Arg Um En T205
A RG U M E N T 205
10 Background information k and k¢. k refers to our total background information, whereas k′ refers to some appropriately chosen background information – for example, for the case of the fi ne-tuning of the constants, k′ is the total background information minus the fact that a particular constant C has a life-permitting value. (See Sections 4.1–4.4 for how to determine k′ for the fi ne-tuning argument.)
11 Other symbols. << will always mean “much, much less than” – for example, P(A|B) << 1
will mean that P(A|B) is very close to zero since P(A|B) cannot have a negative value. ~P(A|B) << 1 will mean that it is not the case that P(A|B) << 1. Wr will usually refer to the width of the life-permitting range of a particular constant C of physics, and WR will usually refer to the width of the “comparison range” for that constant, which we argue is typically the EI region (see Section 4.4). The EI region for a constant will refer to the “epistemically illuminated” region – that is, the region of values for C for which we can determine whether they are life-permitting. A constant is fi ne-tuned if Wr/WR << 1; “BB” stands for “Boltzmann brain” (see Sections 6.3.3
and 6.3.4).
1.3. The basic argument presented: likelihood approach
My basic argument fi rst claims that, given the fi ne-tuning evidence, LPU strongly supports T over the NSU. I call this the core fi ne-tuning argument. After developing this argument in Sections 2 through 5, I then present arguments for preferring T over the multiverse hypothesis (Section 6). Finally, in Section 8, I shall briefl y consider other possible alternative explanations of the fi ne-tuning.
The core fi ne-tuning argument relies on a standard Principle of Confi rmation theory, the so-called Likelihood Principle. This principle can be stated as follows. Let h1 and h2 be two competing hypotheses. According to the Likelihood Principle, an observation e counts as evidence in favor of hypothesis h1 over h2 if the observation is more probable under h1
than h2. Put symbolically, e counts in favor of h1 over h2 if P(e|h1) > P(e|h2), where P(e|h1) and P(e|h2) represent the conditional probability of e on h1 and h2, respectively. Moreover, the degree to which the evidence counts in favor of one hypothesis over another is proportional to the degree to which e is more probable under h1 than h2: specifi cally, it is proportional to P(e|h1)/P(e|h2).2
The Likelihood Principle appears to be sound under all interpretations of probability.
The type of probability that we shall be concerned with is what is called conditional epistemic probability. The conditional epistemic probability of a Proposition A on a Proposition B can be defi ned roughly as the degree to which Proposition B, in and of itself, 2. There are many reasons why the Likelihood Principle should be accepted (e.g. see Edwards 1972; Royall 1997; Forster & Sober 2001; Sober 2002); for the purposes of this chapter, I take what I call the restricted version of the Likelihood Principle (see further discussion) as providing a suffi cient condition for when evidence e supports a hypothesis, h1, over another, h2. For a counterexample to the Likelihood Principle’s being a necessary condition, see Forster (2006). For an application of the Likelihood Principle to arguments for design, see Sober (2005). (I address Sober’s main criticism of the fi ne-tuning argument in Sections 3.2, 5.2, and 7.5.) The Likelihood Principle can be derived from the so-called odds form of Bayes’s Theorem, which also allows one to give a precise statement of the degree to which evidence counts in favor of one hypothesis over another. The odds form of Bayes’s Theorem is P(h |
|
1 e)/P(h2 e) = [P(h1)/P(h2)] × [P(e|h1)/P(e|h2)]. The Likelihood Principle,
however, does not require the applicability or truth of Bayes’s Theorem and can be given independent justifi cation by appealing to our normal epistemic practices.
supports or leads us to expect A to be true. In Section 3.2, I shall explicate this notion of probability in much more detail. Put in terms of epistemic probability, the Likelihood Principle can be reworded in terms of degrees of expectation instead of probability, in which case it becomes what I call the Expectation Principle. According to the Expectation Principle, if an event or state of affairs e is more to be expected under one hypothesis, h1, than another, h2, it counts as evidence in favor of h1 over h2 – that is, in favor of the hypothesis under which it has the highest expectation. The strength of the evidence is proportional to the relative degree to which it is more to be expected under h1 than h2. Rewording the Likelihood Principle in terms of expectation is particularly helpful for those trained in the sciences, who are not familiar with epistemic probability and therefore tend to confuse it with other kinds of probability even when they are aware of the distinction.
Because of certain potential counterexamples, I shall use what I call the restricted version of the Likelihood Principle, although I shall often refer to it simply as the Likelihood Principle. The restricted version limits the applicability of the Likelihood Principle to cases in which the hypothesis being confi rmed is non- ad hoc. A suffi cient condition for a hypothesis being non- ad hoc (in the sense used here) is that there are independent motivations for believing the hypothesis apart from the confi rming data e, or for the hypothesis to have been widely advocated prior to the confi rming evidence. To illustrate the need for the restricted version, suppose that I roll a die 20 times and it comes up some apparently random sequence of numbers – say 2, 6, 4, 3, 1, 5, 6, 4, 3, 2, 1, 6, 2, 4, 4, 1, 3, 6, 6, 1. The probability of its coming up in this sequence is one in 3.6 × 1015, or about one in a million billion. To explain this occurrence, suppose I invented the hypothesis that there is a demon whose favorite number is just the aforementioned sequence of numbers (i.e. 26431564321624413661), and that this demon had a strong desire for that sequence to turn up when I rolled the die. Now, if this demon hypothesis were true, then the fact that the die came up in this sequence would be expected – that is, the sequence would not be epistemically improbable. Consequently, by the standard Likelihood Principle, the occurrence of this sequence would strongly confi rm the demon hypothesis over the chance hypothesis. But this seems counterintuitive: given a sort of commonsense notion of confi rmation, it does not seem that the demon hypothesis is confi rmed.
Now consider a modifi cation of the demon case in which, prior to my rolling the die, a group of occultists claimed to have a religious experience of a demon they called “Groodal,”
who they claimed revealed that her favorite number was 2643156432162441366, and that she strongly desired that number be realized in some continuous sequence of die rolls in the near future. Suppose they wrote this all down in front of many reliable witnesses days before I rolled the die. Certainly, it seems that the sequence of die rolls would count as evidence in favor of the Groodal hypothesis over the chance hypothesis. The relevant difference between this and the previous case is that in this case the Groodal hypothesis was already advocated prior to the rolling of the die, and thus the restricted Likelihood Principle implies that the sequence of die rolls confi rms the Groodal hypothesis.3
3. The restricted version of the Likelihood Principle is not the only way of dealing with the sort of counterexample raised by the fi rst demon case. Another way is to claim that, contrary to our intuitions, in the appropriate technical sense given by Bayes’s Theorem, the sequence really does confi rm the demon hypothesis, but never enough to make it probable since its prior probability is so low. For the purposes of this chapter, however, I am trying to stay as close to our intuitions about evidence as possible and hence prefer restricting the Likelihood Principle to deal with these purported counterexamples.
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Using the restricted version of the Likelihood Principle, the core fi ne-tuning argument can be stated as follows:
(1) Given the fi ne-tuning evidence, LPU is very, very epistemically unlikely under NSU: that is, P(LPU|NSU & k′) << 1, where k′ represents some appropriately chosen background information, and << represents much, much less than (thus making P(LPU|NSU & k′) close to zero).
(2) Given the fi ne-tuning evidence, LPU is not unlikely under T: that is, ~P(LPU|T & k′) << 1.
(3) T was advocated prior to the fi ne-tuning evidence (and has independent
motivation).
(4) Therefore, by the restricted version of the Likelihood Principle, LPU strongly supports T over NSU.
This argument is a valid argument since the conclusion (4) follows from the premises.
Accordingly, the key issues for the argument will be justifying the premises and assessing the signifi cance of the conclusion. Premise (3) seems obviously true since T was advocated long before the fi ne-tuning evidence came to light.4 Supporting Premises (1) and (2) in as rigorous a way as possible will require delving into and resolving several further issues.
