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Re: Re: Re: Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106281] Re: [mg106238] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 7 Jan 2010 02:30:40 -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> <op.u529salwtgfoz2@bobbys-imac.local> <771DE886-36BB-4108-A83C-808109BAA8C3@mimuw.edu.pl> <op.u53a91u1tgfoz2@bobbys-imac.local> <201001061057.FAA14928@smc.vnet.net> <op.u54k94ojtgfoz2@bobbys-imac.local> <86095ED9-9201-4CCE-B9F8-2091CB57BD33@mimuw.edu.pl> <op.u54x9awhtgfoz2@bobbys-imac.local>

It seems to me that this entire discussion has turned into pure 
philosophy and isn't really suitable for this forum. But to put it all 
in a nutshell: I do not see any reason to think that anything that a 
computer can do is in a fundamental way different to what human brain 
does. So, if you claim that "all computer reals are rational" I can't 
see how this is different from the claim that "all reals are rational" - 
since reals surely exist only in mathematics, which is a product of the 
human mind.

Now, as I mentioned earlier, Roger Penrose has tried to argue that the 
human brain is fundamentally different from a computer and that it has 
some sort of access to "real numbers" that a computer cannot achieve (he 
formulates this in terms of Turing machines and computability but 
essentially it amounts to the same thing). This view remains very 
controversial and seems to be a minority one. But anyway, you do not 
seem to be referring to this sort of thing. So put this question to you: 
are you only claiming that "all computer reals are rationals" or are you 
also claiming that "all reals are rationals"? If not, then what is the 
difference between the two? Why can't a computer, in principle of 
course, perfectly simulate the activity of the human brian that we call 
"doing mathematics"?

Andrzej Kozlowski


On 7 Jan 2010, at 08:59, DrMajorBob wrote:

> If I'm told that finite-precision reals are not Rational "because 
Mathematica says so", but that Mathematica success (by some algorithm) 
in finding a Root[...] representation doesn't mean the number is 
algebraic... yet I know that all finite binary expansions ARE both 
rational and algebraic as a matter of basic arithmetic... then I 
question whether Mathematica is saying anything either way.
>
> Perhaps it's just Mathematica USERS holding forth in each direction.
>
> I think the view of reals as monads (a la nonstandard analysis) melds 
with the fact that reals are irrational A.E. and non-algebraic A.E., 
while monads are, of course, consistent with the spirit of Mathematica's 
arbitrary-precision arithmetic (WHEN IT IS USED). The OP posted a number 
far beyond machine precision, so it's reasonable to come at this from 
that arbitrary-precision world-view... in which case you're "right" and 
I'm "wrong".
>
> I called all the reals rational, and you called them monads (or 
equivalent).
>
> Fine.
>
> Bobby
>
> On Wed, 06 Jan 2010 16:46:20 -0600, Andrzej Kozlowski 
<akoz at mimuw.edu.pl> wrote:
>
>>
>> On 7 Jan 2010, at 04:19, DrMajorBob wrote:
>>
>>>> Well, I think when you are using Mathematica it is the designers of
>>>> Mathematica who decide what is rational and what is not.
>>>
>>> Not to repeat myself, but RootApproximant said 100 out of 100 
randomly chosen machine-precision reals ARE algebraic.
>>
>> No, they are not real algebraic. RootApproximant gives algenraic 
approximations to these numbers and in fact it uses a test for what 
makes a good approximation. In never says that these numbers themselves 
are algebraic. You have been completely confused about this. The method 
RootApproximant uses is the LLL method, which finds approximations. 
Because of this it will give you a number of different approximations 
for the same real. For example
>>
>> In[7]:= RootApproximant[N[Pi, 10], 2]
>>
>> Out[7]= (1/490)*(71 + Sqrt[2156141])
>>
>> In[8]:= RootApproximant[N[Pi, 10], 3]
>>
>> Out[8]= Root[37 #1^3-114 #1^2-36 #1+91&,3]
>>
>> So how come N[Pi,10] is equal to two quite different algebraic 
numbers?
>> You should first understand what an algorithm does (e.g. 
RootApproximant) before making weird claims about it. (In fact Daniel 
Lichtblau already explained this but you just seem to have ignored it).
>>
>> Andrzej Kozlowski
>>
>>>
>>> If your interpretation is correct and consistent with Mathematica, 
and if Mathematica is internally consistent on the topic, virtually all 
of those reals should NOT have been algebraic.
>>>
>>> Mathematica designers wrote RootApproximant, I assume?
>>>
>>> Hence, I'd have to say your interpretation is no better than mine.
>>>
>>> Bobby
>>>
>>> On Wed, 06 Jan 2010 04:57:26 -0600, Andrzej Kozlowski 
<akoz at mimuw.edu.pl> wrote:
>>>
>>>> Well, I think when you are using Mathematica it is the designers of
>>>> Mathematica who decide what is rational and what is not.
>>>>
>>>> And when you are not using Mathematica (or other similar software 
which
>>>> interprets certain computer data as numbers), than I can't imagine 
what
>>>> you could possibly mean by a "computer number".
>>>>
>>>> Andrzej
>>>>
>>>>
>>>> On 6 Jan 2010, at 11:45, DrMajorBob wrote:
>>>>
>>>>> Obviously, it DOES make them rational "in a sense"... the sense in
>>>> which I mean it, for example.
>>>>>
>>>>> Bobby
>>>>>
>>>>> On Tue, 05 Jan 2010 20:41:34 -0600, Andrzej Kozlowski
>>>> <akoz at mimuw.edu.pl> wrote:
>>>>>
>>>>>>
>>>>>> On 6 Jan 2010, at 11:13, DrMajorBob wrote:
>>>>>>
>>>>>>> I completely understand that Mathematica considers 1.2 Real, not
>>>> Rational... but that's a software design decision, not an objective
>>>> fact.
>>>>>>
>>>>>> I think we are talking cross purposes. You seem to believe 
(correct
>>>> me if I am wrong) that numbers somehow "exist". Well, I have never 
seen
>>>> one - and that applies equally to irrational and rationals and even
>>>> (contrary to Kronecker) integers. I do not know what the number 3 
looks
>>>> like, nor what 1/3 looks like (I know how we denote them, but 
that's not
>>>> the sam thing). So I do not think that the notion of "computer 
numbers"
>>>> makes any sense and hence to say that all computer numbers are 
rational
>>>> also does not make sense. There are only certain things that we
>>>> interpret as numbers and when we interpret them as rationals they 
are
>>>> rationals and when we interpret them as non-computable reals than 
they
>>>> are just that.
>>>>>> Of course we know that a computer can only store a finite number 
of
>>>> such objects at a given time, but that fact in no sense makes them
>>>> "rational".
>>>>>>
>>>>>> Andrzej Kozlowski
>>>>>
>>>>>
>>>>> --
>>>>> DrMajorBob at yahoo.com
>>>>
>>>>
>>>
>>>
>>> --
>>> DrMajorBob at yahoo.com
>>
>
>
> --
> DrMajorBob at yahoo.com



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