Strength of Isolation

To determine the total cost of a spell, start with the mana cost or alternative cost (such as a madness cost) you're paying, add any cost increases, then apply any cost reductions. The mana value of the spell is determined by only its mana cost, no matter what the total cost to cast that spell was.

Madness works independently of why you're discarding the card. You could discard it to pay a cost, because a spell or ability tells you to, or because you have too many cards in your hand during your cleanup step. You can't discard a card with madness just because you want to, though.

A card with madness that's discarded counts as having been discarded even though it's put into exile rather than a graveyard. If it was discarded to pay a cost, that cost is still paid. Abilities that trigger when a card is discarded will still trigger.