An experimentalist approach is one of the known science approaches in physics (in very crude terms it may be called trial and error method). Another well known approach dominated in the second half of the 20th century is a theoretician approach (new phenomenon is predicted from the analysis of already existing theory and merely verified by experimentalist by using the exactly set experiment - like top quark or Higgs boson discovery). The present day huge crisis in fundamental physics may need for resolution the return back to experimentalist approach [1-3], however, it is the interesting question about methodology - may this method (experimentalist approach) be applied to other disciplines (for example math)? This publication describes an attempt and demonstrates that this is not a good idea.
I tried "experiments" with well known Fibonacci sequence (already 900 years old and well researched). May be something new found through tinkering around (like experiment with electronic devices made from newly synthesized semiconductors or synthesis of new ferromagnetic or potential room temperature "superconductors" [4]). In physics it works well, not like in [4] but like discovery of high-temperature superconductors in 1986 [5].
The original Fibonacci sequence is a linear recurrence relation:
F(n)=F(n-1)+F(n-2)
what if the sign is minus instead of plus? Kind of experiment without looking for general theory, in horizontal direction of knowledge search, not in vertical, without generalizations? And limiting research for whole numbers only?
F(n)=F(n-1)-F(n-2)
The result is unusual - it will generate a simple cycle.
F(n-2)=0
F(n-1)=1
F(n)=F(n-1)-F(n-2)=1-0=1
F(n+1)=F(n)-F(n-1)=1-1=0
F(n+2)=F(n+1)-F(n)=0-1=-1
F(n+3)=F(n+2)-F(n+1)=-1-0=-1
F(n+4)=F(n+3)-F(n+2)=-1-(-1)=-1+1=0
F(n+5)=F(n+4)-F(n+3)=0-(-1)=0+1=1
F(n+6)=F(n+5)-F(n+4)=1-0=1
F(n+7)=F(n+6)-F(n+5)=1-1=0
F(n+8)=F(n+7)-F(n+6)=0-1=-1
F(n+9)=F(n+8)-F(n+7)=-1-0=-1
............
The sequence is obviously a simple cycle:
0,1,1,0,-1,-1,0,1,1,0,-1,-1,.......
This simple cycle is easily proved for any original numbers:
F(n-2)=a
F(n-1)=b
F(n)=F(n-1)-F(n-2)=b-a
F(n+1)=F(n)-F(n-1)=b-a-b=-a
F(n+2)=F(n+1)-F(n)=-a-(b-a)=-a-b+a=-b
F(n+3)=F(n+2)-F(n+1)=-b-(-a)=-b+a=a-b
F(n+4)=F(n+3)-F(n+2)=a-b-(-b)=a-b+b=a
F(n+5)=F(n+4)-F(n+3)=a-(a-b)=a-a+b=b
F(n+6)=F(n+5)-F(n+4)=b-a
F(n+7)=F(n+6)-F(n+5)=b-a-(b)=-a
F(n+8)=F(n+7)-F(n+6)=-a-(b-a)=-a-b+a=-b
F(n+9)=F(n+8)-F(n+7)=-b-(-a)=-b+a=a-b
The sequence is a cycle for any input numbers:
a;b;b-a;-a;-b;a-b;a;b;b-a;-a;-b;a-b.......
So at first glance the experiment leads to something completely new, not associated with Fibonacci sequence at all.
However, quick glance into theoretical approach (checking for general theory using calculus, differential equations, discrete mathematics) quickly explains that this is actually well known result, but not for Fibonacci sequence but for more general linear recurrence relations [6]. Actually there is a big branch of mathematics devoted to recurrence relations. In the simple case of linear recurrence relations of kind
arx(n+r) + ar-1x(n+r-1)+….+a0x(n)=f
if f=0 and modality r=2 the second modality recurrence relation becomes:
a2x(n+2) + a1x(n+1) + a0x(n)=0
And if a2=1, a1=a0=-1 it becomes: x(n+2)-x(n+1)-x(n)=0 or F(n)=F(n-1)+F(n-2) - Fibonacci sequence. And for a1=-1, a2=a0=1 it becomes x(n+2)-x(n+1)+x(n)=0 or F(n)=F(n-1)-F(n-2) cyclic Fibonacci sequence with unusual properties obtained by experiment.
