f



Solve exact problem with approx solution, or solve approx problem with exact solution

Tukey (inventor of FFT) thinks that an approximate solution of the
exact problem is often more useful than the exact solution of an
approximate problem.

I find it hard to argue which one is more important or useful.  Once
you believe in one of them, your belief will lead your research style
to either algorithm-centered or model-construction-centered.

Anybody wants to elaborate on either of these two views?

0
b83503104 (212)
5/15/2006 5:48:17 AM
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b83503104@yahoo.com wrote:
> Tukey (inventor of FFT) thinks that an approximate solution of the
> exact problem is often more useful than the exact solution of an
> approximate problem.
> 
> I find it hard to argue which one is more important or useful.  Once
> you believe in one of them, your belief will lead your research style
> to either algorithm-centered or model-construction-centered.
> 
> Anybody wants to elaborate on either of these two views?
> 


   Interesting background on FFTs
     http://mathworld.wolfram.com/FastFourierTransform.html
     http://en.wikipedia.org/wiki/Fast_Fourier_transform

0
Sam
5/15/2006 5:56:55 AM
b83503104@yahoo.com wrote:
> Tukey (inventor of FFT) thinks that an approximate solution of the
> exact problem is often more useful than the exact solution of an
> approximate problem.
>
> I find it hard to argue which one is more important or useful.  Once
> you believe in one of them, your belief will lead your research style
> to either algorithm-centered or model-construction-centered.

Your question is of central importance to the philosophy of
mathematics, one which can be appreciated only by those who actually
_do_ the mathematics, not just claim they do.

I would categorize the two styles:
  1. Symbolic mathematics
  2. Computational mathematics

In my experience, I cannot decide between the two. The 'most important'
or 'most fundamental' choice depends on the areas of mathematics you
are working in. If you are messing around in number theory, the exact
symbolic solution is usually _the only solution_ that matters (i.e.
"what looks nicest is better"). You can approximate an exact symbolic
solution so many other ways and the symbolisms can be, seamingly,
_irreconcilable_.


However, if you are dealing in computational mathematics, things like
Neural networks, self-organizing maps, cellular automata, then accurate
but computationally simpler approximations seem to be more fundamenta
(i.e  the least complex algorithms)





> Anybody wants to elaborate on either of these two views?

0
Schoenfeld
5/15/2006 7:03:47 AM
In article <1147676627.082933.272060@j33g2000cwa.googlegroups.com>, 
schoenfeld1@gmail.com says...
> 
> b83503104@yahoo.com wrote:
> > Tukey (inventor of FFT) thinks that an approximate solution of the
> > exact problem is often more useful than the exact solution of an
> > approximate problem.
> >
> > I find it hard to argue which one is more important or useful.  Once
> > you believe in one of them, your belief will lead your research style
> > to either algorithm-centered or model-construction-centered.
> 
> Your question is of central importance to the philosophy of
> mathematics, one which can be appreciated only by those who actually
> _do_ the mathematics, not just claim they do.
> 
> I would categorize the two styles:
>   1. Symbolic mathematics
>   2. Computational mathematics
> 
> In my experience, I cannot decide between the two. The 'most important'
> or 'most fundamental' choice depends on the areas of mathematics you
> are working in. If you are messing around in number theory, the exact
> symbolic solution is usually _the only solution_ that matters (i.e.
> "what looks nicest is better"). You can approximate an exact symbolic
> solution so many other ways and the symbolisms can be, seamingly,
> _irreconcilable_.
> 
> 
> However, if you are dealing in computational mathematics, things like
> Neural networks, self-organizing maps, cellular automata, then accurate
> but computationally simpler approximations seem to be more fundamenta
> (i.e  the least complex algorithms)
> 
> 
> 
> 
> 
> > Anybody wants to elaborate on either of these two views?
> 
> 

A complex calculation tend to be very confusing and its very hard for 
people not completely briefed on the subject to accept unless its 100% 
accurate.

So in my experience, all else being equal, the simplest calculation 
method is preferred.


-- 
Be careful, what you predict with the theory of human-caused global 
warming as it will be tested soon enough as we aren't going to reduce 
carbon dioxide emissions. 

Observations of Bernard - No 99

 
0
BernardZ
5/15/2006 9:46:32 AM
BernardZ wrote:
> In article <1147676627.082933.272060@j33g2000cwa.googlegroups.com>,
> schoenfeld1@gmail.com says...
> >
> > b83503104@yahoo.com wrote:
> > > Tukey (inventor of FFT) thinks that an approximate solution of the
> > > exact problem is often more useful than the exact solution of an
> > > approximate problem.
> > >
> > > I find it hard to argue which one is more important or useful.  Once
> > > you believe in one of them, your belief will lead your research style
> > > to either algorithm-centered or model-construction-centered.
> >
> > Your question is of central importance to the philosophy of
> > mathematics, one which can be appreciated only by those who actually
> > _do_ the mathematics, not just claim they do.
> >
> > I would categorize the two styles:
> >   1. Symbolic mathematics
> >   2. Computational mathematics
> >
> > In my experience, I cannot decide between the two. The 'most important'
> > or 'most fundamental' choice depends on the areas of mathematics you
> > are working in. If you are messing around in number theory, the exact
> > symbolic solution is usually _the only solution_ that matters (i.e.
> > "what looks nicest is better"). You can approximate an exact symbolic
> > solution so many other ways and the symbolisms can be, seamingly,
> > _irreconcilable_.
> >
> >
> > However, if you are dealing in computational mathematics, things like
> > Neural networks, self-organizing maps, cellular automata, then accurate
> > but computationally simpler approximations seem to be more fundamenta
> > (i.e  the least complex algorithms)
> >
> >
> >
> >
> >
> > > Anybody wants to elaborate on either of these two views?
> >
> >
>
> A complex calculation tend to be very confusing and its very hard for
> people not completely briefed on the subject to accept unless its 100%
> accurate.
>
> So in my experience, all else being equal, the simplest calculation
> method is preferred.

