From Aristotle down, the laws of logic have been regarded as
fixed and archetypal; and as such that they admit of no conceivable
alternatives. Often they have been attributed to the structure of the Universe
or to the nature of human reason; and in general they have been regarded as
providing Address of the retiring chairman and vice-president an Archimedean
fixed point in the realm of
thought.
[1]Since the time of
the Greeks, philosophers and mathematicians have been interested in defining
truth. What is truth? And how do we discover what is true in the world? It
has always been assumed that these questions could be answered. During the
beginning of the twentieth century, in fact, mathematicians such as David
Hilbert and Bertrand Russell thought that they were making great progress at
answering these questions within the domain of what was called mathematical
formalism. However, Gödel’s incompleteness paper of
1931
[2] necessitated a reassessment of
what truth was. This reassessment should have been discouraging to the
mathematical formalists, because it upset their primary goal for mathematics.
Despite this affront to his worldview, however, Hilbert soon came to embrace
Gödel’s proof as an exposition of formal mathematical principles and
as a proof of ongoing mathematical vitality–the most important goal of
Hilbert’s program.
During the beginning of the twentieth century, there
were two prevailing schools of thought within mathematics. One, the formalist
school was led by Hilbert, Russell, Alfred North Whitehead, and others. The
other, the intuitionist school, was dominated by Luitzen Egbertus Jan Brouwer.
The distinction between these two groups is logical; the formalists put all
faith in logical formalism, and the intuitionists in intuition. Logical
formalism depended on formal systems which contain a finite number of basic
axioms (self-evident rules that are accepted as true) and rules of inference
(concrete methods to deduce further truths). Logical formalism says that the
only way to define mathematical truth is by using this formal system. The
intuitionists, on the other hand, reject logic as the standard of truth within
mathematics. In other words, they said that mathematical objects could not be
defined by a formal system, and that truth, as a mathematical object, is a
construction of the mathematician’s mind, and could therefore only be
determined within the mind.
The Austrian Kurt Gödel, born in 1906, was
educated in the midst of conflicts between these two schools. He clearly worked
within a formalist frame, becoming one of the leaders in set theory, the field
of Bertrand Russell. Showing a strong aptitude, from the very beginning, for
logic, he attended the University of Vienna from 1923 until he finished his
doctoral dissertation in 1929 and took a teaching position there among the
school of logical positivists.
[3]
Then in 1931, he published his first and second incompleteness theorems, for
which he is best known.
[4]
Subsequently, Gödel continued to work and publish within the field of set
theory, but made no discoveries as important as that of his 1931
paper.
[5]As stated above, it is
Gödel’s famous 1931 paper that has earned him the most fame.
Consequently, it was no surprise when it was criticized soon after it had been
published; that is, it was no surprise, considering that his paper came to such
shocking conclusions, by such esoteric methods. The claim of the paper was
that, given any axiomatic system like
Principia Mathematica, Gödel
could construct a sentence within the system that was undecidable, in other
words, a sentence that could be neither proven nor
disproven.
[6] The implications of
this will be explored later.
Gödel’s Incompleteness Theorem had
several necessary precursors without which it would not have been relevant.
Most significantly, it depended on the presence of a book, such as
Principia
Mathematica,
[7] originally
published in 1910, which provided an entire system of arithmetic that rested on
the formalism of the axiomatic system upon which it was based. That is, it
provided a series of axioms or rules that were accepted to be true (for example
if x=y then y=x). These simple statements were then used to derive the rest of
arithmetic.
Principia Mathematica is a good example of an
instrumentalist text, providing “a system of meaningless marks, whose
formulas (or ‘strings’) are combined and transformed in accordance
with stated rules of
operation.”
[8] The
Principia
Mathematica helped greatly in examining the question of consistency, by
providing a notation for the language of arithmetic, as well as for the rules of
inference which are used. The idea, in other words, is that if you accepted the
axioms, or basic premises of the system, then all theorems that follow by the
rules of inference are true, no matter what the variables that you input
represent.
