Justin Brierley recently moderated a debate between Stephen Meyer and Phil Halper on the cosmological evidence for theism. During the debate, Halper, co-author of Battle of the Big Bang, raised some objections that are worthy of consideration or, at least, are likely to create confusion for those unfamiliar with the minutiae of the subject. One of his strategies was to challenge the legitimacy of the probability claims at the heart of fine-tuning arguments. Shortly after the two-hour mark in the debate, Meyer pressed the point that the constants of nature occupy an exceedingly small life-permitting region of parameter space. Halper replied by trying to undermine Meyer’s argument by using analogies based on mysterious dice and jars of beans. His homely examples have surface plausibility and merit a serious answer. Ultimately, Halper’s objections collapse under their own weight.
Halper’s Objections
Halper begins by talking about how the calculation of probabilities with ordinary dice is well-defined. The probability of rolling a six is one in six because dice have six sides that are equally weighted. When more than one die is in play, the joint probabilities for outcomes are also well-defined. But with the constants of nature, he claims, none of this is the case. Around 2:08:04 in the debate he says:
Now think of the constants of nature. Do we know how many sides there are on the dice? No, we don’t. We don’t know how many values are actually possible… We don’t know how many sides are on the dice. Nor do we know if they’re equally weighted, nor do we know how many dice we have.
Halper’s argument invokes what philosophers call the principle of indifference. This principle states that, in the absence of reasons to think otherwise, equal probabilities should be assigned to all possible outcomes. His contention is that for the cosmic constants, the structural conditions for applying the principle are absent. In other words, when considering values the constants of nature might have, we lack appropriate knowledge of the sample space, of the probability distribution over this space, and of the number of trials relevant to getting what is needed. He argues that this pulls the rug out from under claims that various natural constants are improbably fine-tuned. He then presses his point with an example borrowed from Caltech philosopher of science Christopher Hitchcock, in which a configuration of lentils and black beans in a jar that initially looks highly improbable turns out to be unsurprising once the physical mechanism of gravitational sorting is understood.
As Meyer quickly recognized, Halper’s homely examples are versions of the so-called normalizability objection to fine-tuning. This objection was first pressed by Timothy and Lydia McGrew, along with Eric Vestrup, in 2001. This is a substantial objection and it needs a compelling answer. Fortunately, it has one.
Meyer’s Replies
Meyer’s reply to Halper begins around 2:10:55. His introductory remarks take on Halper’s implicit criticism in the bean-jar analogy that the values of the constants might have a mechanism that explains them, then he explicitly cites the normalizability problem and points to two streams of technical work that resolve it.
As we noted, the bean-jar example illustrates how a fundamental physical mechanism can work to sort the beans in the jar into what might otherwise seem to be a highly improbable configuration. What was unexpected becomes expected when the mechanism is understood. The situation with the constants of nature, however, is structurally very different. The fundamental laws, which are contingent mathematical expressions that could conceivably have been different, have various associated constants that govern their interaction strengths; these constants are themselves contingent. Standard physics offers no more fundamental picture or mechanism that undergirds these laws that would explain why the laws have the mathematical form that they do or why the constants take on the values they do. There is no more basic physics in the background that could explain these laws and constants in the way that gravity, as a physical mechanism, explains the sorting of the beans in the jar. Appealing to the laws themselves to explain their own constants, let alone their own form, is a viciously circular non-starter, and making an appeal to metaphysical or mathematical necessity to explain aspects of the universe that are, ex hypothesi, contingent, has no plausibility either. The only option left for a physical mechanism is an appeal to a multiverse of some sort, but as Meyer points out elsewhere in the debate (and Halper acknowledges he needs to think about more), multiverse cosmology does not so much explain the fine-tuning as relocate it. Whatever background dynamics in a multiverse might generate universes with different laws and constants, it would have to function in accordance with contingent metalaws and be finely-tuned itself to produce the right distribution of universes. Halper’s bean-jar analogy simply highlights the fact that, for principled reasons, cosmology lacks the very mechanism he postulates as a deus ex machina.
Turning to Meyer’s discussion of the normalizability objection, one response he mentions draws on the work of Robin Collins and of Luke Barnes focusing on the comparison range that physics itself imposes on the range of values that natural constants can take. One of Collins’s key essays on the subject can be found here. Key contributions of Barnes can be found here, here, and here. The key insight is that evaluating whether the constants of nature are fine-tuned does not require us to consider an unbounded parameter space. The only relevant ranges to consider are bounded subregions within which the physics makes sense and is well enough understood for a parametric sensitivity analysis to be meaningful.
