66
R. Barrett and P. P. Delsanto
two separate divisions, one comprising real numbers and the other imaginary
numbers. However, there are also complex numbers, which have one component that is real and another component that is imaginary. Complex numbers
were for many years regarded with the same suspicion that Pythagoras had
accorded irrational numbers. However, in the 19th Century it was realized
that they were essential tools for the solution of many problems in science
and engineering.
So, at last, our number world is complete. We can perform any mathematical operation we like on complex numbers and still find a result that is
not outside the domain of complex numbers. 3 If you replace “telling a story”
with “solving an equation”, you can now see the analogy between Alice’s story
and the contents of the current section: both cases illustrate the need of going
to a wider system. In the next section, we’ll see what all this has to do with
Gödel, and with some of the current developments of Physics.
4.3 Gödel and Completeness
Kurt Gödel (see Fig. 4.2) is considered by many mathematicians to be the
greatest logician since Aristotle. Einstein had once famously said that his only
reason for going to Princeton was the opportunity to walk there with G×del
and discuss problems with him.To illustrate his fame, it may be appropriate
to recall Einstein’s comments, when he handed out the first Einstein Award,
assigned jointly in 1951 to Julian Schwinger and to Gödel. To the former
he said “You deserve it” and to the latter “You don’t need it”, meaning that
Gödel’s scientific relevance was superior to the importance of the prize itself
[1].
It is curious that something similar had been said thirty years earlier about
Einstein himself, when the Nobel Prize judges hesitated before assigning the
prize to him, since their policy had always been to reward only experimental
discoveries or experimentally verified theories, which was not yet the case
with Einstein’s General Relativity. Eventually a way around the impasse was
found by giving the Nobel Prize to Einstein for his work on the photoelectric effect. On that occasion one of the judges remarked that assigning the
prize to him was an honour more for the prize than for Einstein himself.
However, although Einstein is still something of a popular cult figure more
than 60 years after his death, Gödel’s name and work are largely unknown,
even among many scientists and engineers.
3 The reader might be tempted to suppose that determining the square root ofi would require a
further extension of our number world. Not so. The square root of i is (1 + i)/
√ 2 (i.e., a complex
number), as can be verified by squaring this value, which yields i once more.
R. Barrett and P. P. Delsanto
two separate divisions, one comprising real numbers and the other imaginary
numbers. However, there are also complex numbers, which have one component that is real and another component that is imaginary. Complex numbers
were for many years regarded with the same suspicion that Pythagoras had
accorded irrational numbers. However, in the 19th Century it was realized
that they were essential tools for the solution of many problems in science
and engineering.
So, at last, our number world is complete. We can perform any mathematical operation we like on complex numbers and still find a result that is
not outside the domain of complex numbers. 3 If you replace “telling a story”
with “solving an equation”, you can now see the analogy between Alice’s story
and the contents of the current section: both cases illustrate the need of going
to a wider system. In the next section, we’ll see what all this has to do with
Gödel, and with some of the current developments of Physics.
4.3 Gödel and Completeness
Kurt Gödel (see Fig. 4.2) is considered by many mathematicians to be the
greatest logician since Aristotle. Einstein had once famously said that his only
reason for going to Princeton was the opportunity to walk there with G×del
and discuss problems with him.To illustrate his fame, it may be appropriate
to recall Einstein’s comments, when he handed out the first Einstein Award,
assigned jointly in 1951 to Julian Schwinger and to Gödel. To the former
he said “You deserve it” and to the latter “You don’t need it”, meaning that
Gödel’s scientific relevance was superior to the importance of the prize itself
[1].
It is curious that something similar had been said thirty years earlier about
Einstein himself, when the Nobel Prize judges hesitated before assigning the
prize to him, since their policy had always been to reward only experimental
discoveries or experimentally verified theories, which was not yet the case
with Einstein’s General Relativity. Eventually a way around the impasse was
found by giving the Nobel Prize to Einstein for his work on the photoelectric effect. On that occasion one of the judges remarked that assigning the
prize to him was an honour more for the prize than for Einstein himself.
However, although Einstein is still something of a popular cult figure more
than 60 years after his death, Gödel’s name and work are largely unknown,
even among many scientists and engineers.
3 The reader might be tempted to suppose that determining the square root ofi would require a
further extension of our number world. Not so. The square root of i is (1 + i)/
√ 2 (i.e., a complex
number), as can be verified by squaring this value, which yields i once more.