This system was also sufficiently complex. For example, it
addressed the problem of the theory of types established by Russell. This
problem addressed a paradox that could arise from self-referential systems. It
asked if you had two sets, one that contained precisely all of the sets
containing themselves, and a second containing all sets that do not contain
themselves, then where would you put the second set. If you put it in the first
set, then it would not contain itself and belongs in the second set, but then it
would contain itself and belongs in the first. Whitehead and Russell solved
this problem by defining sets of different types. This way a set could only
contain sets of the previous order. This system was sufficiently complex as to
allow Gödel to go beyond the question of consistency to the question of
completeness.
Gödel’s first incompleteness theorem, proven by his
1931 paper, set out to show that any formal system that is sufficiently complex
is undecidable. The example he uses is called P, using the laws from
Principia Mathematica. For a formal system to be undecidable it must
contain a sentence or formula, A, for which neither A could be proven, nor could
not-A be proven. The formula he came up with is in some ways similar to (and is
often times conflated with) the liar’s paradox (“This sentence is
false.”), in that it is self-referential. However, the sentence does not
refer to truth or falsehood, but instead to provability.
Also, the output of
the Gödel sentence depends more on the action of the sentence than in the
liar’s paradox. A good example is given by Chad Mazzola. Let’s
examine the sentence he gives, “Only a complete fool would take the time
to figure out that he had just written a sentence with exactly eight A's, one B,
four C's, three D's, and [?] E's."
[9]
If we replace [?] with “seventeen” then reexamination of the
sentence would make this false, and the new value for [?]
“twenty-one.” Then, on repeating this process again, we get
“nineteen.” The “correct” value of [?] then proceeds to
oscillate between “eighteen” and “twenty,” never
settling on any one value. However, this sentence does have a correct value for
[?], it just cannot be written, or it changes. The liar’s paradox, on the
other hand, does not have this quality. The Gödel sentence does.
The
Gödel sentence, G, says, “G is not provable.” If the
Gödel sentence is provable, then it is false, and by the
soundness
[10] of our system P, that
is not possible. So, what if the negation of the Gödel sentence is
provable? Again, by soundness, this makes G true (if the negation is provable,
the original sentence cannot be true). This is another contradiction. This is
the kind of sentence that Gödel had set out to find. Also, by closer
examination, we can see that the Gödel sentence is in fact true. It is not
provable. The fact that this truth is not provable within the system leads to
Gödel’s second incompleteness theorem and the subsequent affronts on
formalism. Gödel goes on to prove that this sentence exists in a much more
rigorous fashion, constructing a complex correspondence between formulas and
positive integers, and deriving this sentence logically from the axioms provided
by
Principia Mathematica.
The original response to Gödel’s
paper was misunderstanding by some. The biggest complaints came from Chaim
Perelman. Perelman argues that Gödel’s theorem, the liar’s
paradox, and other simple paradoxes like the question of “who shaves the
barber who shaves every man in town who doesn’t shave himself” are
all equivalent, and justify an approach like that of Brouwer, rejecting formal
systems altogether, in exchange for
intuitionism.
[11] Perelman’s
antinomy, with which he argues against Gödel, is essentially equivalent to
a problem which Russell had identified and addressed previously, with his theory
of types. The faults that Perelman finds with Gödel’s paper are only
applicable to the cursory explanation of Gödel’s proof given in the
introduction to his paper. These complaints are made null and void in the rest
of his paper, when Gödel actually formalizes a language, and a system to
which any sufficiently complex and consistent system can be reduced, and makes a
correspondence between the sets of sentences in this language, and a set of
positive integers which have come to be known as Gödel
numbering.
