This question is one of the topics in the upcoming June 2008 issue of the European Mathematical Society Newsletter. As Science News reports, this subject “has provided fodder for arguments among mathematicians and philosophers” for thousands of years, with no seeming resolution.

On one hand, there are Platonists who believe this:

…[A] mathematician discovers timeless truths independent of human observation and free of the transient nature of physical reality. “The abstract realm in which a mathematician works is by dint of prolonged intimacy more concrete to him than the chair he happens to sit on,” says Ulf Persson of Chalmers University of Technology in Sweden, a self-described Platonist.

But the Platonists are forced to deal with some tricky implications of their views:

Those who espouse discovery note that mathematical statements are true or false regardless of personal beliefs, suggesting that they have some external reality. But this leads to some odd notions. Where, exactly, do these mathematical truths exist? Can a mathematical truth really exist before anyone has ever imagined it?

In contrast, there are those who believe that such talk of an abstract realm is just mystical hogwash:

Brian Davies, a mathematician at King’s College London, writes that Platonism “has more in common with mystical religions than with modern science.” And modern science, he believes, provides evidence to show that the Platonic view is just plain wrong. He titled his article “Let Platonism Die.”

…Reuben Hersh of the University of New Mexico …rejects the Platonic view, arguing instead that mathematics is a product of human culture, not fundamentally different from other human creations like music or law or money.

But the latter school is faced with a different set of intractable questions:

On the other hand, if math is invented, then why can’t a mathematician legitimately invent that 2 + 2 = 5?

…The challenge, [Hersh] admits, is to explain why it is that mathematical statements can be definitively true or false, not subject to taste or whim.

The solution to this millenia-old argument is to abandon both the intrisicist approach of the Platonists and the subjectivist approach of their opponents. Instead, mathematical concepts (like all concepts) are neither intrinsic nor subjective but objective. It is in debates like this where the Objectivist approach to epistemology and concept formation prove their value — in being able to cut through the errors made over the centuries by struggling philosophers and mathematicians.

Of course, properly applying Rand’s theory of concept formation to the philosophy of mathematics is a non-trivial task. Concepts of number are both seemingly self-evident, but also represent feats of tremendous abstraction. But scholars such as Dr. Pat Corvini have made a good start. Her course at the 2007 OCON, “Two, Three, Four and All That“, was on precisely that topic — namely how to apply the Objectivist theory of concept formation to concepts of number:

The concept of number as used in science today is one of man’s greatest achievements: a grand-scale integration capping centuries of effort and enabling a vastly expanded efficacy in all areas of life. But the growth in complexity of the number system has rendered the meaning of number ever more mysterious; number is seen both as a touchstone of certainty and as an arbitrary human construct whose applicability to the real world is a deep mystery. This is because the nature of number has not been properly identified; and as Ayn Rand pointed out, that imprecision is dangerous.

This course clarifies the meaning of “number” by examining it in the light of Miss Rand’s theory of concepts. Recognizing the objectivity of number provides a new framework for resolving both historical and modern debates, and yields a heightened appreciation for the science of mathematics as a whole—further reinforcing the value of Objectivist epistemology.

She is also offering a follow-up course at this year’s 2008 OCON, “Two, Three, Four and All That: The Sequel“:

Science shelves of bookstores are today awash in accounts of modern extensions of the idea of number, including infinity and the continuum, set theory, transfinite numbers, and the like. Many of these ideas, and the “mysteries” that proceed from them, figure prominently in modern philosophy and in popular discussion of the nature and limits of reason.

In this course, Dr. Corvini explains and evaluates some of the most influential of these ideas, using as a frame of reference both their historical context and the view of number as objective developed in her earlier courses. By identifying the fundamental nature of the ideas and of the errors involved, we see again the importance of a proper theory of concepts, and clarify the differences between an objective approach to mathematics and the more traditional views.

I have long had an interest in those topics such as foundations of set theory, the nature of the concept “infinity”, etc. Hence, if her 2008 course is as good as her 2007 course, then it promises to be a real treat. Diana and I have already signed up for it.

Although I have a degree in mathematics (B.S., MIT, 1984), her courses do not require any advanced math background. Dr. Corvini is a very clear and engaging lecturer, and she is excellent at explaining the relevant mathematical concepts to a general audience. If you can count to 10 and you are a normal intelligent adult, then you can follow her lectures.

So if you want to see how the power of the Objectivist theory of concepts can resolve questions that have stumped some of history’s greatest minds for thousands of years, check out her courses!

(I don’t believe that her 2007 course is available yet through the Ayn Rand Bookstore, but I expect that it will be eventually. It was available for purchase by 2007 conference attendees as part of the usual post-conference package, and hence I think it will eventually make it to the main bookstore listing.)