In this latter respect, the Standard Model of particle physics and general relativity provide natural lower and upper limits for most of the key parameters. The Planck scale gives us a rock solid upper bound on energies, masses, and field strengths. There are additional bounds under that ceiling that emerge from the physics of structure formation. For instance, primordial density perturbations must have an amplitude on the order of one part in 10⁵. If it were smaller, matter would be distributed too evenly for galaxies to form, and if it were larger, highly dense regions would collapse into supermassive black holes. The cosmological constant is another example. It must be small enough that gravity can put galactic structures together before the expansion of space spreads things too thin. At the subatomic level, strong nuclear coupling has to fall within a range that permits stable nuclei to form. If it were too weak, the heavier elements could not form, but if it were too strong, helium-2 would be stable and the hydrogen in the universe would burn away on cosmological timescales. For chemistry to be possible, the electromagnetic coupling has to be strong enough to hold atoms together but not so strong that the Coulomb repulsion destabilizes the heavier nuclei necessary to life. These bounds are not arbitrary stipulations. They follow from well-understood gravitational, nuclear, and atomic physics. And it is a fact that within the natural comparison ranges, the life-permitting subregions are astoundingly and exquisitely small. Individually and collaboratively, physicists and cosmologists such as Max Tegmark, Martin Rees, Anthony Aguirre, Frank Wilczek, Fred Adams, and Luke Barnes (see here, here, here, and here) have mapped these subregions in detail by perturbing various constants in the Standard Model and general relativity and watching the universe fall apart on paper as galaxies, stars, atoms, or chemistry become impossible.
Meyer’s second response to Halper’s homespun version of the normalizability problem references more recent work on cosmological fine-tuning by Daniel Díaz-Pachón (University of Miami), collaborating with Ola Hössjer (Stockholm) and Robert Marks (Baylor). Using nonstandard analysis and the hyperreal number system, they have shown how to construct well-defined probability measures on infinite parameter spaces to which standard measure theory would assign only zero or undefined values. This allows the recovery of determinate fine-tuning ratios even on infinite ranges. The upshot is that what was once touted as an in-principle obstacle to fine-tuning calculations has been technically defused. Probabilities on unbounded spaces can be defined coherently after all and determinate fine-tuning ratios can be calculated, even on infinite ranges.
The bottom line is that Meyer’s responses to Halper’s attempt to obfuscate the probabilities associated with fine-tuning were highly effective. He dismantled Halper’s implicit suggestion that there might be some physical mechanism that explained the values of the parameters in question, then took the wind out of Halper’s version of the normalizability objection by alluding to two effective avenues of response that we have just elucidated. The two avenues are particularly well chosen because we show that the bounded-range formulation of fine-tuning works for the comparison-range regime that we actually understand, and the unbounded-range formulation works mathematically even in the infinite limit.1 In short, the cosmic dice are not inscrutable. The number of sides they have, that is, their physically meaningful parameter ranges, the question of how these sides are weighted, that is, the structural information regarding what each parameter regime yields, and how many dice there are, that is, the finite set of relevant fundamental parameters, are all given to us by standard physics. All of this is more than sufficient to make fine-tuning calculations tractable and the answers obtained reasonable. Both the physical and the mathematical resolution of the normalizability problem shut the door on Halper’s argument.
Further Reasons that Halper’s Objections Misfire
Beyond the technical results that Meyer cited, there are other considerations that count against Halper’s objections. It is notable that he doesn’t talk about the comparative likelihood of fine-tuning on naturalism versus theism, but merely tries (unsuccessfully) to obfuscate the fine-tuning probabilities themselves. While we have not focused on this aspect of the fine-tuning argument, the fact remains that the evidence we observe is more expected on the theistic hypothesis than the naturalistic one. When we take Bayesian likelihood ratiosinto account on anobjective degree-of-support reading of epistemic probability, we see that what matters epistemically is the ratio P(E|design) / P(E|chance). On the chance hypothesis required by naturalism, the constants taking precisely life-permitting values is, on any non-pathological measure, vastly improbable. Given intelligent design, on the other hand, it is what we would expect. This likelihood ratio is enormous, and it does not depend on the absolute scale of any prior.