First, we shall need to be more precise on what we mean by “epistemic” probability (Section 3.2) and to develop a preliminary theory of what it is that will be usable for our main argument. Second, we shall need to consider how epistemic probability is justifi ed (Section 3.3). In particular, in Section 3.3.2, I argue for what I call the restricted Principle of Indifference. Third, we shall need to carefully determine what the appropriate background information k′ is (Sections 4.1–4.6). If, for example, we were to include in k′ everything we know about the world, including our existence, then both probabilities in Premises (1) and (2) will turn out to be 1.0 since our existence entails LPU. Determining k′ will also provide a comparison region of law structures (or values for the constants) to which we are comparing the life-permitting region (Section 4.4). Having done this, we shall be ready to complete the argument for Premises (1) and (2) in Sections 5.1 and 5.2 – for example, we shall justify Premise (1) for the case of the fi ne-tuning of the constants by appealing to the restricted version of the Principle of Indifference defended in Sections 3.3.1 and 3.3.2.
Finally, given that we can establish the conclusion, what is its signifi cance? Even if LPU
counts as strong evidence in favor of T over NSU, that does not itself establish that T is likely to be true, or even more likely to be true than NSU. In this way, LPU is much like fi ngerprints found on a murder weapon. Via the Likelihood Principle, a defendant’s fi ngerprints’ matching those on the weapon typically provide strong evidence for guilt because the jury correctly judges that it is very unlikely for this matching to occur if the defendant is not guilty (and claims to have never seen the murder weapon), whereas it is not unexpected if the defendant actually used the murder weapon. Although such a 4. There is one worry about Premise (3) though: T was not advocated prior to the evidence of LPU since the life-permitting character of our universe follows from our existence and all the motivations for T are inter-twined with the fact that we exist. If this is a real diffi culty, one might need to use my alternative version of the fi ne-tuning argument, the method of probabilistic tension (Section 1.4), for which this problem clearly does not arise.
match can provide strong evidence that the defendant is guilty, one could not conclude merely from this alone that the defendant is guilty; one would also have to look at all the other evidence offered. Perhaps, for instance, ten reliable witnesses claimed to see the defendant at a party at the time of the shooting. In that case, by the restricted version of the Likelihood Principle, the matching would still count as signifi cant evidence of guilt, but this evidence would be counterbalanced by the testimony of the witnesses. Similarly, we shall claim that the evidence of fi ne-tuning signifi cantly supports T over NSU; this, however, neither shows that, everything considered, T is probably true, nor that it is the most plausible explanation of existence of the universe, nor even that it is more probable than NSU. In order to show that any hypothesis is likely to be true using a likelihood approach, we would have to assess the prior epistemic probability of the hypothesis, something I shall not attempt to do for T.
Our more limited conclusion is nonetheless highly relevant to the rationality of and justifi cation for belief in God, even though it does not itself establish that, all things considered, T is more probable than NSU. One could argue, for example, that in everyday life and science we speak of evidence for and against various views, but seldom of prior probabilities. For example, if one were to ask most physicists why they (tentatively) believe in General Relativity’s approximate truth (or at least its future empirical fruitfulness), they probably would cite the evidence in favor of it, along with some of Einstein’s motivations.
They probably would not cast these considerations – such as Einstein’s motivations – into talk of prior probability, either epistemic or otherwise. Of course, at the end of the day, some might say things such as “Einstein’s theory is likely to be true,” which is a notion of epistemic probability. But I can imagine them saying, “I have no idea what the prior probability of Einstein’s theory is; all I will say is that Einstein had motivations for considering it and there are at least three strong pieces of empirical evidence in its favor.” Indeed, I think it would be very diffi cult to estimate the prior probability of General Relativity’s approximate truth (or future empirical fruitfulness) in any objective manner, since we should have to weigh incommensurable factors against each other – the simplicity of the mathematical framework of General Relativity against such things as the philosophically puzzling character of the idea of a four-dimensional space-time’s being curved. Arguably, this is analogous to the case of T.
One way of putting my approach in perspective is to note that one of the most common philosophical objections to T is an updated version of one offered by Kant, in which the methods of theoretical reason are restricted to justifying only the existence of natural causes.
Unlike Kant, however, modern atheistic philosophers often reject God as a necessary hypothesis of practical reasoning. Typically, these philosophers claim that by analyzing the notion of explanation, particularly as exemplifi ed in natural science, we fi nd that to explain something involves citing overarching laws, mechanical causes, and the like; thus, they assert, the very notion of explanation involves a restriction to purely naturalistic explanations. The idea of God as providing a complete explanation of all contingent reality is therefore rejected as being empty, an improper extension of the notion of “explain” to places where it cannot apply. Another attack is to argue that even if God could be said to explain contingent reality, God’s own existence would be as much in need of explanation.
Richard Swinburne (2004, chaps. 2–4) has responded to these critics by offering an alternative analysis of ordinary explanation. He claims that these critics have neglected the notion of personal explanation and then goes on to claim that God provides the best personal explanation of everything we know about the universe. His argument rests on the
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dual claim that the simplicity of an explanation is the ultimate criterion of its adequacy and that God provides the simplest explanation of the universe. There are many places one might object to Swinburne’s project, particularly these two claims about simplicity.
Further, by basing his project on the notion of explanation used in everyday life, Swinburne leaves God’s own existence entirely unexplained. Indeed, Swinburne claims that God is the ultimate contingent brute fact and could be said to necessarily exist only in the limited sense that God is without beginning and that, in principle, nothing could explain God’s existence.
The approach I am taking avoids this question of how best to analyze the notion of explanation or whether God ultimately provides the best explanation of the universe.
Rather, it simply attempts to establish the more limited claim that various features of the universe offer strong evidence in favor of T over its major naturalistic alternatives. I believe that establishing this more limited claim in a careful, principled way would alone be a great accomplishment. In Section 7.1, however, I address how this more limited claim fi ts into an overall argument for the existence of God. In that section, I also briefl y address the issue of God’s providing an ultimate explanation and respond to the claim that God is as much in need of a designer as the universe itself. A fuller discussion of this issue, however, is beyond the scope of this chapter.
1.4. Alternative version of argument: method of probabilistic tension
One problem with using simply the Likelihood Principle is that whether or not a hypothesis is confi rmed or disconfi rmed depends on what one builds into the hypothesis. For example, single-universe naturalists could prevent disconfi rmation of their hypotheses by advocating the elaborated naturalistic single-universe hypothesis (NSUe), defi ned as NSU conjoined with the claim that the universe that exists is life-permitting: that is, NSUe = NSU & LPU. Similarly, theists could avoid any question about whether LPU is probable on T by constructing an elaborated theistic hypothesis (Te) which builds in the claim that God desired to create such a universe: Te = T & God desires to create a life-permitting universe.
One could attempt to deal with these sorts of moves by fi nding some principled way of restricting what can be built into the theistic and naturalistic hypotheses – for example, by requiring that the theistic and naturalistic hypotheses be some sort of “bare” theism and “bare” single-universe naturalism, respectively. This, however, is likely to run into diffi culties not only in justifying the principle, but in defi ning what “bare” theism and “bare” single-universe naturalism are supposed to be. A simpler way of addressing the issue is by means of a concept I call probabilistic tension. A hypothesis h suffers from probabilistic tension if and only if h is logically equivalent to some conjunctive hypothesis, h
|
1 & h2, such that P(h1 h2) << 1: that is, one conjunct of the hypothesis is very unlikely, conditioned on the other conjunct. Among other things, a hypothesis h that suffers from probabilistic tension will be very unlikely: since P(h) = P(h1 & h2) =
P(h |
|
|
|
1 h2) × P(h2) = P(h2 h1) × P(h1), it follows that if P(h1 h2) << 1 or P(h2 h1) << 1, then P(h) << 1.
