A constant of physics is fine-tuned for life if the width, Wr, of the range of values of

The Restricted Principle of Indifference

II. What it Means to Vary A Constant of Physics

More Fundamental Theory Problem:

In speaking of the laws in this way, we are already assuming a certain level of description of physical reality. The history of physics is one in which we find that a certain law is only a limiting case of some deeper law, such as Newton’s law of gravity being a limiting case of Einstein’s Equation of general relativity. This is especially relevant in the context of the current search for a grand unified theory. One hope is that such a grand unified theory will have no free parameters. The idea is that higher-level principles of physics – principles such as those that lie at the heart of quantum mechanics and general relativity – will uniquely determine the form of the grand unified theory along with the various fundamental constants of physics that are part of this theory. In this case, given the higher-level principles, it would be impossible for the constants to be any different than they are. Any discussion of fine-tuning must therefore be explicated at a level of description of physical reality below that of a grand unified theory. We will return to this issue below.

This issue also arises even in the context of our current (incomplete) understanding of physics. Consider, for instance, the strength of the strong force that holds neutrons and protons together in the nucleus. This force is actually not fundamental, but is simply a product of a deeper force between the quark constituents of the protons and neutrons, much as the force of cohesion between molecules is not fundamental but rather a product of various electromagnetic (and exclusion principle) forces. This deeper force binding quarks together is given by quantum chromodynamics (QCD), which in current theory has its own set of free parameters. From the perspective of QCD, one cannot simply change the strength of the strong force while keeping everything else the same. Instead, one would have to change one or more of the parameters of QCD, which would in turn change not just the strength of the strong force, but other things such as the masses of the neutron and proton and the range of the strong force. Calculations of these effects are very difficult and hence it is difficult to develop a rigorous FTA at the level of QCD. Thus, for practical purposes, most FTAs need to be developed at a lower level of description, such as at the level of the phenomenological equation governing the strong force between nucleons.

I claim that obtaining probabilities for FTAs at this less than fundamental level is a legitimate procedure. Epistemic probabilities are only useful in conditions of ignorance; they are attempts to generate degrees of expectation, or conditional degrees of expectation, when we do not know everything about a situation. For example, we have an unconditional epistemic probability that a coin will land on heads of 50% (instead of 100% or 0% ) because we are ignorant of the physically determined side on which it will land. That said, we cannot use any model of the phenomena in the domain in question, no matter how physically unrealistic. Rather, we should formulate our expectations using the best models we have for which we can make calculations. Further, when we vary a parameter in these models, we must have confidence that the variation relevantly tracks the corresponding variations of the appropriate parameters in the most fundamental theory of physics. Generally speaking, the larger the variation in our model, the less confidence we will be that it tracks variations in the deeper theory.

As an example of the above points, suppose that although QCD is our most fundamental theory, we use our phenomenological theory, PT, to calculate what values of the strong force constant, gs, are and are not life-permitting over some range R*. For PT to be relevantly reliable means that if we conclude that the life-permitting range of gs is small compared to R*, then there should be some corresponding variation of the fundamental parameters of QCD that roughly correspond to values of gs in R*, such that using the parameters of QCD it also turns out that the life-permitting range is small. The reason for this is that calculations and arguments based on a more fundamental theory always trump calculations and arguments based on a less fundamental theory. Thus we can be confident in our calculations and arguments from the less fundamental theory only insofar as we are confident that those same calculations and arguments would hold good if we performed them using the fundamental theory. Indeed, more generally, these are the conditions of using any approximate theory in physics: we only trust the calculations based on the approximate theory insofar as we are confident they approximately reflect the results we would get if we could perform corresponding exact theory. Often, only heuristic arguments can be given for thinking that the approximate theory relevantly tracks the more exact theory. We expect nothing more in the case of the FTA.

What if our most fundamental theory does not have any free parameters, but rather the structure of the theory itself entails the values of the constants? To begin, I think this is unlikely: the grand unified theories developed so far have not greatly reduced the number of free parameters (for example, the standard model of particle physics has twenty four) and although it was hoped that the various physical principles underlying string theory would dictate only one viable set of constants for physics, this hope has largely faded. Suppose, however, that our most fundamental theory has no free parameters and yet dictates the exact values of the free parameters of some lower level theory, such as that of the standard model in particle physics. In that case, variation of some parameter p in the standard model will correspond to a variation in the fundamental laws of physics themselves. Thus the values for which p is life-permitting will correspond to a certain set of life-permitting law structures, and values for which p is not life-permitting will correspond to another set of law structures. My suggestion is that we then take the parameter p as providing the relevant epistemic measure over these law structures, as long as p is part of the most fundamental theory with free parameters. Consequently, if the life-permitting range of p is small compared to the comparison range R, we could also say that the relevant class of life-permitting law structures are small compared to the relevant class of non-life-permitting law structures.

In this case, therefore, the fine-tuning is not so much of a fundamental parameter of physics as of a set of law structures generated by variations in p. But I do not see that this presents a problem. There is nothing more or less fundamental about varying a law then a parameter. In the strict sense, when we vary the fundamental parameters, we also vary the laws since laws and parameters come as single packages in nature. It is only humans beings that make the distinction. What matters is being able to calculate what happens for nearby law structures, and being able to provide an epistemic measure over the reference class consisting of our law structure and these nearby possible law structures.

