IntroductionProblems with Infinite SetsAlternative PhilosophiesCreativity Versus ObjectivityObjective MathematicsA Creative Objective PhilosophyA Cultural PrescriptionBibliographyWhat is Mathematics About?1Paul BudnikMountain Math [email protected] the Platonic philosophy of mathematics is increasingly beingquestioned, computer technology is able to approach Platonic per-fection in limited domains. This paper argues for a mathematicalphilosophy that is both objective and creative. It is objective in thatit limits the domain of mathematics to questions that are logicallydetermined by a recursively enumerable sequence of events. This in-cludes the arithmetical and hyperarithmetical hierarchies but excludesquestions like the Continuum Hypothesis. This philosophy is creativein recognizing that G¨odel’s Incompleteness Theorem implies one canonly fully explore this mathematics by considering an ever increasingnumber of incompatible possibilities without deciding which is correct.This is how biological evolution created the mathematically capablehuman mind.IntroductionMathematics began with counting and measuring as useful procedures fordealing with physical reality. Counting and measuring are abstract in thatthe same approach applies to different situations. As these techniques weredeveloped and refined, problems arose in connecting highly refined abstrac-tions to physical reality. The circles that exist physically were never the sameas the ideal geometric circle. The length of the diagonal of an ideal squarecould not be expressed in the standard way that fractional numerical valueswere defined as the ratio of two integers. Mathematical thought seemed tobe creating an abstract reality that could never be realized physically.Plato had a solution to this problem. He thought all of physical realitywas a dim reflection of some ideal perfect reality. Mathematics was aboutthis ideal reality that could be approached through the mind. The difficultieswith connecting mathematical abstractions to physical reality often involvedthe infinite. It takes a continuous plane with an infinite number of points toconstruct the ideal circle or diagonal of an ideal square. Plato’s ideal realityseemed to require that the infinite exists. The idea that infinite mathematicalabstractions are an objective Platonic reality became the dominant philoso-phy of mathematics after Cantor seemed to discover a complex hierarchy ofinfinite sets.1Published in Philosophy of Mathematics Education Journal, 22, 2007.1This hierarchy has its origins in Cantor’s proof that there are ‘more’ realsthan integers. Set A is larger than set B if one cannot define a map orfunction that gives a unique member of B for every member of A. Thisis fine for finite mathematics where one can physically construct the mapby pairing off members of A and B. It becomes problematic for infinitesets. If A is larger than B it is said to have larger cardinality than B. Thesmallest infinite set has the cardinality of the integers. Such sets are said tobe countable.Problems with Infinite SetsThe definition of cardinality creates problems because it depends on whatinfinite maps are defined in a mathematical system. Formal mathematicalsystems are, in effect, computer programs for enumerating theorems2andthus can only define a countable number of objects. Because all possiblemaps from integers to reals are not countable, no formal system will con-tain all of these maps. Thus we have the paradoxical L¨owenheim SkolemTheorem. This says that every formal system that has a model must havea countable model. Thus no matter how large the cardinals one can definein a formal system, there is some model of the formal system in which allthese cardinals can be mapped onto the integers. However this cannot bedone within the system itself. But when one looks at the system from out-side one can easily prove this is true because a formal system is a computerprogram for enumerating theorems. Every proof that some set exists comesat a unique finite time and thus the collection of everything that provablyexists is countable.A major question about the hierarchy of cardinal numbers is whetherthe reals are the smallest cardinal larger than the integers. The conjecturethat this is true is called the Continuum Hypothesis. It has been provedthat both the Continuum Hypothesis and its negation are consistent withthe standard axioms of set theory. Thus the question can only be settled byadding new axioms and there is nothing remotely close to agreement abouthow to construct such axioms. On the contrary there is increasing doubt asto whether the Continuum Hypothesis is true or false in any objective sense.Solomon Feferman, the editor of G¨odel’s collected works, observed:I am convinced that the Continuum Hypothesis is an inherentlyvague problem that no new axiom will settle in a convincinglydefinite way. Moreover, I think the Platonic philosophy of math-ematics that is currently claimed to justify set theory and mathe-matics more generally is thoroughly unsatisfactory and that some2It is tedious but straightforward to go from the axioms of a formal system such as settheory and the laws of logic to a computer program that would enumerate every theoremprovable from those axioms and laws of logic. This however is not a practical way togenerate new mathematics because most theorems would be trivial.2other philosophy grounded in inter-subjective human conceptionswill have to be sought to explain the apparent objectivity ofmathematics[4].Alternative PhilosophiesThe Platonic philosophy of mathematical truth is dominant but not univer-sal. Constructivism demands that all proofs be constructive. It disallowsproof by contradiction[1]. The constructivist treats only those mathematicalobjects that he knows how to construct as having an objective mathematicalexistence. Social constructivism has recently been applied to mathematics[3].This approach sees mathematics as a fallible social construction that changesover time. That is an accurate appraisal of the history of mathematics.The dominant Platonic philosophy and the extreme form of social con-structivism are at opposite ends of a spectrum. In Platonic philosophy thereis only absolute truth that must be discovered. In extreme social construc-tivism all truth is relative to some cultural group that creates and recognizes‘truth’ through a cultural process.Constructivism sits between these extremes. It accepts