How many of your loved one's cellphone numbers do you remember? In my case, it's exactly one: the number of the phone my wife and I shared, and which she still has. My brother's, father's, stepmother's, or virtually any of my friends? No. I can't remember my mother's number, and she's on my account– I pay for that phone, and I can't remember it.
It would be more accurate to say that I haven't even tried to memorize it. Why would I? I put it in my cell phone, and put her on speed dial. My mother's not a ten-digit number: she's "hold down the 5 for three seconds."
This kind of thing is not at all unusual: most of the high school or college interns who've worked for me in the last few years confess they program phone numbers, rather than memorize them. Likewise, I don't know anyone's e-mail address; I've got them in my address book. This kind of thing usually inspires hand-wringing about how young people can't remember anything any longer, the arts of memory are being lost, etc. etc. I take a different view, though it's informed mainly by my own experience.
I think the following shift is going on. Those of us who live in a world of cellphones, PDAs, e-mail programs, etc., are spending less mental energy memorizing things that we used to have to remember. For those of us who remember when we used to know our parent's and friends' phone numbers, this feels like a collective, mild case of Alzheimers' syndrome. At the same time, we're better able to remember (or at least recall) specific episodes in the past, single events, places we've visited, etc.
Tools like blogs, Nokia's Lifeblog, Flickr, del.icio.us, and others create records of things that previously would have been quickly forgotten; but the act of using them often helps us fix those things in our memories. The books I remember best are the ones I've taken the best notes of: that's because the act of note-taking helped me commit the book to memory. For me, note-taking is more important as a mnemonic device than as a record-creating tool. Something similar happens when, for example, I geocode pictures on Flickr. Tagging pictures helps me remember taking them, and associate that picture with a place.
So ultimately, the story of computer memory's impact on human memory isn't merely one of "offloading" or externalizing or digital amnesia: it's a story of a shifting of mnemonic resources, and a reconfiguration of the contents of our memories, not a simple shrinking of our memories.