"I know the math is totally against us," Maddon said earlier this week. "I get it, but I don't believe it."
That's where the Rays stand now, nearing the All-Star break amid the best stretch of a mostly disappointing season. They've won 18 of 28 since June 11, pulling within nine games of the American League East-leading Orioles entering Thursday's off-day only a month after owning the Majors' worst record.
Tampa Bay had a 1 percent chance of making the playoffs on June 27, according to Baseball Prospectus. Thursday afternoon, that figure stood at 4 percent. The odds are still stacked against the Rays, but they have improved ever so slightly as Tampa Bay has played more like the team everyone expected this spring.
"We feel like we have a really good team," Rays executive vice president Andrew Friedman said. "Whether it will be good enough at the end of the day remains to be seen, but we're an extremely confident group."
Here are five reasons to be confident in the Rays at this pivotal point in the season.
They still have David Price
For how much longer? Who knows. But remember when the All-Star left-hander waved to the crowd after his "last" start as a Ray at Tropicana Field? That was two weeks ago. He's still here.
Price is also still the top pitcher potentially available on the trade market, so the trade rumors will continue to swirl. And the Rays will have to listen. Bold trades have shaped their current roster and helped sustain their success since 2008, and every team could use an ace like Price.
When outsiders said the Rays had to trade Price this winter, they held on to him. But Tampa Bay surveyed the market at that point, and objectively speaking, the club's chances of reaching the postseason were a lot better then than they are now.
It's possible their recent success may affect where Price pitches the rest of the season, or it may just give the Rays more time to wait for the right offer.
"Specific trades or players, obviously we don't talk about that publicly," Friedman said. "But I would say that our trade philosophy for this July is still evolving. ... The fact that we've played the way we kind of envisioned that we would play this year certainly factors into our thought process."
Numbers never lie
This is the math Maddon likes: The Rays were due for some regression to the mean, especially offensively, and it's happened. Some course correction from what Friedman calls an "imperfect storm" has taken place recently, but there's still room for improvement.
Evan Longoria is slugging .395 with a .735 OPS, for example, but has never posted a slugging percentage lower than .495 or an OPS below .842. Alex Cobb proved last season he's better than his 4.14 ERA. Their defense has performed below its usual high standard, particularly with how few double plays they've turned.
The turnaround has been a team effort, and they're still not whole
Right-hander Chris Archer has grown more confident, and rookie Jake Odorizzi has turned a corner in the rotation. Wednesday night's crushing collapse aside, the bullpen has improved since Maddon opted for a closer-by-committee system led by lefty Jake McGee.
Logan Forsythe, Sean Rodriguez and Brandon Guyer have taken advantage of expanded playing time recently. But the Rays' unexpected star has been Kevin Kiermaier, a former 31st-round pick who's hitting .297/.331/.574 with eight homers while making highlight-reel plays in right field.
"We've gotten a chance to see guys play more, like Kiermaier and Guyer, who have really stepped up and shown us something," Friedman said. "That only portends well for the future."
Right-hander Jeremy Hellickson made his season debut on Tuesday, and shortstop Yunel Escobar is set to return Friday. David DeJesus was quietly one of Tampa Bay's best hitters before his injury, and he should be back in late July or early August. Wil Myers could return by mid-August.
"It's almost like having acquisitions this time of year," Maddon said. "Keep this [race] tight, get a little bit tighter with it and then here comes the latter part of the year, you've got some fresh bodies, man. ... Then you play good in September and see what happens."
The AL East
The Rays are barely ahead of the Red Sox, and they'd have to catch the Yankees, Blue Jays and Orioles in less than 70 games. The AL Wild Card is an even tougher challenge as six teams stand between them and the current leaders, the Angels and Mariners.
The underlying math behind Tampa Bay's focus on winning series -- the kind Maddon doesn't want to hear about -- would put the Rays in line to at least make some noise down the stretch. Fortunately, they might not need 90 wins to do so. The whole division, bunched within 9 1/2 games, is suffering from injuries, underperformance or both.
"Obviously a lot can happen, and a team can get hot and win it with a standard, usual 95 wins," Friedman said. "But there's certainly a real case to be made that the winner of the East will do so with an eight-handle in their number of wins."
They've been here before
The Rays weren't far above .500 last year before a 21-5 July shot them to the top of the division. They famously trailed the Red Sox by nine games on Sept. 4, 2011, and stormed back to reach the postseason.
The circumstances have changed, but at times it seems "Game 162" is as much a part of Tampa Bay's identity as the club's history.
"We've been in this situation many times. We've been in a worse situation than this a couple times," Price said. "There's still a lot of baseball games to be played."
There is a difference, of course, between having been there before and being able to do it again. But the Rays believe it's possible, whether the "math" says so or not.
"If we didn't have the requisite talent, the fact that we've done it before wouldn't mean much to me," Friedman said. "But the fact is we have the requisite talent to put together an extended streak of really good baseball. That's the most important thing."
"I think the guys believe it," Maddon added. "And I think we're on the right track."