A good email client for a business might filter all messages from other members of your company into a given folder, and then only notify for messages in that folder. Notifications for any other messages would just be blocked. Heck, don't even show them in the unread count until the user looks for them in particular.
This problem is easily solved with a filter, no?
And that's why Slack will not be a replacement for e-mail.
Your life doesn't turn around the company you're currently employed in. Or, at least mine doesn't. Company messages are only a small fraction of important messages I get daily.
(Not to mention that Slack killing e-mail would be a pretty dystopian future, with some single profit-seeking company controlling your primary means of communication.)
If your inbox is deluged with spam from outside the company, you need to filter your e-mail by domain.
What I do is simply mute the chatty channels, and go in when I need answers or get invited/DM'd.
Mute is pretty powerful.