Is the Nature of Mathematics a Wicked Question?
by Lisa Kimball
At the Eighth International Conference on Complex Systems in June, an entertaining late-night conversation among participants centered on whether the principles of complexity science are a human invention or some basic truth we’ve simply discovered as we’ve learned more about the world.
This echos a debate that’s been raging in the field of mathematics for years. In the August 2011 issue of Scientific American, Mark Livio explores this question, “Is math an invented set of tools, as Einstein believed? Or does it actually exist in some abstract realm, with humans merely discovering its truths?”
He writes that we accept the view that mathematics is the language of science and expect that it grammar explains experimental results and even predicts novel phenomena. He quotes Albert Einstein who said, “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” Livio describes the way that math captures the natural world as”uncanny.” In the end, he concludes that “mathematics is an intricate fusion of inventions and discoveries.”
How can we credit the power of mathematics to capture critical aspects of the world to help us understand it while appreciating the capacity of mathematicians to produce new knowlesdge that helps us create it?
It’s a wicked question. In The Daily Galaxy, Josh Hill raises questions that challenge both sides of the issue. For those who believe these mathematical truths are purely discoverable, where, exactly, are you looking?
And for those on the other side of the court, why cannot a mathematician simply announce to the world that he has invented 2 + 2 to equal 5? “If a mathematical theory goes undiscovered, does it truly exist? Maybe this will be the next “does a tree falling in the forest make any sound if no one is there to hear it?”
Roger Penrose, one of the world’s most distinguished mathematicians says, “I like to think of mathematics as a bit like geology or archeology, where you’re really exploring beautiful things out there in the world, which have been out there, in fact, for ages and ages and ages, and you’re revealing them for the first time.”
Nobel laureate Frank Wilczek believes that “Inventions have to come from somewhere so they could be inspired by natural phenomena. … You can invent [all kinds of] axioms, but most of them won’t be interesting. And the ones that are interesting are discoveries, so even the inventions have some element of discovery. So as I said, mathematics is more discovered than invented, and this only makes it more so.”
This debate is a great example of what Plexus Advisor, Scott Kelso calls a “complementary pair.” Mathematics – and complexity science – could be seen as both discovered and invented. Kelso has developed an empirically-based scientific theory of how the polarized world and the world in between can be reconciled.
Kelso and his collaborator, David A. Engstrom, coined the term “squiggle sense” to describe the sense of the inextricable, complementary dance of contraries. The notion of the ‘dance’ goes beyond the familiar notion of “both/and” inherent in wicked questions to highlight the importance of the dynamic relationship between the ideas.
This fits with Livio’s conclusion about the nature of mathematics, “A pattern emerges: humans invent mathematical concepts by way of abstracting elements from the world around them – shapes, lines, sets, groups, and so forth – either for some specific purpose or simply for fun. They then go on to discover the connections among those concepts.”