Pure Mathematics and the Actual World

Direction for questions 41 to 45: Read the passage, and answer the questions following.

Not only mathematics is dependent on us and our thoughts, but in another sense, we and the whole universe of existing things are dependent on mathematics. The apprehension of this purely ideal character is indispensable if we are to understand rightly the place of mathematics as one among the arts. It was formerly supposed that pure reason could decide in some respects as to the nature of the actual world: geometry, at least, was thought to deal with the space in which we live. But we now know that pure mathematics can never pronounce upon the questions of actual existence: the world of reason, in a sense, controls the world of facts, but it is not at any point creative of fact, and in the application of its results to the world in time and space, its certainty and precision are lost among approximations and working hypotheses. The objects considered by mathematicians have, in the past, been mainly of a kind suggested by phenomena; but from such restrictions, the abstract imagination should be wholly free. A reciprocal liberty must be accorded; reason cannot dictate to the world of facts, but the fact cannot restrict reason's privilege of dealing with whatever objects its love of beauty may cause to seem worthy of consideration. Here, as elsewhere, we build up our ideals out of the fragments to be found in the world; and in the end, it is hard to say whether the result is a creation or a discovery.

With which of the following would the author agree? I. Mankind relies on mathematics for its very existence. II. Geometry was believed to deal with space in which we live. III. The world of reason has no control over the world of fact.

What does 'indispensable' mean in the context of the passage?