3 Truth and Beauty
45
that came to popular attention when it was seen on a chalkboard belonging
to Homer Simpson in the 1998 episode: “The Wizard of Evergreen Terrace” of
the animated TV comedy, “The Simpsons”. (The strange story behind Homer
Simpson and his chalkboard is discussed in Appendix 3.3.)
Here, each of the three integers 3987, 4365 and 4472 is raised to the
twelfth power. 2 If we try to check the accuracy of this relationship with our
calculators, we will need to be inventive, as the individual terms exceed the
largest integers (2 31 –1) that can currently be stored in a normal computer.
Let us now pose the question: is this equivalence correct, or not?
To a mathematician, the answer is straightforward: the relationship is
wrong because it violates a theorem proposed in the margin of a book in
1637 by Pierre de Fermat, and now known as Fermat’s Last Theorem. It took
358 years for a proof of it to be discovered by Andrew Wiles [3]. The theorem
states that integer relationships, such as the one above, are not possible for
powers greater than 2. A mathematician would not need to carry out any
arithmetic to know that this relationship must be wrong. 3
However, an actual numerical evaluation would reveal that the difference
between the two sides of the equivalence is exceedingly small (about one part
in a hundred billion), and quite negligible compared with the measurement
errors that are observed in the most accurate experiments in a physics laboratory. To most physicists, the equivalence is therefore “true”, and they would
not hesitate to use it, if any of their theories required it. In this sense, near
enough is good enough.
This attitude might seem reprehensible, but the aim of physics is to explain
nature, as far as measurements allow it. Physical laws are constantly being
refined as new experimental information comes to hand. Mind games are
best left to mathematicians.
We have already encountered this different mindset in an example, which
we discussed in Chap. 2. Newton’s Law of Gravity cannot be solved exactly
for a many-body system, such as our solar system, comprising the Sun, Earth,
Moon, other planets and rocket ships. However, extremely accurate approximations to the solutions can be obtained, accurate enough to send space-crafts
to distant planets. These approximations may not be exact solutions, but they
are near enough.
2 3987 12 is a shorthand way of writing 3987 × 3987 × 3987 × 3987 × 3987 × 3987 × 3987 ×
3987 × 3987 × 3987 × 3987 × 3987.
3 For those interested, a simple arithmetical disproof of the relationship is presented in Appendix 3.3,
and the magnitude of the discrepancy between the two sides of the equivalence is estimated.
45
that came to popular attention when it was seen on a chalkboard belonging
to Homer Simpson in the 1998 episode: “The Wizard of Evergreen Terrace” of
the animated TV comedy, “The Simpsons”. (The strange story behind Homer
Simpson and his chalkboard is discussed in Appendix 3.3.)
Here, each of the three integers 3987, 4365 and 4472 is raised to the
twelfth power. 2 If we try to check the accuracy of this relationship with our
calculators, we will need to be inventive, as the individual terms exceed the
largest integers (2 31 –1) that can currently be stored in a normal computer.
Let us now pose the question: is this equivalence correct, or not?
To a mathematician, the answer is straightforward: the relationship is
wrong because it violates a theorem proposed in the margin of a book in
1637 by Pierre de Fermat, and now known as Fermat’s Last Theorem. It took
358 years for a proof of it to be discovered by Andrew Wiles [3]. The theorem
states that integer relationships, such as the one above, are not possible for
powers greater than 2. A mathematician would not need to carry out any
arithmetic to know that this relationship must be wrong. 3
However, an actual numerical evaluation would reveal that the difference
between the two sides of the equivalence is exceedingly small (about one part
in a hundred billion), and quite negligible compared with the measurement
errors that are observed in the most accurate experiments in a physics laboratory. To most physicists, the equivalence is therefore “true”, and they would
not hesitate to use it, if any of their theories required it. In this sense, near
enough is good enough.
This attitude might seem reprehensible, but the aim of physics is to explain
nature, as far as measurements allow it. Physical laws are constantly being
refined as new experimental information comes to hand. Mind games are
best left to mathematicians.
We have already encountered this different mindset in an example, which
we discussed in Chap. 2. Newton’s Law of Gravity cannot be solved exactly
for a many-body system, such as our solar system, comprising the Sun, Earth,
Moon, other planets and rocket ships. However, extremely accurate approximations to the solutions can be obtained, accurate enough to send space-crafts
to distant planets. These approximations may not be exact solutions, but they
are near enough.
2 3987 12 is a shorthand way of writing 3987 × 3987 × 3987 × 3987 × 3987 × 3987 × 3987 ×
3987 × 3987 × 3987 × 3987 × 3987.
3 For those interested, a simple arithmetical disproof of the relationship is presented in Appendix 3.3,
and the magnitude of the discrepancy between the two sides of the equivalence is estimated.