Yes, but you are neglecting the other aspect - the _symbolic_
mathematics. Symbolic mathematics is an approach which emphasizes the
use of symbols to describe the underlying 'structure' of the
mathematics - something not possible with computational mathematics.
For example, the symbol 'pi' represents something essentially
incalculable. Yet, without actually calculating it, it can be proven
that whatever this symbol represents, it carries some logic that allows
it to interact with other things in very strange ways - those other
things can be given symbols too. This to me is the classical approach
to mathematics - the actual 'structure'  being described by these
strings of symbols seems to transcend any logic an algorithm could
possibly describe.

However, were you to actually bring these symbols into reality, that
is, to make sense of them in a meaningful way, you need to evaluate via
an algorithm, and when you do, you lose the infinite (seamingly
'divine') precision and structure originally described. So the question
is, was the original string of symbols independently relevant or they
merely as relevant as the algorithms used to evaluate them? I would
like to know the answer to this question.

>
> --
> Be careful, what you predict with the theory of human-caused global
> warming as it will be tested soon enough as we aren't going to reduce
> carbon dioxide emissions. 
> 
> Observations of Bernard - No 99

0
schoenfeld1
5/15/2006 1:28:00 PM
In article <1147672097.414277.240280@j33g2000cwa.googlegroups.com>,
b83503104@yahoo.com <b83503104@yahoo.com> wrote:
>Tukey (inventor of FFT) thinks that an approximate solution of the
>exact problem is often more useful than the exact solution of an
>approximate problem.

Tukey and Cooley were rediscoverers of the FFT.

Both methods are somewhat dangerous, and Tukey was, alas,
an expositor of both types of excesses.  One has to be
somewhat careful of both, and use the best mathematics
one can manage at both stages, and not just consider
simple alternatives.  It is also likely that both will
have to be done in the same problem.

>I find it hard to argue which one is more important or useful.  Once
>you believe in one of them, your belief will lead your research style
>to either algorithm-centered or model-construction-centered.

NEVER construct a model of a type just because one has
an algorithm for that type of model.  Beware of using
transformations for simplification.  Structural models
are not good regression models, and vice versa.  Know
what you are doing and why, and why it may not be a good
idea at all.

Model building and numerical analysis are both arts;
treat them as such.

>Anybody wants to elaborate on either of these two views?

See my "Commandments" on my web page,

	http://www.stat.purdue.edu/~hrubin/ .

-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
0
hrubin
5/15/2006 7:00:54 PM
b83503104@yahoo.com wrote:
> Tukey (inventor of FFT) thinks that an approximate solution of the
> exact problem is often more useful than the exact solution of an
> approximate problem.
>
> I find it hard to argue which one is more important or useful.  Once
> you believe in one of them, your belief will lead your research style
> to either algorithm-centered or model-construction-centered.
>
> Anybody wants to elaborate on either of these two views?

I'll bite, Tucker's solution, the Exact solution
to the Exact Problem = E(P) = 0.
Ken

0
Ken
5/15/2006 7:54:39 PM
b83503104@yahoo.com wrote:
> Tukey (inventor of FFT)

Tukey did not invent the FFT.

Oscar Buneman, my PhD advisor at Stanford, used the FFT in
1939 during his pioneering WWII research that explains how the
radar klystron works. His PhD students had been using it for years
before I came in 1964.

Imagine our research group's surprise when Cooley and Tukey
published their paper in 1965.

No, Oscar did not invent the FFT either. According to him, it was
a well known technique in Germany during the early 1930s and
was probably discovered well before 1900.

\> exact problem is often more useful than the exact solution of an
> approximate problem.
>
> I find it hard to argue which one is more important or useful.  Once
> you believe in one of them, your belief will lead your research style
> to either algorithm-centered or model-construction-centered.
>
> Anybody wants to elaborate on either of these two views?

I always start with the exact problem. If I can't solve it exactly, I
state,
clearly, the assertions and assumptions that lead to the approximate
problem. Next, I obtain an exact or approximate solution to the
approximate problem. Finally, I go back to see if the  solution is
consistent with the assertions and assumptions.

So, you see, I consider both as just being part of the same solution.

I attended a physics/engineering job applicant seminar at MIT Lincoln
Laboratory. The applicant began his presentation with an approximate
Electromagnetic wave equation. I asked him where the equation came
from and he said it was "well known". Then all hell broke loose. My
colleagues forced him to derive the equation from first principles
(i.e.,
Maxwell's Equations). That took him ~ 1/2 hr out of a 50 minute
seminar (it should have taken him 3 minutes). Needless to say, he was
not offered a job.

Hope this helps.

Greg

0
Greg
5/15/2006 8:58:31 PM
Can you elaborate further?  I believe when you refer to model building
as an "art", then clearly the artist would use one of those two tools,
or both.  Otherwise, if the person is creating an exact solution for an
exact problem formulation then we'd call her a scientist.

0
datamatter
5/15/2006 9:22:42 PM
"Greg Heath" <heath@alumni.brown.edu> wrote in message
news:1147726711.432827.16440@i39g2000cwa.googlegroups.com...
>
> b83503104@yahoo.com wrote:
> > Tukey (inventor of FFT)
>
> Tukey did not invent the FFT.
>
> Oscar Buneman, my PhD advisor at Stanford, used the FFT in
> 1939 during his pioneering WWII research that explains how the
> radar klystron works. His PhD students had been using it for years
> before I came in 1964.