[12]By very rigorous
logical formalism, Gödel takes each sign within the language, and assigns a
numerical value to it. Using these values, he formally defines numbers
(products of ordered prime numbers raised to the power of the numerical
assignment of the sign they represent), that represent each axiom and subsequent
theorem that can be derived. One of the theorems that Gödel derives is a
provability predicate–a formula that tells whether or not the inputted
sentence is provable, when the sentence inputted is a numerical sentences as
defined by the Gödel numbering.
It is actually the intuitionists, some
making the same mistake as Perelman, who ended up understanding and using
implications of Gödel’s incompleteness theorems that Gödel
certainly did not intend. For example, Brouwer uses Gödel’s results
to justify his rejection of non-constructive existence proofs, which are proofs
that follow Hilbert’s ideal reasoning, and which is actually the type of
proof Gödel uses to prove his incompleteness
theorems.
[13] It is not
particularly surprising that intuitionists do not understand Gödel’s
proof, in the same way he intended it, because it depends heavily on formal
mathematical principles, in which he was trained, and the intuitionists
generally were not. In particular, intuitionists rejected the Law of Excluded
Middle, which says that every statement is either true or false. When this is
rejected, proofs by contradiction are no longer valid, so the problem that
arises from Gödel’s incompleteness theorem has no profound impact.
The intuitionists, however, interpreting Gödel’s incompleteness
theorem as the liar’s paradox, assumed that it was a formalized version of
this law, and not what it was, in fact a much more profound statement about the
meaning of proof. The formalists, on the other hand, who should have been
devastated by Gödel’s proof, were instead the first to understand and
embrace it, because of its formal nature.
One of the biggest names associated
with logical formalism is David Hilbert who was an eminent mathematician at the
turn of the century. His work was in various areas including invariant theory,
algebraic number theory, and geometry. His work in Euclidean geometry was
perhaps the most influential. Hilbert’s book ,
Die Grundlagen der
Geometrie[14], or
The
Foundations of Geometry[15],
published in 1899, was a fundamental text in formalist history. It followed
many other contemporary works on Euclidean geometry. Hilbert’s book,
however, went beyond the work of others like Peano, by focusing on metageometry,
that is not the geometry itself, but the metamathematical structure of it. This
was the first attempt to formalize mathematics by separating mathematical
principles from their physical basis in the world. Hilbert believed that even
basic geometry or arithmetic was not purely logical because it depended on the
mathematicians’ intuition or construction of the mathematical objects. He
therefore focused on closely examining the formal logic behind these
mathematical objects.
[16] Hilbert
determined the axioms or fundamental rules that one needed to accept in order to
derive the rest of Euclidean geometry, that is simple geometry of points, lines
and circles in a plane.
Hilbert’s re-axiomatization of Euclidean
geometry, using a much more formal system, was later able to be extended to
other areas of mathematics. These principles were universal, and the idea was
that no matter what the objects of the original axioms were, you could use any
subsequent theorems that had been derived from the axioms. In other words,
while Hilbert defined his objects, points, lines, and planes, he made sure that
these objects were each interchangeable with any other group of objects that had
a set of properties that were assigned to them. Hilbert is often quoted as
having said, “[o]ne must always be able to say ‘tables [sic] chairs,
beer-mugs’ instead of ‘points, lines,
planes’.”
[17] The goal
of the formalists was to establish a system like this to encompass not just
geometry but all of mathematics. Most of the formalists, including Hilbert,
presumed that this formal system would be finite. This is the assumption that
is challenged by Gödel’s incompleteness
theorems.
[18]Because of the
nature of geometry, however, its formalization was not directly affected by
Gödel’s incompleteness
theorem.
[19] Geometry was
originally studied by the Greeks because it required less abstraction than other
mathematics.
[20] Therefore, the
system described is not complex enough to be concerned with proving its own
consistency, a predicate for Gödel’s incompleteness theorem. This is
why problems arise only subsequently, with the publication of
Principia
Mathematica, which attempted to have a much wider scope, for example
incorporating the theory of types, which solved a paradox in set theory that
Russell had discovered, that involved self-referential sets.
