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WillowOne's Positions

Argument from Abstract Objects

Form

The argument runs from Augustine (On Free Choice of the Will, Book II) through Leibniz (Monadology, §§43–46) to Alvin Plantinga's "Two Dozen (or so) Theistic Arguments":

  1. There are necessary, eternal truths: the laws of logic, mathematical truths, true propositions in general. They would be true if no human had ever existed.
  2. Truths are the contents of thoughts; propositions and numbers are abstract objects, and abstract objects either exist in a mind-independent Platonic realm or as the thoughts of a mind.
  3. A mind-independent Platonic realm is untenable: its objects would be causally inert, so we could never know about them, and they would exist uncreated alongside God.
  4. Therefore necessary truths are the thoughts of a necessary, eternal mind.
  5. That mind is God.

The "eternal truths" version stresses step 1 (truths need a thinker who has always been thinking them); the "abstract objects" version stresses step 2 (numbers and propositions need somewhere to live).

My Positions

  1. Premise 2 is a False Dilemma. Platonism and divine conceptualism are not the only accounts of mathematics and logic. Nominalism, fictionalism, structuralism, and the view that logical and mathematical truths are constraints on any coherent description rather than objects at all are each held by serious philosophers and need no mind, divine or otherwise. The argument wins only against an opponent who has already conceded that numbers are things.
  2. The leading apologist has abandoned premise 1. William Lane Craig's God Over All (2016) argues at book length that abstract objects do not exist, precisely because a realm of uncreated necessary entities would compromise God's aseity. If the best-known defender of theism can do without abstract objects, so can I, and the argument has no first premise.
  3. A truth being about something is not a truth being thought by someone. That two and two would make four with no one to count is a fact about any possible collection of things, not a report that someone has always been doing the sum. The argument slides from "true regardless of minds" to "requiring a mind" by reifying propositions into objects and then asking who owns them (Equivocation).
  4. The divine mind would need to think every truth and every falsehood. If propositions exist because God thinks them, then the false ones exist because God thinks them too, along with every truth about every number and every possible world. This is an actually infinite set of thoughts in a single mind, which the Kalam Cosmological Argument told us was impossible. The apologist cannot have both.
  5. Grounding logic in a mind makes logic contingent on that mind. The same objection applies as to the Transcendental Argument: if the law of non-contradiction holds because God thinks it, then it holds in virtue of something, and a thing that could have been otherwise. If it holds necessarily, independent of what any mind thinks, then God is bound by it like everyone else and does not ground it.
  6. Even granted, the conclusion is not God. A necessarily existing "mind" whose only established property is that it contains all truths is a Platonic realm wearing a face. Nothing in the argument gives it will, goodness, agency, or an interest in human affairs, and the step from "a locus of abstracta" to the God of any religion is taken by assertion.

My own view is that logical and mathematical truths are constraints on description, discovered rather than invented, and that the word "exist" is being asked to do too much when applied to them (My Presuppositions). I hold this loosely. What I do not see is any reason to populate the universe with either a realm of Forms or a thinker of Forms.

Sources

  • Augustine, De libero arbitrio (c. 395), Book II
  • Leibniz, G. W., Monadology (1714), §§43–46
  • Plantinga, A., "Two Dozen (or so) Theistic Arguments" (1986 lecture), in Walls, J. and Dougherty, T. (eds), Two Dozen (or so) Arguments for God (2018)
  • Welty, G., "Theistic Conceptual Realism", in Gould, P. (ed.), Beyond the Control of God? (2014)
  • Craig, W. L., God Over All: Divine Aseity and the Challenge of Platonism (2016), the in-house rejection of premise 1
  • Field, H., Science Without Numbers (1980), for nominalism

Also see