I am curious to know what type of Pre-WWII electronics could have used an
FFT.



0
Richard
5/15/2006 10:00:47 PM
"Richard Henry" <rphenry@home.com> writes:

> "Greg Heath" <heath@alumni.brown.edu> wrote in message
> news:1147726711.432827.16440@i39g2000cwa.googlegroups.com...
> >
> > b83503104@yahoo.com wrote:
> > > Tukey (inventor of FFT)
> >
> > Tukey did not invent the FFT.
> >
> > Oscar Buneman, my PhD advisor at Stanford, used the FFT in
> > 1939 during his pioneering WWII research that explains how the
> > radar klystron works. His PhD students had been using it for years
> > before I came in 1964.
> 
> I am curious to know what type of Pre-WWII electronics could have used an
> FFT.

Didn't either Euler or Gauss invent the FFT? Or at least the DFT
computed recursively.

Phil
-- 
The man who is always worrying about whether or not his soul would be
damned generally has a soul that isn't worth a damn.
-- Oliver Wendell Holmes, Sr. (1809-1894), American physician and writer
0
Phil
5/15/2006 10:48:57 PM
In article <1147728162.754529.145990@v46g2000cwv.googlegroups.com>, datamatter@gmail.com writes:
>Can you elaborate further?  I believe when you refer to model building
>as an "art", then clearly the artist would use one of those two tools,
>or both.  Otherwise, if the person is creating an exact solution for an
>exact problem formulation then we'd call her a scientist.
>
It is very exceedingly rare for a scientist to be in a position to 
create an exact solution for an exact problem formulation.  

Mati Meron                      | "When you argue with a fool,
meron@cars.uchicago.edu         |  chances are he is doing just the same"
0
mmeron
5/15/2006 10:57:38 PM
"Greg Heath" <heath@alumni.brown.edu> wrote in news:1147726711.432827.16440
@i39g2000cwa.googlegroups.com:

> 
> b83503104@yahoo.com wrote:
>> Tukey (inventor of FFT)
> 
> Tukey did not invent the FFT.
> 
> Oscar Buneman, my PhD advisor at Stanford, used the FFT in
> 1939 during his pioneering WWII research that explains how the
> radar klystron works. His PhD students had been using it for years
> before I came in 1964.
> 
> Imagine our research group's surprise when Cooley and Tukey
> published their paper in 1965.
> 
> No, Oscar did not invent the FFT either. According to him, it was
> a well known technique in Germany during the early 1930s and
> was probably discovered well before 1900.

The FFT is not much use without electronic calculators. Do you have a cite 
for this "well known technique" prior to 1900 or even the 1930's.

Klazmon.


>SNIP>
0
Llanzlan
5/15/2006 11:10:04 PM
Llanzlan Klazmon <Klazmon@llurdiaxorb.govt> wrote in
news:Xns97C5719B522A6Klazmonllurdiaxorbgo@203.97.37.6: 

> "Greg Heath" <heath@alumni.brown.edu> wrote in
> news:1147726711.432827.16440 @i39g2000cwa.googlegroups.com:
> 
>> 
>> b83503104@yahoo.com wrote:
>>> Tukey (inventor of FFT)
>> 
>> Tukey did not invent the FFT.
>> 
>> Oscar Buneman, my PhD advisor at Stanford, used the FFT in
>> 1939 during his pioneering WWII research that explains how the
>> radar klystron works. His PhD students had been using it for years
>> before I came in 1964.
>> 
>> Imagine our research group's surprise when Cooley and Tukey
>> published their paper in 1965.
>> 
>> No, Oscar did not invent the FFT either. According to him, it was
>> a well known technique in Germany during the early 1930s and
>> was probably discovered well before 1900.
> 
> The FFT is not much use without electronic calculators. Do you have a
> cite for this "well known technique" prior to 1900 or even the 1930's.
> 
> Klazmon.

OK. I looked into this myself. It appears that none other than Carl Gauss
figured out the key result in 1805. 

Klazmon.



> 
> 
>>SNIP>
> 

0
Llanzlan
5/15/2006 11:18:32 PM
In article <oS6ag.3086$KB.904@fed1read08>,
Richard Henry <rphenry@home.com> wrote:

>"Greg Heath" <heath@alumni.brown.edu> wrote in message
>news:1147726711.432827.16440@i39g2000cwa.googlegroups.com...

>> b83503104@yahoo.com wrote:
>> > Tukey (inventor of FFT)

>> Tukey did not invent the FFT.

>> Oscar Buneman, my PhD advisor at Stanford, used the FFT in
>> 1939 during his pioneering WWII research that explains how the
>> radar klystron works. His PhD students had been using it for years
>> before I came in 1964.

>I am curious to know what type of Pre-WWII electronics could have used an
>FFT.


I do not see that electronics are necessary.  By 1939,
there were mechanical and electro-mechanical desk
calculators; I have used such to good advantage.

Fourier did his work in the early 1800s.  Many papers 
before WWII give computations of Fourier transforms
of numerical series.  As far as we can tell, Gauss
invented the FFT, but published it in an obscure 
place, as he could not see much value in it.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
0
hrubin
5/16/2006 12:20:47 AM
Richard Henry wrote:
> "Greg Heath" <heath@alumni.brown.edu> wrote in message
> news:1147726711.432827.16440@i39g2000cwa.googlegroups.com...
> >
> > b83503104@yahoo.com wrote:
> > > Tukey (inventor of FFT)
> >
> > Tukey did not invent the FFT.
> >
> > Oscar Buneman, my PhD advisor at Stanford, used the FFT in
> > 1939 during his pioneering WWII research that explains how the
> > radar klystron works. His PhD students had been using it for years
> > before I came in 1964.
>
> I am curious to know what type of Pre-WWII electronics could have used an
> FFT.

