Can you pay life to cast spells if your life total would go to zero as a result?

Paying life as part of a cost (such as for Phyrexian mana or a card like Sylvan Library) is performed while the spell or ability is being activated or cast. Costs are paid during the casting or activation process, before anyone receives priority again (CR 601.2h, 117.3).

A player only loses the game due to having zero or less life when a player would receive priority — specifically, the game checks state-based actions at that point (CR 704.3, 704.5a). Since casting a spell and paying its costs happen before priority is passed, you finish paying the cost and the spell is placed on the stack before the life-loss can cause you to lose.

However, if your life total is already at zero or below before you try to cast the spell, state-based actions would have already caused you to lose the game. So the condition that matters is: your life total must be above zero when you begin casting, though it may drop to zero or below during payment.

Concrete example: You have 1 life and cast Gitaxian Probe using its Phyrexian mana option, paying 2 life. Your life goes to 0 as part of the cost. The Probe is placed on the stack successfully. Only when a player next receives priority does the game check state-based actions — at which point you lose. Your opponent still cannot respond before you lose, since state-based actions are checked before priority is granted.

Key rules: CR 601.2h (paying costs while casting), CR 117.3 (costs are paid), CR 704.3 (SBAs checked before priority), CR 704.5a (zero life = lose the game).