In one of Tegmark’s essays about the mathematical universe hypothesis, he claims that it can be viewed as a form of radical Platonism. Wikipeida goes further and states baldly that “The MUH is based on the Radical Platonist view that math is an external reality”. In contrast, I contend that one can accept the mathematical universe hypothesis without being a platonist at all. I’ve chosen the name mathematical physicalism to refer to this minimalist version of the MUH.
When I survey the bewildering array of highly technical viewpoints regarding platonism and nominalism in the philosophy of mathematics, I find that I resonate most deeply with the philosophers who urge that “the debate about platonism should be replaced by, or at least transformed into, a debate about truth-value realism”. To me, the important questions are about the objectivity of mathematical statements, not about the existence of mathematical objects. I’m a truth-value realist regarding at least some mathematical facts, and I claim that this modest level of truth-value realism is sufficient to support mathematical physicalism.
Haim Gaifman presents a list of questions for testing degrees of truth-value realism in mathematics. His first question is “What is the largest prime number dividing 2243112609?” Does that question have an objective true answer? I think it must. I don’t even know how to conceptualize the alternative (Gaifman’s thoughts about ultrafinitism notwithstanding). I’m a truth-value realist about this kind of mathematical question.
As I argued in a previous post, asking this kind of number-theoretical question is similar to defining an infinite n-dimensional lattice of cells filled with the digits of some computable number and asking what state some cell would be in after some finite iterations of some computable rule. I’m a truth-value realist about such questions. Mathematical physicalism is simply the contention that these kinds of mathematical facts play the role that “physical existence” is supposed to play in our account of how the world works, and that therefore any patterns of mathematical facts that represent self-aware systems will subjectively perceive their world as “physically real” (and we live in just such a world). One can believe this without believing that abstract objects exist—in other words, without being a mathematical platonist. Mathematical physicalism is the contention that objective mathematical facts are “real enough” to do the job.
What exact level of truth-value realism do I accept? I don’t know! I think there’s an objective fact of the matter about the Twin Prime Conjecture, but I’m too ignorant to have an opinion about the Continuum Hypothesis. Mathematical physicalism doesn’t provide answers to these kinds of questions, but it does make them even more interesting (to me), because they’re now, in a sense, questions about physicality.
What exact form of mathematical platonism, nominalism, or structuralism do I actually endorse? Again I don’t know! One can be a truth-value realist without knowing exactly where to place oneself in this tangled mess of viewpoints. I assume my position will solidify as I continue to study the foundations of mathematics.