Solve exact problem with approx solution, or solve approx problem with exact solution
b835...@yahoo.com
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?
Sam Wormley
Interesting background on FFTs
http://mathworld.wolfram.com/FastFourierTransform.html
http://en.wikipedia.org/wiki/Fast_Fourier_transform
Schoenfeld
b835...@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)
BernardZ
schoe...@gmail.com says...
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
schoe...@gmail.com
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.
Reef Fish
b835...@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'll first take a Data Analyst (Applied Statistician's) view -- an easy
one.
"The most important maxim for data analysis to heed, and on which
many statisticians seem to have shunned, is this: Far better an
approximate answer to the right question, which is often vague, than
an exact answer to the wrong question, which can always be made
precise." (p. 13; Tukey's 1962 AMS "The Future of Data Analysis").
I can give the exact citation because it's in my "Data Analysis"
lecture
notes (unpublished textbook) which I have used since 1970.
I also had a "Theorem", the only one in my book, in Chapter 1,
which states: Nothing in the Real World is Normally DIstributed.
Then I went on to discuss the importance of using normal theory
as a useful approximation, and how to distinguish normal from
nonnormal data via normal probability plots.
In that respect, my Theorem preceded George Box's 1976 JASA
paper on "Science and Statistics" in which we independently
come to the same conclusion:
"the statistician knows, for example, that in nature there never was
a normal distribution, there never was a straight line, yet with
normal and linear transformation, known to be false, he can often
derive results, which match, to a useful approximation, those
found in the real world."
(p. 792, George Box 1976 JASA paper on "Science and Statistics".
The above citation was added to later versions of my Lecture Notes.
Those quotes pretty much summarized my own view about the
question posed. In that respect, one might say BOTH the
statement are wrong since nothing is EXACT in statistics (other
than theory), and theory is to GUIDE rather than command.
"If data analysis is to be well done, much of it must be a matter of
judgment, and 'thoery', whether statistical or non-statistical, will
have to guide, not command." (p. 10, Tukey's 1962 AMS paper.)
What's important in Statistics, is to seek a solution to the RIGHT
problem, rather than solving a wrong problem by making it precise.
-- Bob.
Reef Fish
b835...@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.
>From a Pure Mathematician's point of view, where everything is
axiomatic and every theorem is either PROVEN or not proven, the
mere suggestion of an approximate solution to a "theorem" that
is EXACT is unthinkable!
That's why the Four-Color-Problem (1852) remain unsolved for
over a century, and Fermat's Last Theorem (circa 1630) stood
unproved for nearly 4 centuries even though Sophie Germain
had proved it for n less than or equal to 100, and other
mathematicians extended the correctness of that Conjecture
to many other proven cases.
There are so many theorems in mathematics that one might
say their APPROXIMATE solution had been found -- by having
enumerated large number of cases without finding a
counterexample.
But those are NOT solutions.
In that sense, one could almost say the engineer's proof that
"every odd integer is a prime" an "approximate solution".
"1 is a prime, 2 is a prime, 3 is a prime, 5 is a prime, 7 is
a prime, 9 is an experimental error, 11 is a prime, 13 ..."
thought that would not be a very good approximation. :-)
One might say Goldbach's conjecture (1742) is correct, to an
excellent approximation, because no counterexample has
been found, yet.
In short, there ain't such thing as an approximate solution to
an EXACT mathematical statement.
That leaves only the APPLIED mathematicians.
-- Bob.
Reef Fish
I have already given my view in the shoes of a Pure Mathematician and
a Data Analyst (Applied Statistician). The answers were clear cut.
The question of importance or usefulness ot the statements in question
impacts only the Applied Mathematicians -- in which I have a THIRD
shoe, because some of the worse applied statisticians are applied
mathematicians!
George Box (him again) calls it Mathematistry.
"Mathematistry is characterized by development of theory for theory's
sake, which since it seldom touches down with practice, has a
tendency to redefine the problem rather than solve it."
(p,797, 1976 JASA paper on "Science and Statistics").
That is one example of finding the (wrong) solution to an approximate
problem, by defining (the wrong problem) an Exact problem to solve.
What it leaves, IMHO, is the ONLY valid approach in applied mathematics
(and applied statistics for that matter) is to find an approximate
solution
to an approximate (but correct) problem. And IF an exact solution can
be found for a near-equivalent of the approximate problem, then one
must ensure that the exact solution of the exact problem is a valid
approximation of the approximate problem! :-)
There are many well-defined mathematical problems in partition, such
as those found in "cluster analysis" and in graph theory, where the
exact enumeration of graph structures (in random or deterministic
graphs) is computational prohibitive, and virtually impossible for
large number of "nodes". For those problems, the ONLY feasible
approach is to use known numerical methods to find approximate
solutions.
-- Bob.
Herman Rubin
b835...@yahoo.com <b835...@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
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Ken S. Tucker
I'll bite, Tucker's solution, the Exact solution
to the Exact Problem = E(P) = 0.