A more general shortcoming of Halper’s approach is that he assigns theism such a low prior probability that he seems to think he does not need to worry about its posterior probability in light of fine-tuning. But this is plainly false. Even if we grant uncertainty about measure due to cutoff assumptions, the cosmological constant is fine-tuned anywhere from 60 to 120 orders of magnitude. Only a ludicrously contrived low prior probability on theism could prevent that magnitude of fine-tuning from producing a posterior probability in favor of design. There is no independent rational motivation for assigning theism such a low prior.
Another problem that Halper faces is that, if his objection actually worked, it would prove too much and undermine everything he holds dear. If it were the case that we could not make probability judgments without complete sample-space knowledge, then vast arenas of scientific reasoning would collapse in concert with fine-tuning arguments. And we can see that Halper has a double standard here, since he makes probability judgments throughout the debate. For instance, he cites Weinberg’s anthropic prediction of the cosmological constant; he treats the multiverse as supplying probabilistic resources for life-permitting universes; he deploys survey statistics that use probabilistic reasoning under partial information; and he insists, at one point, that the cosmological constant could be “a hundred times larger” with no harm to life. This last claim involves a probabilistic judgment about fine-tuning in parameter space. The bottom line is that Halper cannot consistently invoke these capacities when they suit him and disavow them when they don’t.
As a final observation, we note that the argument using Hitchcock’s beans pulls the wrong way for Halper’s purposes. The basic idea is that what looks improbable may turn out to be probable once a physical mechanism is identified. The fine-tuning argument has the same structure and, on a suitably general level, can be seen to make the same point. The intuition is that things that are deeply improbable require an explanation. Where cosmological fine-tuning is concerned, there are a number of options. The first is that it is somehow necessary, which has no support in current physics, and Halper concedes this. The second is a multiverse that displaces the fine-tuning to a metalevel where the demand for explanation is reinstantiated. The third is an assertion of brute factuality, the assertion that there is no explanation, which runs afoul of the principle of sufficient reason (that every contingent event has an explanation) that serves as a foundational principle for science. And finally, there is the explanation of intelligent design, which is known to be causally sufficient to the production of finely-tuned systems. What Hitchcock’s example does is motivate the search for a causally sufficient explanation, not move us away from it. Ironically, then, Halper has tried to argue against design using an argument that pushes us toward it.
Beyond Halper’s Obfuscations to the Real Picture
Halper’s appeal to inscrutable dice assumes total ignorance about the physical constraints relevant to fine-tuning calculations. This is nowhere near the truth. The Standard Model, general relativity, dimensional analysis, and the Planck scale provide natural cutoffs and natural orderings on the relevant parameter space. A better metaphor than rolling dice with indeterminate sides would be a dart thrown at a dartboard with a target region that is exceedingly small relative to the board as a whole, even if the ratio is not known with complete precision. Imprecision in the ratio carries over into corresponding uncertainty about the degree of fine-tuning, but it has little impact on the actual magnitude of the fine-tuning, and it comes nowhere close to vacating the argument.
Halper’s principle-of-indifference objection only works against a strawman version of fine-tuning that claims to compute an absolute probability over an unbounded parameter space with a uniform measure. No serious advocate of fine-tuning defends that picture. The actual argument uses well-defined comparison ranges yielding well-grounded comparative likelihoods, and in any case where the bounded version does not apply, the hyperreal framework in nonstandard analysis extends the technical machinery to handle the limiting cases. Ultimately, Halper has argued against a position that no one holds, and his argument as a whole lays claim to the very capacity his objection denies, namely the ability to make probability judgments under partial information. The dice turn out to be not-so-mysterious after all, and Hitchcock’s beans make a good snack for knowledgeable advocates of a finely-tuned cosmos.
Notes
- It is worth noting that the normalizability problem for cosmological fine-tuning that has been resolved by these methods is different from the measure problem in inflationary cosmology. The problem in inflationary cosmology is not that the parameter space is infinite, though it certainly is, but that eternal inflation does not provide us with a unique measure on the resulting set of pocket universes. There are different reasonable cutoff procedures (proper-time, scale-factor, causal-patch, holographic, stationary, light-cone time, and others) that generate mutually inconsistent predictions about what kinds of observers are “typical.” This leaves the debilitating paradoxes (Boltzmann brains, the youngness paradox) unresolved. The problem is thus not normalization, but rather theoretical underdetermination. Hyperreal probabilities can give us coherent ratios once a measure is fixed, but we have no idea which (if any) of the mutually inconsistent measures to fix.