Oscar was of Jewish heritage and left Germany for England in 1933.
When England declared war on Germany, he was interred and worked
as an applied mathematician in an English research lab.

In an effort to understand the physics of the radar klystron, he
simulated, on a mechanical computer, the trajectories of electrons in
a cylindrical geometry under the influence of combined applied and
self electric fields. I don't remember if an applied axial magnetic
field was present. If it was, it was easily incorporated.

The self electric fields were computed in the quasistatic
approximation using Poisson's equation. Since the geometry was
cylindrical, fourier transforming in the axial and azimuthal directions

reduced the partial differential equation in (r,theta,z) to an ordinary

differential equation in radius.

After the war he spent 5 years in nuclear physics research in Canada
before finding his niche in computational plasma physics at Stanford.
As a grad student in the late 60's I used the azimuthal fft to simulate

plasma and electron beams in cylindrical geometry. The simulations
led to the rediscovery of the breakup of hollow beams into rotating
vortex patterns (analgous to Karman vortex streets in hydrodynamics).
My PhD thesis was a theoretical analysis of the instability of the
hollow beam and the resulting stability of the rotating vortex
patterns.

Roger Hockey, another student of Oscar's, pioneered the use of our
simulation techniques in astronomical applications by simulating
the formation of spiral galaxies. He has written a book on
computational simulation which covers a lot of the details.

Hope this helps.

Greg

0
Greg
5/16/2006 12:40:13 AM
> So the question
> is, was the original string of symbols independently relevant or they
> merely as relevant as the algorithms used to evaluate them? I would
> like to know the answer to this question.

The original string of symbols and algorithms can create a framework 
where further systematic discussions is possible, measurements are 
possible and predictions can be made.

For example political science models you often look at a few selected 
factors to access the like hood of an event to occur. 


0
BernardZ
5/16/2006 11:36:18 AM
In article <1147740013.573048.310170@v46g2000cwv.googlegroups.com>,
Greg Heath <heath@alumni.brown.edu> wrote:

>Oscar was of Jewish heritage and left Germany for England in 1933.
>When England declared war on Germany, he was interred and worked
>as an applied mathematician in an English research lab.

You mean "interned", I hope.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
0
israel
5/16/2006 8:31:02 PM
In article <e4dcq6$il5$1@nntp.itservices.ubc.ca>,
 israel@math.ubc.ca (Robert Israel) wrote:

> In article <1147740013.573048.310170@v46g2000cwv.googlegroups.com>,
> Greg Heath <heath@alumni.brown.edu> wrote:
> 
> >Oscar was of Jewish heritage and left Germany for England in 1933.
> >When England declared war on Germany, he was interred and worked
> >as an applied mathematician in an English research lab.
> 
> You mean "interned", I hope.
> 
> Robert Israel                                israel@math.ubc.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel 
> University of British Columbia            Vancouver, BC, Canada

How many others were interred in English research labs, I wonder?
0
Virgil
5/16/2006 8:53:51 PM
b83503104@yahoo.com wrote:
> Tukey (inventor of FFT) thinks that an approximate solution of the
> exact problem is often more useful than the exact solution of an
> approximate problem.
>
> I find it hard to argue which one is more important or useful.  Once
> you believe in one of them, your belief will lead your research style
> to either algorithm-centered or model-construction-centered.
>
> Anybody wants to elaborate on either of these two views?

If you don't specify a problem or a context, "which is better"
discussions are pointless.

Example:
If you want to prove certain properties about a system, it is
often very difficult without a closed form solution. As for
example, convergence or stability.

Example:
If you want to build some hardware, it is often only required
to demonstrate that certain values are within some range.
As for example, you might be interested in pressure or
stress on components that will fail if the value gets too large.
You may be able to prove this with an approximate
solution.

Example:
If your goal is to demonstrate that an approximate solution
method is "close enough" then you need something
authoritative. One such authoritative thing is an exact solution.
And a method of getting one is called "solution generation."
You work backwards from an exact solution that you know
isn't too massively unphysical, and work out what the various
system parameters and conditions would have to be in order
for that solution to be right. Then you compare your method
with that solution. For example, if you wanted to know the
temperature near some heat source, and you knew it was
very roughly parabolic, you could then work backwards from
a parabolic temperature profile, and work out what the
heat diffusion would have to be for that profile. Then you could
apply your numerical method to that system, and then compare
the answers to your exact solution.

So, you need to say what your problem is, and what your
goals w.r.t. that problem are, in order to know whether one
method or another is preferable.
Socks

0
puppet_sock
5/16/2006 9:13:57 PM
Robert Israel wrote:
> In article <1147740013.573048.310170@v46g2000cwv.googlegroups.com>,
> Greg Heath <heath@alumni.brown.edu> wrote:
>
> >Oscar was of Jewish heritage and left Germany for England in 1933.
> >When England declared war on Germany, he was interred and worked
> >as an applied mathematician in an English research lab.
>
> You mean "interned", I hope.

Yes.

It is very difficult to do anything, except decay, when you
are interred.

Greg

P.S. I did a web search to double check. Lo and behold, there are
many out there who make that mistake.