Of course, our choice of nearby law structures must not be biased to yield the results we want. Certainly, however, the above method of choosing to vary the parameters in the most fundamental theory for which we can make calculations is not biased in favor of fine-tuning. What is important for calculating epistemic probability, however, is that our method of selection be non-arbitrary and as far as we can tell, unbiased. This is analogous to calculating statistical probabilities based on samples out of larger classes: the best studies are ones in which scientists choose a sample by a non-arbitrary procedure for which they attempt as much as possible to reduce possible biases.

Concluding Thoughts

Finally, one must keep this whole enterprise in perspective. The ultimate goal is not to provide exact numerical values for the degree of belief one should have that a certain constant falls into the intelligent-life-permitting range. Rather, the goal is to provide as much objective support as possible for the surprise felt at some parameter’s falling into the intelligent-life-permitting range. The goal is to show that this intuitive sense of surprise is not based on some mistake in thinking or perception or on some merely subjective interpretation of the data, but rather can be grounded in a justified, non-arbitrary procedure. Put another way, because we are only considering one reference class of possible law structures (the one we can perform calculations for), it is unclear how much weight to attach to the values of epistemic probabilities one obtains using this reference class. Hence once cannot simply map the epistemic probabilities obtained in this way onto the degrees of belief we should have at the end of the day. What we can do, however, is say that given that we choose our reference class in the way suggested, and assuming our other principles (such as the restricted principle of indifference), we obtain a certain epistemic probability, or range of probabilities. Then, as a second order concern, we must assess the confidence we have in the various components that went into the calculation, such as the representative nature of the reference class. As Peter Achinstein notes for more mundane cases, such as concerns about the quality of the reference class in drug testing, there is yet no way of integrating these second order concerns into the “first order proposals of probability theorists” (1994, P. 349).

One can think of what is going on here as analogous to the use of approximation in physics. Virtually all calculations in physics are made based on idealized circumstances. Then, as a second order concern, one attempts to assess the degree to which one’s calculations apply to the actual circumstances being considered. Similarly, our calculations of epistemic probability only strictly apply to idealized conditions, such as when there is absolutely no concern about possible competing reference classes. Given that there is no algorithm that can get us from these calculations to the actual degree of confidence we actually should have in the theistic hypothesis, or the degree to which the evidence supports the theistic hypothesis over the atheistic single-universe hypothesis, I suggest that the probability calculations should be thought of as simply providing an objective confirmation of the common intuitive sense that the fine-tuning provides evidence for theism over the atheistic single-universe hypothesis.

III. The Procedure for Determining R

Above, we said that determining k’ was the critical step in determining R. Determing k’ will involve two components: Addressing the issue will first involve confronting the problem of old evidence, and then looking at the fact that we can only determine whether a parameter is life-permitting for a local range – what I call the epistemically illuminated (EI) range.

The Problem of Old Evidence

As mentioned above, the problem of old evidence is that if we include known evidence e in our background information k, then even if an hypothesis h entails e, it cannot confirm H under a Bayesian or quasi-Bayesian methodology since P(e/k & h) = P(e/k & -h). But this seems incorrect: general relativity’s prediction of the correct degree of the precession of the perihelion of Mercury (which was a major anomaly under Newton’s theory of gravity) has been taken to confirm general relativity even though it was known for over fifty years prior to the development of general relativity and thus entailed by k. This issue has been extensively discussed in the literature on applying Bayesian methods to scientific inference. This issue confronts the fine-tuning argument since the fact that the constants of physics have life-permitting values follows from our own existence and hence is old evidence.

An attractive solutions is to subtract out our knowledge of old evidence e from the background information k and then relativize confirmation to this new body of information k’ = k – {e}. As Colin Howson explains, “when you ask yourself how much support e gives [hypothesis] h, you are plausibly asking how much knowledge of e would increase the credibility of h,” but this is “the same thing as asking how much e boosts h relative to what else be know” (1991, p. 548). This “what else” is just our background knowledge k minus e. As appealing as this method seems, it faces a major problem: there is no unambiguous way of subtracting e from k. Consider the case of fine-tuning. Let e be the claim that C falls within some range of experimentally determined values x. Now, the fact that C falls into x, along with the laws of physics, the initial conditions of the universe, and the other values for the constants of physics, entails certain facts F₁ about the large-scale structure of the universe, and further renders other such facts F₂ highly probable. So, our question is, when subtracting e from k, do we also subtract F₁ and F₂? If yes, then things get very murky: among other problems, what level of probability counts as probabilistically dependent? That is, to what degree must e render F₂probable in order to say that we should subtract F₂ out? If no, then k will effectively contain e, because elements of F₁ will entail e, and elements of F₂ will make e likely.

For example, consider the case of gravity. The fact, Lg, that the strength of gravity falls into the intelligent-life-permitting range entails the existence of stable, long-lived stars. On the other hand, given our knowledge of the laws of physics, the initial conditions of the universe, and the value of the other constants, the existence of stable, long-lived stars entails Lg. Thus, if we were to obtain k’ by subtracting Lg from our total background information k without also subtracting our knowledge of the existence of long-lived stable stars from k, then P(Lg/k’) = 1.

