4 Gödel, Hawking and the Foundations of Physics
69
As satirist/mathematician, Tom Lehrer, stressed: “the important thing is to
understand what you’re doing, rather than to get the right answer.” New Math
has long been confined to the dustbin of history as another failed experiment by misguided educationalists. However, Set Theory is still an important
component of the foundations of mathematics.
Russell’s Paradox (see Appendix 4.1) is of a self-referential nature, similar
to the Liar’s paradox above. Obviously it is disconcerting to base a formal
analysis of the structure of mathematics on a theory with a paradox at its
heart. Various solutions have been proposed for the paradox. However, what
is of interest to us here is that it inspired a young student by the name of
Kurt Gödel to investigate the logical foundations of mathematics and the
nature of mathematical proofs. His discoveries have shown that the formal
derivation of all of mathematics from a few basic axioms, as attempted by
Whitehead and Russell, is actually impossible. Let us try to be a little more
specific, without going into technical details.
Two things we require from any branch of mathematics are consistency and
completeness. In the case of Whitehead and Russell, it was arithmetic that they
attempted to put on a rigorous formal basis. By consistency, we mean that it
should not be possible, by following two different chains of reasoning from the
same axioms, to prove that something is both true and false. By completeness,
we mean that if something is true, it should be possible to prove that it is true,
starting from our axioms. There should not be any theorems that cannot be
proved. We may not be smart enough to prove them at the moment, but
we should have the hope, sometime in the future, of discovering a proof. It
should certainly not be the case that they can never be proved.
However, what Gödel succeeded in proving is that, if we have a system that
is consistent (i.e. does not contain any contradictions), then it must contain at
least one statement that is true, but which can never be proved. (See Appendix
4.2).
Not a problem, we might say. Let us just include that unprovable statement as an extra axiom, and all will be well. Not so. The inclusion of the
extra axiom will change the system so that somewhere else, another different,
unprovable but true, statement will be discovered. It is like trying to plug
the leaks in a rusty rainwater tank. The stresses of the repair operation
usually cause several new leaks to appear somewhere else in the tank (see
Fig. 4.3). Gödel showed that arithmetic (and presumably other branches of
mathematics) cannot be both complete and consistent simultaneously.
We see from the above why Gödel’s work has severe implications for mathematics. Indeed, debate still rages in philosophical circles as to whether it
demonstrates that there are “ineluctable limits to human reason” [2]. Leaving
69
As satirist/mathematician, Tom Lehrer, stressed: “the important thing is to
understand what you’re doing, rather than to get the right answer.” New Math
has long been confined to the dustbin of history as another failed experiment by misguided educationalists. However, Set Theory is still an important
component of the foundations of mathematics.
Russell’s Paradox (see Appendix 4.1) is of a self-referential nature, similar
to the Liar’s paradox above. Obviously it is disconcerting to base a formal
analysis of the structure of mathematics on a theory with a paradox at its
heart. Various solutions have been proposed for the paradox. However, what
is of interest to us here is that it inspired a young student by the name of
Kurt Gödel to investigate the logical foundations of mathematics and the
nature of mathematical proofs. His discoveries have shown that the formal
derivation of all of mathematics from a few basic axioms, as attempted by
Whitehead and Russell, is actually impossible. Let us try to be a little more
specific, without going into technical details.
Two things we require from any branch of mathematics are consistency and
completeness. In the case of Whitehead and Russell, it was arithmetic that they
attempted to put on a rigorous formal basis. By consistency, we mean that it
should not be possible, by following two different chains of reasoning from the
same axioms, to prove that something is both true and false. By completeness,
we mean that if something is true, it should be possible to prove that it is true,
starting from our axioms. There should not be any theorems that cannot be
proved. We may not be smart enough to prove them at the moment, but
we should have the hope, sometime in the future, of discovering a proof. It
should certainly not be the case that they can never be proved.
However, what Gödel succeeded in proving is that, if we have a system that
is consistent (i.e. does not contain any contradictions), then it must contain at
least one statement that is true, but which can never be proved. (See Appendix
4.2).
Not a problem, we might say. Let us just include that unprovable statement as an extra axiom, and all will be well. Not so. The inclusion of the
extra axiom will change the system so that somewhere else, another different,
unprovable but true, statement will be discovered. It is like trying to plug
the leaks in a rusty rainwater tank. The stresses of the repair operation
usually cause several new leaks to appear somewhere else in the tank (see
Fig. 4.3). Gödel showed that arithmetic (and presumably other branches of
mathematics) cannot be both complete and consistent simultaneously.
We see from the above why Gödel’s work has severe implications for mathematics. Indeed, debate still rages in philosophical circles as to whether it
demonstrates that there are “ineluctable limits to human reason” [2]. Leaving