Sam Gutkind
Bio [2008]
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 Title
Axiomatization and Abstraction--Using Single-Valued Binary Operators
and Related Equational Axiom Systems
Project
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?
Favorite Radius 3/2 Rule
Rule chosen: 2737
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