In Zen and the Art of Motorcycle Maintenance, author Robert Pirsig argues that the basic conceptual tools of science, such as the number system, the laws of physics, and the rules of logic, have no objective existence, but exist in the human mind.  These conceptual tools were not “discovered” but created by the human imagination.  Nevertheless we use these concepts and invent new ones because they are good — they help us to understand and cope with our environment.

As an example, Pirsig points to the uncertain status of the number “zero” in the history of western culture.  The ancient Greeks were divided on the question of whether zero was an actual number – how could nothing be represented by something? – and did not widely employ zero.  The Romans’ numerical system also excluded zero.  It was only in the Middle Ages that the West finally adopted the number zero by accepting the Hindu-Arabic numeral system.  The ancient Greek and Roman civilizations did not neglect zero because they were blind or stupid.  If future generations adopted the use of zero, it was not because they suddenly discovered that zero existed, but because they found the number zero useful.

In fact, while mathematics appears to be absolutely essential to progress in the sciences, mathematics itself continues to lack objective certitude, and the philosophy of mathematics is plagued by questions of foundations that have never been resolved.  If asked, the majority of mathematicians will argue that mathematical objects are real, that they exist in some unspecified eternal realm awaiting discovery by mathematicians; but if you follow up by asking how we know that this realm exists, how we can prove that mathematical objects exist as objective entities, mathematicians cannot provide an answer that is convincing even to their fellow mathematicians.  For many decades, according to mathematicians Philip J. Davis and Reuben Hersh, the brightest minds sought to provide a firm foundation for mathematical truth, only to see their efforts founder (“Foundations , Found and Lost,” in The Mathematical Experience).

In response to these failures, mathematicians divided into multiple camps.  While the majority of mathematicians still insisted that mathematical objects were real, the school of fictionalism claimed that all mathematical objects were fictional.  Nevertheless, the fictionalists argued that mathematics was a useful fiction, so it was worthwhile to continue studying mathematics.  In the school of formalism, mathematics is described as a set of statements of the consequences of following certain rules of the game — one can create many “games,” and these games have different outcomes resulting from different sets of rules, but the games may not be about anything real.  The school of finitism argues that only the natural numbers (i.e., numbers for counting, such as 1, 2, 3. . . ) and numbers that can be derived from the natural numbers are real, all other numbers are creations of the human mind.  Even if one dismisses these schools as being only a minority, the fact that there is such stark disagreement among mathematicians about the foundations of mathematics is unsettling.

Ironically, as mathematical knowledge has increased over the years, so has uncertainty.  For many centuries, it was widely believed that Euclidean geometry was the most certain of all the sciences.  However, by the late nineteenth century, it was discovered that one could create different geometries that were just as valid as Euclidean geometry — in fact, it was possible to create an infinite number of valid geometries.  Instead of converging on a single, true geometry, mathematicians have seemingly gone into all different directions.  So what prevents mathematics from falling into complete nihilism, in which every method is valid and there are no standards?  This is an issue we will address in a subsequent posting.