IEEE Circuits and Systems Magazine - Q4 2022 - 56

recognized in various scientific domains. These analogies
can be explained by the overarching invariants and
symmetries. Analogies are important properties [31]. In
engineering sciences, analogies are used in the analysis
of multidisciplinary systems.
Invariances and symmetries are in fact properties of
reality that make it possible to acquire objective knowledge
and to understand reality. The objects and phenomena
that we can observe are the source of scientific
knowledge. Objective knowledge, however, is impossible
if observers in different places simultaneously and in
the same conditions conducting an experiment do not
observe and measure the same thing. If an observer repeats
an experiment at different times, she/he must also
be able to observe and measure the same.15 If this were
not the case, one would not be able to discover general
relations that may be regarded as laws, and reality would
be incomprehensible. Indeed, all objects and phenomena
would then be experienced as different and unique.
We can however see invariances and symmetries as
properties that characterize the 'deeper nature' of reality.
The invariance for a transformation in space and
time and the scale invariance are conditions that allow
the acquisition of objective knowledge. Reality would not
be intelligible or reproducible without such invariances.
The fact that we can discover in reality these invariances
and that it is possible to derive laws from them points
to the scientifically inexplicable 'intelligibility' of reality.
We also note that mathematics is successfully applied
in many sciences. Especially in the exact sciences,
mathematics is used as a source of models to solve all
kinds of problems. As shown in this paper, we can even
derive Kirchhoff's laws and the analogous and generalized
formulations of these laws directly from the invariance
and symmetry properties of real numbers. This is
reminiscent of Wigner's statement [32], [33] about 'the
unreasonable effectiveness of mathematics in the natural
sciences'. Also in Noether's theorem, conservation
laws for physical systems are derived only from mathematical
considerations. The success of mathematics is
a property of reality that is very mysterious.
The intelligibility of reality is also a great mystery. Reality
turns out to be partially 'unreasonably reasonable'.
This assumption is the basis of the sciences. However, the
fundamental question of what it is ultimately about and
what is actually going on is not yet thoroughly answered.
VIII. Conclusion
The strikingly simple laws of Kirchhoff have a great
potential for generalizations of the electric network
15The theory of relativity is based on the invariance of the speed of
light. The observations are different for other relativistic phenomena.
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IEEE CIRCUITS AND SYSTEMS MAGAZINE
theory. Tellegens' theorem that is based on Kirchoff's
laws, also offers an important collection of interesting
theorems which can be generalized. These generalizations
are grounded on the finding that an abstraction of
physical laws is possible. This leads to a unification of
several branches of science, in particular the engineering
sciences.
Laws in the form of Kirchhoff's laws can also be discovered
in sets of arbitrary real numbers. This is also
the case for sets of measurement results. We have
shown too that Kirchhoff's laws and analogous laws
can be deduced from invariants and symmetries. This
allows a restructuring of sciences with symmetries as
basic axioms. All this leads to a 'physics free' approach
with minimal intellectual effort for maximal benefit.
Problems from various branches of science and engineering
can be considered and treated in the same way.
At first sight, this approach seems only of theoretical
and philosophical importance. However, the analogies are
useful for practical applications in engineering. This is certainly
so for the study of multidisciplinary systems. The
importance of analogies, abstraction, generalization, and
unification should be emphasized in education [7]. After all,
these insights form the basis of engineering sciences and
allow a greater depth of understanding. They also point to
application possibilities in other scientific domains.
It is strange that Kirchhoff's laws can be deduced
from rather philosophical considerations. Moreover, it
is mysterious that symmetries, analogies, duality, and
universality are properties of reality. This is also the
case for the underlying recurring patterns in space and
time, the appearance of simplicity in the complexity of
reality, the possibility to acquire objective knowledge,
the existence of laws and the success of mathematics.
Reality is apparently 'unreasonably reasonable' and
partially intelligible. This intelligibility is a characteristic
of the deeper nature of reality that raises fundamental
questions to which science has no answers.
Acknowledgment
Editor's note: In 2021, Prof. Joos Vandewalle received the
IEEE Gustav Robert Kirchhoff Award " for fundamental
contributions to mathematical foundations of circuits and
systems " . Prof. Vandewalle was a full professor at KU Leuven
in Belgium until 2013. He is a Life Fellow of the IEEE.
To celebrate his Kirckhhoff award, Prof. Vandewalle wrote
an article in collaboration with Dr. Hubert Van Belle on the
broader value and applicability of Kirchhoff's laws.
This work was supported in part by the Mechanical
and Electrical Engineering Departments of the Katholieke
Universiteit Leuven, Belgium. The first author is
the main contributor.
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