CHAPTER 1
Numbers, proof and ‘all that jazz’.
There is a fundamental difference between mathematics and other
sciences. In most sciences, one does experiments to determine laws.
A “law” will remain a law, only so long as it is not contradicted
by experimental evidence. Newtonian physics was accepted as valid
until it was contradicted by experiment, resulting in the discovery of
the theory of relativity.
Mathematics, on the other hand, is based on
absolute certainty
.
A mathematician may feel that some mathematical law is true on
the basis of, say, a thousand experiments.
He/she will not accept
it as true, however, until it is absolutely certain that
it can never
fail.
Achieving this kind of certainty requires constructing a logical
argument showing the law’s validity–i.e. constructing a proof.
There is, however, a problem with the notion that everything
should be proved. If we insist on proving
everything
then we initially
know
nothing
, and, if we know nothing, how can we prove anything?
We have no place to begin.
Clearly, we must have some body of
information that we know to be true on which to base our proofs.
So what can we assume known and what must be proved? Before
the time of Euclid, the answer to this question was personal and
subjective.
You were allowed to assume anything that you could
bully your listener into believing. If I could get you to agree that all
integers are even (which is false) I could use it to prove all sorts of
other wonderful (and equally false) things. This often led to many
mistakes, so much so that it was very difficult to know what was true
and what was not.
Euclid solved this problem for geometry by stating an explicit
collection of “self evident” properties (called axioms) which were as
sumed without proof. Furthermore,
the axioms are the only properties
that were to be assumed without proof.
All other properties must be
proved using either the axioms or their consequences.
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1. NUMBERS, PROOF AND ‘ALL THAT JAZZ’.
Since the time of Euclid, lists of axioms for many fields of math
ematics, such as set theory, logic, and numbers have been compiled.
In these notes, we present one of the standard lists of axioms for the
real numbers, which are the numbers used in calculus. Thus, we are
stating “up front,” those properties that we are allowed to assume
without proof. As will be seen, the list is rather long and will be cov
ered over several sections. We begin with the
field axioms
, which
describe those properties of numbers that do not relate to inequalities.
In principle, every number fact we use should be proved using
only our axioms.
In fact, in these notes, we usually adopt a much
looser standard. As the reader will see, proving everything directly
from the axioms would take so long that we would never progress
beyond this section! It is, however, important that the reader prove
a number of basic number facts using only the axioms in order to
appreciate their significance.
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 Spring '08
 Staff
 Math, Addition

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