I claim that signifi cant probabilistic tension is an epistemic black mark against a hypothesis, and thus offers us a reason to reject it. To see this, consider the fi ngerprint example discussed earlier. We noted that based on the Likelihood Principle, the matching of a defendant’s fi ngerprints with those on a murder weapon often strongly confi rms the guilt hypothesis over the innocence hypothesis. Such matching, however, does not confi rm the
guilt hypothesis over what could be called an “elaborated innocence hypothesis” – that is, an innocence hypothesis constructed in such a way that the matching of the fi ngerprints is implied by the hypothesis. An example of such a hypothesis is the claim that the defendant did not touch the murder weapon conjoined with the claim that someone else with almost identical fi ngerprints touched the weapon. This hypothesis entails that the fi ngerprints will appear to match, and hence by the Likelihood Principle the apparent matching could not confi rm the guilt hypothesis over this hypothesis.
Nonetheless, this elaborated innocence hypothesis suffers from severe probabilistic tension: one conjunct of the hypothesis (that some other person with almost identical fi ngerprints touched the weapon) is very improbable on the other conjunct (that the defendant is innocent) since it is extremely rare for two people to happen to have almost identical fi ngerprints. Given that the guilt hypothesis does not suffer from a corresponding probabilistic tension, the high degree of probabilistic tension of the elaborated innocence hypothesis gives us strong reason to reject it over the guilt hypothesis, even though this elaborated hypothesis is not itself disconfi rmed by the matching of the fi ngerprints.
This idea of probabilistic tension allows us to eliminate any arbitrariness with regard to how we choose the theistic and naturalistic hypotheses that we are confi rming or disconfi rming.5 For example, both the theist and the naturalist can build into their respective hypotheses whatever is necessary for them to entail the relevant data. Then one can apply the method of probabilistic tension to these elaborated hypotheses. Consider, for instance, the NSUe and the Te defi ned earlier: that is, NSUe = NSU & LPU and Te = T & God desires to create a life-permitting universe. Both of these hypotheses entail LPU, and hence neither is confi rmed (via the Likelihood Principle) with respect to the other by LPU.
Now given the truth of Premise (1) of our main argument in Section 1.3, NSUe
clearly exhibits a high degree of probabilistic tension relative to background information k′, since one conjunction of the hypothesis, LPU, is very improbable conditioned on the other conjunct, NSU: that is, P(LPU|NSU & k′) << 1.6 Given the truth of Premise (2), elaborated theism will not suffer any corresponding probabilistic tension. The reason is that according to Premise (2), it is not the case that P(LPU|T & k′) << 1, and hence it follows that it is not unlikely that the God of “bare theism” would desire to create a life-permitting universe. This means there will be no probabilistic tension between
“bare” theism and the claim that God desires to create such a universe. This will be true even if the probability of P(LPU|T & k′) is merely indeterminate, since such a probabilistic tension would exist only if P(LPU|T & k′) << 1. Thus, the fi ne-tuning evidence causes NSUe to suffer from a severe probabilistic tension without causing a corresponding probabilistic tension in Te. Thus, because it creates this probabilistic tension, we could say that the fi ne-tuning evidence (but not LPU itself) gives us strong reason to reject NSUe over Te.
In practice, any suffi ciently elaborated hypothesis will suffer from severe probabilistic tension somewhere. For instance, the elaborated guilt hypothesis mentioned earlier could 5. Later, in Section 4.3, we shall see that the method of probabilistic tension also helps eliminate a potential arbitrariness problem that arises with the choice of the appropriate background information k′.
6. This probabilistic tension, of course, makes NSUe very unlikely. Since P(LPU|NSU & k′) << 1 and P(LPU&NSU) = P(LPU|NSU) × P(NSU), it follows that P(NSUe) = P(NSU & LPU) << 1. One cannot conclude, however, that NSUe is less likely than Te, unless one can determine various prior probabilities, which I am avoiding doing.
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include that the knife used in the murder was green, had a certain shaped scratch mark on its handle, had a weight of 0.15679876675876 kg, and the like. The corresponding elaborated innocence hypothesis would include the same data. Both would suffer
from severe probabilistic tension with respect to each piece of data – for example, the murder knife having a weight of 0.15679876675876 kg is very improbable under both the “bare” guilt and “bare” innocence hypothesis. The lesson here is that only the probabilistic tension that one hypothesis has and that another lacks relative to some specifi ed domain can be used as evidence in favor of one hypothesis over another. In our case, the relevant specifi ed domain causing probabilistic tension is that of the fi ne-tuning data.
Elaborated naturalism might do better in other areas with regard to probabilistic tension, but to say that the fi ne-tuning data counts against elaborated naturalism with respect to elaborated theism, all we have to show is that the fi ne-tuning evidence creates a probabilistic tension within elaborated naturalism without creating a corresponding tension within elaborated theism.
2. The Evidence for Fine-Tuning
2.1. Introduction
The evidence for fi ne-tuning of the universe for life falls into three categories: (i) The
fi ne-tuning of the laws of nature.
(ii) The
fi ne-tuning of the constants of nature.
(iii) The
fi ne-tuning of the initial conditions of the universe.
We shall present examples of each type of fi ne-tuning further in the discussion. Before we begin, we should note that each of the aforementioned types of fi ne-tuning presupposes that a necessary requirement for the evolution of embodied moral agents is that there exist material systems that can sustain a high level of self-reproducing complexity – something comparable to that of a human brain. Given what we know of life on Earth, this seems a reasonable assumption.
2.2. Laws of nature
The fi rst major type of fi ne-tuning is that of the laws of nature. The laws and principles of nature themselves have just the right form to allow for the existence embodied moral agents. To illustrate this, we shall consider the following fi ve laws or principles (or causal powers) and show that if any one of them did not exist, self-reproducing, highly complex material systems could not exist: (1) a universal attractive force, such as gravity; (2) a force relevantly similar to that of the strong nuclear force, which binds protons and neutrons together in the nucleus; (3) a force relevantly similar to that of the electromagnetic force; (4) Bohr’s Quantization Rule or something similar; (5) the Pauli Exclusion Principle.
If any one of these laws or principles did not exist (and were not replaced by a law or principle that served the same or similar role), complex self-reproducing material systems could not evolve. First, consider gravity. Gravity is a long-range attractive force between all
material objects, whose strength increases in proportion to the masses of the objects and falls off with the inverse square of the distance between them. In classical physics, the amount of force is given by Newton’s law, F = G m 1 m 2/ r 2, where F is the force of attraction between two masses, m1 and m2, separated by a distance r, and G is the gravitational constant (which is simply a number with a value of 6.672 × 10−11 N·m2/kg2). Now consider what would happen if there were no universal, long-range attractive force between material objects, but all the other fundamental laws remained (as much as possible) the same. If no such force existed, then there would be no stars, since the force of gravity is what holds the matter in stars together against the outward forces caused by the high internal temperatures inside the stars. This means that there would be no long-term energy sources to sustain the evolution (or even existence) of highly complex life. Moreover, there probably would be no planets, since there would be nothing to bring material particles together, and even if there were planets (say because planet-sized objects always existed in the universe and were held together by cohesion), any beings of signifi cant size could not move around without fl oating off the planet with no way of returning. This means that embodied moral agents could not evolve, since the development of the brain of such beings would require signifi cant mobility. For all these reasons, a universal attractive force such as gravity is required for embodied moral agents.
