The History and Concept of
Mathematical Proof
Steven G. Krantz1
February 5, 2007
A mathematician is a master of critical thinking, of analysis, and of deduc-
tive reasoning. These skills travel well, and can be applied in a large variety
of situations—and in many different disciplines. Today, mathematical skills
are being put to good use in medicine, physics, law, commerce, Internet de-
sign, engineering, chemistry, biological science, social science, anthropology,
genetics, warfare, cryptography, plastic surgery, security analysis, data ma-
nipulation, computer science, and in many other disciplines and endeavors
as well.
The unique feature that sets mathematics apart from other sciences, from
philosophy, and indeed from all other forms of intellectual discourse, is the
use of rigorous proof. It is the proof concept that makes the subject cohere,
that gives it its timelessness, and that enables it to travel well. The purpose
of this discussion is to describe proof, to put it in context, to give its history,
and to explain its significance.
There is no other scientific or analytical discipline that uses proof as
readily and routinely as does mathematics. This is the device that makes
theoretical mathematics special: the tightly knit chain of reasoning, following
strict logical rules, that leads inexorably to a particular conclusion. It is
proof that is our device for establishing the absolute and irrevocable truth
of statements in our subject. This is the reason that we can depend on
mathematics that was done by Euclid 2300 years ago as readily as we believe
in the mathematics that is done today. No other discipline can make such
an assertion.
1It is a pleasure to thank the American Institute of Mathematics for its hospitality and
support during the writing of this paper.
1
Figure 1: Mathematical constructions from surveying.
1 The Concept of Proof
The tradition of mathematics is a long and glorious one. Along with philoso-
phy, it is the oldest venue of human intellectual inquiry. It is in the nature of
the huma