Joshua Helston

Gustavus Adolphus College

May, 2000

Kurt
Gödel put a dagger in the optimistic future of David Hilbert’s vision when he
published, “Über former unentscheidbare
Sätze der __Principia Mathematica__ und verwandter Systeme, I” in 1931. In this remarkable and ingenious creation,
Gödel proves mathematically that any formal axiomatic system that is broad
enough to encompass natural numbers and their properties, such as Bertrand
Russell and Alfred North Whitehead’s __Principia Mathematica__, must
necessarily be either incomplete or inconsistent. Such a conclusion is an enormous blow to the “programme” that
Hilbert and many of his followers sought to pursue. On the other hand, this theorem is not as detrimental to another
route that philosophy of mathematics can take, namely, intuitionism. Unfortunately intuitionism also results in
several necessities that may, or at least should, be unattractive to many
philosophers of mathematics.

David Hilbert[1] was the founder of the formalist approach to mathematics. A formalist view of mathematics notices that mathematics moves from a less abstract and more applied or particular view to one that is more abstract and significantly more generalizable.

This evolution is
exemplified in the study of the history of, for example, geometry. Euclid and the ancient Greeks used geometry
to discover and explain several occurrences in nature. Euclid[2],
in his __Elements__, introduced basic definitions and five axioms from which
he resolved to deduce logically a plethora of useful, interesting and true
statements. In fact, this set of
Euclidean axioms survived for many years and was believed by many to explain
the nature of reality. The relatively
recent discovery of non-Euclidean geometries, such as those by Lobachevski[3],
Riemann[4]
and Bolyai[5]
were viewed skeptically by many at first because of their differences with
Euclidean geometry. Upon further
analysis, however, these geometries can be shown to be plausible or even more
realistic than Euclidean geometry in their application to the nature of
reality.

Hilbert, in __Die
Grundlagen der Geometrie[6]__,
wrote extensively about the relationship between Euclidean geometry and these
other non-Euclidean geometries that were later developed. He was in fact responsible for several important
discoveries about the interrelation of the axioms of Euclidean geometry with
the axioms of other constructed geometries.
The technique he developed in this research undoubtedly influenced
Hilbert’s and his followers’ view of the nature of mathematics. Formalism is thus first a technique used to
investigate the logical interrelation of various fields of mathematics and only
secondarily a philosophy to account for the success of the technique of
formalism.

Mathematics can refer to any
system of objects or members and relationships whose titles can be chosen in
such a way to guarantee that all the axioms of pure mathematics are true of the
objects, members and those relationships.
In other words, mathematics can be likened to a set of hypothetical deductions
based upon largely uninterpreted axioms.
Therefore, meaning to a formalist mathematician consists of
demonstrating the structure of these arbitrary systems. Hilbert’s school conceives the whole of
mathematics in the form of theorems, eloquently symbolized, and deduced from
axioms which are minimally interpreted.
The validity of these statements, deductions and axioms in mathematics
are analyzed and guaranteed by a science of “metamathematics” whose subject
matter “consists of the symbols of mathematics proper, and whose aim it is to
demonstrate the self-consistency of mathematics proper with the help of the
most elementary and indubitably valid arithmetic methods” (pg 149, Black). Thus, if metamathematics is able to complete
its goal, then mathematics can assert its own validity. The symbols and relations in mathematics,
since they are of a completely indeterminate reference frame, can adequately
map the structure of all possible systems.
The mathematical theorems can become, in a sense, true in any reference
frame of any system of interrelated objects.

