I came across an article that is related to several different threads
in this forum. So I decided to start a new one. (01)
The article is a review of a historical book about the imagery that
scientists such as Einstein and Heisenberg used to discover their
radically new theories of relativity and quantum mechanics. (02)
The reviewer is a physicist, David Hestenes, who has himself introduced
some radically different kinds of mathematics (namely geometric
algebras) in order to simplify the way physicists formulate their
theories: (03)
http://modeling.asu.edu/R&E/SecretsGenius.pdf
Secrets of Genius (04)
Some excerpts from this review are copied below. (05)
Note some important points: (06)
1. Imagery is extremely important in physical discoveries. (07)
2. Mathematicians who may be much more skilled in the formalisms
usually don't have the physical background to discover the
fundamental insights; e.g. Poincaré the mathematician discovered
the basic math before Einstein the physicist, but Poincaré did
not have the physical insight to interpret it as Einstein did. (08)
3. It is false that Newtonian mechanics corresponds to normal human
perception of the way the world works, because people who have
not studied physics do not think in terms of Newtonian mechanics.
The socalled paradoxes of quantum mechanics do not conflict with
"common sense", but with the way physicists have been trained. (09)
Although this article is about physical imagery, many of the same
questions can be asked about ontology. How much of the logical
and mathematical formalism really corresponds to socalled "common
sense"? How much of it is the result of the training that the
mathematician or logician received? How much corresponds to
reality  i.e., the world independent of common sense, previous
education, or mathematical and logical formalisms? (010)
John Sowa
______________________________________________________________________ (011)
Scientists and nonscientists alike are fascinated by the creative
processes underlying the great scientific discoveries. We are eager
to know the secrets of genius. Did Einstein possess creative powers
that set him above the ordinary physicist? Or was he privy to some
special heuristics that guided him to his discoveries? We are indebted
to historians of science like Miller for helping us answer such
questions. Recognizing the difficulty of the task, Miller calls for
collaboration between historians and cognitive scientists to study
creative processes in science. He tries to get the process started
in the present book with a historical, epistemological, and cognitive
analysis. His central thesis is that "mental imagery is a key ingredient
in creative scientific thinking." We follow him by focusing attention
on the role of imagery in the creation of the special theory of
relativity and quantum mechanics, two major triumphs of 20th century
physics. But to evaluate the role of imagery we need to know what else
was involved in the creation of these great theories.... (012)
Einstein did not need an elaborate analysis of experimental data to
identify the conflict between Newtonian mechanics and electromagnetic
theory. Both theories are involved in explaining the phenomenon of
electromagnetic induction, which underlies the operation of electric
motors and generators. The essence of the phenomenon is that a magnet
moving relative to a wire loop induces an electric current in the wire.
Einstein observed that the induced current predicted by the theory
depended on whether the wire or the magnet was kept at rest, whereas
the physical phenomenon appeared to depend only on the relative motion
of the magnet and wire. Thus, the theory exhibited an asymmetry which
was not inherent in the phenomena. Einstein removed this asymmetry by
invoking the principle of relativity, which requires that the laws of
physics for an observer at rest must be the same as for an observer
moving with uniform velocity. This principle had been stated for
mechanics by Newton, though not as a basic axiom. Einstein generalized
it to apply to electromagnetic theory as well. Paradoxically, this
required a modification of mechanics rather than electromagnetics... (013)
The greatest remaining mystery is why Poincaré failed to arrive at
the same conclusion and, indeed, to appreciate Einstein's accomplishment
in subsequent years. Miller shows us that Poincaré was well aware of all
the essential facts and ideas. The only thing missing, it seems, was an
appreciation of gedanken experiments. (014)
This case illustrates an important difference between mathematical
and physical thinking which goes a long way toward explaining why so
few mathematicians have made important contributions to physics in
the 20th century. Pure mathematicians do not think about the
equations of physics in the same way as a physicist does. They
are concerned only with the structure of the equations and the
formal rules for manipulating them. But physicists regard the
equations as representations of real things or processes; they are
only partial representations of the physicists' knowledge, so to
improve a representation they may alter the equations in ways that
violate mathematical rules. Both Einstein and Heisenberg were masters
at this. Neither was a mathematical virtuoso. Indeed, in the period
when Einstein was developing his general relativity theory, the
mathematician Hilbert expressed the opinion that Einstein was
mathematically naive. I have heard a similar opinion about Heisenberg
expressed by one of his students in later years. (015)
Mathematics played an essential role in Einstein's thinking, but,
as mathematical physics goes, the mathematics in all his great papers
is comparatively simple. His forte was in analyzing the physical
meaning of the mathematics. Indeed, such analysis is generally
characteristic of the best work in theoretical physics. I have heard
the Nobel laureate Richard Feynman, himself a true mathematical
virtuoso, express this opinion forcefully, asserting that the
value of a paper on theoretical physics is inversely proportional
to the density of mathematics in it.... (016)
The thinking in Einstein’s creation of relativity theory can be
described as theorydriven. As we have seen, it was not directed at
explaining any particular experimental results, but it was nonetheless
empirically grounded in a broad and indirect way. This made empirical
predictions from the theory exceptionally robust. As Miller explains
(p. 118), the empirical data available in 1901 contradicted Einstein’s
theory as well as Lorentz’s theory of electrons. Since Lorentz’s theory
was datadriven, he was ready to abandon it immediately in deference
to the new data. But the rationale for Einstein’s theory was so secure
that he confidently dismissed the data as inaccurate. Strong empirical
confirmation for relativity theory was not available for decades.
Nevertheless, many physicists came to accept it on the basis of its
internal logic.... (017)
The scientists in Miller’s account are unanimous in emphasizing the
crucial role of visualization in scientific thinking along with a
warning that it can be misleading. One place they were misled (along
with Miller and the physics community at large) was in their intuition
that classical mechanics describes what is perceptually given. They
were unaware of the strong cognitive component in their own perception.
It was only by training that classical mechanics came to be integrated
into that perception. Cognitive research has recently established that
the perceptions of people untutored in physics are naturally
inconsistent with classical mechanics in almost every detail (Halloun
& Hestenes, 1985). Thus, Miller’s conclusion (p. 261) that "twentieth
century physicists were forced to liberate their thinking from the
world of perceptions" misses the mark.... (018)
Having recognized the psychological tendency of physicists to confuse
classical physics with perception, we can see more clearly the central
epistemological issue raised by the creation of quantum mechanics.
The conflict between classical and quantum physics had nothing to do
with perception. It arose because physicists were unable to reconcile
the mathematics of quantum mechanics with the classical conception
of reality, so they were forced to construct new "quantum mechanical"
conceptions of reality.... (019)
Anyone involved in the lectures, seminars and informal giveandtake
of creative physicists cannot fail to notice the vivid imagery in
their thinking. Most of this imagery is suppressed in their
publications, partly by conventions concerning the style of scientific
reporting, partly because it is not essential to establishing the
scientific results, and partly because it may be too much trouble
to construct suitable diagrams to express it. This puts severe
limitations on Miller’s historical approach and tells us that the
creative physicist needs to be studied in vivo, while he is alive
and kicking. That is where the cognitive scientist comes in.... (020)
Imagery in physics is a promising domain for cognitive research.
There is a rich lode of physical imagery that has never been mined
systematically. Only a few prospectors like Miller and Simon have
picked up samples. The payoff is likely to be greatest in education,
leading to improvements in the design of images and in the teaching
of imagery skills, thus enhancing creative powers at large. Here
indeed, as Miller suggests, is a domain where historians and cognitive
scientists can work together. But they had better enlist the help of
some physicists. (021)
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