Numbers, sets, laws of logic, and so forth are all abstract objects. What makes something abstract as opposed to concrete is known with some imprecision, but it is generally agreed that one aspect that distinguishes abstract objects from concrete objects is that the former do not stand in causal relations. Now, should we admit that such objects actually exist? Here is one reason to think they do.
Our scientific knowledge of the world can be reduced in some cases to basic "laws," axioms composed of mathematical formulas. For instance, Newton's law of universal gravitation states that: F = G (m1 * m2 / r^2). The distance between the masses is "r" and the axiom requires that r be squared - hence, r^2. The mathematical function of squaring the distance is therefore an indispensable aspect of Newton's law. The same indispensability can be shown for all kinds of abstract objects. This leads us to the following argument:
1. Whatever is indispensable possesses the attribute of indispensability. (Definition)
2. Abstract objects are indispensable. (Premise)
3. Hence, abstract objects possess the attribute of indispensability. (From 1 and 2)
4. Non-existent objects cannot possess any attribute. (Premise)
5. Therefore, abstract objects exist. (From 3 and 4, plus negation)
All that's left is to defend the certainly uncontroversial premise (4). Something non-existent cannot possess any attributes, since attributes are themselves existing qualities. Therefore, at least some abstract objects, at any rate, exist.