- Any two points can be joined by a straight line.
- Any straight line segment can be extended indefinitely in a straight
line. - Given any straight line segment, a circle can be drawn having the
segment as radius and one endpoint as the center. - All right angles are congruent.
- Through any point not on a given straight line, one and only one
straight line can be drawn parallel to the given line through the given
point.
Certain nouns are not defined (point, straight line, etc.), and neither are
certain verbs ( joined, extended indefinitely, etc.), although we all know
what they mean. Once we accept these axioms, in the sense that we agree to
work with them, the game is afoot—derive logical conclusions from them.
That’s all the mathematician has to do.
The theory of electromagnetism starts with Coulomb’s law, which states
that the magnitude of the electrostatic force between two point charges is
directly proportional to the magnitudes of each charge and inversely pro-
portional to the square of the distance between the charges. This law is
analogous to Newton’s law of universal gravitation, which states that the
magnitude of the gravitational force between two point masses is directly
proportional to the mass of each object and inversely proportional to the
square of the distance. The reason the two theories are not identical is
that mass is inherently positive, whereas charge can be either positive or
negative. We accept Coulomb’s law as the starting point because no meas-
urement has ever contradicted it. The game is again to derive logical con-
clusions from it—but that is far from all that the physicist has to do. The
logical conclusions enable the physicist to devise experiments that will
test not the validity of the conclusions—which is all that matters in math-
ematics—but whether the conclusion is consistent with observable real-
ity. Logical conclusions in physics are continually subjected to this reality
check—because the utility of a physical theory is limited by how it ac-
cords with observable reality.
When Two Theories Overlap
Physicists have developed two highly successful theories: relativity, which
does an excellent job of describing the gravitational force, and quantum
mechanics, which does an even better job (at least, from the standpoint of
the precision to which experimentation has confirmed the two theories)
of describing the mechanical and electromagnetic behavior of particles at
34 How Math Explains the World