**Review of **

*Incompleteness: The Proof and Paradox of Kurt Goedel*** byRebecca Goldstein**

** **

**David Guaspari**

*The Weekly Standard***, May 2005**

Rebecca Goldstein's fine book makes
Kurt Goedel the protagonist of a tragic love story. Enamored, at age 20, with the Platonic vision of a realm of
objective truth, he made his life a quest for it. His work in mathematics and logic created new fields of research
and provoked voluminous commentary. Arguably, it shed light on the nature of
the mind and therefore on what it means to be human. But the story ends in bitter irony. Goedel's work was routinely enlisted in a war against the very
possibility of objective truth; and his final years were consumed by what is,
at least figuratively, a disease of reason gone wild: clinical paranoia.

Goldstein, a philosopher and
novelist, presents a moving picture of a passionate life devoted to
"abstruse" concerns and invokes, appropriately, the Platonic theme
that a genuine philosopher hungers for truth with an intensity that is
erotic. _Incompleteness_ is much more
rewarding than the garrulous publishing phenomenon _Goedel, Escher, Bach_,
which labors under the handicap that Goedel, Escher, and Bach have no
nontrivial connections with one another.

What can one plausibly say about the
inner life of someone as guarded and opaque as Goedel? Someone who limited his public statements to
propositions that he could rigorously prove, and was given in private to show-stopping
gnomic utterance--e.g., "I don't believe in natural science."

Goldstein finds a key in Goedel's
deep friendship with Einstein, who said that he went to his office every day
"for the privilege of walking home with Goedel." They were intellectual peers. (Goedel, though unknown to the general
public, was venerated by the world's best mathematicians.) But their personalities could not have been
more different--one furtive and enigmatic, the other with the public face of an
affable secular saint.

What bound them together? Both believed deeply that science is a
search for objective truth, and worked only on problems they believed to be
scientifically and philosophically important.
Most significant, Goldstein suggests, is that these very commitments
made them "intellectual exiles," because the twentieth century saw,
and we are still seeing, an "intellectual revolt against objectivity and
rationality." Goldstein reads
Goedel's work as a direct response. To understand that requires attention to
the surprisingly central role that the philosophy of mathematics has played in
the history of Western thought.

Anyone who wonders about what we know
and how we know it will be struck by the peculiar nature of mathematics. It seems to offer truths that are certain
but not based on any actual experience of the world. Kant summed up this peculiarity in a famous question that he
placed at the center of the _Critique of Pure Reason_--How is pure mathematics
possible?

We'll consider two broad answers that
predate (and postdate) Kant, which I'll call Platonist and anti-Platonist. Plato regarded mathematics as a model of
true knowledge, of apprehending the world's underlying reality undistorted by
its appearance. The anti-Platonists, a
heterogeneous coalition of the unwilling, hold that any supposed
"knowledge" making no appeal to experience can be certain only
insofar as it says nothing at all about the world. "All unmarried men are mortal" is an awfully good bet
but, like any summary of experience, is open to revision. "All unmarried men are bachelors"
is certain precisely because it relies on no facts about men or marriage, but
only on the definitions of the words it contains.

The anti-Platonists could be refuted
by proving mathematical results with philosophical significance. The results would be certain, because
mathematical; yet the very fact that they provoked--better yet, required--a
philosophical response would show that they were most assuredly about
_something_. The heart of Goldstein's
book is an account of such a result, Goedel's First Incompleteness
Theorem.

I will state it anachronistically, by
reference to computing. Euclid presents
geometry as a _theory_: a body of _axioms_, taken as self-evident, from which
further geometrical truths are deduced by _proofs_, which are sequences of
logical steps. By the end of the 19th
century it had become clear that any known field of mathematics (and anything
likely ever to be accepted as mathematics) could, in principle, be expressed in
a formal theory. A theory is _formal_
if a computer can be programmed to recognize whether or not any alleged proof
is valid. Insight may be needed to
devise proofs but, if they are formal, not to check them: a computer can scan a
sequence of symbols and mindlessly apply mechanical rules to determine whether
it represents valid logical steps starting from axioms.

