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Re: Re: Re: Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106283] Re: [mg106238] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 7 Jan 2010 02:31:05 -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> <5A8611E1-4E37-444E-9E26-87D7FFD50F94@mimuw.edu.pl> <op.u542d8vjtgfoz2@bobbys-imac.local>

The important word was "in principle". I have never claimed that 
Mathematica can do topology. I work in topology and  when I do that I do 
not use Mathematica. But Mathematica does or if you prefer "simulates" a 
lot of mathematics that only makes sense under the assumption of 
continuity. In particular things like

Resolve[Exists[x, x^2 == 2], Reals]

True

Mathematica obviously does this "discretely" (so does the human brain) 
but this is a statement about the reals not the rationals. To think in 
any other way just makes no sense to me.

Andrzej


On 7 Jan 2010, at 10:28, DrMajorBob wrote:

> Yes, this discussion is far too philosophical... but it HAS 
illuminated a few real-world Mathematica behaviors.
>
>> are you only claiming that "all computer reals are rationals" or are 
you also claiming that "all reals are rationals"?
>
> The former.
>
>> If not, then what is the difference between the two?
>
> A great deal.
>
> I can imagine the woof and weave (the topology) of real numbers; 
computers can't do that. I can state four assumptions and show that 
every set with these properties is topologically isomorphic to what we 
call "the real line", with NO reference to real numbers, numeric 
representations, or real arithmetic. We did just that in a special 
topics course when I was a sophomore; none of us knew, when we started, 
what the end-goal would be... but that's where we arrived.
>
> The idea that a computer's mimicry of reals is equivalent to that is 
just... absurd.
>
> A computer can't begin to grasp the topology; it begins and ends with 
arithmetic. (That includes smart algorithms such as GroebnerBasis and 
RootApproximant, which are, root and branch, arithmetical.)
>
> Computers can do arithmetic on a finite subset of the reals, it can do 
symbolic algebra faster than a human, and Mathematica's 
arbitrary-precision arithmetic and large integers simulate nonstandard 
analysis in a limited way... but that's very far from understanding 
reals the way a topologist does or fields the way a algebraist does, or 
nonstandard analysis as a mathematical logician does.
>
>> Why can't a computer, in principle of course, perfectly simulate the 
activity of the human brian that we call "doing mathematics"?
>
> In principle of course, human minds ARE computers... but not the kind 
we're likely to build, anytime soon.
>
> You're not claiming that Mathematica simulates the mind of a 
mathematician, I hope?
>
> Show me Mathematica proving topological theorems (beyond FINITE groups 
and graphs)... and you might have something.
>
> Bobby
>
> On Wed, 06 Jan 2010 18:44:15 -0600, Andrzej Kozlowski 
<akoz at mimuw.edu.pl> wrote:
>
>> 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
>>
>>
>
>
> --
> DrMajorBob at yahoo.com



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