Tuesday, August 27, 2019

"God made the integers; all else is the work of man"

A short exchange I had with a philosophical theologian:

Maybe "The Advancement Theory". That captures this idea of the mathematical landscape advancing in God's mind, and then also the advancement of the world, along lines without end.

I don't see how that's supposed to work. Don't mathematical truths involve interlocking systems of entailment relations? If so, all the elements must be in place for the mathematical truth, object, or structure (however we wish to put it) to obtain. Mathematical truths can't be waiting for the coin to drop. Rather, they must exist as a unit or not at all.

Some perhaps. But there may be basic structures that provide a foundation for iterations.

Let's put it another way, do you agree that mathematical truths are paradigm-cases of necessary truths? If so, mathematical truths can't become true. Mathematical truths can't evolve. Of course, if one's a naturalist, then one may try to reduce math to human psychology, in which case it does evolve. But that's bad theology.

A very good question. I'm toying with the idea that only basic mathematical truths are necessary (in the strongest sense).

So you agree with Leopold Kronecker that "God made the integers; all else is the work of man".


  1. Interesting. I can't see how Josh's position could work either, but maybe he will have more to say. Such as what constitutes "basic mathematical truths" as opposed to non-basic mathematical truths.

    1. For example, take pi. Its traditional definition is quite basic in a sense: the ratio of a circle's circumference to its diameter. As such pi (the ratio of a circle's circumference to its diameter) seems to be a fair candidate for a basic mathematical truth.

      At the same time, numerically speaking, pi seems to have no end; it seems to go on forever. If pi continues forever, then perhaps it could be said to evolve. If it evolves, then it might fit Josh's advancement idea. A basic mathematical truth advancing in God's mind.

      However, is not the the ratio of a circle's circumference to its diameter a fixed truth in God's mind? How can it advance in God's mind? And even if pi seems to go on forever to us, doesn't God know pi's end (if any)?

      Granted I'm no philosopher so perhaps these questions are jejune.