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R. Barrett and P. P. Delsanto
of being poisoned. When she became hospitalised for six months in 1977, he
refused to eat, starving himself to death.
Let us leave to psychologists the task of fathoming Gödel’s complex personality. His introvert nature was doubtless much tested by the murder of a friend
during the Nazi period and by the upheaval, both enthusiastic and acrimonious, that followed his discoveries. Instead, let us see if we can obtain an
inkling of the nature of the work that so impressed Einstein (and others),
and the implications that still resound through mathematics and physics.
As we have discussed in Chap. 2, the aim of formal mathematics was
(and still largely is) to begin with a small number of axioms and definitions, and by using the rules of logic, to deduce ever more complex truths, or
theorems. Probably the most extreme example of this approach is the Principia Mathematica of Russell and Whitehead, which we cited in the previous
Chapter.
One of the banes that can occur in logic is the discovery of a paradox,
which may indicate either that we have made a mistake in our reasoning,
or that our logical framework is insufficient for our purposes, as we saw in
the last Section when attempting to find the square root of 2 while limiting
ourselves to rational numbers.
In Chap. 1, we encountered Zeno’s Paradox of Achilles and the Tortoise.
Another famous example, according to tradition, is due to Epimenides of
Crete, who came up with the statement “All Cretans are liars”. 4 Now this
statement presents an intrinsic self-contradiction, since, if it is true, it cannot
be asserted by a Cretan, because a Cretan would lie, and vice versa if it is
false. Since Cretans, like everybody else, are not necessarily always lying or
always truthful, the so-called “Liar paradox” is usually stated more succinctly
as “This statement is false”. If the statement is true, its content indicates
it must be false, and vice versa. Now, since language is essentially a tool of
communication, rather than of logic, it is not surprising that it contains
contradictions or inconsistencies. What is really surprising is that the same
thing also happens in mathematics.
While working on Principia Mathematica, Bertrand Russell uncovered a
paradox in Set Theory, which he was using as the formal mathematical
framework for his ambitious opus. Normally first encountered by mathematics students in advanced University courses, Set Theory enjoyed a brief
period of popularity in the 1960s when it formed the basis of “New Math”
and was taught to primary school students to help them learn arithmetic.
4 All this, of course, has nothing to do with the truthfulness of Cretans, since it is only used as a
dialectic argument. It is curious that St. Paul took the statement literally and used it, together with
other arguments, to conclude that non-Christian Cretans were “evil beasts”.
R. Barrett and P. P. Delsanto
of being poisoned. When she became hospitalised for six months in 1977, he
refused to eat, starving himself to death.
Let us leave to psychologists the task of fathoming Gödel’s complex personality. His introvert nature was doubtless much tested by the murder of a friend
during the Nazi period and by the upheaval, both enthusiastic and acrimonious, that followed his discoveries. Instead, let us see if we can obtain an
inkling of the nature of the work that so impressed Einstein (and others),
and the implications that still resound through mathematics and physics.
As we have discussed in Chap. 2, the aim of formal mathematics was
(and still largely is) to begin with a small number of axioms and definitions, and by using the rules of logic, to deduce ever more complex truths, or
theorems. Probably the most extreme example of this approach is the Principia Mathematica of Russell and Whitehead, which we cited in the previous
Chapter.
One of the banes that can occur in logic is the discovery of a paradox,
which may indicate either that we have made a mistake in our reasoning,
or that our logical framework is insufficient for our purposes, as we saw in
the last Section when attempting to find the square root of 2 while limiting
ourselves to rational numbers.
In Chap. 1, we encountered Zeno’s Paradox of Achilles and the Tortoise.
Another famous example, according to tradition, is due to Epimenides of
Crete, who came up with the statement “All Cretans are liars”. 4 Now this
statement presents an intrinsic self-contradiction, since, if it is true, it cannot
be asserted by a Cretan, because a Cretan would lie, and vice versa if it is
false. Since Cretans, like everybody else, are not necessarily always lying or
always truthful, the so-called “Liar paradox” is usually stated more succinctly
as “This statement is false”. If the statement is true, its content indicates
it must be false, and vice versa. Now, since language is essentially a tool of
communication, rather than of logic, it is not surprising that it contains
contradictions or inconsistencies. What is really surprising is that the same
thing also happens in mathematics.
While working on Principia Mathematica, Bertrand Russell uncovered a
paradox in Set Theory, which he was using as the formal mathematical
framework for his ambitious opus. Normally first encountered by mathematics students in advanced University courses, Set Theory enjoyed a brief
period of popularity in the 1960s when it formed the basis of “New Math”
and was taught to primary school students to help them learn arithmetic.
4 All this, of course, has nothing to do with the truthfulness of Cretans, since it is only used as a
dialectic argument. It is curious that St. Paul took the statement literally and used it, together with
other arguments, to conclude that non-Christian Cretans were “evil beasts”.