Update: Wed. 9:27 am: This still applies thanks to Tuesday's rain out of the Blue Jays - Orioles game. If the Sox defeat the Yankees on Wednesday, even if the Blue Jays lose both games of their doubleheader against the O's, Toronto will become AL East champions.
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The Red Sox (officially) aren't going to make the playoffs in 2015, even if they were to win each of their remaining six games. They do have a few things they can still do however, that can make the season seem like a relative success given their earlier issues.
They can finish in third place after sitting in last in the middle of the season -- that might not sound like much, but back-to-back last-place finishes is disheartening for fans, and much of the reason for their second-half surge lies with key 2016 pieces to boot. The Sox still have a chance, albeit a slim one, to finish with a .500 record, as they sit at 76-80 on the season. More realistically, they can still play spoiler and shape the finish for their division, as a Blue Jays' victory coupled with a Red Sox win on Tuesday against the Yankees would make Toronto the AL East champions.
It's a little thing, sure, but it's still meaningful: it would mean the Yankees haven't won the East for three years -- longer than Boston's current two-year streak -- and it would force New York into the AL Wild Card Game, aka Baseball Thunderdome. They would play in this game without Mark Teixeira, who is out for the year, without Nathan Eovaldi, who is dealing with an elbow injury, and possibly without Masahiro Tanaka, who is starting on Wednesday against the Sox but missed his previous appearance with a hamstring strain. They won't be at full strength, and even if they were, it's still a single-game playoff round: anything can happen, and that anything might send the Yankees back home three hours into the postseason.
If they won the division, they would have home field at least through the AL Division Series, and would have a better chance of moving on to the ALCS and the World Series. The Sox can have a hand in making New York's life more difficult, and if you can't win yourself, that's the next-best thing for a rival.
On top of the present-day implications, defeating the Yankees on Tuesday would also keep them from their 10,000th victory as a franchise. That win is obviously coming, likely this week and within this series, but keeping New York from that shiny round number on the same night they were eliminated from winning the division -- one they were on top of for 85 days with a lead as large as seven games -- would just be an added bonus.
There is a bit of incentive in it for the Sox besides spite, too: as mentioned, Boston can finish 2015 in third place rather than in last. If the Sox win Tuesday and the Jays do as well, Boston will have sole possession of third place in the East, as Toronto is taking on the Orioles, who are currently tied with the Sox for third. The Sox are 32-22 since July 30, a 96-win pace, so finishing strong and moving up through the standings will help keep the chill off this winter, even before Boston starts shopping for a new bullpen.
You would rather be winning the division yourself, obviously, but in lieu of that, playing spoiler -- and against your longest-standing rival -- can fill the void at least a little. The Sox have a chance to do just that on Tuesday, when Rick Porcello takes on Michael Pineda at Yankee Stadium.