N-Person Game Theory: Concepts and Applications
Anatol Rapoport
ISBN 0-486-41455-8 (Dover)
$13 US
[SSZ: If you get this, also get his "2-Person Game Theory" as well.]
Excerpt from "Introduction - Some Mathematical Tools", pp. 11
"Game theory is properly a branch of mathematics. As such it is concerned
with assertions which can be proved to be true if certain other assertions
are true. This way of establishing truth by reference to assertions
previously established as true would lead to infinte regress unless
certain fundamental assertions were simply accepted as true (without
proof). These basic assertions are called axioms. Assertions whose truth
is derived by logical proof are called theorems. Game theory, like any
other mathematical theory, is essentially a collection of theorems derived
from axioms.
The terms (words and phrases) of a mathematical proposition must be
precisely defined. The definitions contain other words, which must also be
defined. Definitions likewise would lead to infinite regress or to
cirularity unless some terms were simply accepted as understood. These
fundamental accepted terms are called primitive terms. Every mathematical
theory must contain some primitive terms. All other terms must be defined
by reference to these.
It is important to keep in mind that a mathematical concept is never
defined "approximately," ie, with a tacit assumption that its meaning is
intuitively clear. A mathematical term is always defined exactly, so that
there can be no dispute about its meaning. Similarly a mathematical
theorem is never approximately or "reasonably" true, or true "with a
high degree of probability." It is always absolutely true (assuming the
axioms to be true). To assert that a theorem is false means to deny
one or more of the axioms. However, to assert that a theorem is true does
not necessarily mean to assert the truth of all axioms. Some theorems
remain true even if some of the axioms of a mathematical theory are
rejected.
To accept the truth of the axioms is simply to agree to assume them to be
true. However, the consistency of the axioms with each other is sometimes
in question. Then, if any two or more axioms of an alleged mathematical
theory are found to be inconsistent with each other, the whole theory
collapses."
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We are all interested in the future for that is where you and I
are going to spend the rest of our lives.
Criswell, "Plan 9 from Outer Space"
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