ScAm - 09.2019

38 Scientific American, September 2019 Illustration by Bud Cook

Throughout the 20th century mathematicians de-

bated whether Zermelo-Fraenkel set theory should be

augmented with an additional rule, known as the axi-

om of choice: If you have infinitely many sets of ob-

jects, then you can form a new set by choosing one ob-

ject from each set. Think of a row of buckets, each con-

taining a collection of balls, and one empty bucket.

From each bucket in the row, you can choose one ball

and place it in the empty bucket. The axiom of choice

would allow you to do this with an infinite row of

buckets. Not only does it have intuitive appeal, it is

necessary to prove several useful and desirable math-

ematical statements. But it also implies some strange

things, such as the Banach-Tarski paradox, which

states that you can break a solid ball into five pieces

and reassemble those pieces into two new solid balls,

each equal in size to the first. In other words, you can

double the ball. Foundational assumptions are judged

by the structures they produce, and the axiom of

choice implies many important statements but also

brings extra baggage. Without the axiom of choice,

math seems to be missing crucial facts, though with it,

math includes some strange and potentially undesir-

able statements.

The bulk of modern mathematics uses a standard set

of definitions and conventions that have taken shape

over time. For example, mathematicians used to regard

1 as a prime number but no longer do. They still argue,

however, whether 0 should be considered a natural

number (sometimes called the counting numbers, natu-

ral numbers are defined as 0,1,2,3... or 1,2,3... , depend-

ing on who you ask). Which characters, or inventions,

become part of the mathematical canon usually de-

pends on how intriguing the resulting play is—observ-

ing which can take years. In this sense, mathematical

knowledge is cumulative. Old theories can be neglected,

but they are rarely invalidated, as they often are in the

natural sciences. Instead mathematicians simply choose

to turn their attention to a new set of starting assump-

tions and explore the theory that unfolds.

DISCOVERY

AS NOTED EARSIER, mathematicians often define objects

and axioms with a particular application in mind. Over

and over again, however, these objects surprise them

during the second stage of the mathematical process:

discovery. Prime numbers, for example, are the build-

ing blocks of multiplication, the smallest multiplica-

tive units. A number is prime if it cannot be written as

the product of two smaller numbers, and all the

nonprime (composite) numbers can be constructed by

multiplying a unique set of primes together.

In 1742 mathematician Christian Goldbach hypoth-

esized that every even number greater than 2 is the

sum of two primes. If you pick any even number, the

so-called Goldbach conjecture predicts that you can

find two prime numbers that add up to that even num-

ber. If you pick 8, those two primes are 3 and 5; pick 42,

and that is 13  +  29. The Goldbach conjecture is surpris-

HOW A HISTORICAL LINGUIST

SEARCHES FOR ANSWERS

Like any scientist, linguists rely on

the scientific method. One of the principal goals of

linguistics is to describe and analyze languages to discover the full range

of what is possible and not possible in human languages. From this, lin-

guists aim to reach their goal of understanding human cognition through

the capacity for human language.

3 ̧îšxßxžäD³øߐx³`āî ̧x† ̧ßîäî ̧lxä`ߞUxx³lD³xßxl§D³øDxäj

to document them while they are still in use, to determine the full range

of what is linguistically possible. There are around 6,500 known human

languages; around 45 percent of them are endangered.

"ž³øžäîäøäxDäÇx`ž‰`äxî ̧…`ߞîxߞDî ̧žlx³îž…āx³lD³xßxl§D³øD-

es and to determine just how endangered a language is: Are children still

learning the language? How many individual people speak it? Is the per-

centage of speakers declining with respect to the broader population?

And are the contexts in which the language is used decreasing?

5šxÔøxäîž ̧³ ̧…ä`žx³îž‰` ̧U¥x`îžþžîāD³lÙîßøîšÚžä` ̧³³x`îxlî ̧

endangered language research. Truth, in a way, is contextual. That

is, what we hold to be true can change as we get more data and evi-

dence or as our methods improve. The investigation of endangered

languages often discovers things that we did not know were possible

in languages, forcing us to reexamine previous claims about the limits

of human language, so that sometimes what we thought was true

can shift.

Lyle Campbell, an emeritus professor of linguistics at the University of Hawaii

at M ̄anoa, as told to Brooke Borel