Instead, it’s more likely an IPO is in the compay’s future. While Dorsey won’t specify when, he echoed Twitter CEO Dick Costolo’s statements from earlier in the day, saying “We’re focused on building a product and a business. Going public is an instrument and tool to help sustain the company. In the future, there may be an IPO, but I don’t know what the timing is.”

Dorsey holds not one but two of the most high-profile jobs in Silicon Valley — at the same time. Each day, he spends 8-10 hours at Twitter where, as cofounder and chairman, he guides decisions on product development and helps think through how to build out the business. Then he walks a few blocks over to the mobile payments company Square where, as cofounder and CEO, he spends another 8-10 hours working with his team to design and build a next-generation payments system. Do the math, and you’ll see that leaves only four to six hours for sleep. No problem, says Dorsey: “Most people have major positions at companies and they’re also raising families. They have two-year-olds. I have it easy,” he says, noting as he often does that he is single and lives just a few blocks from the office.

Imitating life, Dorsey’s interview was split into two distinct areas. First, Dorsey spoke about Square, sharing early stories that revealed his deeply aesthetic approach to design. Dorsey told the audience that Square was originally called Squirrel, but when he traveled to Apple to pitch the payments system, he was dismayed to discover that Apple’s (AAPL) internal payments system was already called Squirrel. Scratch that. (And who’d have thought “Squirrel” would be such a popular name?)

Dorsey also addressed recent newly launched competitors like Google (GOOG) Wallet, saying, “There are ways of going about [web payment systems] that are focused on the payment mechanics (ie: Google), but we don’t think that’s the right way to focus on it. We think there are ways to bring that human experience to everyone.” He compared his aspirations for Square to the Starbucks (SBUX) approach to designing an experience for consumers that makes a simple act of buying from the company lovely.

Then, Swisher turned the interview to Twitter. Dorsey’s energy picked up a bit as he described how he’d like to see the messaging service evolve. “We need to take a lot of the friction out, “ he said. “Two things we announced today go a long way toward doing that. One is photos. Then people are also looking for ways to see the photos. That’s what search is about. It gets people to that value immediately. That’s one of our biggest challenges — getting the people to the product immediately in a consistent way.”

Audience questions focused entirely on Twitter, particularly its role in global uprisings and social movements. Dorsey is a conceptual thinker, and his answers veered toward the theoretical. At one point, he paused and asked, rhetorically, “I mean, what is Twitter?” That question, he noted, is like asking someone to define the world. “Ask 100 people, “what is the world?” and you’ll get a thousand different answers. The same is true for Twitter. It’s different things to different people. Twitter is the world.”

If Dorsey seemed to ramble a bit on this point, he could be forgiven. After all, he hasn’t been getting much sleep lately.