-or-lose games.
The future
Although logic and category theory are limited to a narrow class of two-person win-lose games, game theory is much more general. If category theory can find a way to link game theory and logic more closely, then this relationship could extend to multiple-player games, where there is not always a clear winner and loser. Hofstra has reformulated the mathematical description of games to be more amenable to categorical analysis, allowing category theory to study a much wider class of games. Hofstra’s categorical analysis allows for games with an arbitrary number of players, continuous payoffs, and even chance events. Category theory can now more realistically describe how people make decisions in real-life situations, as these situations often resemble complex games.
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