|[OT] Re: Are you sure ?|
Message #24 Posted by Valentin Albillo on 11 July 2003, 8:54 a.m.,
in response to message #23 by Ernie Malaga
Hi again, Ernesto:
Obviously this discussion could go on a on, and we'll probably end it "agreeing to disagree", but there's one final remark I'd like to bring to your attention: stating that
x / x => 1
for all x, just means (passing the denominator to the right side) that:
x = x
which I'm sure you'll agree it's valid for every conceivable x, be it whatever it may, zero included. That's what the equation and the function are telling us, and trying to see "discontinuities" where there are none, and using "definitions" where none are necessary is just plain silly, a kinf of bureaucratic formalism, that goes against common sense. Also, "defining" that some function has such and such value where no other value is possible at all without inconsistency, is absurd, because if you have no choice, if your are forced, issuing a "definition" is a meaningless act on your part.
That said, don't think I'm blinded to the subleties of mathematical formalism, is just that sometimes I think some of them are pretty stupid, and 0^0 or x/x for that matter just trigger that feeling for me.
On the other hand, I know all too well that you can never prove anything my common sense or intuition alone; logically consistent, formal proofs are always necessary. For instance, assume that you are calculating this infinite sum:
Sum(n = 1, n -> inf, INT(n * tanh(Pi))/10^n)
and the answer you get coincides with 1/81 to 12 decimal places. "How nice!" you think, "I didn't expect this hyperbolic tangent thing to add up to a simple rational fraction as 1/81. Maybe it's just a coincidence ?" and so you duly proceed to compute it to greater accuracy, say 30 decimals.
It still agrees with 1/81 to all 30 decimals. "It's wonderful, let's try that multiprecision package I've got for my 48/49/71 !", you say, and proceed to compute the sum to 100 decimal places. It agress with 1/81. Then to 200 decimal places. It still agrees.
Obviously, common sense would dictate by now that the sum equals exactly 1/81, no doubt about it, right ? 200 decimal places in agreement is more than enough evidence to settle the matter, isn't it ?
Yet common sense would be wrong this time, as the sum does NOT equal 1/81. So much for common sense in mathematics ... :-)
Best regards from V.