Second, consider the strong nuclear force. The strong nuclear force is the force that binds nucleons (i.e. protons and neutrons) together in the nucleus of an atom. Without it, the nucleons would not stay together. It is actually a result of a deeper force, the “gluonic force,”
between the quark constituents of the neutrons and protons, a force described by the theory of quantum chromodynamics. It must be strong enough to overcome the repulsive electromagnetic force between the protons and the quantum zero-point energy of the nucleons. Because of this, it must be considerably stronger than the electromagnetic force; otherwise the nucleus would come apart. Further, to keep atoms of limited size, it must be very short range – which means its strength must fall off much, much more rapidly than the inverse square law characteristic of the electromagnetic force and gravity. Since it is a purely attractive force (except at extraordinarily small distances), if it fell off by an inverse square law like gravity or electromagnetism, it would act just like gravity and pull all the protons and neutrons in the entire universe together. In fact, given its current strength, around 1040 stronger than the force of gravity between the nucleons in a nucleus, the universe would most likely consist of a giant black hole.
Thus, to have atoms with an atomic number greater than that of hydrogen, there must be a force that plays the same role as the strong nuclear force – that is, one that is much stronger than the electromagnetic force but only acts over a very short range. It should be clear that embodied moral agents could not be formed from mere hydrogen, contrary to what one might see on science fi ction shows such as Star Trek. One cannot obtain enough self-reproducing, stable complexity. Furthermore, in a universe in which no other atoms but hydrogen could exist, stars could not be powered by nuclear fusion, but only by gravitational collapse, thereby drastically decreasing the time for, and hence probability of, the evolution of embodied moral agents.
Third, consider electromagnetism. Without electromagnetism, there would be no atoms, since there would be nothing to hold the electrons in orbit. Further, there would be no means of transmission of energy from stars for the existence of life on planets. It is doubtful whether enough stable complexity could arise in such a universe for even the simplest forms of life to exist.
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Fourth, consider Bohr’s rule of quantization, fi rst proposed in 1913, which requires that electrons occupy only fi xed orbitals (energy levels) in atoms. It was only with the development of quantum mechanics in the 1920s and 1930s that Bohr’s proposal was given an adequate theoretical foundation. If we view the atom from the perspective of classical Newtonian mechanics, an electron should be able to go in any orbit around the nucleus.
The reason is the same as why planets in the solar system can be any distance from the Sun
– for example, the Earth could have been 150 million miles from the Sun instead of its present 93 million miles. Now the laws of electromagnetism – that is, Maxwell’s equations
– require that any charged particle that is accelerating emit radiation. Consequently, because electrons orbiting the nucleus are accelerating – since their direction of motion is changing
– they would emit radiation. This emission would in turn cause the electrons to lose energy, causing their orbits to decay so rapidly that atoms could not exist for more than a few moments. This was a major problem confronting Rutherford’s model of the atom – in which the atom had a nucleus with electrons around the nucleus – until Niels Bohr proposed his ad hoc rule of quantization in 1913, which required that electrons occupy fi xed orbitals. Thus, without the existence of this rule of quantization – or something relevantly similar – atoms could not exist, and hence there would be no life.
Finally, consider the Pauli Exclusion Principle, which dictates that no two fermions (spin-½ particles) can occupy the same quantum state. This arises from a deep principle in quantum mechanics which requires that the joint wave function of a system of fermions be antisymmetric. This implies that not more than two electrons can occupy the same orbital in an atom, since a single orbital consists of two possible quantum states (or more precisely, eigenstates) corresponding to the spin pointing in one direction and the spin pointing in the opposite direction. This allows for complex chemistry, since without this principle, all electrons would occupy the lowest atomic orbital. Thus, without this principle, no complex life would be possible.7
2.3. Constants of physics
2.3.1. Introduction
The constants of physics are fundamental numbers that, when plugged into the laws of physics, determine the basic structure of the universe. An example of a fundamental constant is Newton’s gravitational constant G, which determines the strength of gravity via Newton’s law F = G m 1 m 2/ r 2. We will say that a constant is fi ne-tuned if the width of its life-permitting range, Wr, is very small in comparison to the width, WR, of some properly chosen comparison range: that is, Wr/WR << 1. A more philosophically rigorous way of determining this comparison range will be presented in Section 4.4. Here we shall simply use certain standard comparison ranges that naturally arise in physics and hence are used by physicists when they speak of cases of anthropic fi ne-tuning.8
7. The Pauli Exclusion Principle also applies to the nucleus. It prevents an indefi nite number of neutrons from falling into the lowest nuclear shell, thereby putting a limit on the atomic weight of atoms, a limit which appears necessary for life.
8. Physicists often speak of a requirement that there be “no fi ne-tuning” in a way that is independent of considerations regarding the existence of life. Two versions of this requirement are (i) that “all dimensionless parameters should be of the order of unity” (Narkilar 2002, pp. 479–80), and (ii) that the value of a constant should not be much, much smaller than the quantum corrections to its bare value, since otherwise the bare value
There are many examples of the anthropic fi ne-tuning of the fundamental constants of physics. Elsewhere, I have more thoroughly examined six of what I considered the most well-established cases, carefully articulating the physical lines of evidence offered in support of these cases along with correcting some incorrect and often-repeated claims regarding fi ne-tuning (Collins 2003). For purposes of illustration, here I shall explicate in some detail only two constants of physics – the strength of gravity and the cosmological constant.9
2.3.2. Fine-tuning of gravity
Using a standard measure of force strengths – which turns out to be roughly the relative strength of the various forces between two protons in a nucleus – gravity is the weakest of the forces, and the strong nuclear force is the strongest, being a factor of 1040 – or 10
thousand billion, billion, billion, billion times stronger than gravity (Barrow & Tipler 1986, pp. 293–5). Now if we increased the strength of gravity a billionfold, for instance, the force of gravity on a planet with the mass and size of the Earth would be so great that organisms anywhere near the size of human beings, whether land-based or aquatic, would be crushed. (The strength of materials depends on the electromagnetic force via the fi ne-structure constant, which would not be affected by a change in gravity.) Even a much smaller planet of only 40 ft in diameter – which is not large enough to sustain organisms of our size – would have a gravitational pull of 1,000 times that of Earth, still too strong for organisms with brains of our size to exist. As astrophysicist Martin Rees notes, “In an imaginary strong gravity world, even insects would need thick legs to support them, and no animals could get much larger” (2000, p. 30). Based on the evidence from Earth, only organisms with brains of a size comparable to our own have signifi cant moral agency.
Consequently, such an increase in the strength of gravity would render the existence of embodied moral agents virtually impossible and thus would not be life-permitting in the sense that we defi ned.
Of course, a billionfold increase in the strength of gravity is a lot, but compared with the total range of the strengths of the forces in nature (which span a range of 1040), it is very small, being one part in 10 thousand, billion, billion, billion. Indeed, other calculations show that stars with lifetimes of more than a billion years, as compared with our Sun’s lifetime of 10 billion years, could not exist if gravity were increased by more than a factor of 3,000 (Collins 2003). This would signifi cantly inhibit the occurrence of embodied moral agents.
would have to be fi ne-tuned to be almost exactly opposite of the quantum correction to yield such a relatively small value for the constant (see Donoghue 2007, pp. 234–6). It is important, therefore, not to naively equate discussions of fi ne-tuning in the physics literature with anthropic fi ne-tuning, although, as John Donoghue (2007) shows, they are often closely related.
9. Some other examples of fi ne-tuning of the constants are the following: if the mass of the neutron were slightly increased by about one part in 700, stable hydrogen burning stars would cease to exist (Leslie 1989, pp. 39–40; Collins 2003); if the weak force were slightly weaker by one part in 109 of the range of force strengths, then the existence of complex life would be severely inhibited (Collins 2003); fi nally, if the “vacuum expectation value” of the Higgs fi eld were not within a few times its present strength, complex atoms (with atomic numbers greater than hydrogen) could not exist (see Donoghue 2007, pp. 237–8). All these cases, however, need more explication and analysis than can be given here.
