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Re: Re: 0^0 = 1?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg95745] Re: [mg95683] Re: 0^0 = 1?
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Sun, 25 Jan 2009 21:51:57 -0500 (EST)
*References*: <gl7211$c8r$1@smc.vnet.net> <gl9mua$ajr$1@smc.vnet.net> <200901251148.GAA00455@smc.vnet.net>
On 25 Jan 2009, at 12:48, Dave Seaman wrote:
>
> Because it doesn't support your preferred answer?
>
> We define other operations (addition and multiplication, for
> example) by
> starting with the cardinals, then extending to the integers, the
> rationals, the reals, and then the complex numbers. At each stage, we
> want the new definition to be consistent with the old.
That is not the point. The point is that the original reason why the
axiomatic approach to the so called "foundations of mathematics" was
adopted around the turn of the 19 and 20 century (Cantor's theory of
cardinal numbers, Zermelo-Frankel axioms, Peano's axioms for
arithmetic etc) were summed up in Hilbert's first and second problems.
In other words, it was the hope that one could "reduce" all
mathematics to a certain system of axioms and then establish the
consistency of this system and thus the consistency of all of
mathematics. The theorems of Goedel (and to some extent Cohen's
solution of the continuum hypothesis) showed that this hope was
futile. One consequence is that there is no advantage at all in
starting with abstractions such as sets or cardinals or even the "God
given" (according to Kronecker) integers. We are just as well or badly
off if we take real or even complex numbers as the starting point. In
fact this approach has the great advantage of being much closer to our
physical intuition. This is not just my opinion. Most mathematicians
today do not care about "the foundations of mathematics", except as
another area of pure mathematics, or relatively little usefulness. One
of the greatest living mathematicians, V.I. Arnold has been very vocal
in opposing the axiomatic approach to mathematics as very harmful. An
example of his views (quite widely shared) can be seen in this
(somewhat abridged) translation of an essay on the teaching of
mathematics:
http://pauli.uni-muenster.de/~munsteg/arnold.html
Of course this does not mean that Arnold's views have to be viewed as
some kind of holy writ, but only that in your statement "we define..."
etc. the "we" refers to a finite subset of the total set of all
mathematicians, and one of rather small cardinality.
In a computer program of course practical considerations should take
priority. Now, I indeed do not see any problems with 0^0 returning 1.
However, presumably in such cases one would also want 0.^0. to return
1, which actually could cause problems, since such expressions can
appear as a result of cancellations in the base caused by fixed
precision arithmetic. Returning 1. in such situations rather than
Indeterminate could be very misleading. Of course problems of this
type are inevitable with fixed precision arithmetic no matter what
conventions we adopt, but there is no point adding to them for no good
reason.
Andrzej Kozlowski
>
>>> Another way is to notice that 0^0 represents an empty product, whose
>>> value is the identity element in the monoid of the integers (or the
>>> reals).
>
>>> In[1]:= Product[0,{k,0}]
>
>>> Out[1]= 1
>
>> This and the ZF result are arguments for making 0^0 equal to one.
>> They
>> are not in any sense "proofs" that it must be one, given that Power
>> lives in the setting of functions of complex variables.
>
> The ZF result is indeed a proof in the context of the cardinal
> numbers.
> The empty product result applies to any monoid. The complex numbers
> are
> a monoid. The symbol "0" in Mathematica represents the complex
> number 0,
> and as you can see, Mathematica agrees that the empty product yields
> the
> complex number 1.
>
>>> One might also consider the series expansion for Exp[0], which
>>> reduces to
>
>>> 1 = 0^0/0! + (lots of terms that all reduce to zero).
>
>> This point is questionable, since one does not in general encounter
>> the
>> formula with a term x0^0 (where x0 is the point of expansion).
>
> It is not questionable. The series expansion for Exp[x] is
>
> Sum[x^k/k!,{k,0,Infinity}]
>
> whether one normally encounters it in that form or not.
>
>>> Having x^y be discontinuous at (0,0) does not "cause problems" any
>>> more
>>> than having the Sign function be discontinuous at 0 causes problems.
>
>> That's a matter of opinion. Clearly we do not at this time agree with
>> you on this.
>
> But no one has presented an example of an actual "problem" that is
> caused
> by the definition. Discontinuous functions exist. That's the basic
> reason that we need to study limits in the first place.
>
>>> Anyone who works with limits should be aware that you can't just
>>> blindly
>>> assume continuity when evaluating limits. You have to consider the
>>> actual definition of the limit.
>
>> True, but I don't see any relevance to this particular issue. The
>> question at hand is whether or not 0^0 should be assigned a concrete
>> value. If it is given a value, then since limits of x^y depend on how
>> x,y respectively approach 0, they might or might not equal that value
>> (so there is no underlying assumption of continuity). But that is
>> already the case, and the question at hand is whether the undefined
>> value should in fact be defined.
>
> We don't "assign" a value to 0^0. We simply look at the definition
> and
> notice what it says for the case of 0^0. The definition makes no
> statement at all about limits, and therefore thinking that the
> definition
> has implications regarding limits is a misconception.
>
>
>
> --
> Dave Seaman
> Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
> <http://www.indybay.org/newsitems/2008/03/29/18489281.php>
>
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