To solve such problems, Howson says, we should regard k as “in effect, an independent axiomatization of background information and k-{e} as the simple set-theoretic subtraction of e from k” (1991, p. 549). That is, Howson proposes that we axiomatize our background information k by a set of sentences A in such a way that e is logically independent of the other sentences in A. Then k’ would simply consist of the set of sentences A -{e}. One serious problem with this method is that there are different ways of axiomatizing our background information. Thus, as Howson recognizes, the degree to which e confirms h becomes relative to our axiomatization scheme (1991, p. 550). In practice, however, this is not as serious a problem as one might expect, since in many cases our background information k is already represented to us in a partially axiomatized way in which e is logically isolated from other components of k. As Howson notes, “the sorts of cases which are brought up in the literature tend to be those in which the evidence, like the statements describing the magnitude of the observed annual advance of Mercury’s perihelion, is a logically isolated component of the background information” (1991, p. 549). In such cases, when we ask ourselves how much e boosts the credibility of h with respect to “what else we know, ” this what else we know is well defined by how we represent our background knowledge. Of course, in those cases in which there are alternative ways of axoimatizing k that are consistent with the way our background knowledge is represented to us, there will be corresponding ambiguities in the degree to which e confirms h. I agree with Howson that this is not necessarily a problem unless one thinks that the degree of confirmation e provides h must be independent of the way we represent our background knowledge. Like Howson, I see no reason to make this assumption: confirmation is an epistemic notion and thus is relative to our epistemic situation, which will include the way we represent our background information.

In the case of fine-tuning, our knowledge of the universe is already presented to us in a partially axiomatized way. Assuming a deterministic universe, the laws and constants of physics, along with the initial conditions of the universe, supposedly determine everything else about the universe. Thus the set of propositions expressing these laws, constants, and initial conditions constitutes an axiomatization of our knowledge. Further, in scientific contexts, this represents the natural axiomatization. Indeed, I would argue, the fact that this is the natural axiomatization of our knowledge is part of our background knowledge, at least for scientific realists who want scientific theories to “cut reality at its seams.” Furthermore, we have a particularly powerful reason for adopting this axiomatization in this case. The very meaning of a constant of physics is only defined in terms of a particular framework of physics. Saying that the strong force constant has a certain value, for instance, would be meaningless in Aristotelian physics. Accordingly, the very idea of subtracting out the value of such a constant only has meaning relative to our knowledge of the current set of laws and constants, and hence this constitutes the appropriate axiomatization of our relevant background information k with respect to which we should perform our subtraction.

Using Howson’s method, therefore, we have a straightforward way of determining k - {e} for the case of the constants of physics: we let k be axiomatized by the set of propositions expressing the initial conditions of the universe and the laws and fundamental constants of physics within our currently most fundamental theory which we can do calculations. Since the constants of physics can be considered as given by a list of numbers in a table, we simply subtract the proposition expressing the value of C from that table to obtain k’. Thus, k’ can be thought of as including the initial conditions of the universe, the laws of physics, and the values of all the other constants except C. Further, if we want to obtain the appropriate k’ for there existing intelligent-life-permitting values for a set of constants {C1, C2, C3 ... Cn}, we would simply subtract the knowledge that these values fall into the intelligent-life-permitting range from our background information k.

Finally, as mentioned above, we will not include in our original axiomatized background information k the exact value of the constant C that we are considering, but only that it falls into the known intelligent-life-permitting range. The reason is that it makes the argument conceptually simpler. It is legitimate to make this exclusion because all we need to include in k’ is all information relevant to the quasi-Bayesian FTA we are interested in – that is, all information that makes a difference in the ratio P(L_C/T & k’)/P(L_C/AS & k’). Additional information regarding the experimental value of C – or more precisely, the range of values for C within experimental error – presumably will not affect our judgment of this ratio. Thus, for the purposes of this argument, it can be excluded from our background information, just as other irrelevant information, such as the location in which I live, can be excluded. On the other hand, even if it were relevant, we would have arrived at an identical k’ and hence an identical comparison range by initially including the experimentally determined value (within experimental error) of C in k, and then subtracting it out to determine k’. If we followed this procedure, we would have had to consider the experimental value of C as our relevant evidence, instead of the mere fact that it fell into the intelligent-life-permitting range. Although one could run quasi-Bayesian FTA this way, it seems less natural to do so.

