Why circumcision?

Why was circumcision a covenant sign? Because the Bible doesn't actually explain the symbolism, readers are left to speculate. 

On the face of it, this seems like an odd sign. Given that Jews normally frowned on public nudity, what's the purpose of an invisible sign? Isn't that paradoxical? 

i) That depends in part on who it's designed to signal. Although it can't function as a public witness, it might be a reminder to Jewish males or Jewish couples. It would be visible to them, if not to the general public. 

Even if that's the case, it doesn't explain what it's meant to remind people of. 

ii) One traditional explanation is that it symbolizes the excision of guilt and pollution (Louis Berkhof). The only thing that explanation has going for it is the possible symbolism of cutting. But to connect that to cutting away guilt and pollution is arbitrary. I'd say that's a makeshift explanation. 

On a related note, O. Palmer Robertson thinks is represents purification, based on the hygienic value of circumcision. However, he offers no evidence that Jews practice circumcision for hygienic reasons. And that seems to be anachronistic. 

iii) As far as that goes, there might still be a play on words, where you have an association between cutting the flesh and the idiom of cutting a covenant. But by itself, that's a pretty thin connection.

iv) On a related note, it might be a play on words, where you have an association between cutting the flesh and those who are "cut off" from the covenant community (Gen 17:14). That potentially makes it signify a curse sanction. To be circumcised cuts you into the covenant while to be uncircumcised cuts you out of the covenant. That explanation has rather more going for it than (ii-iii).

iv) Another explanation trades on the reproductive function of the penis. On this view, circumcision signifies the consecration of Jewish posterity to God. They are set apart for God. In addition, God's promise extends from one generation to the next. Moreover, Jewish posterity will become a source of blessing for Gentiles.

That explanation has a lot going for it. It certainly dovetails with the context of Gen 17:1-8. 

v) The permanence of the mark may signify the perpetuity of the covenant. 

It's possible that God's choice of circumcision was intended to trigger multiple associations. The symbolism is multifaceted. 

vi) Here's another explanation I haven't seen discussed. The effect of circumcision is to reduce the sensitivity of the penis. Removing the foreskin removes many nerves that would otherwise be stimulated by sexual intercourse. Indeed, I've read that premature ejaculation is sometimes treated by circumcision, because reducing the sensitivity of the penis means it takes longer to achieve sexual climax. 

For normal men, sexual intercourse is a regular highpoint of life. Yet circumcision diminishes the physical sensation, which diminishes the physical pleasure. 

So that may indicate God's claim on our total devotion. A sacrifice to God of something that means the most to people (in this case, men). 

(I think circumcision is defunct under the new covenant. It makes sense for Christians to discontinue the custom.)