Difference between revisions of "Pluralism"

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Some links are suggesed at http://en.wikipedia.org/wiki/Occam's_razor
  
Some links are suggesed at http://en.wikipedia.org/wiki/Occam's_razor
   
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The criterion of simplicity allows to tolerate many different scientific concepts.
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==[[Superfunctions]]==
  
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This criterion indicates, which of them should be applied first.
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There is no need to declare other concepts as "wrong"; often, it is sufficient to say, that they are "more complicated" in the description of the same object.
   
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==[[Tetration]] and other [[Superfunctions]]==
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The [[Pluralism]] as idea of simplicity, had been used to guess the asymptotic behaviour of the natural [[tetration]] at $\pm \mathrm i \infty$
  
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<ref>
  
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The [[Pluralism]], as idea of simplicity, had been used to guess the asymptotic behaviour of [[tetration]] and [[arctetration]] at \(\pm \mathrm i \infty\)
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http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br>
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<ref>http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html<br><!--
 
http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf <br>
  
http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf <br>
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http://mizugadro.mydns.jp/PAPERS/analuxp99.pdf
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http://mizugadro.mydns.jp/PAPERS/analuxp99.pdf<br>!-->
 
D.Kouznetsov. Solutions of F(z+1)=exp(F(z)) in the complex z plane. Mathematics of Computation, 78 (2009) 1647-1670
  
D.Kouznetsov. Solutions of F(z+1)=exp(F(z)) in the complex z plane. Mathematics of Computation, 78 (2009) 1647-1670
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</ref>.
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</ref>. Then, the same idea is applied to other [[superfunctions]]
  
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  ⚫
The same idea can be applied to other [[superfunctions]]
 
<ref>
  
<ref>
 
http://www.ils.uec.ac.jp/~dima/BOOK/443.pdf D.Kouznetsov. [[Superfunctions]]. 2017.<br>
  
http://www.ils.uec.ac.jp/~dima/BOOK/443.pdf D.Kouznetsov. [[Superfunctions]]. 2017.<br>
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</ref>.
  
</ref>.
   
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Certain postulates about of simplicity of the [[superfunction]] and [[abelfunction]] lead to uniqueness of the solution. In particular, [[tetration]] not only exist, but also is unique super exponential, that have simple behaviour at infinity.
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Certain postulates about of simplicity of the [[superfunction]] and the [[abelfunction]] lead to uniqueness of the solution.
  +
In particular, [[tetration]] not only exist, but also is unique super exponential, that have simple behaviour at infinity.
 
<ref>
  
<ref>
 
http://www.springerlink.com/content/u7327836m2850246/
  
http://www.springerlink.com/content/u7327836m2850246/
 
http://mizugadro.mydns.jp/PAPERS/2011uniabel.pdf
  
http://mizugadro.mydns.jp/PAPERS/2011uniabel.pdf
 
H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)
  
H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)
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</ref><ref>
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http://dx.doi.org/10.11568/kjm.2016.24.1.81
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William Paulsen.
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Finding the natural solution to \(f(f(x)) = \exp(x)\)
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Korean J. Math. Vol 24, No 1 (2016) pp.81-106.
 
</ref><ref>
  
</ref><ref>
 
http://myweb.astate.edu/wpaulsen/tetration2.pdf
 
http://myweb.astate.edu/wpaulsen/tetration2.pdf
 
http://link.springer.com/article/10.1007/s10444-017-9524-1
 
http://link.springer.com/article/10.1007/s10444-017-9524-1
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William Paulsen and Samuel Cowgill. Solving $F(z+1)=b^{F(z)}$ in the complex plane. Advances in Computational Mathematics, 2017 March 7, p. 1–22
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William Paulsen and Samuel Cowgill. Solving \(F(z+1)=b^{F(z)}\) in the complex plane. Advances in Computational Mathematics, 2017 March 7, p. 1–22.
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.. we study the fractional iterates of the exponential function. This is an unresolved problem, not due to a lack of a known solution, but because there are an infinite number of solutions, and there is no agreement as to which solution is "best.'' We will approach the problem by first solving Abel's functional equation \(\alpha(e^x) = \alpha(x) + 1\) by perturbing the exponential function so as to produce a real fixed point, allowing a unique holomorphic solution. We then use this solution to find a solution to the unperturbed problem. However, this solution will depend on the way we first perturbed the exponential function. Thus, we then strive to remove the dependence of the perturbed function. Finally, we produce a solution that is in a sense more natural than other solutions.
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</ref>
  