0
Greg
5/17/2006 12:47:58 AM
Virgil wrote:
> In article <e4dcq6$il5$1@nntp.itservices.ubc.ca>,
>  israel@math.ubc.ca (Robert Israel) wrote:
>
> > In article <1147740013.573048.310170@v46g2000cwv.googlegroups.com>,
> > Greg Heath <heath@alumni.brown.edu> wrote:
> >
> > >Oscar was of Jewish heritage and left Germany for England in 1933.
> > >When England declared war on Germany, he was interred and worked
> > >as an applied mathematician in an English research lab.
> >
> > You mean "interned", I hope.
> >
> > Robert Israel                                israel@math.ubc.ca
> > Department of Mathematics        http://www.math.ubc.ca/~israel
> > University of British Columbia            Vancouver, BC, Canada
>
> How many others were interred in English research labs, I wonder?

While surfing for unambiguous definintions of interred and interned,
I ran into usenet postings discussing US, Canadian, German, and
Japanese relocation, internment and concentration camps.

I don't think it would be too difficult to hone in on sources which
would give statistics for the British camps. However, I'm not sure
what categories would have been chosen for statistical summarization.

Hope this helps.

Greg

0
Greg
5/17/2006 1:00:10 AM
Llanzlan Klazmon wrote:

> OK. I looked into this myself. It appears that none other than Carl Gauss
> figured out the key result in 1805.

Wikipedia cites the following:

Carl Friedrich Gauss, "Nachlass: Theoria interpolationis methodo nova
tractata," Werke band 3, 265-327 (K=F6nigliche Gesellschaft der
Wissenschaften, G=F6ttingen, 1866). See also M. T. Heideman, D. H.
Johnson, and C. S. Burrus, "Gauss and the history of the fast Fourier
transform," IEEE ASSP Magazine 1 (4), 14-21 (1984).

Well, color me surprised.

0
Gene
5/17/2006 3:07:56 AM
Greg Heath wrote:
> Robert Israel wrote:
> > In article <1147740013.573048.310170@v46g2000cwv.googlegroups.com>,
> > Greg Heath <heath@alumni.brown.edu> wrote:
> >
> > >Oscar was of Jewish heritage and left Germany for England in 1933.
> > >When England declared war on Germany, he was interred and worked
> > >as an applied mathematician in an English research lab.
> >
> > You mean "interned", I hope.
>
> Yes.
>
> It is very difficult to do anything, except decay, when you
> are interred.

On the other hand, Jean Leray came up with the Leray spectral sequence
and other neat stuff in a POW camp in Austria. Many people have been
interred since then trying to understand it all. It's worked out well
in other fields also;  Olivier Messiaen wrote Quatuor pour la fin du
temps while a guest of the German government during World War II.

0
Gene
5/17/2006 3:18:07 AM
"Schoenfeld" <schoenfeld1@gmail.com> wrote in message 
news:1147676627.082933.272060@j33g2000cwa.googlegroups.com...
>
> b83503104@yahoo.com wrote:
>> Tukey (inventor of FFT) thinks that an approximate solution of the
>> exact problem is often more useful than the exact solution of an
>> approximate problem.
>>
>> I find it hard to argue which one is more important or useful.  Once
>> you believe in one of them, your belief will lead your research style
>> to either algorithm-centered or model-construction-centered.
>
> Your question is of central importance to the philosophy of
> mathematics, one which can be appreciated only by those who actually
> _do_ the mathematics, not just claim they do.
>
> I would categorize the two styles:
>  1. Symbolic mathematics
>  2. Computational mathematics
>
> In my experience, I cannot decide between the two. The 'most important'
> or 'most fundamental' choice depends on the areas of mathematics you
> are working in. If you are messing around in number theory, the exact
> symbolic solution is usually _the only solution_ that matters (i.e.
> "what looks nicest is better"). You can approximate an exact symbolic
> solution so many other ways and the symbolisms can be, seamingly,
> _irreconcilable_.
>
>
> However, if you are dealing in computational mathematics, things like
> Neural networks, self-organizing maps, cellular automata, then accurate
> but computationally simpler approximations seem to be more fundamenta
> (i.e  the least complex algorithms)
>
>
>
>
>
>> Anybody wants to elaborate on either of these two views?
>
++++++++++++++++++++++++++++++++++++++++++++++++++++
This has been a very interesting thread.

Now where does the process of reducing data by statistical methods fit in? 
Is it 1 or 2.?

As Bob L and others have pointed out, statistics uses approximations of 
conceptual reality. This conceptual reality,(i.e randomness exists as a 
mathematical model) is not clear. How can randomness be constructed, Does it 
only exist in computational mathematics? What is an exact mathematical model 
of randomness?

How do the mathematicians deal with the randomness of "quarks"? and the 
possibility of objects moving from a finite mathematical structure to an 
unknown, incomplete entity with undefined boundaries?

Are our statements about the interval that a population mean lies within, 
strictly computational mathematics?

What is the meaning of an exact solution in computational mathematics, given 
the fact that the realized computational process is finite and limited? 
Gentile clearly said that computers do no do exact mathematics. Is a 
computer output then an approximate solutions to an exact problem?

The more I think about this, the more I understand the inability to frame 
inquires such that they are "well structured" for logical combinations and 
results (e.g. The Oxford school).

Other than the above, I thought this thread was excellent, and all 
participants are to be thanked for their contribution.

DAH



0
David
5/17/2006 3:30:20 AM
Gene Ward Smith wrote:
> Greg Heath wrote:
> > Robert Israel wrote:
> > > In article <1147740013.573048.310170@v46g2000cwv.googlegroups.com>,
> > > Greg Heath <heath@alumni.brown.edu> wrote:
> > >
> > > >Oscar was of Jewish heritage and left Germany for England in 1933.
> > > >When England declared war on Germany, he was interred and worked
> > > >as an applied mathematician in an English research lab.
> > >
> > > You mean "interned", I hope.
> >
> > Yes.
> >
> > It is very difficult to do anything, except decay, when you
> > are interred.
>
> On the other hand, Jean Leray came up with the Leray spectral sequence
> and other neat stuff in a POW camp in Austria. Many people have been
> interred since then trying to understand it all. It's worked out well
> in other fields also;  Olivier Messiaen wrote Quatuor pour la fin du
> temps while a guest of the German government during World War II.

