Waitress: Yeah, yeah, it appears.

It's like a virtual credit card.

Lara Logan: You don't take your wallet out, you don't even take your phone out of your pocket.

Jack Dorsey: I could.

Lara Logan: It just pops up on the screen.

Jack Dorsey: Yeah.

The receipt shows up in an email. It's seamless and simple. But like his other invention, Twitter, Jack Dorsey thinks Square fundamentally changes how people interact. And how they feel.

Jack Dorsey: Money touches every single person on this planet. And at one point in their life they feel bad about it. It feels dirty sometimes. It never feels great. But it does feel great when it disappears. It feels like you're taken care of. It feels like the world is just working.

Everything seems to be working for Jack Dorsey. The Wall Street Journal Magazine named him Technology Innovator of the Year. Accepting that award brought him back to New York, a city he fell in love with when he worked for the dispatch company he hacked into. Dorsey may spend most of his life in the virtual world, but he's still fascinated by real world systems and Manhattan, with its subways, is the biggest system on earth. It may seem like an unrealistic ambition for such a reserved person, but Dorsey not only wants to move here someday, he wants to run for mayor.

Jack Dorsey: What I love about New York is just the electricity I feel right away. I mean just look at us in this station. There's just people walking everywhere and everyone.

Lara Logan: Chaos?

Jack Dorsey: It's chaos. It's kinda like being in a car in the middle of a thunderstorm, right. Everything is raging around you, but you're safe inside that car. So New York feels very much to me like that.

Jack Dorsey knows it helps to be a billionaire if you run for mayor in New York. It's worked before. But if he's serious about running, and he says he is, it will be interesting to see just how he communicates with the voters.

Lara Logan: Do you find it easier to communicate with people via Twitter or face-to-face?

Jack Dorsey: I guess my natural state would be through mediation of letters or through texts, all those mediums, I definitely find ease with. But do I appreciate as much as face-to-face communication? No. Do I feel like I'm an expert in having a normal conversation face-to-face? Absolutely not. That's just not my natural state. I would rather be walking Land's End and thinking about things.

Lara Logan: And tweeting.

Jack Dorsey: And tweeting.