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The case of fi ne-tuning of gravity described is relative to the strength of the electromagnetic force, since it is this force that determines the strength of materials – for example, how much weight an insect leg can hold; it is also indirectly relative to other constants – such as the speed of light, the electron and proton mass, and the like –
which help determine the properties of matter. There is, however, a fi ne-tuning of gravity relative to other parameters. One of these is the fi ne-tuning of gravity relative to the density of mass-energy in the early universe and other factors determining the expansion rate of the Big Bang – such as the value of the Hubble constant and the value of the cosmological constant. Holding these other parameters constant, if the strength of gravity were smaller or larger by an estimated one part in 1060 of its current value, the universe would have either exploded too quickly for galaxies and stars to form, or collapsed back on itself too quickly for life to evolve.10 The lesson here is that a single parameter, such as gravity, participates in several different fi ne-tunings relative to other parameters.
2.3.3. The cosmological constant
Probably the most widely discussed case of fi ne-tuning for life is that of the cosmological constant. The cosmological constant, Λ, is a term in Einstein’s equation of General Relativity that, when positive, acts as a repulsive force, causing space to expand and, when negative, acts as an attractive force, causing space to contract. Einstein’s equation implies that if the vacuum – that is, space-time devoid of normal matter – has an energy density, then that energy density must act in a mathematically, and hence physically, equivalent way to a cosmological constant. The seeming need for fi ne-tuning of the cosmological constant arises from the fact that almost every fi eld within modern physics – the electromagnetic fi eld, the Higgs fi elds associated with the weak force, the infl aton fi eld hypothesized by infl ationary cosmology, the dilaton fi eld hypothesized by superstring theory, and the fi elds associated with elementary particles – each contributes to the vacuum energy far in excess of the maximum life-permitting amount. These contributions to the vacuum energy can be either negative or positive. If the total effective cosmological constant is positive and larger than some positive value Λ+max, or negative and smaller than some negative value Λ−max, then the universe would have expanded (if positive), or collapsed (if negative), too quickly for stars or galaxies to form. Thus, for life to occur, the cosmological constant must be between Λ−max and Λ+max. I shall let Λmax designate the larger of the two absolute values of Λ−max and Λ+max. Since the absolute values of Λ−max and Λ+max are within one or two orders of magnitude of each other, I shall explain the cosmological constant problem for the case in which Λ is assumed to be positive, but the same analysis will apply for the case in which Λ is negative.
10. This latter fi ne-tuning of the strength of gravity is typically expressed as the claim that the density of matter at the Plank time (the time at which we have any confi dence in the theory of Big Bang dynamics) must have been tuned to one part in 1060 of the so-called critical density (e.g. Davies 1982, p. 89). Since the critical density is inversely proportional to the strength of gravity (Davies 1982, p. 88, eqn. 4.15), the fi ne-tuning of the matter density can easily be shown to be equivalent to the aforementioned claim about the tuning of the strength of gravity.
Of course, if one cites this fi ne-tuning of gravity, one cannot then treat the fi ne-tuning of the force of the Big Bang or matter density of the Big Bang as an independent fi ne-tuning. (See Section 5.1.1 for how to combine cases of fi ne-tuning.)
Einstein originally hypothesized the existence of the cosmological constant so that his theory would imply a static universe. Thus, the original cosmological constant that Einstein postulated was not associated with contributions to the vacuum energy of the various fi elds of physics. If we let Λvac represent the contribution to the cosmological constant from the vacuum energy of all the fi elds combined, and Λbare represent the intrinsic value of the cosmological constant apart from any contribution from these fi elds, then the total value, Λtot, of the cosmological constant is Λtot = Λvac + Λbare. The contributions to Λvac can be further divided into those contributions arising from various forms of potential energy, V, as the universe passes through different phases along with those arising from the zero-point energies of the vacuum fl uctuations of the quantum fi elds of the fundamental forces and elementary particles.
Finally, there have been various proposals for a new and highly speculative type of energy, called quintessence, whose defi ning feature is that it acts like a cosmological constant in the way it causes space to expand or contract, but unlike the cosmological constant it can change with time as the universe evolves from the Big Bang onward. Consequently, we can defi ne the total effective cosmological constant as the sum of all these contributions that function in the same way as the cosmological constant with regard to causing space to expand or contract: that is, Λeff = Λvac + Λbare + Λq, where Λq designates the contribution resulting from quintessence. The fi ne-tuning problem can now be stated as follows: without fi ne-tuning or some new principle of physics, L eff is expected to be at least 1053 to 10120 times larger than the maximum life-permitting value L max. The smallness of the cosmological constant compared to its non-fi ne-tuned, theoretically expected value is widely regarded as the single greatest problem confronting current physics and cosmology.
To understand this fi ne-tuning problem more fully, it will be helpful to consider the three major types of contribution to the vacuum energy term, Λvac, of the cosmological constant in modern cosmology. Following standard terminology, we will let ρvac designate the vacuum energy density, and ρmax the maximum vacuum energy density compatible with the existence of life, given that the vacuum energy is the only contribution to the cosmological constant.11 The fi rst contribution we shall consider arises from the Higgs fi eld postulated as part of the widely accepted Weinberg–Salem–Glashow electroweak theory.
According to this theory, the electromagnetic force and the weak force acted as one force prior to symmetry breaking of the Higgs fi eld in the very early universe when temperatures were still extremely high. Before symmetry breaking, the vacuum energy of the Higgs fi eld had its maximum value V0. This value was approximately 1053 ρmax. After symmetry breaking, the Higgs fi eld fell into some local minimum of its possible energy density, a minimum which theoretically could be anywhere from zero to 1053 ρmax, being solely determined by V0 and other free parameters of the electroweak theory.12
Now either this local minimum is less than ρmax, or it is greater than ρmax and the other contributions to the cosmological constant offset its contribution to Λeff so that Λeff < Λmax.
In either case, the fi ne-tuning would have to be in one part in 1053. In the former case, for instance, the local minimum of energy would have to be between zero and ρmax, which would be one part in 1053 of its possible range of values.
11. Choosing units in which the speed of light, c, is equal to one (as is commonly done), it follows from Einstein’s equation that ρvac = 8πGΛvac and ρmax = 8πGΛmax, where G is Newton’s constant of gravity. Hence, the vacuum energy and the cosmological constant are strictly proportional to each other.
12. See Collins (2003, p. 196, endnote 9) for more analysis of this case.
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The second contribution to the vacuum energy is the postulated infl aton fi eld of infl ationary cosmology. Infl ationary universe models hypothesize that the infl aton fi eld had an enormously high energy density in the fi rst 10−35 to 10−37 seconds of our universe, resulting in an effective cosmological constant that caused space to expand by a factor of around 1060 (Guth 1997, p. 185). By around 10−35 seconds or so, however, the value of the infl aton fi eld fell to a relatively small value corresponding to a local minimum of its energy.13 Now, in order to start infl ation, the initial energy density of the infl aton fi eld, ρi, must have been enormously larger than ρmax: ρi >> ρmax. Theoretically, however, the local minimum of the infl aton fi eld could be anything from zero to ρi (see Sahni & Starobinsky 1999, sec. 7.0; Rees 2000, p. 154). The fact that the effective cosmological constant after infl ation is less than ρmax requires an enormous degree of fi ne-tuning, for the same reason as the Higgs fi eld mentioned – for example, neglecting other contributions to the cosmological constant, the local minimum of energy into which the infl aton fi eld fell must be between zero and ρmax, a tiny portion of the its possible range, zero to ρi.
The fi nal contribution results from the so-called zero-point energies of the fi elds associated with forces and elementary particles, such as the electromagnetic force and electrons and protons. If we calculate this contribution using quantum fi eld theory and assume that space is a continuum, the contribution turns out to be infi nite. Physicists, however, typically assume that quantum fi eld theory is valid only up to a certain very large cutoff energy (see Section 4.5), in which case the contribution turns out to be extraordinarily large, but fi nite, with the actual value depending on the cutoff energy below which quantum fi eld theory is taken as valid. The so-called Plank energy is often assumed to be the energy scale that quantum fi eld theory breaks down, in which case the energy contribution of the zero-point energy for the various fi elds would be expected to be 10120 ρmax (see Sahni & Starobinsky 1999, p. 44). To reduce this contribution to within an order of magnitude of the life-permitting range (thus eliminating any signifi cant fi ne-tuning) would require an extremely low cutoff energy, which most physicists consider very unlikely (Donoghue 2007, p. 236).