Sorry, my reply was intended to be clever. I was so clever that I
failed
to clarify the main point:

"Internment" is synonymous with "confinement"

whereas

"Interrment" is synonymous with "burial" !

Hope this helps.

Greg

0
Greg
5/17/2006 8:27:04 AM
In article <lNwag.131$Fw1.184417@news.sisna.com>,
David A. Heiser <daheiser@gvn.net> wrote:

>"Schoenfeld" <schoenfeld1@gmail.com> wrote in message 
>news:1147676627.082933.272060@j33g2000cwa.googlegroups.com...

>> b83503104@yahoo.com wrote:

			......................

>> However, if you are dealing in computational mathematics, things like
>> Neural networks, self-organizing maps, cellular automata, then accurate
>> but computationally simpler approximations seem to be more fundamenta
>> (i.e  the least complex algorithms)





>>> Anybody wants to elaborate on either of these two views?

>++++++++++++++++++++++++++++++++++++++++++++++++++++
>This has been a very interesting thread.

>Now where does the process of reducing data by statistical methods fit in? 
>Is it 1 or 2.?

>As Bob L and others have pointed out, statistics uses approximations of 
>conceptual reality. This conceptual reality,(i.e randomness exists as a 
>mathematical model) is not clear. How can randomness be constructed, Does it 
>only exist in computational mathematics? What is an exact mathematical model 
>of randomness?

Randomness cannot be "constructed".  While the ideas of
probability and randomness might have originated from
repeated events under "identical" conditions, the
fundamental concepts in probability are those of the 
unrepeatable event, and related ideas such as random
variable.  These can be REPRESENTED as subsets of a 
measure space and measurable functions on such a space,
but this is a representation, not the concept itself.

>How do the mathematicians deal with the randomness of "quarks"? and the 
>possibility of objects moving from a finite mathematical structure to an 
>unknown, incomplete entity with undefined boundaries?

The representation here is in terms of quantum processes,
which are far worse than random processes.  However, the
observations form a random process; it is what goes on
between the observations which is not too well understood.

>Are our statements about the interval that a population mean lies within, 
>strictly computational mathematics?

>What is the meaning of an exact solution in computational mathematics, given 
>the fact that the realized computational process is finite and limited? 
>Gentile clearly said that computers do no do exact mathematics. Is a 
>computer output then an approximate solutions to an exact problem?

Usually by "exact" solution we mean withing acceptable
computational error.

>The more I think about this, the more I understand the inability to frame 
>inquires such that they are "well structured" for logical combinations and 
>results (e.g. The Oxford school).

>Other than the above, I thought this thread was excellent, and all 
>participants are to be thanked for their contribution.

>DAH





-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
0
hrubin
5/18/2006 7:57:30 PM
b83503104@yahoo.com wrote:
> Tukey (inventor of FFT) thinks that an approximate solution of the
> exact problem is often more useful than the exact solution of an
> approximate problem.

In most cases the problem is what is the problem.

>
> I find it hard to argue which one is more important or useful.  Once
> you believe in one of them, your belief will lead your research style
> to either algorithm-centered or model-construction-centered.

I wonder if you think there are algorithmic solutions to problems that
do not require a model of the problem to be solved.

>
> Anybody wants to elaborate on either of these two views?

You views are the outcome of a totally distorted and wrong view of how
science (and engineering) works.

Mike

0
Mike
5/18/2006 8:32:29 PM
Most models, if carefully structured, are by definition, �exact� (see 
class A, below), with the qualifications of Godel's 1931 "Incompleteness 
Theorem".

On the other hand, all empirical statements are approximations, at best. 
(See class B.)

David Hume, �An Enquiry Concerning Human Understanding� (1777 edition)
<http://www.etext.leeds.ac.uk/hume/ehu/ehupbsb.htm> first addressed this 
problem with clarity:

He divided all statements of language into two classes:

A. Statements that concern relations only among words, ideas or symbols: 
[Those relations are the propositions, definitions and rules of 
language, mathematics and deductive logic; they�re social contracts of 
convenience, designed to keep us on the same page.] Such statements are 
absolutely true or false (or undecidable, if some critical definition or 
premise is missing), only as a result of prior agreement about how 
they�re to be used. And,

B. empirical statements, which are the province of inductive �logic�, 
concern �matters of fact�. [Inductive reasoning is defined as the 
process of extrapolating or interpolating from observation.] Empirical 
statements depend upon sets of data that are always incomplete, partial 
samplings of an as yet unobserved whole. Hume was the first to note that 
there�s no logical way to guarantee that future observations will track 
those of the past. This was his 'incompleteness theorem'.

The overall task of science can be modeled as the problem of determining 
which class A statements (idealistic hypothetical models) best fit 
well-established class B statements (the realistic facts). This sub-set 
of class A statements provides our most confident description of 
physical and biological reality. Most remaining class A statements may 
have other values (e.g., aesthetic, emotional, logical, mathematical, 
ideological, religious) but generally fall outside the realm of science.