Following his
work in geometry, Hilbert had a plan for mathematics that has come to be known
as Hilbert’s Program.
[21]
Hilbert embraced a particular strain of mathematical instrumentalism.
Instrumentalism is the examination of mathematical objects as instruments,
rather than actual constructions. This goes back to Hilbert’s famous
quote, on the preceding page. As opposed to others who studied instrumentalism,
Hilbert emphasized two main points. One is that he focused his attention on
ideal proofs, which are clear concise proofs that include only the signs of
formulas needed for the proof, with no explanation of the proof’s
significance.
[22] The French
mathematician Henri Poincaré argued that this method could grow to be
self-referential, but his argument is
weak.
[23]The second particular
to Hilbert’s program is his answer to the dilution problem. This refers
to controlling the quality, by which I mean the strength, of the results of a
given proof. Hilbert emphasized the use of a restricted mathematical
instrumentalism that was finite in order to avoid the problem of
dilution.
[24] Gödel’s
second incompleteness theorem subsequently raises issue with this, by showing
that an infinite system is needed for completeness. Hilbert’s emphasis on
strength in mathematics is a sign of his love of the subject. This explains his
hope to keep the subject alive in the future, which will be explained later in
this paper.
Another implication of Gödel’s first incompleteness
theorem is that the Gödel sentence is true, even if it is not provable
within the system P. This means that Gödel’s second incompleteness
theorem is easily derived. Gödel’s second incompleteness theorem
talks about the consistency of system P (or any sufficiently complex system).
The heart of this proof lies in being able to formalize the statement of
Gödel’s first incompleteness
theorem.
[25] This theorem defines a
formal predicate (Con) that denotes a statement of consistency of the system P.
It then goes on to show that the statement of consistency, (Con), cannot be
derivable in P unless P is in fact inconsistent. However, the system P can be
strengthened to include a proof of (Con), but it is now a new system, P’
for which we must derive a new statement of consistency (we could call it
(Con)
P’), which is again not derivable in P’ unless the
system is in fact inconsistent. This can be done over and over again, providing
more and more complete systems, but never one that is complete enough to prove
its own consistency.
It is this implication that makes the formalist program
to find a finite axiomatic system to encompass all of mathematics impossible.
We can see from Gödel’s second incompleteness theorem that there is
no entirely complete and consistent system, because we can always add an axiom
to strengthen a system, providing a transfinite number of axioms. Following
Hilbert’s lead in geometry, the formalists had presumed that there existed
a finite number of axioms that would define all of mathematics.
Then, when
Hilbert published his
Foundations of Mathematics or
Grundlagen der
Mathematik,
[26] he was
forced to acknowledge Gödel’s Second Incompleteness Theorem, and was
in fact the first to give a formal proof of
it.
[27] This shows that Hilbert had
understood Gödel’s original proof (because Gödel’s second
incompleteness theorem follows directly from the first). It also shows that he
embraced the proof, despite understanding its implications to his program. This
follows Hilbert’s policy of seeking out knowledge, because he thought that
there were no limitations on what one could
learn.
[28] This is seen in a quote
that is now seen on Hilbert’s tombstone, “
Wir müüssen
wissen. Wir werden wissen.” That is, “We must know. We will
know.”
[29]There are many
scholars who argue that Gödel’s incompleteness theorems were a
profound development and were extremely detrimental to the further development
of formalist mathematics. As Morris Kline, a mathematician famous for his
textbooks on the history of mathematics put it, “Gödel’s
theorems produced a
debacle.”
[30] He goes on to
refer to the Gödel paper as a “Pandora’s
box.”
[31] Similarly, another
mathematician who has spent time studying Gödel’s incompleteness
theorems, John Dawson argues that it is only
natural to invoke geological
metaphors to describe the impact and the lasting significance of
Gödel’s incompletness theorems. Indeed, how better to convey the
impact of those results–whose effect on Hilbert’s Programme was so
devastating and whose philosophical reverberations have yet to
subside–than to speak of tremors and shock waves? The image of shaken
foundation is
irresistible.
