One of the flagship features of iOS 9 is its proactivity. This new OS can scan through your email looking for phone numbers and invitations. If you're not expecting these two features, they can seem slightly creepy at first. Here's how to manage these features, turned on by default in iOS 9 after installation.
1. Contacts in Email. What happens here is that you may have, in the past, received an email from someone who is not in your Contacts list. That email may contain a phone number. Later, when you get a phone call from that number, instead of showing up as "Unknown," iOS 9 will suggest that "maybe" the name associated with that phone number (in that previous email) is calling you.
This can be very convenient for some, and there's really no downside except for one thing. Some users have become accustomed ot the fact that "Unknown" is a tipoff that the caller doesn't have a privileged place in their Contacts. Putting a name there could be momentarily deceptive. It also invites people to send you email so that they can weasel their way into a hastily answered phone call..
If you like this feature, it's on by default after the iOS 9 install. However, to turn it off, go to Settings > Mail, Contacts, Calendars > Contacts section and uncheck "Contacts Found in Mail."
2. Events in Email. There's another similar feature to the one above. iOS 9 will look through the iPhone's email searching for invitations that are iOS compliant. If it finds one, it will place the event in the Calendar app.
This could be great if one receives a lot of email and doesn't want to miss an important invitation to a meeting. On the other hand, it opens to door for people to over-invite the recipient and clutter up their calendar.
Just as in the Contacts case above, there's a fine line between letting the iPhone do a lot of proactive work vs. the traditional feeling of explicitly controlling one's calendar.
To turn this one off, go also to Settings > Mail, Contacts, Calendars > Calendars section and uncheck "Events Found in Mail."
Finally, as with any new proactive feature, what seems creepy at first often turns out to be helpful with judicious, informed use. If you find yourself, at first, surprised by the actions of these two iOS 9 features, you can turn them off for now and open the door later as needed and as time has proven their worth.