EPILOGUE 3oi
is, they are self-referential. For example, statements like the following
are paradoxical:
This sentence is false.
I am a liar.
This statement cannot be proven.
In the first case, if the sentence is true, it means it is false. If the
sentence is false, then the statement is true. Likewise, if I am telling the
truth, then I am telling a lie; and if I am telling a lie, then I am telling
the truth. In the last case, if the sentence is true, then it cannot be
proven to be true.
(The second statement is the famous liar's paradox. The Cretan
philosopher Epimenides used to illustrate this paradox by saying, "All
Cretans are liars." However, Saint Paul missed the point entirely and
wrote, in his epistle to Titus, "One of Crete's own prophets has said it,
'Cretans are always liars, evil brutes, lazy gluttons' He has surely told
the truth.")
The incompleteness theorem builds on statements such as "This
sentence cannot be proven using the axioms of arithmetic" and creates
a sophisticated web of these self-referential paradoxes.
Hawking, however, uses the incompleteness theorem to show that
a theory of everything cannot exist. He claims that the key to Gôdel's
incompleteness theorem is that mathematics is self-referential, and
physics suffers from this disease as well. Since the observer cannot be
separated from the observation process, it means that physics will always refer to itself, since we cannot leave the universe. In the final
analysis, the observer is also made of atoms and molecules, and hence
must be an integral part of the experiment he is performing.
But there is a way to avoid Hawking's criticism. To avoid the paradoxes inherent in Gôdel's theorem, professional mathematicians today
simply state that their work excludes all self-referential statements.
They can then circumvent the incompleteness theorem. To a large degree, the explosive development of mathematics since Gôdel's time has
is, they are self-referential. For example, statements like the following
are paradoxical:
This sentence is false.
I am a liar.
This statement cannot be proven.
In the first case, if the sentence is true, it means it is false. If the
sentence is false, then the statement is true. Likewise, if I am telling the
truth, then I am telling a lie; and if I am telling a lie, then I am telling
the truth. In the last case, if the sentence is true, then it cannot be
proven to be true.
(The second statement is the famous liar's paradox. The Cretan
philosopher Epimenides used to illustrate this paradox by saying, "All
Cretans are liars." However, Saint Paul missed the point entirely and
wrote, in his epistle to Titus, "One of Crete's own prophets has said it,
'Cretans are always liars, evil brutes, lazy gluttons' He has surely told
the truth.")
The incompleteness theorem builds on statements such as "This
sentence cannot be proven using the axioms of arithmetic" and creates
a sophisticated web of these self-referential paradoxes.
Hawking, however, uses the incompleteness theorem to show that
a theory of everything cannot exist. He claims that the key to Gôdel's
incompleteness theorem is that mathematics is self-referential, and
physics suffers from this disease as well. Since the observer cannot be
separated from the observation process, it means that physics will always refer to itself, since we cannot leave the universe. In the final
analysis, the observer is also made of atoms and molecules, and hence
must be an integral part of the experiment he is performing.
But there is a way to avoid Hawking's criticism. To avoid the paradoxes inherent in Gôdel's theorem, professional mathematicians today
simply state that their work excludes all self-referential statements.
They can then circumvent the incompleteness theorem. To a large degree, the explosive development of mathematics since Gôdel's time has