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Re: Re: Re: Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106277] Re: [mg106234] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 7 Jan 2010 02:29:50 -0500 (EST)
  • References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net> <201001050647.BAA24123@smc.vnet.net> <201001061056.FAA14855@smc.vnet.net> <op.u538nypttgfoz2@bobbys-imac.local>

It is not at all beside the point. There is no natural phenomenon in the 
world where an actual choice is being made out of an uncountable number 
of possibilities, but we constantly model reality with continuous 
distributions. What Mathematica does it the same thing. If you like, it 
simulates the behaviour of continuous distribution. For example
RandomReal[NormalDistribution[0, 1], {10}]
is simulating a choice of 10 values from a normal distribution which 
have to be non-computable reals for mathematical reasons.
The mathematical structure of a continuous distribution is quite 
different from a discrete distribution, although what actually happens 
in the computer involves of course a finite number of objects.
The essential point is that the mathematics of what is being simulated 
is quite different.


Andrzej Kozlowski



On 6 Jan 2010, at 23:46, DrMajorBob wrote:

> We've already noted that RandomReal[] outputs are not only of measure 
zero... but also finite.
>
> Hence, arguing that computable numbers have measure zero and hence 
can't be RandomReal[] outputs, seems beside the point.
>
> Bobby
>
> On Wed, 06 Jan 2010 04:56:38 -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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