 Occupying space (taḥayyuz) was for the Muslim theologians the necessary result of any thing’s being a substance, and it was this attribute of occupying space that confirmed the existence of substances; for occupying space was held to be an attribute that was directly perceived by the senses and as such could not be gainsaid. Thus that substances exist was a matter of direct knowledge. Most Muslim theologians additionally maintained that these substances also have an atomic structure. This is not to say that they thought humans could directly perceive an individual atom taken in isolation—they did not; however, they believed that one could demonstrate that the ultimate material constituents of the world must be discrete rather than continuous. In general, their arguments for atomic rather than continuous substances used paradoxes involving the infinite. For example, it was argued that if a body were continuous, then it would be divisible infinitely, in which case the body would have an actually infinite number of parts that all adhere together to form the body and that correspond with the infinite number of divisions. The existence of an actual infinite is absurd, however, and so the division of the body must terminate at a finite number of indivisible parts or atoms. It does no good to say that the body has only a potentially infinite number of parts corresponding to the divisions rather than an actually infinite number; for it was argued that if something is truly potential, then some agent, such as God, must have the power to bring about or actualize that potentiality. Thus, let God bring about the purportedly potentially infinite number of divisions, in which case there would be an actually infinite number of parts, which again is absurd. Similarly, it is of no avail to say that the divisions are neither actual nor potential, but only imagined, since what is truly impossible cannot be imagined, and the existence of an actual infinite, it was believed, is truly impossible. Another argument for the atomic structure of matter was a variation on Zeno’s dichotomy paradox, couched in terms of an ant’s creeping over a sandal. If a body were continuous, the argument runs, then should an ant attempt to cross a sandal, it would first need to reach the halfway point on the sandal, but before it could reach the halfway point, it would need to reach its halfway point and so on infinitely. Since an infinite cannot be traversed, the ant could never cross the sandal, but of course a body such as a sandal can be crossed, and thus bodies cannot be continuous, but must be composed of a finite number of atomic parts. Yet another argument observed that differences in size are accounted for by difference in the number of constituent parts that make up the body. Now if bodies were continuous and so composed of an infinite number of parts, then the number of parts constituting a mustard seed and the number of parts constituting a mountain would be the same, namely, infinite. Consequently the two bodies should not differ in size; for one infinite cannot exceed another, and yet such a conclusion is patently false. In the end, most Islamic speculative theologians concluded that the matter constituting the cosmos must be discrete rather than continuous.