The general solution of recurrence relations of second modality a2x(n+2) + a1x(n+1) + a0x(n)=0
is determined by the characteristic equation [6]:
a2λ2+a1λ+a0=0
(for the other modalities that would be arλr+ar-1λr-1+…. + a0=0 )
For Fibonacci sequence the characteristic equation is λ2-λ-1=0 with both roots are real - it will follow the solution described in [6].
For cyclic Fibonacci the equation is λ2-λ+1=0 and both roots are complex numbers - similar to linear differential equations the solution is oscillating (periodic). If the roots are degenerate (for characteristic equation λ2-2λ+1=0 both roots are equal to 1 the Fibonacci sequence becomes:
F(n)=2F(n-1)-F(n-2)
And this is the third type of "Fibonacci" sequence - neither exponential growth nor periodic but linear: 0,1,2,3,4,5,6,7,8,9
Strictly speaking only original sequence F(n)=F(n-1)+F(n-2) may be called Fibonacci, the rest is simply linear recurrence relations [6], there are infinite number of them and analysis is very easy if characteristic equation is used - if you use theoreticians approach. You may choose what period you need, figure out the necessary roots and arrive to linear recurrence relation without "trial and error" method. The difference may be illustrated by the scheme:
So in the case of mathematics the experimentalist approach is kind of useless and theoretician (mathematician) approach is the best. What is the problem with physics in this consideration? Why it is not working perfectly exactly like described?
The answer is that mathematics is a formal science, not natural science. The general solution found on upper level works for any object on lower level without limitations. But physics is different [7,8] - the nature is unpredictable and impossible to guess. It is always "orthogonal" to what may be expected.
Suppose Einstein never appeared with General Relativity. From experimental observations of Tycho Brahe Kepler and Newton developed upper level of planets motion theory - Newton physics. It predicted discovery of planets Neptune, Pluto successfully and thus was considered as a general theory that must work everywhere (like general theory of recurrence relations in mathematics). Since Mercury has a problem with orbit, the Vulcan planet was predicted [9]. It was searched for decades with zero success but since the general Newton theory tells it must be present and in the absence of Einstein what would be the final conclusion? This is a dark planet! Because it must be present and because it is so close to Sun it is somehow invisible, only reveal itself by gravity (because the general Newton theory must hold). That sounds very familiar in the beginning of 21st century because it is exactly what modern physics is looking for: dark matter. This is where the physics deviates so much from mathematics - mathematical laws are absolute, works for any numbers and objects, physical laws are merely approximations, they have limitations sooner or later discovered.
If mathematics would be also non-exact, the experimentalist approach would be vindicated. For example, the extensive search of linear recurrence relations may eventually lead to discovery of still "Fibonacci" sequence like constant or logarithmic "Fibonacci" sequences or whatever it turned out. It would be like mathematics established (greatly exaggerated) that 1*1=1 and 2*2=4 but 3*3=8.999 and 10*10=99.3 or even sometimes 99.3, sometimes 100.2. Impossible for mathematics but sometimes happened in physics.
Possibly the huge crisis in modern fundamental physics is exactly because of this - modern fundamental physics is theoretician (mathematicians) approach, while real physics needs both theoreticians and experimentalists approaches to move forward.
References.
1. Dmitriy S. Tipikin “The quest for new physics. An
experimentalist approach. Where to find new physics?”// LAP Lambert Academic
Publishing, 2021.
2011.0172v1.pdf (vixra.org) or https://www.researchgate.net/publication/353523212_The_quest_for_new_physics_An_experimentalist_approach
2. Dmitriy S. Tipikin “The quest for new physics. An
experimentalist approach. Vol.2 Reflections on tired light hypothesis versus
Big Bang” // LAP Lambert Academic Publishing, 2022.
2212.0058v1.pdf (vixra.org) or
3.Dmitriy S. Tipikin "The quest for new physics. An experimentalist approach. Vol.3 The new cosmology" // Published on Internet on January 7 2026.
https://vixra.org/pdf/2601.0030v1.pdf
5.High-temperature superconductivity - Wikipedia
https://www.math.uci.edu/~ndonalds/math180b/4recurrence.pdf
9.Vulcan (hypothetical planet) - Wikipedia
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