Thursday, February 3, 2011

The Indispensability Argument for the Existence of Abstract Objects

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.


  1. Putting it this way makes it very Cartesian in spirit; the Cartesians run arguments like this all the time (using their version of (4), the principle, "nothing has no properties," which Regis goes so far as to make the fundamental principle of philosophy).

  2. hi doug, On the face of it, 2 says either says:

    p. Ia

    q. (x)(if x=a and Ix, then Ia)

    If p, then you're begging the question since it is true only if abstract objects exist. Heck, it's a presupposition of the claim, Ia. Remember: Ia → ∃x(x = a).

    If it's q, then 3 is properly said as:

    3. Whatever is an abstract object has the property of indespensibility.

    4 seems like a plausible ontological thesis, but notice that it cannot derive your conclusion since your interpretation of 3 is denied.

    i really don't understand what 2 says if not p or q.

  3. The real question is whether abstract objects *are* indispensible. For example, there may be possible ways to express what the equation denotes without actually using mathematical "objects". (I usee scared quotes because fictionalists don't think numbers actual exist) Check out Hartry Field's Mathematics Without Numbers, if you can find it.

  4. Brandon, yes, I think the argument has a Cartesian air about it. It's one of the reasons I find arguments involving absolute certitude so fascinating.

    Mickey, you make a good point. My guess is that the realist could plausibly argue transcendentally. (2) is arguably circular, but is its negation demonstrably absurd?

    Alfredo, thanks for the suggestion. If one prefers to think about an abstract object other than numbers or mathematical objects, there are always the laws of logic, etc.