Re: .999... ?= 1
From: Robin Chapman (rjc_at_ivorynospamtower.freeserve.co.uk)
Date: 06/10/04
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Date: Thu, 10 Jun 2004 12:03:51 +0100
Eckard Blumschein wrote:
>>
>> Why are you bothering mathematicians then?
>
> See above. I acknowledge you explained to me why generalized functions
> and distributions were introduced. I am still uncertain whether or not
> the distributions would also make sense with the restriction to IR+.
What do you mean by IR+ anyway? The usual notation for the real
numbers is (boldface) R (no I). Some authors use R^+ to denote
the positive reals, some to denote the nonnegative reals
(and some to denote the additive group of real numbers :-) )
I find it convenient to use interval notation: (0,infinity)
for the set of positive reals and [0,infinity) for the set
of nonnegative reals.
Now one could consider the dual space of C_infinity
compactly supported functions on (0,infinity) or
the dual space of C_infinity
compactly supported functions on [0,infinity) or
the space of distributions with support contained in
(0,infinity) or in [0,infinity).
> Of course, I would not expect such "IR+ distribution" to provide the
> derivative of Dirac delta.
The derivative of dirac's delta takes a test function f to -f'(0).
Its support is {0}.
> To some extent I also ask for clarity how to deal with zero. It is still
> my gut feeling that as far as real numbers represent real time,
I really have no desire to hear of the feeling of your gut ....
I think digust would be an appropriate reaction to them :-(
> exclusion of zero like a neutral position between IR+ and IR- does not
> make sense. In principle, the whole issue is of no relevance for an
> elapsed-time based analysis just within IR+ because in this case there
> is no stationary value at zero. Imagine the function of time permanently
> moving towards larger elapsed time and getting new values at its origin.
There we are: "time" is not a mathematical notion. If that is
your concern, you may find more relevance in physics or philosophy
newsgroups.
> Admittelly, this notion of time is quite uncommon. I merely have to
> explain how the analysis in IR+ relates to Heaviside's trick buoying
> generations up with seeming symmetry.
Dunno what that is.
> Let me add why I do not completely trust in Abraham Robinson's
> hyperreals. He reached a lot of improvement. However, he did not abandon
> the relations larger equal and smaller. So he cannot release Buridan's
> donkey
That poor beast again. But his fate has nowt to do with maths.
> because this would require to accept that integers and other
> discrete values are of no special weight (in the sense of distributions)
> within a continuum.
Now, Z has Lebesgue measure zero. Is this the mathematical fact that
you are trying to enunciate with your mystical claptrap?
-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html "Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9" Francis Wheen, _How Mumbo-Jumbo Conquered the World_
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