What is the central purpose for the poem "A Shady Friend for Torrid Days" by Emily Dickinson?
First, you must understand that we can never truly know the answer to this question. Why? Because we can't ask Emily. Further, no one ever asked her during her lifetime, either. Therefore, your guess is as good as mine. And, thus, we can at best take notice of what Dickinson scholars have said about the poem.
This being said, I believe that the central purpose of Emily's poem is to ruminate upon two things: different kinds of people and, perhaps more importantly, why they exist. It is in taking this central purpose further where my own ideas differ from that of more common interpretations. Let's begin by looking at the first stanza:
A shady friend for torrid days
Is easier to find
Than one of higher temperature
For frigid hour of mind.
A less controversial interpretation of this stanza is that the speaker notes that it's easier to find someone to bring you down than it is to find someone to cheer you up. I, however, like to consider the speaker to be Emily herself and, knowing how she has often pined for love, I like to take the word "torrid" to mean more than just "hot," but also passionate. Therefore, this first stanza can be interpreted that it's easier to find someone to cool your passions (due, perhaps to their indifference?) than one that will ignite your passions. This was certainly true of Emily herself.
The second stanza has to do with the different kinds of people in the world who the speaker speaks of as different types of cloth:
The vane a little to the east
Scares muslin souls away;
If broadcloth breasts are firmer
Than those of organdy,
Muslin and organdy are lighter types of cloth and, therefore, represent those people who can't take much passion or much heat or much coldness or much of ANYTHING. It is only broadcloth that is "firmer." These are the people who persevere and are wonderful to keep around. Note that the speaker negatively suggests that there are more of the former type of people than the latter.
The last stanza leaves the speaker questioning why people are this way. The speaker asks, "Who is to blame?"
Who is to blame? The weaver?
Ah! the bewildering thread!
The tapestries of paradise!
So notelessly are made!
If people are cloth, then "the weaver" is the one who made the cloth: God. Interestingly enough, the speaker never answers her own question. She simply follows it up with her exclamations of confusion about the cloths (people) on the earth as well as the "tapestries of paradise" or the cloths in heaven. Of course, if the cloths are all made "notelessly," then we can be assured that the speaker thinks little of her God (or perhaps that her God thinks little of the people in the world).
In conclusion, what I said at the beginning of my answer remains firm: at best this poem is a rumination about the people and the "whys" of this world. There are no answers, only thoughts.