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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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