The Epistemically Illuminated Region

The second step in determining k’ involves what I call the “epistemically illuminated” (EI) range. This range is defined above as the range of values for a constant C such that, given our current physical theories, we can make a reasonable judgement as to whether a particular value of the constant in that range is (intelligent) life-permitting. In some sense, it is obvious why we must limit our range R to the EI range, but we will offer some further justification for it by looking at a mundane example involving a dart-board. Suppose we had a dart board that extended far into the distance, but the entire dart board was not illuminated: only some finite region around the bull’s eye was illuminated, with the rest of the dart board in darkness. We thus neither know how far the dart board extends nor whether there are other bull’s eyes on it. If we saw a dart hit the bull’s eye in the illuminated (IL) region, and the bull’s eye was very, very small compared to the IL region, we would take that as evidence that the dart was aimed, even though we cannot say anything about the density of bull’s eyes on other regions of the board. One way of providing a quasi-Bayesian reconstruction of the inference to include the fact that the dart fell into the IL region as part of the information being conditioned on: that is, include it into the background information k’. The confirmation theory argument therefore would proceed as follows: given that as part of our background information k’ we know that the dart has fallen into the illuminated region , it is very unlikely for it to have hit the bull’s eye by chance but not if it was aimed, and thus its falling in the bull’s eye confirms the aiming hypothesis. That is, BI strongly confirms A over C because P(BI/C & k’) << P(BI/A & k’), where BI is the hypothesis that the dart hit the bull’s eye, C is the chance hypothesis, A is the aiming hypothesis, and k’ includes the information that it landed in the illuminated region. Similarly, for the case of fine-tuning, we should include the fact that the value of a constant is within the EI region as part of our background information k’ being conditioned on.

Is including EI in k’ an adequate procedure? The case of the dartboard, I believe, shows that it is not only a natural procedure to adopt, but arguably the only way of providing a quasi-Bayesian reconstruction of the inference in this sort of mundane case. First, it is clearly the ratio of the area taken up by the bull’s eye to the illuminated region around the bull’s eye that leads us to conclude that it was aimed. Second, one must restrict the comparison range to the illuminated range (that is, include IL in k’) since one does not know how many bull’s eyes are in the unilluminated portion. Thus, if one expanded the comparison range outside the illuminated range, one could make no estimate as to ratio the area of the bull’s-eye regions to the non-bull’s-eye regions, and thus could not get a confirmation theory argument off the ground. Yet, it seems intuitively clear that the dart’s hitting the bull’s eye in this case does confirm the “aimed” hypothesis over the chance hypothesis.

In general, as applied to the fine-tuning, one can also think of k’ as determining the reference class of possible law structures to be used for purposes of estimating the epistemic probability of Lc under As: the probability of Lc given k’ & As is the relative proportion of law structures that are life-permitting in the class of all law structures that are consistent with k’ and As. (The measure over this reference class is then given by the restricted principle of indifference.) Thinking in terms of reference classes, the justification for restricting our reference class to EI region is similar to that used in science: When testing an hypothesis, we always restrict our reference classes to those for which we can make the observations and calculations of the frequencies or proportions of interest– what in statistics is called the sample class. This is legitimate as long as we have no reason to think that such a restriction produces a relevantly biased the reference class. Tests of the long-term efficacy of certain vitamins, for instance, are often restricted to a reference class of randomly selected doctors and nurses in certain participating hospitals, since these are the only individuals that they can reliably trace for extended periods of time. As discussed in section II, the justification for varying a constant, instead of varying the mathematical form of a law, in the fine-tuning argument is that in the reference class of law structures picked out by varying a constant we can make some estimate of the proportion of life-permitting law structures, something we probably could not do if our reference class involved variations of mathematical form. The same sort of justification underlies restricting the class to the EI range. [Add footnote about alternative to old evidence view in terms of reference classes??]

Including EI in k’ provides a quasi-Bayesian reconstruction of John Leslie’s “fly on the wall” analogy, which he offers in response to the claim that there could be other distant values for the constants of physics, or unknown laws, that allow for life:

If a tiny group of flies is surrounded by a largish fly-free wall area then whether a bullet hits a fly in the group will be very sensitive to the direction in which the firer’s rifle points, even if other very different areas of the wall are thick with flies. So it is sufficient to consider a local area of possible universes, e.g., those produced by slight changes in gravity’s strength, . . . . It certainly needn’t be claimed that Life and Intelligence could exist only if certain force strengths, particle masses, etc. fell within certain narrow ranges . . . . All that need be claimed is that a lifeless universe would have resulted from fairly minor changes in the forces etc. with which we are familiar. (1989, pp. 138-9).

As Leslie stresses, what matters for the FTA is the comparison between the life-permitting range and the local region around it. Including EI in k’ is just a way of casting Leslie’s response in terms of conditional epistemic probabilities.

Defining R

We are now ready to define R. We define R as the region of possible values for our constant C allowed by our background information k’. Typically, within this region, k’ & As will give us no reason to prefer one value of C over any others. Given this, the restricted principle of indifference implies that under AS and k’, we should assign equal probabilities to equal ranges within R. This implies that P(L_C/AS & k’) = W_r/W_R, and hence that if W_r/W_R << 1 then P(L_C/AS & k’) << 1. Since in almost all, if not all, cases the EI range will be the major delimiting factor in determining the values allowed by k’, not the background theories and values of the other constants included in k’, the comparison range R will simply be the range EI.

