Does Stony Silence prevent you from tapping artifacts for mana?

Stony Silence reads: 'Activated abilities of artifacts can't be activated.' Tapping an artifact for mana — such as tapping a Mana Vault or a Sol Ring — is an activated ability. Activated abilities are defined in CR 602.1 as abilities written in the form 'Cost: Effect.' The tap symbol (T) is a cost, so these mana-producing abilities are activated abilities and are fully shut off by Stony Silence.

It doesn't matter that these are mana abilities (CR 605.1). While mana abilities have special timing rules and generally don't use the stack, they are still activated abilities when written in cost:effect form. Stony Silence's blanket prohibition on activating artifact activated abilities applies to them just the same (CR 602.4).

Note that triggered abilities of artifacts — those that begin with 'When,' 'Whenever,' or 'At' — are NOT affected by Stony Silence. Only activated abilities are shut off.

Concrete example: You control a Sol Ring and your opponent controls Stony Silence. You cannot tap Sol Ring to add {C}{C} to your mana pool, because that tap ability is an activated ability of an artifact. Sol Ring simply sits on the battlefield doing nothing while Stony Silence is in effect.