One solution to the cosmological constant problem is to claim that God, or some other intelligence, fi ne-tuned the various contributions to the cosmological constant, Λvac + Λbare + Λq, in such a way that Λeff < Λmax. Another much discussed solution is an appeal to multiple universes and some anthropic selection effect, which we shall discuss in Section 6. Is there a nondivine, nonanthropic solution to the fi ne-tuning of the cosmological constant? Physicist Victor Stenger, the leading critic of the data appealed to by advocates of the fi ne-tuning argument, claims that there is. According to Stenger:
. . . recent theoretical work has offered a plausible non-divine solution to the cosmological constant problem. Theoretical physicists have proposed models in which the dark energy is not identifi ed with the energy of curved space-time but rather with a dynamical, material energy fi eld called quintessence. In these models, the cosmological constant is exactly 0, as suggested by a symmetry principle called supersymmetry. Since 0 multiplied by 10120 is still 0, we have no cosmological constant problem in this case. The energy density of quintessence is not constant but evolves along with the other matter/energy fi elds of the universe. Unlike the cosmological constant, quintessence energy density need not be fi ne-tuned. (2004, p. 182) 13. Although the Higgs fi elds and the other fi elds of physics could contribute to infl ation, for various technical reasons infl ationary cosmology requires a distinct energy fi eld that helps create a very large effective cosmological constant in the very early universe. There is currently no candidate for this fi eld that is “deeply rooted in well-established theories of fundamental physics” (Albrecht 2004, p. 381).
Although in a later publication, Stenger (2007, pp. 151–3) does not mention supersymmetry, he still claims that a hypothesized new form of energy, quintessence, could solve the cosmological constant problem and that it “requires no fi ne-tuning” (Stenger 2007, p. 152).
Stenger’s proposal can be summarized in three steps: (1) postulate some natural symmetry or principle that requires that the cosmological constant, Λtot = Λvac + Λbare, be zero; (2) postulate some additional quintessential fi eld to account for what appears to be a small positive value of the effective cosmological constant today;14 and (3) postulate that there is some natural equation that implies that Λq < Λmax in the early universe, an equation which itself does not require fi ne-tuning. Since Λeff = Λvac + Λbare + Λq, those three steps would guarantee in a natural way that Λeff < Λmax.
A well-known proposal that would go part way to making Λtot = 0 is the appeal to the speculative hypothesis of supersymmetry. Supersymmetry requires that for each bosonic fi eld there exist a corresponding fermionic fi eld, where bosonic fi elds are those associated with spin-1 particles, such as the photon, and fermionic fi elds are those associated with spin-½ particles, such as electrons and protons. It further requires that the positive zero-point energy contribution associated with each bosonic fi eld is exactly canceled by the negative zero-point energy contribution associated with the corresponding fermionic fi eld.
Consequently, it requires that the total zero-point energies associated with the various fi elds of physics sum to zero, resulting in a net contribution of zero to the total cosmological constant. This solution faces a major diffi culty: even if supersymmetry exists, it is presently a broken symmetry and thus cannot solve the cosmological constant problem. As astrophysicist John Peacock notes, “supersymmetry, if it exists at all, is clearly a broken symmetry at present day energies; there is no natural way of achieving this breaking while retaining the attractive consequence of a zero cosmological constant, and so the Λ problem remains as puzzling as ever” (1999, p. 268).
Further, even if some other symmetry could be discovered that would force the contributions of the bosonic or fermionic fi elds to cancel each other out, the fi rst two contributions to the cosmological constant mentioned earlier would remain – that is, those arising from the Higgs fi eld and the infl aton fi eld. In order to get a zero cosmological constant, one would have to postulate some law, symmetry, or other mechanism that forced the sum of all contributions to the cosmological constant to be zero. In order to get this suggestion to work, physicists would have to either (a) abandon infl ationary cosmology, which requires that the effective cosmological constant be initially very large and then fall off to near zero, or (b) invoke some special law, symmetry, or “mechanism” that selectively requires that the cosmological constant be zero at the end of the infl ationary period. If options (a) and 14. The assumption that Λeff > 0 is regarded by most physicists as the best way of accounting for the evidence, based on redshifted light from distant supernovae, that the universe is accelerating. An effective cosmological constant is not the only way of explaining the cosmic acceleration, however. Olga Mena and José Santiago (2006) have recently developed a modifi cation of Einstein’s theory of gravity that explains the acceleration of the universe without appealing to an effective cosmological constant. Their model, however, still must assume some nonmaterial substance in addition to normal matter. As stated in their abstract, “the inverse-curvature gravity models considered cannot explain the dynamics of the Universe just with a baryonic matter component.” If a successful non- ad hoc and non-fi ne-tuned modifi cation of General Relativity could be developed that would account for the acceleration of the universe, it could serve as an alternative to steps (2) and (3) in an attempt to avoid the fi ne-tuning of the cosmological constant.
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(b) are both rejected, one will be left with the fi ne-tuning problem generated by a large effective cosmological constant required for infl ation that must drop off to near zero after infl ation in order for life to exist.
Further, supposing that option (a) or (b) are chosen, steps (2) and (3) are still required to account for the small, nonzero effective cosmological constant today. In typical models of quintessence, Λq “tracks” the matter and radiation density of the universe – that is, Λq is some function of these densities. One problem here is that unless the function is both natural and simple, without any adjustable parameters needed to make Λq < Λmax, the problem of fi ne-tuning will simply re-arise: if that function is not simple or natural, or such a parameter is needed, then the question will arise as to why that function or parameter is such that the value of the effective cosmological constant is within the life-permitting range instead of falling outside the life-permitting range. So far, no such natural function has been found, and it is widely argued that current models of quintessence require fi ne-tuning, especially when combined with infl ationary cosmology.15 Further, quintessential energy must have special characteristics to act as an effective cosmological constant. As noted by physicists Robert R. Caldwell and Paul J. Steinhardt (2000), “The simplest model proposes that the quintessence is a quantum fi eld with a very long wavelength, approximately the size of the observable universe.” The long wavelength of the fi eld means that its energy is dominated by potential energy, which in turn allows for it to act as an effective cosmological constant.
In sum, it is conceivable that by postulating the right set of laws–symmetries–
mechanisms, physicists will be able to explain the fi ne-tuning of the effective cosmological constant in a non- ad hoc way. Nonetheless, two points should be made. First, any such explanation will require the hypothesis of just the right set of laws. At best, this will merely transfer the fi ne-tuning of the cosmological constant to that of the laws of nature; even if those laws of nature are deemed “natural,” one would still have to have the right set of laws to eliminate the fi ne-tuning of the cosmological constant. Consequently, other than eliminating the ability to quantify the degree of fi ne-tuning, it is unclear how much such a move undercuts the need for some anthropic explanation. Second, it is unclear at what point we should continue searching for such an explanation in terms of physical laws when no plausible candidates have been forthcoming in the last 20
years. Atheists such as Stenger claim that we should continue searching until we can be absolutely sure that no scientifi c explanation can be found. In Section 2.5.2, where I consider the “God of the gaps” objection, I argue that such a requirement begs the question against T.