IV. Determining R: Force Strengths and Other Constants

Let us now see how the procedure outlined above delimits R in the case of force strengths and other constants. The EI range effectively determines the range R for the force strengths. The EI range is determined by the fact that our current physical theories have known limitations of applicability with regard to energy levels. In the past, we have found that physical theories are limited in their range of applicability – for example, Newtonian mechanics was limited to medium size objects moving at speeds relatively slow compared to the speed of light. For fast objects, we require special relativity; for massive objections, general relativity; for very small objects, quantum theory. When the Newtonian limits are violated, these theories predict completely unexpected and seemingly bizarre effects, such as time dilation in special relativity or the phenomenon of tunneling in quantum mechanics. There are good reasons to believe that current physics is limited in its domain of applicability. The most discussed of these limits is that of energy scale. The current orthodoxy in high-energy physics and cosmology is that our current physics is either only a low-energy approximation to the true physics that applies at all energies or the low-energy end of a hierarchy of physics, with each member of the hierarchy operating at its own range of energies. The energy at which any particular current theory is no longer to be considered approximately accurate for the purposes under consideration is called the cutoff energy, though despite its name we would typically expect a continuous decrease in applicability, not simply a sudden change from applicability to non-applicability.

In contemporary terminology, our current physical theories are to be considered effective field theories. The limitation of our current physics directly affects thought experiments involving changing force strengths. Although in everyday life we conceive of forces anthropomorphically as pushes or pulls, in current physics forces are conceived of as interactions involving exchanges of quanta of energy and momentum. This idea has developed naturally from the definition in classical mechanics of work – that is, the amount of energy expended – as being proportional to the force applied in the direction of motion. Further, the change in momentum is proportional to the force applied times the amount of time over which it is applied, by F = ma. The strength of a particular force, therefore, can be thought of as proportional to rate of exchange of energy-momentum, expressed quantum mechanically in terms of probability cross sections. Drastically increasing the force strengths, therefore, would drastically increase the energy- momentum being exchanged in any given interaction. Put another way, increasing the strength of a force will involve increasing the energy at which the relevant physics takes place. So, for instance, if one were to increase the strength of electromagnetism, the binding energy of electrons in the atom would increase; similarly, an increase in the strength of the strong nuclear force would correspond to an increase in the binding energy in the nucleus.

The limits of the applicability our current physical theories to below a certain energy scales, therefore, translates to a limit on our ability to determine the effects of drastically increasing a value of a given force strength – for example, increasing the strong nuclear force by a factor of 10¹⁰⁰⁰. If we naively applied current physics to that situation, we would conclude that no complex life would be possible because atomic nuclei would be crushed. If a new physics applies, however, entirely new, and almost inconceivable effects could occur that make complex life possible, much as quantum effects make the existence of stable atomic orbits possible whereas such orbits were inconceivable under classical mechanics.

Further, we have no guarantee that the concept of a force strength itself remains applicable from within the perspective of the new physics at such energy scales, much as the concept of a particle having a definite position and momentum, or being infintitely divisible, lost applicability in quantum mechanics, or the notion of absolute time lost validity in special relativity, or gravitational “force” (versus curvature of space-time) in general relativity. Thus, by inductive reasoning from the past, we should expect not only entirely unforseen phenomena at energies far exceeding the cutoff, but we should even expect the loss of the applicability of many of our ordinary concepts, such as that of force strength.

The Plank scale is one often assumed cutoff for the applicability of the strong, weak, and electromagnetic forces. This is the scale at which unknown quantum gravity effects are suspected to take place thus invalidating certain foundational assumptions current quantum field theories are based on, such a continuous space-time. (For example, see Sahni and Starobinsky, 1999, p. 44; Peacock, p. 275). The Plank scale occurs at the energy of 10¹⁹ GeV ( billion electron volts), which is roughly 10²¹ higher than the binding energies of protons and neutrons in a nucleus, which are at the most around 10 MeV per nucleon. This means that we could expect a new physics to begin to come into play if the strength of the strong force were increased by more than a around a factor of 10¹⁹ . Another commonly considered cutoff is the GUT (Grand Unified Theory) scale, which occurs around 10¹⁵ GeV. (Peacocke, 1999, pp. 249, 267). The GUT scale is the scale at which physicists expect the strong, weak, and electromagnetic forces to be united. From the perspective of GUT, these forces are seen as a result of symmetry breaking of the united force that is unbroken above 10¹⁵ GeV, where a new physics would then come into play. Effective field theory approaches to gravity also involve general relativity being a low energy approximation to the true theory. One common proposed cutoff is the Plank scale, though this is not the only one ( See for example, Burgess, 2004, p. 6).

Where these cutoff’s lie, and the fundamental justification for them, are controversial issues. The point of the above discussion is that most likely the limits of our current theories are finite, but very large since we know that our physics does work for a enormously wide range of energies. Accordingly, if the life-permitting range for a constant is very small in comparison, then Wr/WR << 1, and then there will be fine-tuning. Rigorously determining the degree of fine-tuning, however, is beyond the scope of this paper.

Almost all other purportedly fine-tuned constants also involve energy considerations: for example, the masses of particles, such as the electron, neutron, proton, and electron correspond to energies ranging from around 1 MeV to a 1 GeV and the cosmological constant is now thought of as corresponding to the energy density of empty space. Thus, the considerations of energy cutoff mentioned will play a fundamental role in defining the epistemically illuminated region, and hence of our comparison range R.

