3 Truth and Beauty
43
foundation, on which the towering edifice of any field of mathematics is
constructed.
In the case of Euclidean geometry, the axioms reflect the reality of the
terrestrial space in which we live our daily lives. However, it is not necessary
for the axioms to have any connection with physical reality at all. Many fields
of mathematics are highly abstract, and some mathematicians consider only
mathematics that has no practical applications to be “pure”. G. H. Hardy,
whom we will meet again later in this Chapter, made a distinction between
“real” mathematics, “which has permanent aesthetic value”, and the remainder,
“the dull and elementary parts of mathematics”, that have practical use [2].
Hardy’s aloofness provoked retorts, such as the following, from those of a
more practical bent: A group of physicists and engineers were enjoying a flight in
a balloon until sudden gusts of strong wind compelled them to seek a safe landing
place. After they had set down, a physicist in the party leaned out of the basket and
asked a passing local where they were. “In a balloon”, came the reply. The physicist
turned back to his companions. “That man is obviously a mathematician,” he
declared. “His answer is totally correct, but absolutely useless.”
Let us now see how the axiomatic structure of mathematics compares with
the nature of physics. In physics, we start with definitions of quantities, such
as energy, momentum, velocity, mass, length, time, force etc., that are analogous to the definitions of Euclid. Some of these are more fundamental than
others: for instance, velocity is defined as the length travelled in a particular direction divided by the travel time. Then, instead of axioms, we have
physical “laws”, which are inferred through observations and experiments. For
example, Newtonian Mechanics, which describes the motion of objects in the
everyday world, is based on three laws (see Appendix 3.2). From them and
from the basic definitions, the interactions of bodies, both in the laboratory
and in the heavens, can be calculated using the rules of logic, as embodied in
mathematics.
Although the analogy between physics and mathematics is clear from
the above arguments, there is an important difference. The axioms of
mathematics are in a sense a priori, or originating from the mind of mathematicians. 1 There may be a correspondence between these axioms and the
physical world, but it is by no means necessary. Likewise, Riemann developed his own geometry where parallel lines meet, not to describe travels on
the Earth, but as a kind of mathematical game. When Einstein developed
1 We have discussed in Chap. 2 the ongoing philosophical debate on the a priori versus a posteriori nature of mathematics (Platonism versus Empiricism). However, since practising mathematicians
formulate their axioms with little or no regard for the physical reality, we presume axioms to be a
priori for our purposes here. In Part 3 we shall reconsider this issue.
43
foundation, on which the towering edifice of any field of mathematics is
constructed.
In the case of Euclidean geometry, the axioms reflect the reality of the
terrestrial space in which we live our daily lives. However, it is not necessary
for the axioms to have any connection with physical reality at all. Many fields
of mathematics are highly abstract, and some mathematicians consider only
mathematics that has no practical applications to be “pure”. G. H. Hardy,
whom we will meet again later in this Chapter, made a distinction between
“real” mathematics, “which has permanent aesthetic value”, and the remainder,
“the dull and elementary parts of mathematics”, that have practical use [2].
Hardy’s aloofness provoked retorts, such as the following, from those of a
more practical bent: A group of physicists and engineers were enjoying a flight in
a balloon until sudden gusts of strong wind compelled them to seek a safe landing
place. After they had set down, a physicist in the party leaned out of the basket and
asked a passing local where they were. “In a balloon”, came the reply. The physicist
turned back to his companions. “That man is obviously a mathematician,” he
declared. “His answer is totally correct, but absolutely useless.”
Let us now see how the axiomatic structure of mathematics compares with
the nature of physics. In physics, we start with definitions of quantities, such
as energy, momentum, velocity, mass, length, time, force etc., that are analogous to the definitions of Euclid. Some of these are more fundamental than
others: for instance, velocity is defined as the length travelled in a particular direction divided by the travel time. Then, instead of axioms, we have
physical “laws”, which are inferred through observations and experiments. For
example, Newtonian Mechanics, which describes the motion of objects in the
everyday world, is based on three laws (see Appendix 3.2). From them and
from the basic definitions, the interactions of bodies, both in the laboratory
and in the heavens, can be calculated using the rules of logic, as embodied in
mathematics.
Although the analogy between physics and mathematics is clear from
the above arguments, there is an important difference. The axioms of
mathematics are in a sense a priori, or originating from the mind of mathematicians. 1 There may be a correspondence between these axioms and the
physical world, but it is by no means necessary. Likewise, Riemann developed his own geometry where parallel lines meet, not to describe travels on
the Earth, but as a kind of mathematical game. When Einstein developed
1 We have discussed in Chap. 2 the ongoing philosophical debate on the a priori versus a posteriori nature of mathematics (Platonism versus Empiricism). However, since practising mathematicians
formulate their axioms with little or no regard for the physical reality, we presume axioms to be a
priori for our purposes here. In Part 3 we shall reconsider this issue.