God and Necessary Truths

Christianity maintains that only God is self-existent. He is the sovereign Creator, who freely creates everything else. God exists necessarily, in that it is impossible for God not to exist, God's non-existence is inconceivable. The created universe, in contrast, is merely contingent, in that it could have been different, and need not have existed at all. But what about abstract objects, such as the laws of logic and math, which seem to be necessary truths? How do they relate to God? 

Consider, for example, the equation 3 + 4 = 7. Was there ever a time when three plus four did not equal seven? Surely this was always true.  So, how could this proposition have been created?

Moreover, is it not necessarily true? If so, does this not suggest that it exists independently of God? That would challenge God’s sovereignty. Further, if necessary abstract objects depend on God, was God forced to create or uphold them? That would challenge God’s freedom to create.

In fact, even God’s attributes seem to be closely connected with necessary truths. For example, God’s character has a logical aspect. God’s word is truth (John 17:17); God never lies (Titus 1:2) and is always faithful (Ps. 117:2). God means what he says, not the opposite. These all entail that the law of non-contradiction holds. Also, God’s identity is eternally the same; hence the logical law of identity must be eternally true. Thus the very nature of God seems to entail the eternal validity of the laws of logic.

God’s character also has a numerical aspect: God is tri-une, consisting of three distinct persons – Father, Son, and Holy Spirit. Since God’s trinity is eternal, so, it would seem, are numbers.

So how do necessary truths relate to God? There are two main views, associating abstract objects, such as mathematical or logical truths, either with God’s mind or with his creation.

1. Mathematics in God’s Mind

Augustine placed mathematical objects as ideas or concepts in God’s mind. He associated mathematics with “wisdom”, the divine Word (logos) that was “in the beginning,” and was both “God” and, also, “with God” (John 1:1). Thus, mathematical objects are eternal, necessary, directly connected to God, and independent of human minds. This is called “theistic conceptualism”. More recently, a similar position has been defended by Steven Boyer and Walter Huddell III.[1]

A closely related view is that mathematical objects are products of the divine mind (e.g., counting numbers, collecting sets, etc.). This is known as theistic activism. It has been defended by Christian philosophers Alvin Plantinga and Christopher Menzel.[2]

A variant of this has abstract objects depend on God’s awareness of his ability to plan and create, where the prime mathematical structures are found in God’s plan for the universe in Christ.[3]

The details of these variants need not concern us. Augustine thought the exact relation between mathematics and wisdom was shrouded in mystery, since creatures, including humans, cannot comprehend the divine mind.

Necessary truths, eternally associated with God’s mind, do not in any way limit God’s sovereignty, or his freedom to act according to his will. For example, God cannot sin, for, being perfectly good, such is contrary to his will. But that self-imposed constraint does not hamper God's plan. Likewise, since God is rational, he knows and upholds all necessary truths. God acts according to his character, and God’s character determines even what is necessary.

Alvin Plantinga argues that, since God is necessarily all-knowing, necessary propositions (e.g., 3 + 4 = 7) are necessarily always known to God, who thus affirms their existence. The abstract objects of logic and mathematics exist as ideas in the mind of God. They pose no threat to God since they are merely inert ideas that depend on God for their existence. Plantinga writes:

According to Kronecker God created the natural numbers and men created the rest...Kronecker was wrong on two counts. God hasn't created the numbers; a thing is created only if its existence has a beginning, and no number ever began to exist. And secondly, other mathematical entities (the reals, for example) stand in the same relation to God...as do the natural numbers. Sequences of numbers, for example, are necessary beings and have been created neither by God nor by anyone else. Still, each such sequence is such that it is part of God's nature to affirm its existence.[4]

Here Plantinga refers to the German mathematician Leopold Kronecker (1823-1891), who believed that, in mathematics, only the natural numbers had a real, objective existence. Plantinga believes that, in exploring mathematics, one is exploring the nature of God's rule over the universe ... and the nature of God Himself. He concludes, "mathematics thus takes its proper place as one of the loci of theology".[5]