V. The Purported Problem of Infinite Ranges

Timothy McGrew, Lydia McGrew, and Eric Vestrup (2000) have recently argued that the comparison ranges for the various cases of fine-tuning are all infinite, and that such infinite ranges pose two fatal problems for the FTA: the probabilities in the FTA are not normalizable and what they call “the coarse-tuning argument” is valid. In developing their argument they assume that the comparison range just is the range of values that the parameters conceivably could have, and since we can conceive of the values being anything, this range is infinite.

As argued above, the comparison range should not be taken as the conceivable range, but rather will be effectively delimited by what I called the epistemically illuminated region. For the sake of the argument and in case a future physics develops a theory in which EI is infinite, let us suppose that for some of the constants the comparison ranges are infinite. I will argue, however, that even if the proper comparison range is infinite, the McGrew/Vestrup objection to the FTA fails.

The Normalizability Problem

The McGrews and Vestrup argue that if the comparison range is infinite, and if we consider all sub-ranges of equal width equally probable – which is, they claim, the only possible non-arbitrary assignment of probability – then the probability assignments are not normalizable:

Probabilities make sense only if the sum of the logically possible disjoint alternatives adds up to one – If there is, to put the point more colloquially, some sense that attaches to the idea that the various possibilities can be put together to make up "one hundred percent" of the probability space. But if we carve an infinite space up into equal finite-sized regions, we have infinitely many of them; and if we try to assign them each some fixed positive probability, however small, the sum of these is infinite. (2003, p. 203).

An immediate response to this objection is that we can assign each finite sub-range a zero probability. To see this, note that any infinite range can be divided into a countably infinite set of finite sub-ranges of equal length that cover the entire range. If we apply the restricted principle of indifference, we must assign a probability of zero to the parameter landing in any given sub-range.

To this, the McGrews and Vestrup could argue that such an assignment of probability violates the axiom of countable additivity, an issue they unfortunately do not address. The axiom of countable additivity says that the countable sum of mutually exclusive classes of events must be equal to the probability of an event occurring in the union of the classes. That is, å_i=1^¥P(x_i) = P(x₁ or x₂ or x₃ ....), where the x_i are mutually exclusive events. In the case of fine-tuning, if the ranges are mutually exclusive and exhaustive, then the probability of a parameter’s falling somewhere within the entire range should be the sum of the respective probabilities for each of the infinite number of sub-ranges. Since the parameter must have some value, however, the probability for the entire infinite range is one, whereas the sum of the probabilities of each range is zero.

One response here is to save countable additivity simply by assigning each region an infinitesimal probability such that the countable sum of these infinitesimals adds up to one; this issue is explored in detail in [author unnamed]’s paper in this symposium. This technical solution aside, the “normalizability problem” rests on confusing the mathematical definition of probability with the sort of probability used in science and everyday life, particularly epistemic probability, which is the type of probability I am claiming occurs in the fine-tuning argument. The McGrews and Vestrup never tell us what type of probability they are working with. Is it relative frequency? Logical probability? Epistemic probability? The answer to this question is crucial to determining whether countable additivity applies. As mentioned above, epistemic probability has to do with rational degrees of belief. It seems to be rational to be certain that any given member of an infinite class of events will not occur, and at the same time believe that one of the events will nonetheless occur. Or, at the very least, such a set of beliefs is not obviously irrational.

Consider a situation in which there is an infinite number of discrete alternatives a_i that are mutually exclusive and exhaustive, and for which we are epistemically indifferent between the alternatives. Under this circumstance, it certainly seems rational to be certain, for each alternative a_i, that it will not be the case: that is, to assign each a_i zero epistemic probability. Indeed, in such a situation, this is the only non-arbitrary degree of belief one could have in any given a_i. Any uniform finite degree of belief over the a_i would violate the axiom of finite additivity. Countable additivity, on the other hand, would demand that we either assign some a_i different probabilities than other ones, or claim that there is no probability in this case. Clearly, however, it cannot be rationally obligatory to assign different a_i different rational degrees of belief when there is no epistemic distinction between them. As Williamson notes (1999, p. 408), de Finetti objected that such a principle would require one to treat as epistemically unequal what one regarded as epistemically equal: “even if I think they are equally probable, [...] I am obliged to pick a convergent series which, however I choose it, is in absolute contrast to what I think.” Worse, picking such a convergent distribution would force one to be practically certain (» 99.9999% epistemic probability) of the occurrence of a member of some finite set of alternatives when one believed that the set was epistemically on a par with each of an infinite class of sets of other alternatives. As de Finetti remarked, “Should we force him, against his own judgement, to assign practically the entire probability to some finite set of events, perhaps chosen arbitrarily?” (quoted in Williamson, 1999, p. 407). It is difficult to see how such a principle could be rationally obligatory!