Any "useful" theory can be
made formal (because all mathematics can), must be consistent (not allow the
deduction of contradictory results), and will incorporate at least the ability
to perform elementary arithmetic (with rules for addition, multiplication, and
simple algebra, such as x+y=y+x).

The First Incompleteness Theorem,
slightly cleaned up, says that any such "useful" theory is
incomplete--i.e., there are propositions of elementary arithmetic that the
theory can neither prove nor disprove. In fact, we can find such a proposition
that is essentially the statement of an elaborate algebraic rule. And someone who understands what that
proposition means will recognize that it is true--because, under a clever
interpretation, it means that it cannot be proven. Were we to add this proposition as a new axiom, the
Incompleteness Theorem would immediately provide another truth unprovable in
this beefed-up theory. There is no
escape.

This is head-spinning. How, for example, can we
"recognize" that some arithmetical rule is true if we cannot prove it
from beloved axioms that hitherto seemed to say all we knew about numbers? And just how does one reason to a conclusion
about the limits of reasoning?

It is also an affront. David Hilbert, one of history's greatest
mathematician, testified to an ancient faith when he said that, "In
mathematics, there is no _ignorabimus_."
When we seek an answer, "We must know. We will know."

Goldstein presents a sketch of
Goedel's reasoning that should be accessible to a general reader. Some small technical miscues are not
seriously misleading. (For the
cognoscenti: The fixed-point lemma is misstated; and the demonstration that the
"Goedel sentence" is unprovable appeals unnecessarily to its meaning,
and thereby relies on a stronger hypothesis than consistency.)

Goedel announced his results in one
terse paragraph at the end of a three-day technical meeting--and made not a
ripple. Only the great scientific
polymath John von Neumann realized that something important had happened. But what?

Goedel's interpretation is
startling. It follows, he says, that at
least one of the following two things must be true (and he, in fact, believed
both): "Either the human mind cannot be reduced to the working of the
brain" or "mathematical objects and facts ... exist objectively and
independently of our mental acts and decisions." (These quotations come from _Logical Dilemmas_, John Dawson's biography
of Goedel. Goldstein cites opinions a
bit less provocative.) Here's my guess
at what he meant: If my mind is only my brain, it's some kind of computer, so
all its purely deductive workings are captured within some "useful"
theory. But my mind can also prove the
First Incompleteness Theorem, allowing it to recognize some mathematical truths
that the theory does not imply. That
act of recognition can't be a deduction, so must be a perception--of an
independently existing mathematical realm.

Imagine Goedel's dismay, if not
despair, when many drew the opposite conclusion--that mathematics had lost its
certainty and been revealed as a social practice based on arbitrary decisions
about what axioms to choose. Or when, as Goldstein says, "Wittgenstein
never accepted that Goedel had proved what he provably did prove"--since
it would have contradicted Wittgenstein's deep belief that logic was
necessarily empty, and could contain no surprises.

Goedel's last years became a perverse
_reductio ad absurdum_ of his sustaining belief that everything has a rational
explanation. What had been intermittent episodes of paranoia--and
depression--became chronic. He feared
that his food might be poisoned, and that dangerous gasses were seeping from
his radiators and ice box. And he
starved to death.

Goldstein does not avert her eyes
from this obscene ending, but does not allow it the final word. She concludes with an elegant novelistic
turn. Goedel pursued philosophical questions about time in his characteristic
way, by seeking results that could be established mathematically. He produced a surprising solution of
Einstein's gravitational field equations, a hypothetical universe in which it
is possible to travel in time. If time
does loop back on itself, Goldstein says, "then a young Goedel will once
again sit in a college classroom in Vienna, transfigured by the notion of the
infinite eternal verities ... He will dream, silently and audaciously of
proving a mathematical theorem the likes of which has never before been seen, a
mathematical theorem that will illuminate the nature of mathematics
itself.

"And then he will do
it."