Ken
Greg Heath
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
datam...@gmail.com
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.
Richard Henry
"Greg Heath" <he...@alumni.brown.edu> wrote in message
news:1147726711....@i39g2000cwa.googlegroups.com...
>
> b835...@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.
Phil Carmody
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
mme...@cars3.uchicago.edu
create an exact solution for an exact problem formulation.
Mati Meron | "When you argue with a fool,
me...@cars.uchicago.edu | chances are he is doing just the same"
Llanzlan Klazmon
@i39g2000cwa.googlegroups.com:
>
> b835...@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>
Llanzlan Klazmon
news:Xns97C5719B522A6Kl...@203.97.37.6:
OK. I looked into this myself. It appears that none other than Carl Gauss
figured out the key result in 1805.
Klazmon.
>
>
>>SNIP>
>
Herman Rubin
In article <oS6ag.3086$KB.904@fed1read08>,
Richard Henry <rph...@home.com> wrote:
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.
Greg Heath
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
John Doe
news:1147676627.0...@j33g2000cwa.googlegroups.com...
>
> 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)
Good topic.
As a practicing engineer, I consider myself a mathemetician on occassion.
Math is purely a means to my end, which is to obtain a solution to the
problem at hand. I do attempt symbolic solutions, but these solutions are
almost always the answer to a simplified version of the problem. Symbolic
partial solutions are helpful to verify the full computational solution is
in the right neighborhood, but usually not much more. I would put
computational vs. symbolic results at a 20:1 ratio for my work. I do prefer
the computational approach.
BernardZ
> 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.
Jerry Dallal
If two people working together produce something better than they could
working alone, it's science.
If two people working together produce something less than they could
working alone, it's art.
Reef Fish
Cook's (NOT Dennis Cook of course <g>) Generalization:
If N Cooks, N > 2, working together to produce a dish less
appetizing than what each of them could have produced
working alone, it's called Cook's (Spoilt) Broth.
-- Bob.
Robert Israel
Greg Heath <he...@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 isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Virgil
isr...@math.ubc.ca (Robert Israel) wrote:
How many others were interred in English research labs, I wonder?
puppe...@hotmail.com
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
Greg Heath
Robert Israel wrote:
> In article <1147740013.5...@v46g2000cwv.googlegroups.com>,
> Greg Heath <he...@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.
Greg Heath
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
Gerry Myerson
Jerry Dallal <gda...@nospam.tufts.edu> wrote:
So, Gilbert & Sullivan were doing science? and Rogers & Hammerstein?
and Torvill & Dean?
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Gene Ward Smith
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 (Königliche Gesellschaft der
Wissenschaften, Göttingen, 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.
Gene Ward Smith
Greg Heath wrote:
> Robert Israel wrote:
> > In article <1147740013.5...@v46g2000cwv.googlegroups.com>,
> > Greg Heath <he...@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.
David A. Heiser
"Schoenfeld" <schoe...@gmail.com> wrote in message
news:1147676627.0...@j33g2000cwa.googlegroups.com...
>
>> 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.
>
> 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)
>
>
>
>
>
>
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
Greg Heath
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
Jerry Dallal
(1) While there are some very nice pieces with two composers or two
lyricists involved, they generally sound more formulaic than when genius
comes from a single individual. (BTW, I'll take Hart over Hammerstein on
most days, Old Man River notwithstanding.)
(2) What is the sound of one hand clapping? :-)
Some things by definition--art, science, whatever--involve two people.
The rule kicks in after only after the minimum requirement is met. What
most people don't know is that Torvill & Dean started out as a
trio--Torvill, Dean, and (their American friend Scott) Hamilton. One
day, they were sitting around wondering why things were going so poorly
when all of a sudden it struck Jayne, "You know, boys, I hate to say it,
but I think this would go better if there were only two of us. You'll
have to draw straws." Scott had to admit that the trips to England for
practice were taking their financial toll and offered to try a solo
career. The rest is history!
(3) When I was growing up in the early '50s, my mom would try to get me
to eat by telling me that "children were starving in Europe". One day,
I challenged her to, "Name *one*!" When she couldn't, I had time to
ponder this as I sat in a corner and discovered a universal truth: When
one hears a sweeping statement to which no exceptions can be named or to
a short list is offered in refutation--that is, if the the statement is
not lame enough to be judged false *without* the need for any
counterexamples--then it's probably within hyperbole of being true!
Herman Rubin
David A. Heiser <dahe...@gvn.net> wrote:
>"Schoenfeld" <schoe...@gmail.com> wrote in message
>news:1147676627.0...@j33g2000cwa.googlegroups.com...
>> b835...@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
--
Mike
b835...@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
Leonard Ornstein
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
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
Leonard Ornstein
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
Edward Green
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".
jmfb...@aol.com
"Edward Green" <spamsp...@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