Re: [hackers-il] [Fwd: The Unreasonable Effectiveness of Mathemat ics]
- On Sun, Feb 11, 2001, Chen Shapira wrote about "RE: [hackers-il] [Fwd: The Unreasonable Effectiveness of Mathemat ics]":
> Correct me if I'm wrong but its pretty common in all areas of mathematics toThis is true, but you have to watch for two things:
> have more than one definition, each definition can be proved as a theorem
> from the others and so one can choose his definition according to his needs
> and prove the rest as theorems.
> Do you take Axiom of Choice as an axiom or theorem provable from Zorn's
> Are Sin and Cos defined as properties of triangles or infinite sums?
1. When you teach a part of mathematics, or write a book about it, you must
be consistant in the order of your definitions. By that I mean that you
can't define A assuming the reader already knows the definitions of B,
and then define B assuming the reading already knows the definitions A.
Not only will this confuse a beginner that doesn't have previous knowledge
of the subject, but most likely that using such a flawed exposition you
can "prove" pretty much anything.
This is why, if you teach the function e^x in a beginning of a calculus
course, you cannot define it based on taylor series or integrals.
Later, when you teach those subjects (taylor series and integrals) you're
free (and indeed expected) to show that a new definition, based on these
new concepts, is equivalent to the original definition. The benefit of
seeing equivalent defintions is that sometimes the alternative defintion
is easier to use in practice in many cases. E.g., except in the beginning
of the calculus course, nobody actually _uses_ the "lim (1+1/n)^n"
definition of e.
2. When something has an original definition (e.g., sin as a ratio of
triangle's sides), and you later come up with a new definition that you
show is equivalent (e.g., the taylor series), you sometimes find that the
new definition can suddenly be extended to a bigger domain, such as to
define sin() for complex numbers. If you choose to extend the domain, you
must be aware that this is in fact a new definition: some of the the old
properties may still be true (e.g., sin''=-sin is true also for complex
arguments), but some are not (e.g., sin of complex numbers has nothing to
do with triangles).
Whether this new extension becomes useful depends on whether it keeps the
_important_ properties of the original definition. For example, e^x is
important because e'=e, not because, say, e^x>0 forall x (the latter is
no longer true for complex: e^(pi*i)=-1, but nobody cares about that).
Nadav Har'El | Sunday, Feb 11 2001, 18 Shevat 5761
Phone: +972-53-245868, ICQ 13349191 |Disclaimer: The opinions expressed above
http://nadav.harel.org.il |are not my own.