Subject: RE: [hackers-il] [Fwd: The Unreasonable Effectiveness of Mathemat ics]

From: Chen Shapira <[email-masked]>
Date: Sun, 11 Feb 2001 12:09:28 +0200

> Depends a lot of what the definition of e^x is. It's no good to say
> "the usual power definition", because usually, "a^b" is defined as 
> "e^(ln(a)*b)".
> 
> The usual definition is
>        oo
> e^x = sigma x^n*(1/n!)
>        n=0

Correct me if I'm wrong but its pretty common in all areas of mathematics to
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
Lemma?
Are Sin and Cos defined as properties of triangles or infinite sums?

There are countless examples for this, I think the usual name is
"Equivalence".

---

From: "Nadav Har'El" <[email-masked]>
Date: Sun, 11 Feb 2001 15:28:06 +0200

> Are Sin and Cos defined as properties of triangles or infinite sums?

This is true, but you have to watch for two things:

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
[email-masked]             |-----------------------------------------
Phone: +972-53-245868, ICQ 13349191 |Disclaimer: The opinions expressed above
http://nadav.harel.org.il           |are not my own.