By the early 20th century, the world's great mathematicians knew they had a big problem. The very foundations of logic, held sacred since the time of Aristotle, were under siege after the discovery of troubling paradoxes by Cantor, Frege and Russell. It was unclear whether things could ever be set aright.

It was against this backdrop that David Hilbert, the most prominent mathematician of the age, created a program to get things back on track. Unfortunately, just the opposite happened. First came Kurt Gödel's famous incompleteness theorems and then the work of Alan Turing. Logic, as everyone knew it, was dead.

Nevertheless, out of the ashes of Aristotle's logic arose the foundations of modern computing. As it turns out, solving a fundamental problem, even if the solution isn't what you thought you wanted, can unlock enormous value. What's more, in every field and industry, there are countless problems just waiting to be solved. Here's how to go about working them out.

## 1. Develop Deep Domain Expertise

Hilbert's call wasn't put out to the general public. In fact, you would have had to have significant training even to understand what he was talking about. So it was no accident that Gödel and Turing were well versed in mathematical logic specifically. To solve a problem you first need to understand the issues involved.

We tend to think about creativity as an exercise in divergent thinking, but in fact decades of research has found that deep domain expertise is absolutely essential. You need experts who understand the contours of the problem and what would constitute a truly novel solution.

Many problems are tough because they are in highly sophisticated domains that are inscrutable to most people. Figuring out something like nuclear fusion takes more than common sense and and creativity. That's why the International Thermonuclear Experimental Reactor being built in southern France is staffed with scientists who've devoted their lives to solving that particular problem.

Yet Gödel and Turing weren't just any mathematicians. They were also very young -- both about 25 -- when they made their pathbreaking discoveries. That's no accident either. Many problems not only require expertise to solve them, but also a bit of fresh thinking.

## 2. Cast A Very Wide Net

Innocentive is a platform designed to solve tough problems. It was first established at Eli Lilly as an experiment to see what would happen if they posted problems that had been around for years. They offered a small reward -- usually $25,000 or less -- and, amazingly, within six months, a third came back successfully solved.

Yet as the early results came in, the executives at Eli Lilly noticed something interesting. None of the problems were solved by people working in the domain in which they arose, but some adjacent field. So chemistry problems, for example, weren't solved by chemists but by physicists or biologists.

It soon also became clear that the more problems they posted, the more solvers they attracted to the platform. So the firm began opening it up so that others could post problems as well and eventually spun it off as separate company. Today, it has a network of 380,000 solvers who have completed more than 2000 challenges.

So while domain expertise is absolutely essential for defining problems and evaluating solutions, bringing in a wider diversity of perspectives and approaches can often identify new, more fruitful paths to a solution. Separate research has found that the most highly cited scientific work combines experts in a particular field with a bit of insight from some other domain.

## 3. Get Out Of The Building

One of the things that lean startup guru Steve Blank continually stresses is how important it is to get out of the building, because when we spend all of our time with close colleagues, we tend to get tunnel vision. We limit our discussions to questions already asked, rather than looking for new questions and new points of view.

In an interview in *Harvard Business Review*, he recounts how when the Apple Store in Palo Alto first opened, he used to see Steve Jobs there all the time, just standing in the corner observing. This was after Apple had already gone through dozens of prototypes and opened several stores, but Jobs still wanted to how customers interact with the experience.

When Richard Feynman was still a young wunderkind, fresh from working on the Manhattan Project after the war, he hit a slump and was unable to do serious physics. Soon after, he observed a spinning plate in the cafeteria and became intrigued with its motion. The incident sent him in new directions and helped lead to one of his greatest discoveries and a Nobel Prize.

Just as interacting with specialists from other fields can take us in new directions, seeking out new experiences ourselves can give us new perspectives and lead us in new directions. It is often a random insight that can finally crack a really tough problem.

## Tackling Grand Challenges

We spend most of our time trying to get better at things we already do reasonably well and we can accomplish a lot that way. Much like compounding interest, incremental improvements can create a massive amount of value over time. Incremental improvements in computer chips, after all, are what made the digital revolution possible.

Clearly, when Hilbert started his famous program to reexamine the foundations of logic, mathematicians were already engaged in important work. Still, his challenge inspired young scholars like Gödel and Turing to set off in completely new directions and we are all the better for it. It set off ripples that still reverberate today.

I have come to believe that every enterprise needs to pursue grand challenges for exactly this reason. While there is much to be said about continuous improvement, the fact is that eventually every square-peg business meets its round-hole world and loses relevance. There's never any guarantee that the needs of the present will be relevant in the future.

The key to solving these kind of fundamental challenges is to go both wider and deeper. We need to understand problems on a fundamental level and then go in search of new perspectives, both from experts in other fields and seeking out new experiences ourselves. Innovation is never about individual modes. It's always about networks.