One might respond that when the only non-arbitrary distribution of degrees of belief violates the axiom of countable additivity, the most rational alternative is to remain agnostic. After all, one need not assign epistemic probabilities to all propositions. I do not believe this is an adequate response, since I think in some cases it would be irrational to remain agnostic. To illustrate, suppose that God creates an infinite lottery by creating a truly random number generator that randomly picks the winning number from the integers. (Never mind for now whether you think the creation of such a generator is possible.) Further suppose that you have no reason to believe that this number falls into one range instead of any other and that each ticket sells for $20, with a $100,000 jackpot. Finally, suppose that your sister decides to buy lottery tickets, spending $20 a month on them. I am sure you would try to persuade her not to. You would not merely be agnostic about her winning; you would be sure that she would lose. And this certainty could be substantiated with significant arguments. For example, you might reason that if the lottery had a trillion, trillion, trillion tickets, it would be a waste of money to buy tickets, since the chance of winning would be so small you could say with confidence that she could be certain of not winning. Moreover, the larger the number of tickets, the more you would be certain that she won’t win. Certainly, in the limit as the number of tickets becomes infinite, you should not merely become agnostic about whether she will win. You would be absolutely certain she would lose. Compare this with the case in which there is a lottery, but you are given no idea of how many tickets there are. In such a case, you would be truly agnostic about your sister’s winning. You might simply hope she would win, without any idea of what her chances are. In the infinite lottery case, it seems clear that you should have no hope whatsoever of her winning, which is not the attitude of agnosticism.

Here is another argument against requiring agnosticism in these cases. Suppose you adopted the agnostic alternative. Then for any finite range [M, M + N], you should be agnostic about whether the winning number is in that range, where M and N are positive integers. Even though agnosticism does not commit you to a degree of belief, to be agnostic is to be less than certain. Now consider some given number k that is in the range [0, M], and let k* be the proposition that k is the winning number. If you were certain that the number was in the range [0, M], but did not have any information about which member of the range it was, then you would assign every member of this range an epistemic probability of 1/M for being the winning number. If you were agnostic about whether it fell into that range, then your probability would obviously be less. Thus, you could reason as follows: “If I were sure k was in that range, then k* would have an epistemic probability of 1/M. But, since I don’t even know that the winning number is in the range, I have even less reason to believe the winning number is k – that is, my epistemic probability P(k*) should be less than 1/M. Now, for any positive integer N, k will be a member of some range [0, M + N]. Thus, for every N, the probability of k being the winning number will have to be less than 1/(M + N). The only consistent value I can assign to the probability of k*, therefore, is zero or an infinitesimal, since P(k*) is less than 1/(M + N) for all N.”

Of course, one might object that no such random number generator could ever be constructed, even by God. As Williamson notes, however, this objection is irrelevant [1999, p. 407]. The issue is not whether some objective random number generator could be made, but whether the agent in question believes that such a random number generator exists, since epistemic probability is a relation between propositions held by some agent in a set of specified epistemic circumstances. Further, many of the arguments would go through even without having a random number generator: one could simply believe that God picked some finite number, and that God picked the number independently from any foreknowledge of what number one would guess, and that one’s choice was causally independent of what God chose.

Given the formidable set of arguments above for rejecting countable additivity for epistemic probability, those who (like the McGrews and Vestrup) insist on countable additivity need to show that distributions that violate this axiom are rationally incoherent.

The Coarse-tuning Argument

The second argument presented by the McGrews and Vestrup goes back to a similar argument presented by theoretical physicist Paul Davies (1992), and also repeated by Manson (2000). According to this argument, which McGrew and Vestrup call “the Coarse-tuning Argument” (CTA), if the comparison range is infinite, then no matter how large the range r of intelligent-life-permitting values, as long as it is finite, the ratio of it to the conceivable range will be zero. This means that the narrowness of the range becomes irrelevant to our assessment of degree of fine-tuning. The mere finitude of the intelligent-life-permitting range would confirm design or many-universes. The McGrews and Vestrup conclude that “if we are determined to invoke the Principle of Indifference regarding possible universes, we are confronted with an unhappy conditional: if the FTA is a good argument, so is the CTA. And conversely, if the CTA is not a good argument, neither is the FTA” (2003, p. 204). Davies presents a similar worry:

From what range might the value of, say, the strength of the nuclear force... be selected? If the range is infinite, then any finite range of values might be considered to have zero probability of being selected. But then we should be equally surprised however weakly the requirements for life constrain those values. This is surely a reductio ad absurdum of the whole argument. [1992, pp. 204-205]

One response is to agree that we should be “equally surprised however weakly the requirements for life constrain those values.” If we truly found for some constant C that (i) the comparison range R as determined by our method R is infinite, and (ii) the range of intelligent-life-permitting values is finite, and (iii) we should be epistemically indifferent over the range of possible values, then we should think the fact that C falls into the intelligent-life-permitting range does strongly confirm design (or multiple universes) over the atheistic single-universe hypothesis. That is, we should conclude that the CTA is a sound argument. I see nothing odd or counterintuitive about this. The CTA is certainly not obviously absurd. Physics would still be required in constructing the CTA – for example, to show that the intelligent-life-permitting range was indeed finite and that the total theoretically possible range was infinite, and to show that we have no physical reason for preferring one value of a parameter over any other in the range.

The main reason Davies has trouble with the CTA is that he claims that we are more impressed when “the requirements of life are more restrictive” [Davies, p. ] According to Davies, it is the smallness of the range of intelligent-life-permitting values that appears to ground our temptation to infer design or many universes. If, however, we had good reason to believe that the comparison range for C is truly infinite, and that we should be epistemically indifferent over that range, then we would just have to conclude that our initial impressions were wrong that it was the smallness of the range, not its finiteness, that gave the FTA its force. Although there is a slight presumption in favor of such initial impressions, they are by no means indefeasible.

