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Re: Re: Re: algebraic numbers
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106261] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Wed, 6 Jan 2010 06:02:07 -0500 (EST)
*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net> <201001050647.BAA24123@smc.vnet.net> <E44EA2F2-1274-43E8-93DE-DC5BD31884A5@mimuw.edu.pl> <op.u52ai6jwtgfoz2@bobbys-imac.local> <504E0A05-61DB-4A43-9637-68216076623C@mimuw.edu.pl>
Just one more comment, I hope my last one on this subject. Obviously
RandomReal make it choices out of a countable set of entities. One would
have to be insane to claim otherwise and I am not that yet.
But, Mathematica does not regard these entities as rational numbers and
so they are not that. If you call them rationals the it does not make
*mathematical* sense (because rationals have measure 0). So, if
Mathemaitca does not regard them as rationals they are not rationals.
How could they be that ? Until they are interpreted by Mathematica, they
are not numbers at all but just some data stored in computer memory -
which are not numbers of any kind. Mathematica interprets them as
non-computable irrationals in order to make mathematical sense when
returning them while simulation a real distribution, because all other
numbers have measure 0.
This is all about "simulating mathematics" - numbers do not live in any
sense inside computers. To say that "all computer numbers are rational"
is weird - there is no such things as "computer numbers". Numbers exist
only and (probably) exclusively in the human mind.
To say that 1.2 is rational in Mathematica even if Mathematica says
Element[1.2, Rationals]
False
does not make any sense at all.
Andrzej Kozlowski
On 6 Jan 2010, at 07:04, Andrzej Kozlowski wrote:
> Well, you are obviously misunderstanding what I am trying to explain
but I have no desire to spend any more time on it. I give up.
>
> Perhaps you should try to explain yourself why Mathematica gives
>
> In[1]:= Element[1.2, Rationals]
>
> Out[1]= False
>
> In[2]:= Element[1.2, Reals]
>
> Out[2]= True
>
> and you might also read
>
> http://en.wikipedia.org/wiki/Computable_number
>
> (but that's the last time I posting anything to do with any logic or
mathematics here.)
>
> Andrzej Kozlowski
>
>
>
>
>
> On 5 Jan 2010, at 22:31, DrMajorBob wrote:
>
>> RandomReal[] returns numbers from a countable set of rationals.
>>
>> Or call them reals, if you must; it still selects from a countable
set of possibilities... not from the uncountable unit interval in the
reals.
>>
>> The range of RandomReal[] is a set of measure zero, just like the
algebraic numbers.
>>
>> Bobby
>>
>> On Tue, 05 Jan 2010 02:08:24 -0600, Andrzej Kozlowski
<akoz at mimuw.edu.pl> wrote:
>>
>>>
>>> On 5 Jan 2010, at 15:47, DrMajorBob wrote:
>>>
>>>> If computer reals are THE reals, why is it that RandomReal[{3,4}]
can
>>>> never return Pi, Sqrt[11], or ANY irrational?
>>>
>>> It can't possibly do that because these are computable real numbers
the set of computable real numbers if countable and has measure 0.
Computable numbers can never be the outcome of any distribution that
selects numbers randomly from a real interval.
>>>
>>> The most common mistake people make about real numbers is to think
that numbers such as Sqrt[2] or Pi as being in some sense typical
examples of an irrational number or a transcendental number but they are
not. They are very untypical because they are computable: that is, there
exists a formula for computing as many of their digits as you like. But
we can prove that the set of all reals with this property is countable
and of measure 0. So Sqrt[2] is a very untypical irrational and Pi a
very untypical transcendental. So what do typical real look like? Well,
I think since a "typical" real is not computable we cannot know all of
its digits and we cannot know any formula for computing them. But we can
know a finite number of these digits. So this looks to me very much like
the Mathematica concept of Real - you know a specified number of
significant digits and you know that there are infinitely many more than
you do not know. It seems to me the most natural way to think about
non-computable reals.
>>>
>>> Roger Penrose, by the way, is famous for arguing that our brain is
somehow able to work with non-computable quantities, although of course
not by using digital expansions. But this involves quantum physics and
has been the object of a heated dispute since the appearance of "The
emperor's New Mind".
>>>
>>
>>
>> --
>> DrMajorBob at yahoo.com
>
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