Finally, one way of avoiding the fi ne-tuning of the cosmological constant is to claim that one day the current framework of modern physics will be superseded in such a way as to avoid the fi ne-tuning problem. For example, one could claim that Einstein’s theory of General Relativity will be superseded by a future theory that retains the verifi ed predictions of General Relativity but does not predict that vacuum energy will cause space to 15. For example, physicists Christopher Kolda and David H. Lyth note that “an alternative to a cosmological constant is quintessence, defi ned as a slowly-varying scalar fi eld potential V(ϕ). . . . In contrast with ordinary infl ation, quintessence seems to require extreme fi ne tuning of the potential V(ϕ″)” (1999, abstract). Further, as physicist Gabriela Barenboim notes, models that combine infl ation and quintessence “require signifi cant ad hoc tuning to simultaneously produce the features of infl ation and quintessence” (2006, p. 1).
expand.16 Or one could claim that this prediction is an artifact of General Relativity that is not to be taken realistically, in analogy to how most physicists reject waves that travel backward in time, even though such waves are one mathematically legitimate solution to the equations of electromagnetism. Such moves, however, will involve giving up infl ationary cosmology or radically reformulating it (since it depends on this prediction of General Relativity), and they would present diffi culties for explaining the compelling evidence that the expansion of the universe is accelerating. Further, such moves clearly will not work for the fi ne-tuning of other constants, since many of them depend on facts so basic that they certainly will not be superseded. For example, the fi ne-tuning of the strength of gravity, as discussed in Section 2.3.2, only depends on the fact that bodies with masses typical of planets and stars attract each other with a force approximately given by Newton’s law, and that if the gravitational pull of a planet is too large, most organisms would be crushed.
Thus, as a general strategy, this way circumventing the fi ne-tuning of the cosmological constant is of limited value.
2.4. Initial conditions of the universe
One other fundamental type of fi ne-tuning should be mentioned, that of the initial conditions of the universe. This refers to the fact that the initial distribution of mass-energy – as measured by entropy – must fall within an exceedingly narrow range for life to occur. Some aspects of these initial conditions are expressed by various cosmic parameters, such as the mass density of the early universe, the strength of the explosion of the Big Bang, the strength of the density perturbations that led to star formation, the ratio of radiation density to the density of normal matter, and the like. Various arguments have been made that each of these must be fi ne-tuned for life to occur (see e.g. Rees 2000; Davies 1982, chap. 4). Instead of focusing on these individual cases of fi ne-tuning, I shall focus on what is arguably the most outstanding special initial condition of our universe: its low entropy.
According to Roger Penrose, one of Britain’s leading theoretical physicists, “In order to produce a universe resembling the one in which we live, the Creator would have to aim for an absurdly tiny volume of the phase space of possible universes” (Penrose 1989, p. 343).
How tiny is this volume? According to Penrose, if we let x = 10123, the volume of phase space would be about 1/10x of the entire volume (1989, p. 343). This is vastly smaller than the ratio of the volume of a proton – which is about 10−45 m3 – to the entire volume of the visible universe, which is approximately 1084 m3. Thus, this precision is much, much greater than the precision that would be required to hit an individual proton if the entire visible universe were a dartboard! Others have calculated the volume to be zero (Kiessling 2001).
Now phase space is the space that physicists use to measure the various possible confi gurations of mass-energy of a system. For a system of particles in classical mechanics, this phase space consists of a space whose coordinates are the positions and momenta (i.e. mass × velocity) of the particles, or any other so-called “conjugate” pair of position and momenta variables within the Hamiltonian formulation of mechanics. Consistency requires that any probability measure over this phase space remain invariant regardless of which conjugate positions and momenta are chosen; further, consistency requires 16. Apart from General Relativity, the absolute value of the energy has no physical consequences, only the relative differences in energy from one space-time point to another.
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that the measure of a volume V(t0) of phase space at time t0 be the same as the measure that this volume evolves into at time t, V( t), given that the laws of physics are time-reversal invariant – that is, that they hold in the reverse time direction.17 One measure that meets this condition is the standard “equiprobability measure” in which the regions of phase space are assigned a probability corresponding to their volume given by their position and momenta (or conjugate position and momenta) coordinates. Moreover, if an additional assumption is made – that the system is ergodic – it is the only measure that meets this condition.18 This measure is called the standard measure of statistical mechanics, and forms the foundation for all the predictions of classical statistical mechanics. A related probability measure – an equiprobability distribution over the eigenstates of any quantum mechanical observable – forms the basis of quantum statistical mechanics.
Statistical mechanics could be thought of as the third main branch of physics, besides the theory of relativity and quantum theory, and has been enormously successful. Under the orthodox view presented in physics texts and widely accepted among philosophers of physics, it is claimed to explain the laws of thermodynamics, such as the second law, which holds that the entropy of a system will increase towards its maximum with overwhelming probability.
Applying this measure to the initial state of the universe, we could then say that under the standard measure, it is enormously improbable, having a probability equal to the minute portion of phase space compatible with it. Indeed, in discussions of the issue, it is typically assumed that this state is enormously improbable. The probability here is not the probability of the particles’ (or fi elds’) being in the exact microstate that they were in; that is always zero. Rather, it is the state specifi ed by the requirement that the entropy be low enough for the occurrence of stars and ultimately life. An infi nite number of microstates meet this requirement, but they all must be in that tiny region of phase space that Penrose mentions. Finally, it is important to note that the standard measure of statistical mechanics must imply a corresponding epistemic probability measure. The reason is that statistical mechanics is supposed to tell us what to expect a system’s behavior to be. For instance, the calculation that an increase in entropy for a physical system in the next 5 minutes is enormously improbable leads us to be almost certain that it will not occur – that is, it generates a very low epistemic probability for its occurrence. Thus, applying the standard measure to the initial condition of our universe implies that it has an enormously low unconditional epistemic probability of occurring.
Many points could be disputed in the aforementioned argument and I cannot ade-
quately enter into the debate here. Rather, I shall just summarize three of the major 17. The reason that the statistical mechanics measures are assumed to be time-invariant is easy to see, given that the laws of physics are deterministic and time-reversal invariant. Let C(t0) be some class of possible initial states of a system. Each member m(t0) of C(t0) will evolve into some particular state m(t) at time t. Call C(t) the class of all such states m(t). If the laws of physics are time reversal invariant, any state m*(t) ∈ C(t) could have come from only one state m*(t0) ∈ C(t0) at time t0. Thus, there is a one-to-one correspondence between microstates in C(t0) and microstates in C(t). Consequently, if the probability of some system having a microstate in a class C(t0) is x%, then it must be the case that the probability of the system having a microstate in the corresponding class C(t) of states at time t is x%.
18. An ergodic system is one in which in the limit as time goes to infi nity, the average proportion of time a system spends in a given region of phase space with fi nite volume is the same as the standard equiprobability measure for that region.
ways to avoid assigning an extremely low epistemic probability to the initial state of the universe. First, as suggested by David Albert (2000, pp. 150–62), one could attempt to ground the standard probability measure of statistical mechanics in postulated indeterministic quantum processes that have just the right probability distribution to yield this measure. As Albert (2000, p. 160–1) recognizes, such a procedure would still require the postulation of an enormously special, low-entropy macroscopic state at the beginning of the universe but would not require that one postulate a probability measure over the states of its phase space. Second, one could simply restrict all statements of probability in statistical mechanics to statements of conditional probability, where the statement being conditioned on is that that the universe started out in this special macro-state of exceedingly low entropy. Then one could treat the measure over phase space as applying only to those scattered points in phase space consistent with this initial assumption. Doing this would recover all the predictions of statistical mechanics, although it might seem an arbitrary restriction imposed to avoid treating the initial state as improbable.
Third, one could point out, as John Earman (2006) has, that no one has been able to develop an even near-adequate mathematical measure for the degrees of freedom of the gravitational fi eld, which is thought to play an essential role in the low entropy of the initial state of the universe.
In light of these responses, how should we view the purported improbability of the initial state? First, we can point out that the improbability assigned by the standard measure is the same as the one that we should assign using the method outlined in Sections 3.3.2
and 3.3.3, in which we argue that scientifi c confi rmation requires that we place an epistemic equiprobability measure over the natural variables in a theory. In statistical mechanics these natural variables turn out to be the position and momenta, or other conjugate variables, used to construct phase space. Thus, it demonstrates a certain consistency in the method we proposed for arriving at epistemic probabilities. Second, all these responses admit that the initial state is in some sense enormously special in some way, while denying the degree of “specialness” can be quantifi ed or physically explained. This leaves us with a strong qualitative, although nonquantifi able, form of fi ne-tuning.
