Why precise definitions are important:

Example 1:  Confusion E:

	Economist E1: economy grew by 1%
	Economist E2: economy grew by 2%

The debate between E1 and E2 can not be resolved unless there is a precise
definition of "economy".

If E1 thinks "the economy" = "amount of goods/services produced" but
but E2 thinks "the economy" = "amount of goods/services sold" then
they won't compute the same number.

So we need to agree on a precise definition of "the economy" that E1 and E2
can use to calculate the growth. That is the only way that Confusion E
can be resolved.





Example 2:  Confusion B:  (what does it mean to say A is bigger than B?)

	Mathematician M1: The sets N and Q are equally big but the set (0,1) is bigger!
	Mathematician M2: You are an idiot. N is a proper subset of Q so Q is bigger than N.

	[Note: at first, Cantor received pushback similar to M2].

Mathematician M2 is perfectly reasonable: no mathematician should tolerate
ambiguous statements. For a statement like M1 to be accepted, we should
first have precise definitions. That's the only way to resolve Example 2.
And once we have precise definitions, we should no longer give ambiguous
statements like M1. We should only give precise statements like:




Example 3:

	Definition: cardinality (equality, cardinal comparison, etc).

	M1: The sets N and Q have the same cardinality. The set (0,1) has a larger cardinality.
	M2: Precise definitions allow me to verify the proofs.
	    The proofs are correct so I agree with you.

Again mathematician M2 is perfectly reasonable.  Once we have precise statements, we can
do proofs, and when we have a proof, then every mathematician must accept the result.




Math is about understanding things; I don't like telling math students to memorize things.
But the only way to resolve confusion E and B is to use precise definitions and not an
intuitive interpretation. Writing things in terms of vague intuitive notions leads to
statements that can't be proved/disproved, and thus, shouldn't be accepted.

We can use an intuitive understanding to guide us towards a proof. But when you write
down that proof: omit the untuitive understanding and use precise definitions instead.

So do not make a drawing for "injective function" in your proofs, those belong on the
scratch paper that you don't turn in. Instead, use the correct definition.
It is important that you know the precise definitions, as long as there is any
doubt on that, there will be quiz/test/final questions like: "Write down the
definition of an injective function" (with no credit for imprecise answers)
(a logically equivalent statement such as the contrapositive will also count
as correct).  I certainly don't like asking definition questions like that,
but it is very important that you have the right definition.



Example 4: Convergence, continuity, open sets:

Imprecise definition C1:
	A sequence of points x1, x2, ...  converges to a point p if the sequence
	gets closer and closer to p.

Imprecise definition C2:
	A function f is continuous at p means that if x gets close to p then f(x)
	gets close to f(p).

Imprecise definition C3:
	A set U is open when, any time you have a point in U, all nearby points
	are in U as well.


These C1, C2, C3 are not precise enough to do proofs. To settle debates/conflicts like
in Examples 1 and 2, we need definitions that are more precise.  In C1, what exactly
does it mean "gets closer and closer"?  In C2, what does it mean "gets close"?  And
what does "nearby" mean in C3?

We can make these statements precise by using quantifiers "for all" "there exists".
Only then can we write rigorous proofs.
This will look technical at first, and takes some getting used to. But once you know
the writing proofs guidelines, how to compute the negation of statements involving
quantifiers, you will see how to handle this. The exercises in Chapter 7 will be an
excellent source of examples.


In summary:  Mathematicians reject any ambiguous statement, no matter how plausible
it may sound.  But precise statements, once proven, will be accepted even
when they are counter-intuitive (like infinite sets with different cardinalities!)