Within this model, there will always be some class A statements that lie 
in limbo; theoretical constructions, seemingly empirically verifiable, 
but so far neither supported nor refuted by �direct� observation. Based 
on what may be deduced from the �rest of reality�, these currently seem 
either quite possible (Higgs boson, gravity waves, String Theory) or 
improbable (impenetrable shields against ballistic missiles, caches of 
Iraqi WMD, extraterrestrial intelligence, etc.). They�re science fiction 
today, but perhaps science fact tomorrow.

Hope this helps.

Len Ornstein


Mike wrote:
> b83503104@yahoo.com wrote:
>> Tukey (inventor of FFT) thinks that an approximate solution of the
>> exact problem is often more useful than the exact solution of an
>> approximate problem.
> 
> In most cases the problem is what is the problem.
> 
>> I find it hard to argue which one is more important or useful.  Once
>> you believe in one of them, your belief will lead your research style
>> to either algorithm-centered or model-construction-centered.
> 
> I wonder if you think there are algorithmic solutions to problems that
> do not require a model of the problem to be solved.
> 
>> Anybody wants to elaborate on either of these two views?
> 
> You views are the outcome of a totally distorted and wrong view of how
> science (and engineering) works.
> 
> Mike
> 
0
Leonard
5/19/2006 5:28:42 PM
Leonard Ornstein wrote:
> Most models, if carefully structured, are by definition, 'exact' (see
> class A, below), with the qualifications of Godel's 1931 "Incompleteness
> Theorem".
>
> On the other hand, all empirical statements are approximations, at best.
> (See class B.)
>
> David Hume, "An Enquiry Concerning Human Understanding" (1777 edition)
> <http://www.etext.leeds.ac.uk/hume/ehu/ehupbsb.htm> first addressed this
> problem with clarity:

Philosophy has progressed a lot since them. Hume's analyis was very
simplistic although he was right about the problem of induction from a
philosophical perspective only.

>
> He divided all statements of language into two classes:
>
> A. Statements that concern relations only among words, ideas or symbols:
> [Those relations are the propositions, definitions and rules of
> language, mathematics and deductive logic; they're social contracts of
> convenience, designed to keep us on the same page.] Such statements are
> absolutely true or false (or undecidable, if some critical definition or
> premise is missing), only as a result of prior agreement about how
> they're to be used. And,


of ourse, he said nothing new, nothing that Aristotle has not said
already with his categorical logic.

>
> B. empirical statements, which are the province of inductive 'logic',
> concern "matters of fact". [Inductive reasoning is defined as the
> process of extrapolating or interpolating from observation.] Empirical
> statements depend upon sets of data that are always incomplete, partial
> samplings of an as yet unobserved whole. Hume was the first to note that
> there's no logical way to guarantee that future observations will track
> those of the past. This was his 'incompleteness theorem'.

The problem of induction was known since antiquity. But there is what
is called "crying evidence".


> The overall task of science can be modeled as the problem of determining
> which class A statements (idealistic hypothetical models) best fit
> well-established class B statements (the realistic facts). This sub-set
> of class A statements provides our most confident description of
> physical and biological reality. Most remaining class A statements may
> have other values (e.g., aesthetic, emotional, logical, mathematical,
> ideological, religious) but generally fall outside the realm of science.
>

No. The problem of science today is to come up with class A statements
that generate new class B statements which in turn corroborate those
class A statements.

> Within this model, there will always be some class A statements that lie
> in limbo; theoretical constructions, seemingly empirically verifiable,
> but so far neither supported nor refuted by 'direct' observation. Based
> on what may be deduced from the 'rest of reality', these currently seem
> either quite possible (Higgs boson, gravity waves, String Theory) or
> improbable (impenetrable shields against ballistic missiles, caches of
> Iraqi WMD, extraterrestrial intelligence, etc.). They're science fiction
> today, but perhaps science fact tomorrow.
>

> Hope this helps.

It is too simplistic and antiquated to add anything of value.

Mike



>
> Len Ornstein
>
>
> Mike wrote:
> > b83503104@yahoo.com wrote:
> >> Tukey (inventor of FFT) thinks that an approximate solution of the
> >> exact problem is often more useful than the exact solution of an
> >> approximate problem.
> >
> > In most cases the problem is what is the problem.
> >
> >> I find it hard to argue which one is more important or useful.  Once
> >> you believe in one of them, your belief will lead your research style
> >> to either algorithm-centered or model-construction-centered.
> >
> > I wonder if you think there are algorithmic solutions to problems that
> > do not require a model of the problem to be solved.
> >
> >> Anybody wants to elaborate on either of these two views?
> >
> > You views are the outcome of a totally distorted and wrong view of how
> > science (and engineering) works.
> > 
> > Mike
> >

0
Mike
5/19/2006 11:53:44 PM
Mike:

Mike wrote:
> Leonard Ornstein wrote:
>> Most models, if carefully structured, are by definition, 'exact' (see
>> class A, below), with the qualifications of Godel's 1931 "Incompleteness
>> Theorem".
>>
>> On the other hand, all empirical statements are approximations, at best.
>> (See class B.)
>>
>> David Hume, "An Enquiry Concerning Human Understanding" (1777 edition)
>> <http://www.etext.leeds.ac.uk/hume/ehu/ehupbsb.htm> first addressed this
>> problem with clarity:
> 
> Philosophy has progressed a lot since them. Hume's analysis was very
> simplistic although he was right about the problem of induction from a
> philosophical perspective only.