That said, we can rationally reconstruct why we are impressed with the relative smallness of the intelligent-life-permitting region instead of merely its finiteness. The reason we are impressed with the smallness, I suggest, is that we actually do have some vague finite range to which we are comparing the intelligent-life-permitting range. We do not, as a matter of fact, think that all values for a parameter are equally likely, just those in some limited range around the actual value of the parameter in question. This vague sense of a comparison range is given a rigorous foundation when we actually apply my method above. If we ever developed a theory with free parameters which we are confident has no limit of applicability, and the sum of the life-permitting ranges were known to only take up a finite width, then the CTA would be the correct argument to apply, not the FTA. Since in today’s physics EI is finite, the FTA is the correct argument and thus the stress in the literature on the relative smallness, versus simply finiteness, of the life-permitting range is appropriate.

Finally, rejecting the CTA for the reasons the McGrews and Vestrup give is counterintuitive. Assume that the FTA would have probative force if the comparison range were finite; although they might not agree with this assumption, making it will allow us to consider whether having an infinite instead of finite comparison range is relevant to the cogency of the

FTA. Now imagine increasing the width of this comparison range while keeping it finite. It seems that the more W_R increases, the stronger the FTA gets. Indeed, if we accept the restricted principle of indifference, as W_R approaches infinity, P(L_C/AS) will converge to zero, and thus P(L_C/AS) = 0 in the limit as W_R approaches infinity. Accordingly, if we deny the CTA has probative force because W_R is purportedly infinite, we must draw the counterintuitive consequence that although the FTA gets stronger and stronger the larger W_R gets, magically when W_R becomes actually infinite, the FTA loses all probative force.

References

Achinstein, Peter. “Stronger Evidence,” Philosophy of Science, 61, September 1994., pp. 329 -350.)

Burgess, Cliff P. (April, 2004). Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory. Living Reviews in Relativity, Published by the Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Germany (http://www.livingreviews.org/lrr-2004-5, pdf format).

Cao, Tian Yu. (1997). Conceptual Developments of 20th Century Field Theories, Cambridge: Cambridge University Press.

Collins, Robin. (2003). “Evidence for Fine-tuning.” In Neil Manson, ed., God and Design: The Teleological Argument and Modern Science, London: Routledge, pp. 178-199.

Davies, Paul (1992) The Mind of God : The Scientific Basis for a Rational World (New York: Simon and Schuster).

Gillies, Donald (2000) Philosophical Theories of Probability (London and New York: Routledge).

Hacking, Ian (1987) “Coincidences: Mundane and Cosmological,” reprinted in Origin and Evolution of the Universe: Evidence for Design, ed. John M. Robson (Montreal: McGill-Queen’s University Press), pp. 119 – 138.

Howson, Colin (1991) “Discussion: The Old Evidence Problem,” The British Journal for the Philosophy of Science 42, pp. 547 - 555.

Leslie, John. Universes. New York, Routledge, 1989.

Keynes, J. M. A Treatise on Probability. London: Macmillan, 1921.

Manson, Neil A. (2000) “There is No Adequate Definition of ‘Fine-tuned for Life’,” Inquiry 43, pp. 341-52.

McGrew, Timothy, McGrew, Lydia, and Vestrup, Eric (2001) “Probabilities and the Fine-tuning Argument: A Skeptical View,” Mind 110, pp. 1027-38. Reprinted in God and Design: The Teleological Argument and Modern Science, ed. Neil A. Manson (London: Routledge, 2003), pp. 200-208.

Oberhummer, H., Csótó, A. and Schlattl, H. (2000a) “Fine-Tuning of Carbon Based Life in the Universe by Triple-Alpha Process in Red Giants,” Science 289, no. 5476 (7 July 2000), pp. 88-90.

––––– (2000b) “Bridging the Mass gaps at A = 5 and A = 8 in nucleosythesis,” http://xxx.lanl.gov/abs/nucl_th/0009046 (18 September 2000).

Peacock, John. (1999). Cosmological Physics. Cambridge: Cambridge University Press, 1999.

Plantinga, Alvin. (1993). Warrant and Proper Function. Oxford: Oxford University Press.

Sahni, Varun and Starobinsky, Alexei. (1999) “The Case for a Positive Cosmological Λ-term,” at http://xxx.lanl.gov/abs/astro-ph/9904398, (28 April 1999).

Teller, Paul. (1988). “Three Problems of Renormalization.” In Rom Harré and Harvey Brown, eds., Philosophical Foundations of Quantum Field Theory, Oxford: Clarendon Press, pp. 73-89.

von Plato, Jan (1994) Creating Modern Probability: Its Mathematics, Physics and Philosophy in Historical Perspective (Cambridge, U.K.: Cambridge University Press).

Wald, Robert. (1984). General Relativity, Chicago, IL: University of Chicago Press.

Williamson, J.O.D. (1999) “Countable Additivity and Subjective Probability,” British Journal for the Philosophy of Science 50, pp. 401-416.

Zee, A. (2003). Quantum Field Theory in a Nutshell. Princeton, NJ: Princeton University Press.