Updates to this story
He told the Daily Telegraph that he did not really have an answer to the question "what is the point of Twitter" and he is trying to "get some clarity on that question".
We have to applaud any tech boss who does not stand in front of the media and say that their product is "super", "cool" or "magical", but Costolo's moment of honesty caught us by surprise.
He said that he was trying to define what Twitter's purpose is in the long term. "We will be able to be more specific on that answer in the near future."
To be fair Costolo only took over as chief executive last month, after Evan Williams, Twitter's co-founder, stepped down. But he has been chief operating officer so he must know a fair bit about what the outfit does.
For four years the outfit has been successfully peddling the idea of micro-blogging and now get 370,000 people joining Twitter every day.
But it is to IT what Big Brother was to television. It takes up a lot of time and is no earthly use to anyone. Yet the wnalysts say that it must start making money and expand globally.
As the Telegraph points out that task is considerably harder if nobody knows what Twitter is for.
Costolo said that he is focusing on trying to ensure Twitter retains the culture of the successful tech start-up it began life as.
But he adds that Twitter has yet to reach its true potential. When Twitter reaches its potential, it will be successful. It has not done that yet.
"It will have only done so when billions of people around the world are using Twitter on a daily basis to consume the majority of their information in real time.," he said.
But what is the point? No we don't know either.