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Re: Re: Re: Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106287] Re: [mg106238] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 7 Jan 2010 02:31:53 -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> <19ECAF1E-F291-4761-911F-D9B953B1843C@mimuw.edu.pl> <op.u5447ck3tgfoz2@bobbys-imac.local>

I really think we are all the time talking cross purposes. Obviously 
everything mathematica does is "algebra" in a very wide sense. But a 
great many human topological arguments are also "algebra" in the same 
sense. The same applies to arguments used in analysis. Essentially they 
are just "discrete" deductive steps, based on certain statements which 
are taken as facts - and all of that is perfectly reproducible by a 
computer, although it may be hard. There is no reason why one should not 
be able to store in a computer's memory a vast number of facts about the 
topology of, say, three manifolds and then get the computer, by purely 
discrete means, deduce from them new facts. The computer would then be 
doing topology and the statements that it discovered would be 
interpreted  as statements about objects whose existence depends on the 
the continuum.

There is only one difference between this process and what actually 
happens in the human mind - we have a geometric intuition to guide us in 
selecting what might be true and also the steps that we need in a proof 
- a computer has no geometric intuition at all. This is the crux of the 
matter. This is also what all these books of Penrose are really about. 
The issue is exactly this "intuition". Formal, rigorous mathematics is 
not, in principle, different from what computer can do. Intuition may or 
may not be. We do not know whether intuition is not simply based on 
having a vast amount of knowledge stored in our memory.
Computers now can play chess on the level of the strongest grandmasters. 
The way this was achieved was not by making than try to compute all the 
possibilities in any given situation - this is the way to nowhere, but 
by storing vast amounts of human knowledge and making computers imitate 
the way humans play chess. Now, it is pretty hard, to discover just by 
studying the moves in a chess game, whether one or both players are 
computer programs or not. A human chess player uses above all his 
intuition so that he does not need to consider the great majority of 
possible moves in a given situation because his intuition tells him they 
are bad moves. A computer has no intuition, but it can learn to act as 
if it had one, simply by finding in its stored database of games an 
identical or very similar position and choosing a move stored in its 
memory.
There is no obvious reason why computers should not some day be able to 
do topology as well as they can play chess. They would be making use of 
human knowledge in doing that. To a large extent this is what 
Mathematica already does. That's my entire point.

Andrzej Kozlowski






On 7 Jan 2010, at 11:29, DrMajorBob wrote:

>> 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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