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Re: Re: Ansatz?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg101158] Re: [mg101124] Re: Ansatz?
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Thu, 25 Jun 2009 07:15:18 -0400 (EDT)
*References*: <h1nf0q$91h$1@smc.vnet.net> <h1qcub$7ul$1@smc.vnet.net> <200906241035.GAA23330@smc.vnet.net>
On 24 Jun 2009, at 19:35, magma wrote:
> On Jun 23, 12:05 pm, "M.Roellig" <markus.roel... at googlemail.com>
> wrote:
>> Hi,
>>
>>> Somewhere I've picked up the idea that "ansatz" can also be used to
>>> indicate the "form" or the "approach" -- more specifically,
>>> something
>>> like the choice of coordinates and variables and equations -- the
>>> "geometry and notation" so to speak -- in which one sets up a
>>> problem o=
> r
>>> a calculation.
>>
>> I would say, that this is the common understanding of ansatz in
>> science (at least for a native german speaker). An example would be
>> the
>> german word Loesungsansatz, meaning the initial choice of how to
>> approach
>> (and solve) a given problem,
>> e.g. the starting point of a mathematical proof or the set of initial
>> assumptions.
>>
>>> Wolfram MathWorld says:
>>
>>> An ansatz is an assumed form for a mathematical statement
>>> that is not based on any underlying theory or principle.
>>
>>> SEE ALSO: Conjecture, Hypothesis, Principle, Proposition
>>
>> So, to assume something without any prior derivation could be an
>> ansatz, but
>> usually an ansatz would be based on some reasonable assumptions or
>> additional
>> knowledge, so "not based on ANY underlying theory or principle"
>> sounds
>> too
>> much like a crystal ball.
>>
>> Cheers,
>>
>> Markus
>
> I agree with Markus.
> An Ansatz is an assumption, that is a statement which is considered
> (at least temporarily) true.
> The description given in wikipedia "not based on ANY underlying theory
> or principle" sounds to me more like the definition of "Axiom", which
> is a statement considered true permanently (at least within a given
> theory)
> During the course of a demonstration an Ansatz could turn out to be
> self-contradictory and therefore false (this happens in "reductio ad
> absurdum" proofs), while an Axion should never turn out self-
> contradictory (unless your theory is incoherent).
>
Could you give an reference to any use of "Ansatz" (in English
language papers) in which "an Ansatz could turn out to be self-
contradictory and therefore false" ? In my experience, in English,
the word Ansatz is never used in this sense. The most common use of
Ansatz occurs in the context of solving PDE's and boundary value
problems where it takes the following form: you assume that there is
a solution having some special form with some unknown parameters, you
substitute this in the equation and then find the parameters. Once you
have found the parameters the form is justified - a priori there was
no reason to assume that a solution of this form must exist.
Of course this approach could fail in principle but it never does - if
it did nobody would call this an "Ansatz".
This kind of meaning of ansatz seems to be most common in physics and
applied mathematics (for example the Bethe Ansatz. By the way, not
that here
http://en.wikipedia.org/wiki/Bethe_ansatz
Wikipedia suggests the words "conjecture" and "guess". Actually
"educated guess" seems to be more like it).
There are also some named Ansatzes in mathematics: for example the
Langlands Ansatz in the theory of elliptic curves, the Sinai-Chernov
Ansatz in the ergodic theory of billiards, which are actually theorems
which are used as starting point of some procedure.
However, I have never heard of an Ansatz that fails, i.e. an incorrect
guess. Of course many such would-be ansatzes are invented every day
but they generally to not survive long enough to be called an ansatz.
Andrzej Kozlowski
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