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Cauchy Integral Experiment
(10/08/2009, 07:02 AM)mike3 Wrote: ...
So did you have some sort of intuitive idea as to what it might look like before you actually went and ran the equation?
Yes. The idea is simple: <b>I MAKE THE SCIENCE</b>.

I see most of people, who make the science (or pretend that they make it), do not know, what is Science.

<b>Science</b> is special kind of human knowledge, human achievements, and human activity related to elaboration of the specific language, id est, system of concepts and notations for usable description of reproducible phenomena of any origin, characterized in the following:

[0] Applicability. The language is built up of the concepts. Each concept has limited range of applicability. For example, concepts that operate with "theory of everything" or "set of all possible sets" are not allowed.

[1] Verifiability. Each concept allows the verification: Within the terms of already accepted language, some experiment with some specific result, that confirms the concept, can be described.

[2] Negability. Each concept allows the negation. Within terms of this concept, some experiment with some specific result, that negates the concept, can be described.

[3] Self-consistence. No internal contradiction for the concept are known.

[4] Correspondence. If the area of applicability of a new concept evolves that of another already established and verified concept, then, in this area, the new concept either reproduces the results of the already verified concept, or indicate a way to negate the previously established concept.

[5] Pluralism and simplicity. Co-existence of mutually-contradictive concepts, satisfying criteria [0-4] above is allowed. If two contradicting concepts are successful in the same range of applicability, then the simplest one has priority and is considered as main.


From the definition it <b>follows</b>, that the concept of tetrational, that is almost constant in the most of complex plane, is <b>main</b>, and should be considered <b>first</b>.

Quote:As for why they didn't see it before, well the lack of powerful computers may have been a problem. In addition, I'm not sure if anyone knew the formula before, e.g. how would they have known that it decayed to the fixed points at imaginary infinity?
The researchers had to understand, that is sicence. (See above.) Then it becomes straightforward.

Unfortunately, the students in the USSR and, then, Russia, consider SCIENCE as set of dogmas and formulas they have to remember in order to pass the exams.

The students in the USA and Japan and some other countries consider Science as a tool to make a good business.

With such an ideology, they do not have much opportunity to make the real Science.
So then you got it via the scientific process -- you first hypothesized that it should decay to a fixed point, as that would be a simple type of behavior. Then combined with the postulate of the holomorphism, the tetrational can be recovered via the Cauchy integral. And that provides a test: construct it via the integral given the supposed behavior, and see what happens. And the test was done, and it worked. So now as this is mathematics, I guess the next step would be the formal proof of the existence and uniqueness of the solution.
(10/10/2009, 01:05 AM)mike3 Wrote: I guess the next step would be the formal proof of the existence and uniqueness of the solution.

exactly Wink

On the other hand there is often a situation where you work with a certain law without having a proof that it is always true (e.g. map publisher relied long before having a proven basis that a map needs at most 4 colors).

Or in other words I think such proofs are really difficult to find.

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