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</ref>:<br><i>
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.. we study the fractional iterates of the exponential function. This is an unresolved problem, not due to a lack of a known solution, but because there are an infinite number of solutions, and there is no agreement as to which solution is "best.'' We will approach the problem by first solving Abel's functional equation \(\alpha(e^x) = \alpha(x) + 1\) by perturbing the exponential function so as to produce a real fixed point, allowing a unique holomorphic solution. We then use this solution to find a solution to the unperturbed problem. However, this solution will depend on the way we first perturbed the exponential function. Thus, we then strive to remove the dependence of the perturbed function. Finally, we produce a solution that is in a sense more natural than other solutions.
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</i>
   
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Some examples of search for simplest concept (and to choose it) are mentioned in article [[Place of science in the human knowledge]].
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Several examples of search for simplest concept (and to choose it) are mentioned in article [[Place of science in the human knowledge]]
 
<ref>
  
<ref>
 
http://budclub.ru/k/kuznecow_d_j/2010mestoe.shtml D.Kouznetsov. Place of Science in the human knowledge (2010). <br>
  
http://budclub.ru/k/kuznecow_d_j/2010mestoe.shtml D.Kouznetsov. Place of Science in the human knowledge (2010). <br>
 
http://ufn.ru/tribune/trib120111 Д.Кузнецов. Место науки и физики в человеческом знании. Трибуна УФН, 2010–2011, No.110. (In Russian) <!--<br>
  
http://ufn.ru/tribune/trib120111 Д.Кузнецов. Место науки и физики в человеческом знании. Трибуна УФН, 2010–2011, No.110. (In Russian) <!--<br>
 
http://budclub.ru/k/kuznecow_d_j/2010mestor.shtml Д.Кузнецов. Место науки (и физики) в человеческом знании, in Russian !-->
  
http://budclub.ru/k/kuznecow_d_j/2010mestor.shtml Д.Кузнецов. Место науки (и физики) в человеческом знании, in Russian !-->
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</ref>
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</ref>.
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==More equal than others==
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The ideas of pluralism of certain objects (scientific concepts, animals, etc.) and their discrimination for some criteria is described in the [[fairy tale]] by George Orwell ("Animal's farm"): <b>All animals are equal, <i>but some of them are more equal than other.</i></b>.
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The same can be applied to concepts: <br>
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1. Concepts, efficient in description of some class of phenomena, are "equal" in their right to exist, providing the [[pluralism]].<br>
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2. Some of then are "more equal", as I keep my right to use the simplest one for the application in the practical activity.
   
 
==References==
  
==References==

Latest revision as of 18:48, 30 July 2019

Pluralism (плюрализм) is the last axiom among the TORI axioms set [1][2][3] Pluralism, postulated as compulsory for the scientific concept, declares, that mutually-contradicted concepts may coexist in science, but specifies their subordination:

If two concepts satisfying first 5 TORI axioms, have some common range of validity, then, in this range, the simplest of them has priority.

Concepts, that use complicated formalism to describe simple phenomena, have low priority. The simple concepts, that are applicable to the case, should be considered first.

Occam razor

The ideas of pluralism and simplicity are not new.

They are discussed and applied during centuries.

Some links are suggesed at http://en.wikipedia.org/wiki/Occam's_razor

The criterion of simplicity allows to tolerate many different scientific concepts. This criterion indicates, which of them should be applied first. There is no need to declare other concepts as "wrong"; often, it is sufficient to say, that they are "more complicated" in the description of the same object.

Tetration and other Superfunctions

The Pluralism, as idea of simplicity, had been used to guess the asymptotic behaviour of tetration and arctetration at \(\pm \mathrm i \infty\) [4].

The same idea can be applied to other superfunctions [5].