Kurt Gödel[7]
in 1931 published a paper that brought serious undermining doubts to Hilbert’s
“programme.” The theorem of
incompleteness proved mathematically that any axiomatic system that is broad
enough to include the natural numbers and their properties is necessarily
either incomplete or inconsistent. By
incomplete, a mathematician means that there exists at least one statement
about the subject matter in question that is not provable in the system. In other words, there exists a statement (F)
such that neither (F) nor (not F) are provable in the system. On the other hand, when a system is
inconsistent this implies that there is some statement such that both it and
its negation are true. In other words
there exists a statement (F) such that both (F) and (not F) are true in the
system. Obviously both of these
properties are unattractive to any mathematician, in particular a formalist. The latter more so because it directly
entails a contradiction, thus any statement in such a system will be
necessarily true and the system will be unbelievable.

Gödel was interested in the
mathematical logic that composed the foundations of mathematical deduction and
understanding. Gödel concentrated his
study on systems of axioms that were rich enough to describe the natural
numbers and rules that govern them such as addition and multiplication. One prime example of such a system is
Russell[8]
and Whitehead’s[9] __Principia
Mathematica__.[10] The formulas of this system when interpreted
in the way they were intended speak of natural numbers and their
properties. By a technique that is now
called Gödel numbering, Gödel was able to demonstrate how some theorems that
lie within the system must necessarily reflect particular metamathematical
assertions about the system itself. In
other words, he was able to correlate formulas of a certain system describing
natural numbers with metamathematical statements about the entire system. Under the numbering scheme that Gödel developed,
each formula under “normal” interpretation is true if and only if the
corresponding metamathematical statement is also true.

To prove the incompleteness
theorem, Gödel assigned each elementary sign within the system a certain
particular natural number. This number
is defined as the symbol’s “Gödel number.”
Using these primitive symbols and their Godel numbers, it is then
possible to define what numerical properties a number must have in order to be
the Gödel number of some formula. The
Gödel number becomes a numerical function of the Gödel numbers of the
elementary symbols that appear in the formula, in the order in which they
appear. Thus, given any Gödel number,
one can derive the original formula to which it must correspond. The Gödel number for a string of formulas
can then be defined in a similar way.
In the end, this numbering can eventually be drawn out to the point that
one can describe the Gödel number of a proof.
The Godel number of a proof would be the Gödel number of a sequence of
formulas, each of which is either an axiom or can be derived from earlier
statements in the sequence by using the transformation rules of the
system. Therefore, it is possible to
define what properties a natural number must have to be called the Gödel number
of a theorem, in other words, the last formula in a proof. Consequently, it is also possible to
describe what numerical properties a number must have in order to be the Gödel
number of a formula that is not a theorem in the system.

Gödel carefully constructed
these definitions so that assertions about Gödel numbers will be in true if and
only if the metamathematical statements corresponding to them are true
also. Conversely, the Gödel number of a
non-theorem of the system will have certain numerical properties which make it
impossible to construct a proof of the formula with the Gödel number in
question. Therefore, assertions
concerning Gödel numbers are correlated with formulas of the system, in
particular the formula which, when the Gödel number is interpreted using
“normal” methods, will portray that assertion.
Thus, some formulas of the system are directly correlated with
assertions about Gödel numbers. Those
are, in turn, correlated with metamathematical assertions. In this way there does exist a direct
correlation between formulas of the system and the metamathematical statements
about them.

The punch of Gödel’s
incompleteness theorem is formulated as follows. Suppose we have a formula that is correlated with the claim that
a particular natural number has the numerical properties that make it the Gödel
number of a well-formed statement that is not a theorem of our system. This statement makes a claim about a number,
so there must be some numeral in it or at least some sequence of signs that,
under the “normal” interpretation, would represent a number. The formula in Gödelian language is a Gödel
number to which there corresponds a particular number, say, (X). Now let the numeral in the formula be the
very numeral that corresponds to the Gödel number of the formula itself,
(X). Gödel, in his publication,
demonstrated how to construct such a formula.

Notice that this special
Gödelian formula, when normally interpreted, expresses a true statement about
natural numbers if and only if its associated metamathematical statement is
true. But this corresponding
metamathematical statement is precisely the statement that this very formula is
not a theorem of the system. This is a
contradiction, for there exists a formula that is not a theorem if it is true
about the natural numbers, and is a theorem if it expresses a falsity about the
natural numbers.

If this method of
constructing the Gödelian formula is confusing, then consider the following
alternative interpretation. The
following construction of an undecideable proposition, f, is discussed in
Körner’s book, __Philosophy of Mathematics[11]__. Suppose some system F is a formalization of
elementary arithmetic. Then the
integers and properties of the integers must have formal counterparts in the
language of F. Let the properties of
the integers, J, be represented by G(j) with different properties being
subscripted by different subscripts on G, and j being an integer in J. For example, if G_{0}(x) is the
formal counterpart of ‘x is a prime number’ then G_{0}(5) is the
counterpart of ‘5 is a prime number’ and G_{0}(6) of ‘6 is a prime
number.’ While this set of all formal
properties could be ordered in several ways, let us consider one, say,

G_{1}, G_{2},
G_{3}, …

Now, to construct the naughty aforementioned formal
proposition, formulate first any formal proposition that is made by inserting
the subscript of G into its corresponding property. So these propositions are G_{1}(1), G_{2}(2), G_{3}(3)
… and so on. Now pick out a particular
one, such as G_{5}(5). This
proposition may or may not be a theorem of F.
For example, suppose G_{5}(5) is not a formal theorem of F. Then ‘G_{5}(5) is not a formal
theorem of F’ is not a formal proposition of F, but it is an actual proposition
about a formal proposition. This is
what Hilbert would call a metastatement used in metamathematics, the
metalanguage that is used to refer to F.

In
this scheme it is now possible to use a Gödelian recursive process to define a
Gödel number to correspond to elementary symbols, axioms, statements, proofs
and theorems in F. In particular, we
could demarcate in this manner a collection T of all formal propositions that
are formal theorems in F. For example,
if the statement ‘(p) or (not p)’ is a formal theorem of F, then it can be
equivalently expressed by some constant c in T such that c is the Gödel number
of {(p) or (not p)} in F. Then it is
possible to define a function K from the Cartesian product N ´ N[12]
to T which takes (n, p) to the appropriate Gödel number corresponding to G_{n}(p).

Therefore
saying ‘G_{5}(5) is not a formal theorem of F’ is equivalent to saying
‘K(5,5) is not an element of T.’ In
general, saying ‘G_{n}(n) is not a formal theorem of F’ is equivalent
to claiming ‘K(n,n) is not an element of T.’

Clearly
‘K(n,n) is not an element of T’ is a property of natural numbers that belongs
to elementary arithmetic. Consequently
there must exist some formalization in F somewhere in the sequence G_{1},
G_{2}, G_{3}, … Suppose
that G_{q} formalizes this statement, in the sequence of G_{i}’s.

Note
the formal property G_{q} takes natural numbers as arguments, so it
could take on q as an argument.
Consider closely the formal proposition G_{q}(q). G_{q}(q) could be roughly translated
as ‘the number q has the property formalized by G_{q} which is the
arithmetical property that K(n,n) is not an element of T.’ In other words, G_{q}(q) is not a
theorem of F.

Therefore, if G_{q}(q)
is a formal theorem in F then it is correlated to, and represents, a false
statement in mathematics. But if (not G_{q}(q))
is a formal theorem in F then (G_{q}(q)) must represent a true
statement about arithmetic. But then
(not G_{q}(q)) would be formalized by a formal theorem of F. Notice, however, if F is consistent then
neither of these can happen. Thus, if F
is consistent then G_{q}(q) is undecideable and hence F is
incomplete.

Notice then, that if there
is a formalization that is broad enough to at least encompass natural numbers
and their properties, then it is necessarily either incomplete or
inconsistent. That is to say, if the
system is consistent then, as demonstrated above in Gödel’s incompleteness
theorem, there must be some undecideable well-formed statement in the system;
hence the system is incomplete. If,
however, the system is complete then another implication of Gödel’s theorem is
that we have both some statement and its negation as formal theorems in the
system. Thus the system is
inconsistent. So the only way a system
can be complete is by paying the price of being inconsistent, and furthermore
it can only maintain its consistency by paying the undesirable price of being
incomplete.

Another way to summarize this consequence is
to say that any consistent axiomatization of any system rich enough to include
natural numbers and their properties fails to capture all truths about the
natural numbers, for in any such system it is possible to construct Gödel
numbers. Although different
axiomatizations may attain various different degrees of success[13],
no consistent axiomatization can possibly hope to get at every truth in the
system. This is a rather decisive blow
against the foundational ideas of Hilbert’s formalism, namely that mathematical
truth can be made equivalent to a set of deductions from axioms that are
assumed true.

Perhaps surprisingly, the
concept of possible necessary undecideable mathematical statements was not new
when Gödel published this amazing result.
Before Gödel published his famous theorem in 1931 the mathematical
intuitionism movement had already begun.
Mathematical intuitionism considers alternatives to the didactic mindset
of classical mathematicians and logicians.
Leopold Kronecker[14]
once said, “the integers were made by God; all else is the work of man.” Although this statement was probably little
concerned with Kronecker’s philosophy of mathematics[15],
it can still serve as a succinct summary of a typical intuitionist
position.

Although an intuitionist’s
view of mathematics is most commonly associated with the Danish mathematician
Luitzen Brouwer[16], the
fundamentals originated as early as the late nineteenth century. Kronecker, the son of prosperous Jewish
parents, had a well-documented affinity towards the natural numbers as
evidenced by the after-dinner remark above.
This, coupled with his knack for arithmetic, distaste for geometry and
the philosophical influence of his father and Ernst Kummer[17],
gave rise to a “doubting Thomas” view of any non-constructive proofs in
mathematics. Despite living in an era
with other famous mathematicians such as Karl Weierstrass[18]
who made great use of the concepts of infinite sequences, quantities and
infinitesimally small quantities, Kronecker would only accept the mathematics
that explicitly constructed the items to which they referred. Furthermore, he viewed mathematics as valid
only if the constructed entities could be made in a finite number of steps.

As a result of this
skepticism towards non-constructive proofs in mathematics, Kronecker advocated
a primitive intuitionism that was founded upon four main ideas:[19]

1)
Natural
numbers and addition are a secure foundation for mathematics because they are
intimately tied to our intuition.

2)
Any
definition or proof should be ‘constructive’ in the sense that it should start
from our basic intuitions concerning natural numbers and construct the
resulting entity as a culmination after a finite number of steps.

3)
Logic
differs from mathematics, so logic may use modus tollens and be consistent with
its set of rules, but it may not count as valid mathematics.

4)
It
is not possible to consider infinities as actual or completed. The only infinity to be considered is a
potential infinity when a set is contemplated in which numbers are added
without limit.

Although Kronecker was ferociously opposed
non-constructivist proofs in mathematics, he did not often write about his
philosophy of mathematics. He did,
however, hold powerful positions within the academic world of the time, such as
the editor of prominent mathematical periodicals, and was in a position to
control, at least partially, the content of the mathematical journals that he
edited. This mathematical position of
intuitionism was laughable to many mathematical geniuses of the time, such as
Weierstrass, Riemann and Lebesgue[20],
but also came into conflict with far more intense and paranoid personalities,
such as Georg Cantor[21]
who discovered transfinite arithmetic.

This first wave of
intuitionist mathematics versus classical mathematics, personified by the
conflict between Cantor and Kronecker, proved to be the first in at least a
two-part series. The next era was
characterized by an almost comical set of relations between the father of
modern mathematical intuitionism, Luitzen Brouwer, and the aforementioned David
Hilbert. Hilbert, who led the formalist
movement of the early twentieth century, viewed Cantor and his work reverently
and saw Kronecker’s, and thus Brouwer’s, philosophy as an imminent danger to
the entire future of mathematics.

Brouwer first clarified his intuitionist doctrine in his doctoral thesis written in 1907. To distinguish his stance from Kronecker’s, Brouwer referred to his doctrine as ‘neo-intuitionism’ but soon after gave up this attempt to distinguish between the two doctrines. The first chapter of Brouwer’s dissertation, entitled “The Construction of Mathematics,” illustrated the constructivist programme that he advocated. It, like Kronecker’s mathematical intuitionism, was centered and founded on the human intuitions concerning the natural numbers. His thesis, however, was further strengthened by the solution of various mathematical problems that were thus far unsolved by the methods of the classical mathematicians. Much of the remainder of Brouwer’s thesis did not do actual mathematics, but rather drew out his and others’ philosophical stances about mathematics.

Perhaps
the fundamental difference between intuitionist mathematics and classical mathematics
(such as formalist mathematics) is that the intuitionist mathematician does not
accept the law of the excluded middle as a truth in mathematics. The law of the excluded middle, to the
classical logician, is the tautology {(p) or (not p)}. In other words, either something is the case
or it is not. It is either raining or
it is not. I am either six feet tall or
I am not. You are reading this paper or
you are not, and so forth. It, along
with the law of identity[22]
and the law of non-contradiction[23],
are what Aristotle[24]
first established as the laws of logic that eventually became associated with
the laws of thought.

Gödel’s
incompleteness theorem insinuates that mathematics can never be complete, that
is to say, there can never be an underlying system of axioms that will be able
to consistently determine the truth-value of any statement. Therefore there will be a statement (f) such
that neither (f) nor its negation will be provable in the system. To the intuitionist mathematician, this is
not a surprise. The foundations of
intuitionistic mathematics, as illustrated by Brouwer in his thesis, presuppose
a logic that is not two-valued. In
other words, statements are not necessarily true or false, but may also have
other alternative truth-values. Since
proofs have to be constructible to the intuitionist, these other truth-values
are similar to undecideable values.
Therefore, if it is not possible to construct either a statement or its
negation by means of a finite number of constructive steps, the statement is
neither true nor false. It is
undecideable. A good example of such a
sentence is ‘Either God exists or He does not.’ Many have argued that it is impossible to prove either God exists
or that he does not.[25] So, to an intuitionist mathematician, all
Gödel’s theorem implies is that given an axiomatic system there indeed exist
statements within it that take on truth-values other than merely true or
false. In our current understanding of
mathematics, some examples of these “undecideable” theorems are the axiom of
choice and the continuum hypothesis.

Although the fashion in
which an intuitionist mathematician does mathematics may be appealing to many
that worry about the Gödelian implication, it does not come without its own
problems. The results of Brouwer’s
claim that the only valid mathematics is that which can be discovered by a
finite number of constructive steps are great and far-reaching. It results in a mathematics that is far
smaller in extent, more limited in power, and far more predictable than the
traditional mathematics that employs the traditional two-valued logic. Mathematical truths to the intuitionist
mathematician are merely a subset of all the mathematical truths to a
traditional mathematician. Not any
truth of traditional mathematics is necessarily a truth of intuitionist
mathematics, but every truth of intuitionist mathematics would certainly be a
truth of classical mathematics; well, except the denial of the law of
non-contradiction I guess.

For example, dependent on how strong a case of intuitionist
mathematics is purported, discussion about certain infinities becomes
nonsense. The intuitionist mathematics
that Brouwer presented in his thesis does not endorse the concept of actual
infinities, but rather the idea of a potential infinity in which a set is
continuously adjoined with additional elements without any end. Thus the entire realm of transfinite
arithmetic[26] discovered
by Cantor is at least incomprehensible, and perhaps even ridiculous, to
intuitionist mathematicians.

Furthermore, the loss of the
law of excluded middle results in the loss of reductio ad absurdum proofs
following the logical law of modus tollens.[27] This proof technique is extremely powerful
to the classical mathematician and often viewed as an elegant fashion by which
to prove theorems. An example of a
classical elegant proof that employs the modus tollens technique is Euclid’s
proof of the infinite number of primes.
This proof can be summed up eloquently as, “if there is a finite number
of primes, p_{1}, p_{2}, p_{3}, … p_{n}, then
one of them has the impossible task of dividing the number (p_{1}p_{2}p_{3}…p_{n}
+ 1).” Obviously, all the intuitionist
mathematician can conclude from such a proof is that it is not the case that
there is a finite number of primes.
Unfortunately this does not imply anymore, as it did to classical
mathematicians, that the number of primes is infinite but rather just not
finite. Relegating indirect proofs to a
position of mere curiosity, as opposed to valid mathematics, severely cuts down
the number of theorems that would be viewed as valid.

Additionally, since
intuitionist mathematics cannot make use of actual infinities, limits or
infinitesimally small quantities, calculus becomes more difficult to understand
in this philosophy of mathematics. All
techniques of integration, which is an unquestionably useful and important
facet of mathematics, make use of either infinitesimally small chunks of space
(such as Lesbesgue integration[28])
or limits towards infinity (as in Riemannian integration[29]). Calculus, although frustrating to most
undergraduate freshmen, is not something that many people would be willing to
dismiss out of hand simply because it deals with concepts that are not
constructible. The results and
processes are still far-reaching and extremely potent.

Finally, some other sets of
mathematics that would be irreversibly affected by the advent of intuitionistic
mathematics as the primary and preferred mathematical philosophy are particular
kinds of analysis. In particular, real
analysis, topology and the foundational analysis of the calculus employ
concepts of infinity regularly[30]. By dismissing these concepts because they
are not rigorously constructible in a finite number of steps, many fascinating
and important results from these fields would have to be abandoned in lieu of a
set of mathematical truths that is far more limited in scope.

Clearly the goal of
mathematicians is to make mathematics that is more universal and generalizable
so that the scope of mathematics may become as broad as possible. By limiting mathematics in the way that
Brouwer proposes, many fascinating, important and interesting elements of our
current mathematical outlook will have to be abandoned. By abandoning these methods and results, not
only is the field of mathematics holding itself to a far smaller doxastic
repertoire than is potentially attainable, but also eliminating many fields
that have not yet been fully realized.
For example, in transfinite arithmetic there is an important and open
question concerning how the cardinality, or size, of the reals compares to the
cardinality of the natural numbers.
Cantor proved, using a diagonizable argument and the concept of an
infinite construction, that there are many more real numbers even in the
interval from 0 to 1 than there are total natural numbers.[31] He also established that the cardinality of
the set of all subsets of the natural numbers is also greater than the
cardinality of the natural numbers, and further, was equal to the cardinality
of the reals. The “continuum
hypothesis” claims that the cardinality of the reals is the “next biggest”
cardinal number after the cardinality of the natural numbers. One should notice that to the intuitionist
mathematician such questions concerning infinite quantities and their
properties, though possibly useful, are nonsensical, never answerable and do
not warrant being considered.

Therefore,
intuitionist mathematics is appealing because it has more than two truth-values
and can thus avoid the Gödelian problem of indeterminate claims because it
considers the “undecideable” category to be a viable truth-value that can be
assigned to a particular statement. On
the other hand, because of this additional set of truth-value options, indirect
proofs that arise through the logical process of modus tollens are no longer
considered valid. This results in the
loss of a number of valid mathematical proof techniques that will clearly cause
mathematics to lose a certain amount of its previous scope.

All
truths discovered by an intuitionist's logic would also be truths in the
classical sense of mathematics. In
fact, these proofs sometimes take an otherwise unexplored and unexpected path
to truth because of their limited reserve of techniques. Still, however, the set of all truths in intuitionist
mathematics is only a minor island off a coast in the globe of truths in
classical mathematics.

Furthermore,
the truths of classical mathematics are formed in such a way that they are
definitely as valid, despite the reference frame, as logic itself. In many cases, such as those of induction, a
classical mathematician can, using the axiom of choice and the axiom of
infinity, prove certain statements about all natural numbers. Such proofs are difficult to recreate in an
intuitionist's mathematics because mathematicians with this philosophy can only
construct their proofs in a finite number of steps without using the concept of
infinity. Claims about a certain number
of finite cases are impossible to expand to claims about all possible cases because,
even if it is possible to prove something about a billion cases of an infinite
number of cases, there is still the same number left to prove. For example, attempts to find a “magical
equation” that would generate prime numbers alone have often been made. One such example is “n^{2} – 79n +
1601”[32]
which outputs a prime number for any integer n between 1 and 80. When this is extended to 81, however, the
“magical equation” fails. It is
conceivable to discover some such equation that would generate particular
numbers or results for any number of finite cases, but the next one and next
one and next one will always need to be observed. The only way to avoid these undecideable theorems is to have some
concept of mathematical induction as a proof technique.

Therefore,
it is clear that mathematical intuitionism results in a mathematics that is
both too ideal and far too limited to be considered viable. By allowing for pure existence theorems in
mathematics, many useful mathematical techniques can be formulated and applied
without necessarily having to construct the mathematical entities in
question. The incompleteness theorem of
Gödel only implies that there must necessarily exist undecideable statements in
any consistent system. This doesn’t
imply, however, that mathematics or its philosophy should be reworked in such a
drastic fashion that everything that is even the slightest bit complex is
thrown to the wind. There are many
important, interesting and useful facets of our current understanding of mathematics
that are not necessarily constructible and are hence incomprehensible and
unallowable to the intuitionist mathematicians.

Acknowledging
the existence of these undecideable statements does not seriously undermine our
current system of mathematics. Of
course, it would be wonderful to have a perfect axiomatization of mathematics
that is both consistent and complete, but such a system has been mathematically
shown to be unattainable and hence too ideal.
What we can have, however, is a system that can correctly assess the
surroundings of the everyday world to a certain degree of high accuracy. If a religion is a system of belief that has
necessarily undecideable statements, such as concerning God’s existence, then
perhaps mathematics is the only religion that can rigorously prove itself to be
one!

__References:__

Barker, Stephen. __Philosophy
of Mathematics__. New Jersey, USA:
Prentice-Hall, Inc., 1964.

Barrow, John D.
__Pi in the Sky: Counting Thinking Being__. Boston, MA, USA: Little Brown and Company, 1992.

Black, Max. __The
Nature of Mathematics__. Totowa, New
Jersey, USA: Littlefield, Adams and Company, 1965.

Feferman, Soloman.
__Kurt Gödel Collected Works II (1938-1974)__. Oxford, England: Oxford University Press,
1990.

Gödel, Kurt.
__Kurt Gödel Collected Works I (1929-1936)__. Ed. Soloman Feferman. Oxford, England: Oxford University Press,
1986.

Halmos, Paul R.
__Naïve Set Theory__.
Princeton, New Jersey, USA: D Van Nostrand Company, Inc., 1960.

Hofstadter, Douglas R. __Gödel, Escher, Bach: An Eternal Golden Braid__. New York, New York, USA: Vintage Books
Edition, 1989.

Körner, Stephan. __The Philosophy of Mathematics:
An Introduction__. New York, New
York, USA: Harper Touchbooks, 1960.

Moore, R. L.
__Foundations of Point Set Theory__.
Providence, Rhode Island, USA: American Mathematical Society, 1962.

Nagel, Ernest and James R. Newman. __Gödel’s Proof__. London, England: Routledge & Kegan Paul
Ltd., 1958.

Neale, Stephen. “The Philosophical Significance of
Gödel’s Slingshot.” __Mind__ October
1995, v104, n416, p761.

Oppy, Graham.
“The Philosophical Insignificance of Gödel’s Slingshot.” __Mind__ January 1997, v106, n421, p121.

Pourciau, Bruce.
“The Education of a Pure Mathematician.” __The American Mathematical Monthly__ October 1999, v106, n8,
p720.

Shapiro, Stewart.
“Induction and indefinite extensibility: the Godel sentence is true, but
did someone change the subject?” __Mind__
July 1998, v107, n427, p597.

[1] David Hilbert was a German mathematician who lived from1862 to 1943. His name is closely tied to the formalist philosophy of mathematics.

[2] Euclid was an ancient Greek mathematician who probably flourished about 300BC.

[3] Lobachevski was a Russian mathematician who lived from 1792 to 1856.

[4] Riemann was a German mathematician who lived from 1826 to 1866.

[5] Bolyai was a Romanian mathematician who lived from 1775 to 1856.

[6] Published in two volumes, 1934 and 1939.

[7] Gödel was a Czechoslovakian mathematician who lived from 1906 to 1978.

[8] Russell was an English mathematician, philosopher and historian who lived from 1872 to 1970.

[9] Whitehead was an English mathematician who lived from 1861 to 1947.

[10] The __Principia
Mathematica__ was worked on between 1910 and 1913. The mathematical community viewed it with great confidence as a
giant leap in the promotion of Hilbert’s “programme,” at the time.

[11] The form of this explanation in Körner was borrowed from Mostowski.

[12] Recall the Cartesian product from N ´ N is an ordered pair consisting of two elements, both from a set N.

[13] Dependent on which axioms are adopted, there are clearly different theorems that can be derived.

[14] Kronecker was a Polish mathematician who lived from 1823 to 1891.

[15] This
comment was an after-dinner remark according to John D. Barrow in __Pi in the
Sky__, page 188. Hence, it most
likely did not bear on his “actual” philosophy of mathematics.

[16] Brouwer was a Danish mathematician who lived from 1881 to 1966. This name is most closely tied to the intuitionist philosophy of mathematics.

[17] Kummer was a German mathematician who lived from 1810 to 1893.

[18] Weierstrass was a German mathematician who lived from 1815 to 1897.

[19] These are
mentioned in Barrow’s __Pi in the Sky__, page 200.

[20] Lebesgue was a French mathematician who lived from 1875 to 1941.

[21] Cantor was a Russian mathematician who lived from 1845 to 1918.

[22] The law of identity is that A is always equivalent to itself.

[23] The law of non-contradiction deals with the impossibility of a statement and its negation both being true at the same time.

[24] Aristotle was an ancient Greek philosopher and scientist who lived from about 384BC to 322BC. He is credited with the development of a logic that was considered a ‘closed’ subject until the advent of predicate logic.

[25] Because “God,” by definition, is of an infinite nature and some argue that humans, since we are finite, cannot truly grasp “God’s” nature adequately enough to describe it, much less prove or disprove it.

[26] Transfinite
arithmetic performs arithmetical operations on the infinite cardinal numbers À_{0},
À_{1}
… Without the concept of actual
infinities, such manipulation becomes nonsense.

[27] That is, if (not p) entails a contradiction, then we can conclude (not (not p)) which implies p.

[28] Lebesgue’s integration technique uses infinitesimally small chunks of space to find an integral.

[29] Riemannian integration uses the areas of rectangles, which limit a partition of an interval into lengths of 1/n, as n goes towards infinity.

[30] Such as in the integration techniques explained above. Furthermore, a concept of infinity is needed in reference to sets that are used in many fields of analysis and topology.

[31] The argument is essentially an attempt to “list the real numbers” and then construct a real number along the diagonal of the list that differs from each number in the list in at least one spot. Thus, if we assume to have listed all reals (in other words they are countably infinite) then the “new” real number that is constructed provides a contradiction.

[32] I found
this example in Barrow’s __Pi in the Sky__, page 236.