[32]These
implications especially caused problems for the formalists. For example Mary
Tiles, a philosopher of logic and mathematics, in her book,
Mathematics and
the Image of Reason, argues that “The results of both Gödel and
Church
[33] demonstrate that Hilbert
was mistaken in thinking that it might be possible, by suitable selection of
notation, to have a formal system which was both sufficiently complex to be
capable of being interpreted as a formalization of classical arithmetic and yet
in which theorems would be recognizable as such on the basis of possession of
some relatively simple morphological
characteristic.”
[34] Another
scholar, Jeremy Gray, stated that the work of the 1930's (including Gödel,
Turing, etc.) had
by and large dashed Hilbert’s hopes. . . It is not
possible to set up all of mathematics along the finitistic lines Hilbert had
hoped for, but perhaps something nearly as good could be achieved. . . It
certainly seemed as if Gödel’s incompleteness theorem ruled out any
natural starting point for a formalised mathematics, but perhaps Hilbert’s
ideas could be re-interpreted, or his aims realised in some unexpected
way.
[35]This asserts that
Gödel’s incompleteness theorem had changed the possible scope of
Hilbert’s program.
Another scholar, J. Fang, addresses the issue of why
Hilbert was so interested in foundational questions. “Hilbert’s
intimately personal experience of the activity in mathematical creation was
ultimately and entirely responsible for his plunging into the darkness of the
foundational problems. The most fundamental, and the most urgent, of them all
was the proof of consistency from the very beginning: A proof must be given, for
any branch of mathematics, that no contradictory theorems can ever arise from
the permitted procedures of
demonstration.”
[36] If Fang
is correct, that consistency was Hilbert’s priority, then
Gödel’s results would in fact have been very significant for Hilbert
and his outlook on mathematics. However, perhaps Hilbert cared more about the
success of mathematics in the bigger picture.
It is clear, especially before
the publication of Gödel’s incompleteness theorems, that this
finitist-formalist position had been very important to Hilbert. We can see this
in his 1927 lecture on “The Foundations of
Mathematics.”
[37] Here he
says, for example: “Mathematics proper becomes an inventory of
formulas.”
[38] “This
formula game is carried out according to certain definite rules, in which the
technique of our thinking is expressed. These rules form a closed system
that can be discovered and definitively
stated.”
[39] and throughout
this paper, explaining a proof of
finiteness.
[40] From these
quotations, we can see that in 1927 before any ideas of incompleteness, Hilbert
was totally convinced that his program was viable. Then how do we explain his
change of heart, with the inclusion of the proof of Gödel’s second
incompleteness theorem in his later publication with Bernays,
The Foundations
of Mathematics?
Perhaps this proof of Gödel’s second
incompleteness theorem was included by Bernays. Bernays seemed to emphasize the
strength of the implications of Gödel’s incompleteness theorems on
formal mathematical systems even more than Gödel himself. For example,
even in his original paper, Gödel seems to doubt the scope of his paper
saying,
I wish to note expressly that Theorem XI (and the corresponding
results for M and A) do not contradict Hilbert’s formalistic viewpoint.
For this viewpoint presupposes only the existence of a consistency proof in
which nothing but finitary means of proof is used, and it is conceivable that
there exist finitary proofs that cannot be expressed in the formalism of P (or
of M or A).
[41]This is a
note that refers to one of the steps in the proof of Gödel’s Second
Incompleteness Theorem that involves a step that some have argued is weak.
However Gödel was soon (by 1958, anyway) convinced by Bernays that this was
an unnecessary caution.
[42] In any
case, the inclusion of the proof in Hilbert and Bernays’ book basically
quieted all opposition to Gödel’s paper and asserted that it held
appreciable developments.