Could you elaborate on "from a philosophical perspective ONLY"?
> 
>> He divided all statements of language into two classes:
>>
>> A. Statements that concern relations only among words, ideas or symbols:
>> [Those relations are the propositions, definitions and rules of
>> language, mathematics and deductive logic; they're social contracts of
>> convenience, designed to keep us on the same page.] Such statements are
>> absolutely true or false (or undecidable, if some critical definition or
>> premise is missing), only as a result of prior agreement about how
>> they're to be used. And,
> 
> 
> of course, he said nothing new, nothing that Aristotle has not said
> already with his categorical logic.
> 
>> B. empirical statements, which are the province of inductive 'logic',
>> concern "matters of fact". [Inductive reasoning is defined as the
>> process of extrapolating or interpolating from observation.] Empirical
>> statements depend upon sets of data that are always incomplete, partial
>> samplings of an as yet unobserved whole. Hume was the first to note that
>> there's no logical way to guarantee that future observations will track
>> those of the past. This was his 'incompleteness theorem'.
> 
> The problem of induction was known since antiquity. But there is what
> is called "crying evidence".
>
Do you mean that the evidence for the incompleteness of induction was 
crying out to be recognized? But, by raising the case from the implicit 
to explicit level, Hume changed the world for the rest of us.

> 
>> The overall task of science can be modeled as the problem of determining
>> which class A statements (idealistic hypothetical models) best fit
>> well-established class B statements (the realistic facts). This sub-set
>> of class A statements provides our most confident description of
>> physical and biological reality. Most remaining class A statements may
>> have other values (e.g., aesthetic, emotional, logical, mathematical,
>> ideological, religious) but generally fall outside the realm of science.
>>
> 
> No. The problem of science today is to come up with class A statements
> that generate new class B statements which in turn corroborate those
> class A statements.

Certainly you're right; for science to keep moving ahead, new 
theoretical science needs to be generated to stimulate new experimental 
science. That small correction doesn't undo the model.
> 
>> Within this model, there will always be some class A statements that lie
>> in limbo; theoretical constructions, seemingly empirically verifiable,
>> but so far neither supported nor refuted by 'direct' observation. Based
>> on what may be deduced from the 'rest of reality', these currently seem
>> either quite possible (Higgs boson, gravity waves, String Theory) or
>> improbable (impenetrable shields against ballistic missiles, caches of
>> Iraqi WMD, extraterrestrial intelligence, etc.). They're science fiction
>> today, but perhaps science fact tomorrow.
>>
> 
>> Hope this helps.
> 
> It is too simplistic and antiquated to add anything of value.
> 
The aim was to remind that theory can be exact, but our knowledge of 
reality never can be; and in the end, it's confident understanding of 
reality and an ability to 'predict' outcomes that's science's main job.

Algorithm-centered and model-construction-centered theories should both 
lead to observations and experiments. How productive the experiments 
turn out to be is really what counts. Sometimes the first will be more 
productive; sometimes the second. And sometimes we have to wait an 
awfully long time to find out whether a particular model isn't just 
science fiction. Is that too simplistic?

Len

> Mike
> 
> 
> 
>> Len Ornstein
>>
>>
>> Mike wrote:
>>> b83503104@yahoo.com wrote:
>>>> Tukey (inventor of FFT) thinks that an approximate solution of the
>>>> exact problem is often more useful than the exact solution of an
>>>> approximate problem.
>>> In most cases the problem is what is the problem.
>>>
>>>> I find it hard to argue which one is more important or useful.  Once
>>>> you believe in one of them, your belief will lead your research style
>>>> to either algorithm-centered or model-construction-centered.
>>> I wonder if you think there are algorithmic solutions to problems that
>>> do not require a model of the problem to be solved.
>>>
>>>> Anybody wants to elaborate on either of these two views?
>>> You views are the outcome of a totally distorted and wrong view of how
>>> science (and engineering) works.
>>>
>>> Mike
>>>
> 
0
Leonard
5/20/2006 1:21:01 AM
Greg Heath wrote:

> Gene Ward Smith wrote:
> > Greg Heath wrote:
> > > Robert Israel wrote:
> > > > In article <1147740013.573048.310170@v46g2000cwv.googlegroups.com>,
> > > > Greg Heath <heath@alumni.brown.edu> wrote:
> > > >
> > > > >Oscar was of Jewish heritage and left Germany for England in 1933.
> > > > >When England declared war on Germany, he was interred and worked
> > > > >as an applied mathematician in an English research lab.
> > > >
> > > > You mean "interned", I hope.
> > >
> > > Yes.
> > >
> > > It is very difficult to do anything, except decay, when you
> > > are interred.
> >
> > On the other hand, Jean Leray came up with the Leray spectral sequence
> > and other neat stuff in a POW camp in Austria. Many people have been
> > interred since then trying to understand it all. It's worked out well
> > in other fields also;  Olivier Messiaen wrote Quatuor pour la fin du
> > temps while a guest of the German government during World War II.
>
> Sorry, my reply was intended to be clever. I was so clever that I
> failed
> to clarify the main point:
>
> "Internment" is synonymous with "confinement"
>
> whereas
>
> "Interrment" is synonymous with "burial" !
>
> Hope this helps.

Hoary joke has Beethoven continuing to work while interred.

Researcher opening tomb finds great man sitting up at sarcophagus,
erasing sheet after sheet of music manuscript.  "What are you doing
sir!", the startled investigator asks.  "Decomposing".

0
Edward
5/20/2006 7:28:45 PM
In article <1148153325.061905.224880@u72g2000cwu.googlegroups.com>,
   "Edward Green" <spamspamspam3@netzero.com> wrote:
<snip setup>

>Hoary joke has Beethoven continuing to work while interred.
>
>Researcher opening tomb finds great man sitting up at sarcophagus,
>erasing sheet after sheet of music manuscript.  "What are you doing
>sir!", the startled investigator asks.  "Decomposing".

<GROAN>

/BAH

0
jmfbahciv
5/21/2006 10:59:10 AM
Reply: