Joshua Helston

May, 2000

Metamathedogmatics

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 G0(x) is the formal counterpart of ‘x is a prime number’ then G0(5) is the counterpart of ‘5 is a prime number’ and G0(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,

G1, G2, G3, …

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 G1(1), G2(2), G3(3) … and so on.  Now pick out a particular one, such as G5(5).  This proposition may or may not be a theorem of F.  For example, suppose G5(5) is not a formal theorem of F.  Then ‘G5(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 Gn(p).

Therefore saying ‘G5(5) is not a formal theorem of F’ is equivalent to saying ‘K(5,5) is not an element of T.’  In general, saying ‘Gn(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 G1, G2, G3, …  Suppose that Gq formalizes this statement, in the sequence of Gi’s.

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

Therefore, if Gq(q) is a formal theorem in F then it is correlated to, and represents, a false statement in mathematics.  But if (not Gq(q)) is a formal theorem in F then (Gq(q)) must represent a true statement about arithmetic.  But then (not Gq(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 Gq(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, p1, p2, p3, … pn, then one of them has the impossible task of dividing the number (p1p2p3…pn + 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 “n2 – 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 À0, À1  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.