[43]An
alternative interpretation is that perhaps, once Hilbert had read and understood
Godel’s paper, he recognized that it was an example of what he considered
a great problem. It was easily stated, and then had a rigorous proof.
What’s more, its proof left the field open for an infinite number of
questions that could subsequently be derived from it. This meant a perpetually
alive mathematics for Hilbert.
This idea of mathematics being alive is
perhaps unintuitive for those people who studied mathematics in school only when
they had to. However, as far as Hilbert and many other mathematicians were and
are concerned, mathematics is alive. Hilbert asserted this vivacity in a
lecture on Wednesday August 8
th, 1900, at the International Congress
of Mathematicians, in Paris.
[44] At
this lecture he posed the twenty-three problems that he posed in a lecture for
which he is most well-known.
[45] If
a similar assertion were made today, of the mathematical problems facing the
next generation, some of Hilbert’s original problems would remain (namely
Goldbach’s conjecture), but many more would have originated from the
development of Gödel’s incompleteness theorem.
Hilbert defined
problems that he thought were good problems. He judged good problems by the
fact that their solutions opened the mind to a wider view of mathematics, and
perhaps lead to new theorems and new fields of research. It was also very
important for Hilbert’s problems to be “characterized by their
clarity and ease of
comprehension.”
[46] These
problems are a good example of the intimate relationship between the mathematics
of Gödel and Hilbert. Gödel had a hand in solving both of
Hilbert’s first two problems, although he did not solve them in the way
that Hilbert had anticipated.
Hilbert’s first problem was on the
continuum hypothesis. This question was originally posed by Georg Cantor, and
was partly solved by Gödel in his 1940
book,
[47] and then completely solved
by one of his students, Paul Cohen. However, rather than deriving an answer
from concrete suppositions and inferences, they helped to prove that the First
problem was undecidable (Gödel proved that the continuum hypothesis was not
disprovable, and Cohen proved that it was not
provable).
[48] Hilbert’s
second problem was on the compatibility of the axioms of
arithmetic.
[49] It was this second
problem that helped Gödel on the road to developing his incompleteness
theorems, by providing the sort of concrete finite system against which
Gödel argues.
[50]In his
1900 lecture Hilbert expressed that problems within a discipline are a sign that
it is alive. He said, “As long as a branch of science offers an abundance
of problems, so long is it alive; a lack of problems foreshadows extinction or
the cessation of independent
development.”
[51] During the
past century, many of Hilbert’s original 23 problems have been solved.
Some have, as he hoped they would, generated more questions, and a wider
understanding of the mathematics they involve. There is no doubt that
Gödel’s incompleteness theorem helped to ensure an abundance of
mathematical problems for a long time to come.
In conclusion, although
Hilbert’s program was, according to most scholars, Hilbert was not as
discouraged by this as he might have been. Hilbert and Gödel respected
each other as scholars, even working together as is seen by Hilbert’s
influence on Gödel, Gödel’s help in solving Hilbert’s
first and second problems, and Hilbert’s publication of the proof of
Gödel’s second incompleteness theorem. These two mathematicians were
working toward the same goal, affirming mathematical vitality.
As the German
mathematician Hermann Weyl,
[52] who
studied under Hilbert, although he later joined the intuitionist school, said,
“mathematics, in spite of its age, is not doomed to progressive sclerosis
by its growing complexity, but is still intensely alive, drawing nourishment
from its deep roots in mind and
nature.”
[53] This idea was
ensured by Gödel’s incompleteness theorem, which had two very
important implications in mathematics. One is that it made Hilbert’s
dream of establishing a complete and consistent system that would encompass all
truth impossible. The second is that it set certain limits on the scope of
mathematics, allowing for problems that may be true, but have no formal
axiomatic proof, for example, meaning that there will always remain problems
within mathematics to solve. As one of Hilbert’s dreams, in posing his
twenty-three famous problems, there is an interesting irony to this story by
which Gödel has saved Hilbert’s world, while at the same time,
destroying his world view.
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