#935 Tegmark’s Mathematical Universe Hypothesis and Natural Theology

Dr. craig’s response

Thank you for your kind remarks on my Systematic Philosophical Theology, Jordan! I’m glad you’re enjoying it.

It’s worth noting in passing that Alexander tacitly recognizes the irrelevance of MUH for the moral argument for God existence. Moreover, Tegmark’s work is strikingly supportive of the kalām cosmological argument and the fine-tuning argument if we think, contrary to MUH, that the concrete universe exists. For he argues that we should abandon the “fundamentally flawed assumption” of infinite quantities and, hence, the beliefs that “space can have an infinite volume, that time can continue forever, and that there can be infinitely many physical objects.”[1] This is almost right. We should hold that space cannot have an infinite volume, that time cannot have continued forever (though it can continue potentially infinitely toward the future), and that there cannot be an actually infinite number of physical objects.

I deal with Tegmark’s MUH in volume IIb in my Excursus on Natural Theology in the chapter on the argument from the applicability of mathematics, which is the target of Tegmark’s hypothesis. I point out that Eugene Wigner saw that the question of the applicability of mathematics to physical phenomena is not why there are mathematical concepts and structures which are applicable to physical phenomena, for the question is not about the fecundity of the mathematical realm. Quite the reverse, the question is why the physical world exhibits a structure that is so amenable to mathematical description of its natural laws. As Wigner states, there is nothing about the mathematical formalism discerned in the laws of nature that renders its instantiation inevitable.

In discussing the response of the mathematical realist to Wigner’s problem, I point out that about the only way for the realist to avoid this problem is to deny that there really are two realms at issue, one abstract and one concrete. Rather the concrete realm is unreal; only the abstract, mathematical realm is real! Thus there is no gap to be bridged, and so no problem of “applicability” can arise. Max Tegmark has labeled this view the Mathematical Universe Hypothesis (MUH) and offers it as his solution to Wigner’s problem. “The MUH provides this missing explanation. It explains the utility of mathematics for describing the physical world as a natural consequence of the fact that the latter is a mathematical structure, and we are simply uncovering this bit by bit.”[2] In other words, there are no concrete objects!

The MUH is so outrageously implausible that we should do everything we can to avoid it. On this view we are victims of the most gigantic sort of illusionism. For example, since the mathematical realm is timeless, it follows that time is unreal and our experience of time and change is illusory. Since abstract objects are causally unconnected to one another, our experience of causal connections between things is illusory. Since abstract objects are immaterial, our perceptions of the physical world are wholly illusory. Perhaps most outrageous, since reality is entirely mathematical, either no persons exist or else I am an abstract object, which is absurd. We shall always have better reason to trust our experience than to think that the concrete realm is unreal, so that such a radical view is incapable of rational affirmation.

Moreover, such a view seems to be logically incoherent, since it presupposes a sort of plenitudinous Platonism, according to which mathematically incompatible structures all exist. One could try to escape the contradiction by sealing off the competing structures in worlds of their own, but then, as cosmologist Donald Page points out, the set of universes would itself be a larger mathematical structure. “At the ultimate level, there can be only one world and, if mathematical structures are broad enough to include all possible worlds or at least our own, there must be one unique mathematical structure that describes ultimate reality. So I think it is logical nonsense to talk of . . . the co-existence of all mathematical structures.”[3]

Tegmark argues that if we accept that there exists an external reality completely independent of us humans, what Tegmark calls the External Reality Hypothesis, then we are logically committed to the MUH.[4] Observing that since the time of Wigner’s writing ever more examples of the unreasonable effectiveness of mathematics have been discovered, Tegmark asserts, “I know of no other compelling explanation for this trend than that the physical world really is completely mathematical.”[5] He thereby overlooks entirely a theistic explanation of the otherwise unreasonable effectiveness of mathematics. Theism is a far more plausible explanation of the applicability of mathematics than a hypothesis that denies that the concrete world exists. MUH thus constitutes a poignant illustration of the dead end to which secularism drives us.