Sam Gutkind has always been interested in the way human beings can use theories to describe, predict and understand the world around them. His interest in mathematics and its foundations began in 2006 when he first learned about Gödel’s incompleteness theorems, which show that there is more to mathematics than simply solving problems by following a set of rules. This fall, he will return for his final year at Pittsburgh Allderdice High School.
Project: Axiomatization and Abstraction–Using Single-Valued Binary Operators and Related Equational Axiom Systems
Goals are to investigate the properties of axiom systems, starting with those that constrain one or more binary operators that operate on truth values; to investigate the necessary complexity of such an axiom system necessary to select “useful” operators; and, if possible, to extend this to constant, unary, ternary, etc. operators. This will be used to investigate the ways in which systems based on simple programs (or that obey simple but nonrestrictive constraints, like the ability to formulate the notion of space) are able to represent abstract truth of the kind studied in mathematics.
First studied will be the process of taking a set of selected models of an axiom system (such as binary operators) and finding the smallest axiom system that allows a set of models as close as possible to the selected ones.
Next investigated will be processes that allow systems to progress from the ability to represent abstract facts about their environments to the ability to make general statements about abstract concepts. How do we progress from the notion of a quantity to formalized arithmetic, or from the notion of a set to formal set theory? Both of these formalizations allow us to make much more general statements than would ever be possible otherwise, but how have we figured out how to make accurate generalizations that govern the infinite, drawing from finite experience? Does this process happen differently for different systems, and how easily does it occur?
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