40 Scientific American, September 2019about a soda can or a doughnut; it refers to an ab-
stract mathematical circle, where distances are ex-
act and the points on the circle are infinitesimally
small. Such a perfect circle is causally inert and
seemingly inaccessible. So how can we learn facts
about it without some type of special sixth sense?
That is the difficulty with realism—it fails to
explain how we know facts about abstract math -
ematical objects. All of which might cause a math -
ematician to recoil from his or her typically realist
stance and latch onto the first step of the math -
ematical process: invention. By framing math-
ematics as a purely formal mental exercise or a
complete fiction, antirealism easily skirts prob-
lems of epistemology.
Formalism, a type of antirealism, is a philosoph-
ical position that asserts that mathematics is like a
game, and mathematicians are just playing out the
rules of the game. Stating that 7 is a prime number
is like stating that a knight is the only chess piece
that can move in an L shape. Another philosophical
position, fictionalism, claims that mathematical
objects are fictions. Stating that 7 is a prime num-
ber is then like stating that unicorns are white.
Mathematics makes sense within its fictional uni-
verse but has no real meaning outside of it.
There is an inevitable trade-off. If math is sim-
ply made up, how can it be such a necessary part of
science? From quantum mechanics to models of
ecology, mathematics is an expansive and precise
scientific tool. Scientists do not expect particles to
move according to chess rules or the crack in a din-
ner plate to mimic Hansel and Gretel’s path—the
burden of scientific description is placed exclusive-
ly on mathematics, which distinguishes it from oth-
er games or fictions.
In the end, these questions do not affect the
practice of mathematics. Mathematicians are free
to choose their own interpretations of their profes-
sion. In The Mathematical Experience, Philip Davis
and Reuben Hersh famously wrote that “the typical
working mathematician is a Platonist on weekdays
and a formalist on Sundays.” By funneling all dis-
agreements through a precise process—which em-
braces both invention and discovery—mathemati-
cians are incredibly effective at producing disci-
plinary consensus.
MORE TO EXPLORE
Logicomix: An Epic Search for Truth. Apostolos Doxiadis and Christos H.
Papadimitriou. Art by Alecos Papadatos and Annie Di Donna.
Bloomsbury USA, 2009.
Where Proof, Evidence and Imagination Intersect. Patrick Honner
in Quanta Magazine. Published online March 14, 2019.
FROM OUR ARCHIVES
Why Isn’t 1 a Prime Number? ÿy ̈Ă ́" D®Uè3`y ́ï` ®yà` D ́Î`¹®j
published online April 2, 2019.
scientificamerican.com/magazine/saIllustration by Brook VanDevelderOUR INNER
UNIVERSES
REALITY IS CONSTRUCTED BY THE BRAIN,
AND NO TWO BRAINS ARE EXACTLY ALIKEBy Anil K. Seth
“We do not see things as they are, we see
them as we are.”
—from Seduction of the Minotaur,
by Anaïs Nin (1961)On the 10th of April this year Pope Francis, Presi-
dent Salva Kiir of South Sudan and former rebel
leader Riek Machar sat down together for dinner
at the Vatican. They ate in silence, the start of a two-
day retreat aimed at reconciliation from a civil war
that has killed some 400,000 people since 2013.
At about the same time in my laboratory at the Uni-
versity of Sussex in England, Ph.D. student Alberto
$Dß ̧§DÿDäÇøîî³îx³ä³î ̧ø`xäî ̧D³xÿ
experiment in which volunteers experience being
in a room that they believe is there but that is not.
In psychiatry clinics across the globe, people arrive
complaining that things no longer seem “real” to
them, whether it is the world around them or their
own selves. In the fractured societies in which
we live, what is real—and what is not—seems to
be increasingly up for grabs. Warring sides mayexperience and believe in different realities. Perhaps eating together in
silence can help because it offers a small slice of reality that can be agreed
on, a stable platform on which to build further understanding.NEUROSCIENCE