Certain postulates about of simplicity of the superfunction and the abelfunction lead to uniqueness of the solution. In particular, tetration not only exist, but also is unique super exponential, that have simple behaviour at infinity. [6][7][8]:
.. we study the fractional iterates of the exponential function. This is an unresolved problem, not due to a lack of a known solution, but because there are an infinite number of solutions, and there is no agreement as to which solution is "best. We will approach the problem by first solving Abel's functional equation \(\alpha(e^x) = \alpha(x) + 1\) by perturbing the exponential function so as to produce a real fixed point, allowing a unique holomorphic solution. We then use this solution to find a solution to the unperturbed problem. However, this solution will depend on the way we first perturbed the exponential function. Thus, we then strive to remove the dependence of the perturbed function. Finally, we produce a solution that is in a sense more natural than other solutions.

Several examples of search for simplest concept (and to choose it) are mentioned in article Place of science in the human knowledge [9].

More equal than others

The ideas of pluralism of certain objects (scientific concepts, animals, etc.) and their discrimination for some criteria is described in the fairy tale by George Orwell ("Animal's farm"): All animals are equal, but some of them are more equal than other..

The same can be applied to concepts:
1. Concepts, efficient in description of some class of phenomena, are "equal" in their right to exist, providing the pluralism.
2. Some of then are "more equal", as I keep my right to use the simplest one for the application in the practical activity.

References

  1. ↑ http://www.scirp.org/journal/PaperInformation.aspx?PaperID=36560 http://mizugadro.mydns.jp/PAPERS/2013jmp.pdf D.Kouznetsov. TORI axioms and the applications in physics. Journal of Modern Physics, 2013, v.4, p.1151-1156.
  2. ↑ http://pphmj.com/abstract/5076.htm D.Kouznetsov. Support of non-traditional concepts. Far East Journal of Mechanical Engineering and Physics, 1, No.1, p.1-6 (2010)
  3. ↑ http://ufn.ru/tribune/trib120111 D.Kouznetsov. Place of science and physics in the human knowledge. Physics-Uspekhi, v.181, Трибуна, p.1-9 (2011, in Russian)
  4. ↑ http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
    D.Kouznetsov. Solutions of F(z+1)=exp(F(z)) in the complex z plane. Mathematics of Computation, 78 (2009) 1647-1670
  5. ↑ http://www.ils.uec.ac.jp/~dima/BOOK/443.pdf D.Kouznetsov. Superfunctions. 2017.
    http://mizugadro.mydns.jp/BOOK/444.pdf D.Kouznetsov. Superfunctions. 2018. (41742154 bytes)
  6. ↑ http://www.springerlink.com/content/u7327836m2850246/ http://mizugadro.mydns.jp/PAPERS/2011uniabel.pdf H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)
  7. ↑ http://dx.doi.org/10.11568/kjm.2016.24.1.81 William Paulsen. Finding the natural solution to \(f(f(x)) = \exp(x)\) Korean J. Math. Vol 24, No 1 (2016) pp.81-106.
  8. ↑ http://myweb.astate.edu/wpaulsen/tetration2.pdf http://link.springer.com/article/10.1007/s10444-017-9524-1 William Paulsen and Samuel Cowgill. Solving \(F(z+1)=b^{F(z)}\) in the complex plane. Advances in Computational Mathematics, 2017 March 7, p. 1–22. .. we study the fractional iterates of the exponential function. This is an unresolved problem, not due to a lack of a known solution, but because there are an infinite number of solutions, and there is no agreement as to which solution is "best. We will approach the problem by first solving Abel's functional equation \(\alpha(e^x) = \alpha(x) + 1\) by perturbing the exponential function so as to produce a real fixed point, allowing a unique holomorphic solution. We then use this solution to find a solution to the unperturbed problem. However, this solution will depend on the way we first perturbed the exponential function. Thus, we then strive to remove the dependence of the perturbed function. Finally, we produce a solution that is in a sense more natural than other solutions.
  9. ↑ http://budclub.ru/k/kuznecow_d_j/2010mestoe.shtml D.Kouznetsov. Place of Science in the human knowledge (2010).
    http://ufn.ru/tribune/trib120111 Д.Кузнецов. Место науки и физики в человеческом знании. Трибуна УФН, 2010–2011, No.110. (In Russian)

Keywords

TORI, TORI axiom, Philosophy, Place of science in the human knowledge, Religion, Science

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