42
R. Barrett and P. P. Delsanto
related to that of mathematics, which undoubtedly explains why mathematics is so entwined with physics. Modern physical theories are formulated
in abstract mathematics, which is not the case in most other disciplines. This
gives physics its own aura that discourages those with scanty mathematical
talent.
To make the correspondence between physics and mathematics a little
clearer, let us revisit our high school days and our first introduction to geometry. For most students, this is their first encounter with the logical structure
of mathematics. Euclid founded his geometry upon a number of definitions
and axioms. The definitions tell us what we mean when we talk of points,
lines, circles, etc. The axioms, which are detailed in Appendix 3.1, are sometimes called “self-evident truths”, and are assumed without any attempt to
prove them.
All Euclid’s axioms seem to be very reasonable, and few among high school
students, or their teachers, question them. From these humble beginnings
the whole of Euclidean geometry can be derived by strict application of
elementary logic. We have discussed in Chap. 2 the nature of logic, and its
origins.
The fifth of Euclid’s axioms states that parallel lines never meet. Over the
centuries the question was raised as to whether this axiom was really necessary,
or indeed, true. From a purely mathematical point of view, mathematicians
attempted to deduce the axiom logically from the other four (in which case
the fifth would have lost its status as an axiom, and become a theorem),
but had no success. We live on a spherical planet, which, because of its size,
appears to us to be totally flat locally (except for hills, etc.). Hence, we might
expect from our experience that the fifth axiom could be valid locally, but not
globally.
Let us, for instance, imagine two-dimensional animals living on a large
sphere, analogous to the Earth. If two such individuals set off from different
points on the equator on parallel trajectories heading due north, they will
meet each other at the North Pole because of the curvature of their spherical
domain. In their world, parallel lines do indeed meet. Eventually, nineteenth
Century German mathematician, Bernhard Riemann, developed a geometry
for such a world. As we shall see in Chap. 7, in Einstein’s General Theory of
Relativity, where space–time is not flat, Euclid’s geometry does not apply and
Riemannian geometry is essential.
To summarise, mathematics uses logic to deduce complex “truths” from
underlying assumed simpler truths. In some branches of mathematics, these
basic truths may not be all that intuitive. However, they represent the
R. Barrett and P. P. Delsanto
related to that of mathematics, which undoubtedly explains why mathematics is so entwined with physics. Modern physical theories are formulated
in abstract mathematics, which is not the case in most other disciplines. This
gives physics its own aura that discourages those with scanty mathematical
talent.
To make the correspondence between physics and mathematics a little
clearer, let us revisit our high school days and our first introduction to geometry. For most students, this is their first encounter with the logical structure
of mathematics. Euclid founded his geometry upon a number of definitions
and axioms. The definitions tell us what we mean when we talk of points,
lines, circles, etc. The axioms, which are detailed in Appendix 3.1, are sometimes called “self-evident truths”, and are assumed without any attempt to
prove them.
All Euclid’s axioms seem to be very reasonable, and few among high school
students, or their teachers, question them. From these humble beginnings
the whole of Euclidean geometry can be derived by strict application of
elementary logic. We have discussed in Chap. 2 the nature of logic, and its
origins.
The fifth of Euclid’s axioms states that parallel lines never meet. Over the
centuries the question was raised as to whether this axiom was really necessary,
or indeed, true. From a purely mathematical point of view, mathematicians
attempted to deduce the axiom logically from the other four (in which case
the fifth would have lost its status as an axiom, and become a theorem),
but had no success. We live on a spherical planet, which, because of its size,
appears to us to be totally flat locally (except for hills, etc.). Hence, we might
expect from our experience that the fifth axiom could be valid locally, but not
globally.
Let us, for instance, imagine two-dimensional animals living on a large
sphere, analogous to the Earth. If two such individuals set off from different
points on the equator on parallel trajectories heading due north, they will
meet each other at the North Pole because of the curvature of their spherical
domain. In their world, parallel lines do indeed meet. Eventually, nineteenth
Century German mathematician, Bernhard Riemann, developed a geometry
for such a world. As we shall see in Chap. 7, in Einstein’s General Theory of
Relativity, where space–time is not flat, Euclid’s geometry does not apply and
Riemannian geometry is essential.
To summarise, mathematics uses logic to deduce complex “truths” from
underlying assumed simpler truths. In some branches of mathematics, these
basic truths may not be all that intuitive. However, they represent the