If the World Science Festival's panel on string theory tackled the question of whether the math behind it could be a reliable guide to reality, its panel on the Limits of Understanding seemed to question whether math could be reliable at all. As the panel presented matters, math is coming up short when faced with some of the biggest scientific challenges around. Even dizzyingly complex physical systems can be simplified down to the point where they're tractable to meaningful computations, but the simplifications needed to model even a single cell seem to result in math that is completely divorced from the underlying reality.

If the practical failure is bad, however, it may pale in comparison with some of the theoretical concerns raised by the work of Kurt Gödel, a mathematician and personal friend of Albert Einstein. Gödel's most famous work involved showing that formal systems that can be used to perform math are necessarily incomplete, raising the possibility that all our systems for doing math are limited.

All of this really bothers Gregory Chaitin, a prominent mathematician who was part of the Limits of Understanding panel. For Chaitin, mathematical systems aren't developed so much as discovered, and the math itself seems to have an underlying reality, one that, so far at least, has helped us understand physical reality. For him, the fact that biology appears to be too complex to be handled easily by math is intensely frustrating (his fellow panelist, astrophysicist Mario Livio, referred to the "unreasonable ineffectiveness of math in biology"). But what's really got him unnerved are the implications of Gödel's incompleteness theorems.

As explained by philosopher Rebecca Goldstein, Gödel's work describes formal, rule-based systems. For any of these systems sufficiently complex to handle mathematical transformations, his theorems demonstrate that you can express something according to the rules that cannot be proven; and that you can't prove whether the system is consistent (meaning that a statement and its converse won't both evaluate to true).

As Chaitin put it, "80 years later, we still don't know what the hell Gödel proved." Many areas of math appear to be formal systems. For example, the square root of negative one is meaningless under some systems, while the use of imaginary numbers in other systems allows for some very interesting math; the same thing applies to breaking some of the rules of Euclidian geometry. But, if all of math is a formal system, then we don't necessarily know which ideas will end up being unprovable, or whether we've accidentally proven two things that might contradict each other.

So, where does Gödel leave us? Nobody's entirely sure. Livio, Goldstein, and Chaitin all suggested there might be some sort of "pure math" that wasn't a formal system, but none of them seemed especially happy about that. A few other ideas were floated around—it's possible that we've only developed the sorts of tools we've needed so far, and attacking biology with math is a relatively new field, for example.

There were a number of examples that suggested this might be the case. Mario Livio said that Galileo ended up stuck when it came to motion because he used identical terms for what are now recognized as momentum and energy, and that, while Indian and Chinese mathematicians worked with prime numbers, they never appear to have recognized that they represent a distinct and interesting category. Similarly, Marvin Minsky, an artificial intelligence expert, said that consciousness is at least 26 distinct problems, and we were making a mistake by viewing it as one.

Collectively, these suggest that the math to tackle some of the problems we view as intractable might actually be out there, undiscovered (as Chaitin might put it). But, at the same time, they also indicate how strongly math is tied into social factors, such as our ability to recognize when there's more than one problem involved—which implies that it's a formal system, and subject to the limitations Gödel identified.

So, we've reached a point where math can't answer many questions in biology, but the most promising path for advancing physics (string theory) remains trapped in the realm of pure math. Is this a cue for panic? Maybe not, as illustrated in an exchange between two panelists: "You're not upset because you're not a mathematician," Chaitin told Livio, "you don't care because you're a physicist."

"We know there's problems with quantum mechanics, but has that stopped anything?" Livio countered.

It's not just quantum mechanics. Biology may have resisted easy quantification, but it has hardly slowed the field down. If math turns out to be just a tool (and a tool with some substantial limits), that may disappoint mathematicians, but it won't necessarily slow down our ability to understand and model the natural world. This may be my background as a scientist talking, but that seems like the most important consideration, and I'm willing to live with a community of disappointed mathematicians in order to get there.

Listing image by NSF