September 2019, ScientificAmerican.com 39ing because although primes were designed to be mul-
tiplied together, it suggests amazing, accidental rela-
tions between even numbers and the sums of primes.
An abundance of evidence supports Goldbach’s
conjecture. In the 300 years since his original obser-
vation, computers have confirmed that it holds for all
even numbers smaller than 4 × 1018. But this evidence
is not enough for mathematicians to declare Gold-
bach’s conjecture correct. No matter how many even
numbers a computer checks, there could be a counter-
example—an even number that is not the sum of two
primes—lurking around the corner.
Imagine that the computer is printing its results.
Each time it finds two primes that add up to a specific
even number, the computer prints that even number.
By now it is a very long list of numbers, which you can
present to a friend as a compelling reason to believe
the Goldbach conjecture. But your clever friend is al-
ways able to think of an even number that is not on
the list and asks how you know that the Goldbach
conjecture is true for that number. It is impossible for
all (infinitely many) even numbers to show up on the
list. Only a mathematical proof—a logical argument
from basic principles demonstrating that Goldbach’s
conjecture is true for every even number—is enough
to elevate the conjecture to a theorem or fact. To this
day, no one has been able to provide such a proof.
The Goldbach conjecture illustrates a crucial dis-
tinction between the discovery stage of mathematics
and the proof stage. During the discovery phase, one
seeks overwhelming evidence of a mathematical fact—
and in empirical science, that is often the end goal.
But mathematical facts require a proof.
Patterns and evidence help mathematicians sort
through mathematical findings a nd d ecide w hat t o
prove, but they can also be deceptive. For example, let
us build a sequence of numbers: 121, 1211, 12111, 121111,
1211111, and so on. And let us make a conjecture: all the
numbers in the sequence are not prime. It is easy to
gather evidence for this conjecture. You can see that
121 is not prime, because 121 = 11 × 11. Similarly, 1211,
12111 and 121111 are all not prime. The pattern holds for
a while—long enough that you would likely get bored
checking—but then it suddenly fails. The 136th ele-
ment in this sequence (that is, the number 12111... 111,
where 136 “1”s follow the “2”) is prime.
It is tempting to think that modern computers can
help with this problem by allowing you to test the con-
jecture on more numbers in the sequence. But there
are examples of mathematical patterns that hold true
for the first 10^42 elements of a sequence and then fail.
Even with all the computational power in the world,
you would never be able to test that many numbers.
Even so, the discovery stage of the mathematical
process is extremely important. It reveals hidden con-
nections such as the Goldbach conjecture. Often two
entirely distinct branches of math are intensively
studied in isolation before a profound relation be-
tween them is uncovered. A relatively simple example
is Euler’s identity, ei π + 1 = 0 , which connects the geo-
metric constant π with the number i, defined algebra-
ically as the square root of –1, via the number e, the
base of natural logarithms. These surprising discover-
ies are part of the beauty and curiosity of math. They
seem to point at a deep underlying structure that
mathematicians are only beginning to understand.
In this sense, math feels both invented and discov-
ered. The objects of study are precisely defined, but
they take on a life of their own, revealing unexpected
complexity. The process of mathematics therefore
seems to require that mathematical objects be simul-
taneously viewed as real and invented—as objects
with concrete, discoverable properties and as easily
manipulable inventions of mind. As philosopher Pe-
nelope Maddy writes, however, the duality makes no
difference to how mathematicians work, “as long as
double-think is acceptable.”REAL OR UNREAL?
MATHEMATICAL REALISM is the philosophical position
that seems to hold during the discovery stage: the ob-
jects of mathematical study—from circles and prime
numbers to matrices and manifolds—are real and ex-
ist independently of human minds. Like an astrono-
mer exploring a far-off planet or a paleontologist
studying dinosaurs, mathematicians are gathering in-
sights into real entities. To prove that Goldbach’s con-
jecture is true, for example, is to show that the even
numbers and the prime numbers are related in a par-
ticular way through addition, just like a paleontolo-
gist might show that one type of dinosaur descended
from another by showing that their anatomical struc-
tures are related.
Realism in its various manifestations, such as Pla-
tonism (inspired by the Greek philosopher’s theory of
Platonic forms), makes easy sense of mathematics’
universalism and usefulness. A mathematical object
has a property, such as 7 being a prime number, in the
same way that a dinosaur might have had the property
of being able to fly. And a mathematical theorem, such
as the fact that the sum of two even numbers is even, is
true because even numbers really exist and stand in a
particular relation to each other. This explains why
people across temporal, geographical and cultural dif-
ferences generally agree about mathematical facts—
they are all referencing the same fixed objects.
But there are some important objections to realism.
If mathematical objects really exist, their properties
are certainly very peculiar. For one, they are causally
inert, meaning they cannot be the cause of anything,
so you cannot literally interact with them. This is a
problem because we seem to gain knowledge of an ob-
ject through its impact. Dinosaurs decomposed into
bones that paleontologists can see and touch, and a
planet can pass in front of a star, blocking its light
from our view. But a circle is an abstract object, inde-
pendent of space and time. The fact that π is the ratio
of the circumference to the diameter of a circle is not