How Mathematicians Convince Each Other or
“The Kingdom of Math is Within You”
Introduction and abstract
There is a problem about mathematical proof—actual mathematical proof,
done by actual mathematicians. Typical samples of mathematicians’ proof hardly
resemble formal proof.
Formal proof has been well described and analyzed by logicians. Mathemati-
cians’ proofs are different. We rarely cite the rules of logic. (Many of us don’t
know much about logic.) Our “proofs” don’t usually start from axioms. If they
include rule-governed calculations, these calculations are embedded in noncalcula-
tive reasoning—a combination of verbal argument and citation of literature. Nev-
ertheless, these “informal” proofs work—they compel agreement. That is what
mathematicians mean by a proof—an argument that compels agreement.
Mathematicians and logicians both end by claiming that a certain statement
has been “proved.” But the steps that reach this claim are very different.
Recognizing this discrepancy, Brendan Larvor has asked the following question:
“What qualifies the usual informal mathematical arguments as ‘proof’?”
Before answering Larvor’s question, I begin by posing and answering a different
question. Not “what qualifies them?”, but rather, “What makes informal proof
work?” How does it compel agreement?
Mathematicians’ arguments are glaringly incomplete from the point of view of
formal logic, yet they actually do compel agreement. This “working”, this com-
pelling agreement, is stronger in mathematics than in any other human endeavor.
Indeed, it is the defining quality of mathematics. “Mathematics is the science that
draws necessary conclusions” (Benjamin Peirce).
How does it “work”? How does it compel assent, inconclusive as it is by the
standards of formal logic?
Mathematician’s proof works for the same reason that ordinary empirical sci-
ence works. Different people observing the stars see the same thing, because they
are looking at the same thing. Different mathematicians observing a sporadic group
or a stochastic process “see” the same thing, because they also are “looking” at
the same thing. That is how, starting from established mathematics, we establish
a new result, which then becomes part of established mathematics.
This claim is argued below, supported by a classic description of Isaac Newton’s
thinking, and by half a dozen famous mathematicians, describing their experience
of this internal reality.