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Re: Re: Re: Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106285] Re: [mg106238] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
  • From: DrMajorBob <btreat1 at austin.rr.com>
  • Date: Thu, 7 Jan 2010 02:31:28 -0500 (EST)
  • References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>
  • Reply-to: drmajorbob at yahoo.com

> But Mathematica does or if you prefer "simulates" a lot of mathematics  
> that only makes sense under the assumption of continuity.

Continuity of a function does NOT depend on completeness in the domain,  
and I suspect that

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

True

succeeds based on algebra, not topology.

You or I might (MIGHT) treat it as a topological problem, but I doubt  
Resolve can do so.

A better example might be

Reduce[Exists[x, Exp[x^7 + 3 x - 11] + x - 6/10 == 0], Reals]

True

A human might have great difficulty solving the equation, but he might  
easily establish that the LHS is negative for some value and positive for  
another, hence a solution exists in between.

Yet, since this works:

FindInstance[ Exp[x^7 + 3 x - 11] + x - 6/10 == 0, x]

{{x -> Root[{-3 + 5 E^(-11 + 3 #1 + #1^7) + 5 #1 &,
      0.599896128076431511686789719766}]}}

I suspect that algebra and root-search was used in Resolve.

Unless a developer can confirm that Resolve didn't find a solution, merely  
proved that one could be bracketed?

To do that, Resolve would have to know the LHS is continuous on the real  
line, and haven't we found, frequently, that Mathematica CAN'T identify  
continuous functions?

And what does THIS mean?

0.5998961280764315116867897197655402817356291002252018609367`30. // \
RootApproximant

Root[1 - #1 + #1^3 - 7 #1^4 + 10 #1^5 - 16 #1^6 + 15 #1^7 - 15 #1^8 +
    15 #1^9 - 16 #1^10 + 12 #1^11 - 10 #1^12 + 11 #1^13 -
    2 #1^14 + #1^15 + 8 #1^16 &, 3]

(Note the constant included in the output from FindInstance.)

Did FindInstance (and Resolve) generate and solve a series approximation  
to -3 + 5 E^(-11 + 3 #1 + #1^7) + 5 #1 & ?

Or is the RootApproximant result a pure accident?

Bobby

On Wed, 06 Jan 2010 19:42:51 -0600, Andrzej Kozlowski <akoz at mimuw.edu.pl>  
wrote:

> 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
>


-- 
DrMajorBob at yahoo.com


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