Similarly, logic is not above God but derives from God's constant and non-contradictory nature. Reformed philosopher Gordon Clark notes,

the law of contradiction is not to be taken as an axiom prior to or independent of God…the law is God thinking.[6]

Clark views truth and logic as attributes of God. Many other Christian thinkers concur. For example, theologian John Frame writes:

Does God, then, observe the law of non-contradiction? Not in the sense that this law is somehow higher than God himself. Rather, God is himself non-contradictory and is therefore himself the criterion of logical consistency and implication. Logic is an attribute of God, as are justice, mercy, wisdom, knowledge.[7]

Hence, in this view, necessary truths such as logic or math are intimately connected with God's very nature. 

2. Mathematics as part of creation

Some Christians reject such a high view of necessary truths. Christian philosopher Roy Clouser, for example, considers logical and mathematical truths to be part of creation, and not eternal.[8] Creation, in his view, consists of (1) concrete things and (2) laws (including logic and mathematics) governing these things.

According to Clouser, God stands above such created logic and mathematics; God has taken on his logical and numerical characteristics only for the sake of covenantal fellowship with us. Had God wanted to, he could have taken on quite different characteristics. Clouser contends that God accommodates himself to our creaturely limitations. God's uncreated, unrevealed being is unknowable to us.[9]

In short, Clouser argues that mathematics is created, so that mathematical (including logical) truths are not necessary truths. In particular, they need not apply to God.

This raises deep, subtle questions about God's essential nature. Might it be the case that God, in his essential nature is not triune? Or not faithful, just, or good? 

It seems implausible that God's unrevealed being would conflict with his revealed being. Nowhere in his revealed Word does God give any hint of that. The Bible gives no indication that God's logic is any different from ours. Rather, genuine human wisdom appears to be part of the same wisdom that informs God (see Proverbs 8).

One might speculate about God's unrevealed nature, but how can we ever truly know anything about God other than that which he has revealed to us? 

Moreover, it seems incoherent to claim that we can know nothing about God's essential nature and, at the same time, that normal logic need not apply to it. This implies that we do know something about God's essential nature, namely, that it is unknowable and above logic. But, again, how could we know this to be true, if God has not revealed it to us? 

The more prudent course is to accept that God really is as he reveals himself to us in his Word. To think otherwise places undue trust on extra-biblical, human assumptions.

Conclusion

In summary, although necessary truths, such as those in math and logic, have close ties with God, they pose no challenge to God's sovereignty or freedom. Necessary truths are established and upheld by God, ultimately deriving from God's very character.

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[1] Steven Boyer and Walter Huddell III 2015. “Mathematical Knowledge and Divine Mystery: Augustine and his Contemporary Challengers”, Christian Scholar’s Review 44:3, 207-236.

[2] Christopher Menzel 2001. “God and Mathematical Objects,” in Russell W. Howell and W. James Bradley (eds.) Mathematics in a Postmodern Age: A Christian Perspective. Grand Rapids, MI: Eerdmans, p.73.

[3] Walter Schultz 2014. “The Actual World from Platonism to Plans,” Philosophia Christi 16: 81–100.

[4] Plantinga, Alvin 1980. Does God have a Nature? Marquette University Press, Milwaukee, p. 142.

[5] Ibid., p. 144.

[6] Clark, Gordon 1968. The Philosophy of Gordon Clark: A Festschrift, Ronald H. Nash (ed.), Presbyterian & Reformed, Philadelphia, pp.64-70, p. 67.

[7] Frame, John M. 1987. The Doctrine of the Knowledge of God. Phillipsburg, N.J: Presbyterian & Reformed, p. 253.

[8] Roy Clouser is an advocate of the Cosmonomic Philosophy. For a discussion of this philosophy, as well as other Christian philosophies of mathematics, see Vern S. Poythress 2015. Redeeming Mathematics: A God-Centered Approach. Wheaton, IL: Crossway Books. Appendix B. See also my post Dooyeweerd's Legacy.

[9] Clouser, Roy A. 1991. The Myth of Religious Neutrality, Notre Dame: University of Notre Dame Press, p. 183.

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