Thursday, October 04, 2018

The argument from numbers

However, if it’s a divine idea, then we could all be talking about the same idea, God’s idea. Divine ideas would have to be relevantly different from our own. They’re at least different in that they can secure the infinity and necessity of numbers. 

It could be that God would always have had the ideas of 2 and 4, and it could be that God would always have had the idea that 2 + 2 = 4, no matter what. But perhaps it could be that God would not necessarily have the ideas. Why should he have them? If God could have had quite different ideas (or none at all), it would be a cosmic coincidence for him to choose just those ideas, no matter what.

Let’s focus on the main subject of Leibniz’s argument first: the necessary truths of mathematics. Whenever we try to ground some domain in the divine, there’s a Euthyphro-style dilemma lurking. A comparison here between morality and mathematics might be illuminating. The traditional Euthyphro dilemma: Does God command it because it’s obligatory? Or is it obligatory because he commands it? If the former, then there’s some morality independent of God’s say-so. If the latter, then God could with as much reason command murder as he could forbid it. Does God think that 2 + 2 = 4 because 2 + 2 = 4? Or does 2 + 2 = 4 because he thinks it? If the former, then the truth is independent of God’s say-so. If the latter, then God could with as much reason have decided that 2 + 2 = 5.

Does God have the idea of 2 because it exists? Or does it exist because he has the idea? If the former, then the number is independent of God’s intellect. If the latter, then God could with as much reason never have had the idea of 2, so that it never existed.

But I wonder: Why would God’s rational nature ensure that 2 + 2 = 4? Unless there’s something intrinsically rational about 2 + 2 = 4, unless there’s something about the numbers that God’s rationality is tracking and that isn’t up to anyone, God could have dreamt up something else. But, if God’s rational nature is tracking something, then that is what mathematics is about. That is where the numbers live. It might be a luminous Platonic realm. It might be next to nothing at all. There might even be something divine about it.

This poses some interesting challenges to the theistic foundations of math. 

1. I don't know what he means by saying God might not have certain ideas. 

2. Suppose we can't explain what makes mathematical truths necessarily true. That would still leave intact an argument based on God to embed mathematical infinities. 

If, likewise, numbers are ultimately mental entities, and only a timeless divine mind can ground them, then that argument remains whether or not we can explain what makes mathematical truths necessarily true. 

3. Is 2+2=4 necessarily true in isolation, or in relation to an interlocking system of mathematical relations? If the latter, then it's necessarily true within that system. 

4. Maybe there's no room for 2+2≠4 because the interlocking system is exhaustive. An infinite, transfinite totality that edges out any alternatives. Divine reason takes that as far as it can go, to ultimate internal closure. 

In order for 2+2≠4, that requires a systematic readjustment to all other equations. Maybe there can be no structure in which 2+2≠4 because there's no infinitely consistent, mutually entailing alternative. 

5. Suppose there is an alternate system, like non-Euclidean geometries. The equations would be necessarily true within those alternate system, rather than contingent or arbitrary. 


  1. If part of God's omniscience is His knowledge of all true propositions, then would that mean that God knows all the numbers of infinity and the infinity of sets within sets? But is that even possible? Could it be that there's an upper limit to God's knowledge of numbers, sets and mathematical formulas? In which case, would we then have to attenuate the definition of God's omniscience similar to open theists who claim to hold to omniscience because the future cannot be known and so is not an object of knowledge?

    1. If I recall, even Gordon Clark denied God's infinity, in the sense quantitative infinity. Though, theologians like William Lane Craig affirm God's qualitative infinity. I haven't read Craig's books on God's relation to abstract entities, but I wonder if the implications of the question above in my previous post is part of the reason why Craig appears to side with some form of non-realism.

    2. Gordon Clark operated with the pre-Cantorian definition of infinity as a potential infinity rather than an actual infinity. A potential infinite is an actual finite. Moreover, a potential infinite suggests an uncompleted process.

    3. Craig is a fictionalist. If infinite numerical sets are actually infinite, then that's a completed totality, so that's an object of knowledge. Not to mention if numbers are in fact constituted by divine cognition.

    4. For Clark, the infinite never ends. No last number. Always another number after that in the series. So he had a problem with defining divine omniscience in that sense.

      One problem is that that way of conceiving the inviting is too indebted to the process of counting up.