ADVANCES
14 Scientific American, October 2019MATH AND ART
Beautiful
Truths
Nonmathematicians agree on
what makes proofs pleasing
Scientists and mathematicians often
describe facts, theories and proofs as
“beautiful,” even using aesthetics to help
guide their work. Their criteria might seem
opaque to nonexperts, but new research
finds that novices can consistently assess
a proof’s beauty or ugliness.
A mathematician and a psychologist
analyzed responses from about 200 online
participants for each of three experiments
in their study, published in August in Cogni-
tion. Most had attended college but had
not studied math beyond university calcu-
lus. In each experiment, they read four
simple mathematical arguments and were
tested for comprehension. (Two included
diagrams; see the graphic for an example.)
The subjects rated each argument’s
“similarity” to each of four landscape paint-
ings, and the results were clearly consis-
tent: people generally agreed on which
arguments matched which paintings—and
their choices roughly aligned with those
made by eight mathematicians. (The argu-
ment pictured was most strongly matched
to a Yosemite landscape by Albert Bier-
stadt, seen here.) The second experiment
produced a similar result with classical pia-
no music in place of paintings.
For the third experiment, subjects rated
the arguments and paintings on 10 adjec-tives, including “beautiful.” Again, results
were consistent, and elegance, followed by
profundity and clarity, was the biggest fac-
tor in judging beauty for both the math and
the art. Samuel Johnson, a psychologist at
the University of Bath in England, and one
of the paper’s co-authors, says he was
most surprised that those qualities could
predict the first group’s pairings of ideas
and paintings—indicating that the math-This can be understood by dividing up
a square with total area 1.Argument:^1 ⁄ 2 +^1 ⁄ 4 +^1 ⁄ 8 +^1 ⁄ 16 +^1 ⁄ 32 +... = 1
(^1) ⁄ 2
(^1) ⁄ 4
(^1) ⁄ 8
(^1) ⁄ 16
(^1) ⁄ 32
(^1) ⁄ 64
Graphic by Jen Christiansen
ALAMY
( painting
); SOURCE: “INTUITIONS ABOUT MATHEMATICAL BEAUTY: A CASE STUDY IN THE AESTHETIC EXPERIENCE OF IDEAS,”
BY SAMUEL G.
B. JOHNSON AND STEFAN STEINERBERGER, IN
COGNITION,
VOL.
189; AUGUST 2019 (
mathematical argument
)
demic officer of the Cleveland Clinic Lerner
College of Medicine, a program in collabo-
ration with Case Western Reserve Universi-
ty that opened a new cadaverless campus
this summer. “But as technology advanced
and as knowledge increased, there came a
push to do things better and faster and give
students a more appropriate representation
of human anatomy.”
Young, who studied medicine in the
1970s, experienced a “massive disconnect”
between his own anatomy education and
what he saw during clinical training in car-
diology. When he tried to access organs in
living patients, looking at imaging results
or footage from tiny inserted cameras,
he found the inside of human bodies did
not match what he had seen in cadavers.
“They’re totally different,” Young says. “The
embalmed cadaver has a very flat, com-
pressed organ presentation. The colors are
not the vibrant colors of a living human.” The
difference can distract from learning, he says.
Virtual anatomy tools, in contrast, pro-
vide a more faithful view of living organs,
helping students form a foundational under-
standing of the body’s structures, Young
and other medical educators say. By don-
ning VR headsets or augmented-reality
goggles, which show digital imagery super-
imposed on the real world, students can
examine an organ from all angles. They can
connect structure with function by watch-
ing a beating heart or moving joints. They
can also select views that add other organs
or the entire circulatory and nervous sys-
tems to better see relations among struc-
tures. “I was amazed,” says Mark Schuster,
dean of Kaiser Permanente School of Medi-
cine in Pasadena, Calif., which will welcome
its first class of medical students in 2020.
“I wished I had that when I’d been learning
anatomy. It really helped make it all come
together.” His program’s first-year students
will have a cadaverless curriculum.
Adopting high-tech alternatives makes
sense for brand-new medical programs
that have neither the tradition nor the
facilities for cadaver dissection, but even
some existing ones are adopting digital
tools to supplement their anatomy courses.
“The big advantage I see is that the visuals
are very clean,” says Darren Hoffman, an
assistant professor of anatomy and cell
biology, who uses interactive 3-D anatomy
software in his courses at the University of
Iowa Carver College of Medicine. “That
helps building your 3-D mind’s eye of the
body, so that when you look at a patient’s
ankle, you know what’s underneath the
surface and how it’s all related.”
Besides the educational advantages,
going cadaverless is an economic decision
for new programs. It costs several million
dollars to build a cadaver laboratory, which
requires a lot of space, as well as safety
measures that meet legal regulations. And
although cadavers are donated, medical
schools still pay for preparation, mainte-
nance and, eventually, burial. These costs
are an even bigger challenge for schools in
less wealthy nations, Young says. What is
more, many countries still face a shortage
of donations and rely on unclaimed bodies
for dissection, according to a 2018 study.
Cadaverless anatomy education has
its drawbacks. It may be hard to develop
a perception of depth in a virtual body, and
students will miss out on seeing bodies’ nat-
ural anatomical variations, according to
Hoffman. Students may also lose the emo-
tional, even philosophical impact of working
with a cadaver, commonly seen as a doc-
tor’s first patient. “There’s a sort of awe and