Jack Dorsey, CEO of Twitter, at the 2010 Time 100. Photo courtesy of David Shankbone.

The following script is from "The Innovator" which aired on March 17, 2013. Lara Logan is the correspondent. Tom Anderson, producer. View the segment at the end of this story.

Jack Dorsey is one of the biggest and most ambitious innovators of our time. His name doesn't resonate like Jobs or Bezos or Zuckerberg, but his innovations do. His low profile may have a lot to do with his personality.

Dorsey describes himself as extraordinarily reserved and shy, which is ironic considering he's the man who created Twitter and changed the way people communicate around the world.

His latest creation is a company called Square, which is helping to transform the way we pay for things.

Square is still a start-up but we were surprised to learn that Jack Dorsey already has his eye on the next job he wants - mayor of New York City - an unlikely role for a man who calls himself a loner and spends a lot of time dreaming and thinking.

Lara Logan: Forbes Magazine said you're more of a nerd than Steve Jobs.

Jack Dorsey: Which I found insulting. I'm more of a nerd than Steve Jobs. I think the reference was because I was a programmer. So if that is the nerdy way, then guilty. I'm a nerd.

Many believe Jack Dorsey is the intellectual successor to Jobs. He created the code for Twitter, with messages that -- unlike e-mails - can be read by anyone in the world.

Lara Logan: Twitter is so instant. It's that moment.

Jack Dorsey: Yeah, it's so instant. And now I get to see the entire world and how they're thinking and how they're feeling and what they're doing and what they care about and where they're going.

When he created Twitter seven years ago, Jack Dorsey had no idea how big it would become - that 200 million people would be sending more than a billion tweets every three days. That young revolutionaries would use it to help communicate the strategies that fueled the Arab Spring. That it would force a congressman to resign. Twitter has become a marketing tool for Hollywood and big business. We use it at 60 Minutes. The president tweets. Benedict XVI tweeted his final message as pope.

Lara Logan: What are you most proud of with Twitter?

Jack Dorsey: I'm most proud of how quickly people came to it and used it in a million different ways. They're all over the world. And Twitter enables them to take a $5 cell phone and, wherever they are, communicate with the world for free.

It was a revolutionary new way for people to connect --- something Dorsey admits he's not very good at in person.

Lara Logan: What do you think your weaknesses are?

Jack Dorsey: I do have a tendency to really think about things by myself and decide things.