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Visualization as a Tool for
Scientific Understanding
Henk W. de Regt
Lorentz Fellow, NIAS 2009-2010
Faculty of Philosophy, VU University Amsterdam
Lorentz Workshop Understanding and the
Aims of Science
Leiden, 4 June 2010
Schrödinger on Heisenberg
“I naturally knew about
his theory, but was
discouraged, if not
repelled, by what
appeared to me as very
difficult methods of
transcendental algebra,
and by the lack of
Anschaulichkeit.”
Heisenberg on Schrödinger
“The more I think of the
physical part of
Schrödinger’s theory, the
more abominable I find it.
What Schrödinger writes
about Anschaulichkeit
makes scarcely any sense,
in other words I think it is
crap.”
Outline
1. Introduction
2. The loss of visualizability in atomic physics
3. The debate about Anschaulichkeit
4. A theory of scientific understanding
Classical physics and
Anschaulichkeit
C.F. von Weiszäcker (1979):
“Physicists identify anschaulich
with ‘classical’ because
classical physics describes all
physical phenomena as states
of quantities in threedimensional, Euclidean space
and as changes of these states
in one-dimensional, objective
time.”
Bohr’s atomic model (1913)
A partially visualizable, semi-classical model:
– discrete electron orbits  visualizable
– ‘quantum jumps’  non-visualizable
The waning of visualizability
• Wave-particle duality
– of light (Einstein 1905) and matter (De Broglie
1923)
– no unambiguous visualization
• Reality of electron orbits disputed
– Pauli’s fourth quantum number (1924)
– atoms completely non-visualizable
Two new quantum theories
• Heisenberg’s matrix mechanics (1925)
– only relations between observable quantities
– no (visualizable) model of atomic structure
• Schrödinger’s wave mechanics (1926)
– atomic structure is complex wave phenomenon
– no quantum jumps; no wave-particle duality
– a promise of visualization
Visualizing Schrödinger’s atom
Schrödinger on Anschaulichkeit
“The aim of atomic research is to fit our empirical
knowledge concerning it into our other thinking.
All of this other thinking, so far as it concerns the
outer world, is active in space and time.”
“We cannot really alter our manner of thinking in
space and time, and what we cannot comprehend
within it we cannot understand at all.”
The epistemic power of
visualization
Examples:
• Wave mechanics yielded many more applications
than matrix mechanics.
• Electron spin (Goudsmit and Uhlenbeck, 1925):
visualizing Pauli’s fourth quantum number.
• After WW II: Feynman diagrams
Electron spin
Wolfgang Pauli
(1900-1958)
Pauli on Anschaulichkeit
“We should not want to clap the atoms into the
chains of our preconceptions […] but we must on
the contrary adjust our ideas to experience.”
“Even though the demand […] for Anschaulichkeit is
partly legitimate and healthy, still this demand
should never count in physics as an argument for
the retention of fixed conceptual systems. Once
the new conceptual systems are settled, then also
these will be anschaulich.”
Heisenberg’s 1927 paper: Über den
anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik
• Reinterpretation of Anschaulichkeit
“We believe we understand [anschaulich zu Verstehen] a
physical theory when we can see its qualitative
experimental consequences in all simple cases and
when at the same time we have checked that the
application of the theory never contains inner
contradictions”
• Uncertainty relations: Δp.Δq ≥ ½(h/2π)
Outcome of the debate
• Schrödinger’s attempted visualization of atomic
structure failed
• Heisenberg admitted visualizable concepts
• Result: new ‘quantum mechanics’ that combined
both theories
Conclusions of case study
• Visualizability problematic in quantum domain 
can’t be a necessary condition for intelligibility.
• But visualization remained a useful tool for
understanding, even in the quantum era.
The epistemic significance of
intelligibility
• Thesis: intelligibility is epistemically significant.
• Intelligibility = pragmatic understanding of theory,
ability to work with the theory.
• Explanations (understanding of phenomena)
require intelligible theories.
Explanation and pragmatic
understanding
• Deductive-nomological (Hempel)
– Example: flying jets
– Deductive reasoning requires skill
• Model-based (Cartwright)
– No deduction, but tinkering (approximation,
idealization)
– Skills and judgment needed
Intelligibility
Positive value that scientists attribute to the
cluster of theoretical qualities that facilitate
use of the theory
– No intrinsic property, but related to skills 
context-dependent
– Examples of valued qualities: visualizability,
causality, unifying power, simplicity, etc.
A possible test for intelligibility
(esp. physical science)
A scientific theory is intelligible for
scientists if they can recognize qualitatively
its characteristic consequences without
performing exact calculations.
Conclusion: understanding as a
means and an end
• Epistemic aim of science: explanations that
provide understanding of phenomena.
• Understanding phenomena requires
understanding theories – intelligibility!
• Intelligibility: not a matter of ‘feeling good’
about a theory, but of being able to use it.
If you want more:
H.W. de Regt, S. Leonelli & K. Eigner (eds),
Scientific Understanding: Philosophical
Perspectives. University of Pittsburgh Press, 2009.
H.W. de Regt & D. Dieks, ‘A contextual approach to
scientific understanding’, Synthese 144 (2005)
137-170.
H.W. de Regt, ‘Making sense of understanding’,
Philosophy of Science 71 (2004) 98-109.