In math the difference between exploration (theorizing) by operations, and proof

In math the difference between exploration (theorizing) by operations, and proof, is narrow. But in argument the difference between exploration using operations on meaning (models) which we call ‘reason’, and operations in reality (those that are existentially possible) ‘actions’, can diverge greatly. For the same reasons that mathematicians call functions numbers – aggregates of operations are easier to work with than many discreet operations….