by Eric Steinhart
Leibniz’s version of the cosmological argument (his Sufficient Reason Argument) runs from the continency of our universe to the existence of some necessary being. This necessary being is the ground of our universe. The ground isn’t part of our universe – it stands in no spatial, temporal, or causal relation to any thing in our universe.
The ground doesn’t even come close to matching any of the descriptions of any of the gods of mythology or even philosophy. So it’s hard to see why affirming the existence of the ground would be offensive to atheists. And yet, looking at the various responses to my last post, it seems that many atheists object. Why?
The ground is not mysterious. Leibniz has given us a great deal of information about it. First, it exists necessarily. Second, the very fact that the ground is not involved in physical relations is informative. To say that the ground is not involved in physics means that the ground is an abstract object. Abstract objects are objects that don’t participate in any spatial, temporal, or causal relations. Mathematical objects are abstract.
Mathematical objects include things like sets, numbers, vectors, functions, and so forth. If mathematical objects exist, then they aren’t physical. And, just to be clear, they aren’t concepts. Numbers aren’t things in your brain. There are infinitely many numbers; but you don’t have infinitely many things in your brain. If numbers exist, then they have an entirely objective existence that doesn’t depend on you or your thoughts at all. If you think math is all in your head, then you can’t do math.
Mathematical objects obviously play roles in science (especially in basic physics, which is intensely mathematical). The Quine-Putnam Indispensability argument says that since mathematical objects are needed for science, they exist. If you believe in quarks and gravity, you ought to believe in math. If there are no numbers, what sense would it make to use equations to describe the physical world? And, of course, you can’t just say you believe in some mathematical objects and not others. You get the whole system or none of it. And, looking back to Leibniz, mathematical objects exist necessarily.
These ideas about mathematics are known in metaphysics as platonism. Platonists affirm the objective reality of a world of mathematical entities. And, even though we can’t see or touch mathematical objects, we obviously know a lot about them. Indeed, math is the most stable and enduring part of human knowledge. Math involves proof – everything else is uncertain. Of course, platonism, like every part of philosophy, is controversial. But platonism comes with enormous benefits. Why not use them?
How could the ground be a mathematical object? What does that even mean? To start to answer this question, consider a cellular automaton like the game of life. Such cellular automata can have incredibly rich physical content. Any instantaneous stage of some game of life can be encoded in a bit string – a sequence of 0s and 1s. Another way of putting this is to say that the stage supervenes on the bit string. A game of life is a series of stages; so any game of life supervenes on a series of bit strings. The bit strings aren’t arbitrary. They are generated by the iteration of a function which encodes the causal laws for the game of life. The function is Turing-computable. So every game of life supervenes on a sequence of bit strings generated by the iteration of an abstract Turing machine. The abstract Turing machine is itself just a function from numbers to numbers.
Many writers have thought hard about the possibility that our universe supervenes on the iterations of some abstract Turing machine. This is the computable universe hypothesis. (See, for instance, the work of Ed Fredkin or Jurgen Scmidhuber). I suspect our universe is too complex to supervene on the iterations of a Turing machine. But there are far more complex abstract machines. So the hypothesis that the ground is a kind of mathematical object is perfectly intelligible.
It’s easy to see why a theist might object to the mathematical ground – it directly competes with the theistic god! If the mathematical ground exists, then the theistic god doesn’t exist (or at the very least, that god is cosmologically unemployed).
Should atheists object to the mathematical ground? If so, why? Platonism gives atheists an enormously powerful metaphysics – a world of abstract, eternal, transcendental, necessary objects. But none of them are gods. And that world is knowable by reason (it’s the very peak of rationality). You’d expect atheists to embrace that.
So I’m wondering: can atheists do math?Guest Contributor Eric Steinhart is an associate professor of philosophy at William Paterson University. Many of his papers can be found here .