2.5. Stenger’s objections
As mentioned when we discussed the fi ne-tuning of the cosmological constant, in the last 15 years Victor Stenger has emerged as one of the leading critics of the evidence for fi ne-tuning. In the next two subsections, we shall look at two of his objections.
2.5.1. Stenger’s “Monkey God” objection
One major way in which Stenger has attempted to cast skepticism on fi ne-tuning arguments is by constructing a computer program that shows that random selections of the constants of physics generally produce viable, life-permitting stars. He calls his computer program “Monkey God.” Based on his program, Stenger concludes that:
No basis exists for assuming that a random universe would not have some kind of life. Calculations of the properties of universes having different physical constants than ours indicate that long-lived stars are not unusual, and thus most universes should have time for complex systems of some type to evolve. (2000, p. 50)
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Stenger calculates the lifetime of stars using the equation ts = (α2/αG) (mp/me)2 h
¯ /(mpc2)−1,
where α is the dimensionless electromagnetic interaction strength, αG is the dimensionless gravitational binding energy, mp is the mass of the proton, me is the mass of the electron, h
¯ is Plank’s constant divided by 2π, and c is the speed of light.
Using this equation and a program that randomly selects values for the relevant parameters in the aforementioned equation, Stenger concludes that long-lived stars are not unusual among these randomly selected universes and takes this to count as evidence against claims of fi ne-tuning. The fi rst criticism of his approach is that he does not address the question of whether these universes would have other life-inhibiting features relative to ours. For example, if one decreases the strength of the strong nuclear force by more than 50 percent (while keeping the electromagnetic force constant), carbon becomes unstable, and with a slightly greater decrease, no atoms with atomic number greater than hydrogen can exist (Barrow & Tipler 1986, pp. 326–7). This would make it virtually impossible for complex life forms to evolve. That Stenger ignores these other life-inhibiting features is clear from his equation for the lifetime of a star (which is unaffected by changes in the strong nuclear force, since none of the parameters he uses depends on this strength), and is also obvious from what he says elsewhere regarding his “Monkey God”
calculations:
I fi nd that long-lived stars, which could make life more likely, will occur over a wide range of these parameters. . . . For example, if we take the electron and proton masses to be equal to their values in our universe, an electromagnetic force any stronger than its value in our universe will give a stellar lifetime of more than 680 million years. The strength of the strong interaction does not enter into this calculation. (Stenger 2004, pp. 179–80)
Obviously, if we increased the electromagnetic force by very much while keeping the strong interaction the same, the nuclei of atoms other than hydrogen would break apart due to the increased electromagnetic repulsion of the protons in the nuclei. In this case, there could be no nuclear fusion in stars, and hence no stars.
Second, the equation he uses is based on a simple star model of stellar evolution. The equation does not take into account the complexities of a stellar evolution, such as whether the energy transport from the center of the star to the surface is by convection or radiative diffusion. More importantly, it assumes that the star is made mostly of hydrogen, which would not be the case if the strong force were increased beyond a small amount (see Collins 2003, p. 192 and references therein); further, it does not take into account the effects on star stability of quantum degeneracy, which require much more sophisticated codes to take into account. No simple equation could incorporate these sorts of complexities. As I have shown elsewhere (Collins 2003, pp. 192–3), using a simple star model, one can increase the strength of gravity a million- or a billionfold, and still obtain stable, long-lived stars with around the same surface temperature as our Sun. When one takes into account quantum degeneracy effects, however, one can only increase the strength of gravity by around a thousandfold before signifi cantly decreasing the lifetime of stars (Collins 2003, pp. 193–4).
Presumably, if one also changed one of the other constants, one could increase the strength of gravity by more than 3,000-fold and still obtain a stable, long-lived star, since it would change when electron degeneracy kicks in. In sum, life-prohibiting effects related to stellar lifetimes and stability only come to light when one begins to consider the complexity of the physics involved in stellar evolution, something Stenger has not done.
2.5.2. Stenger’s “God of the gaps” objection
Another common objection to the fi ne-tuning argument is that it is a variation of the “God of the gaps” argument, and so it should be rejected. Victor Stenger raises this objection with regard to the fi ne-tuning of the cosmological constant. According to Stenger: While quintessence may not turn out to provide the correct explanation for the cosmological constant problem, it demonstrates, if nothing else, that science is always hard at work trying to solve its puzzles within a materialistic framework. The assertion that God can be seen by virtue of his acts of cosmological fi ne-tuning, like intelligent design and earlier versions of the argument from design, is nothing more than another variation on the disreputable God-of-the gaps argument. These rely on the faint hope that scientists will never be able to fi nd a natural explanation for one or more of the puzzles that currently have them scratching their heads and therefore will have to insert God as the explanation. As long as science can provide plausible scenarios for a fully material universe, even if those scenarios cannot be currently tested they are suffi cient to refute the God of the gaps. (2004, p. 182)
Elsewhere, Stenger claims that one would be justifi ed in invoking God only if “the phenomenon in question is not only currently scientifi cally inexplicable but can be shown to forever defy natural description” (2007, pp. 13–4). As he recognizes, this requirement of proof is exceptionally strong. Although he qualifi es his assertion regarding God as a scientifi c hypothesis, the question that arises is the level of proof that we need regarding the nonexistence of a plausible scientifi c explanation before we are justifi ed in invoking God as an explanation of the fi ne-tuning, regardless of whether it is considered a scientifi c or a metaphysical explanation.
To answer this latter question, we must consider the reasons for thinking that a “God of the gaps” sort of explanation is in principle something to be avoided. The reasons partly depend on whether one is a theist or an atheist; and if one is a theist, it will depend on how one understands the nature of divine action. Many theists will claim that ultimately we should avoid a God of the gaps explanation because it is bad theology. According to these theists, God would be greater if God created a material order that could function on its own without God’s needing to intervene and fi ll various gaps. If these theists are correct, then for theological reasons one should strenuously avoid appealing to divine intervention in the natural order to explain phenomena that science has not yet explained and instead trust that God has created a material world with its own integrity. Such theological reasons, however, will not apply to the structure of the cosmos itself – its basic laws, its initial conditions, and the values of its constants – since these do not require any intervention in the natural order. Other theists, such as intelligent design theorists, will be even more lenient concerning those cases in which it is appropriate to invoke God as an explanation.
Of course, atheists who are fully committed to a naturalistic account of the cosmos will always claim that it is illegitimate to appeal to God, since God does not exist. In order for the God of the gaps objection to avoid begging the question against the theist, however, it has to be framed in such a way as to carry force even on theistic assumptions. Such an argument does carry force, at least on the assumption of many theists, when God or some other transcendent designer is invoked to explain, for instance, seemingly irreducibly complex biological systems, since in such cases it implies that nature lacks suffi cient integrity to produce the systems on its own. Stenger and others have not shown that it carries any non-question-begging force for the case of the structure of the universe as a whole.
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One might object to this response by claiming that the history of science provides independent grounds for rejecting any appeal to God to fi ll in the apparent gaps left by science. The failure of such appeals, however, can be explained as well by the theist as the naturalist: for example, many theists would claim that Newton’s famous invocation of God to keep the planetary orbits stable implies a less than satisfactory picture of a constantly intervening God. The key question is how one inductively extrapolates from these historical incidences, and that all depends on one’s background assumptions – that is, whether one is an atheist or a theist, and what kind of theist one is. In themselves, these incidences can tell us nothing about whether we can be justifi ed in appealing to God for explaining the fi ne-tuning.
But what about the scientifi c strictures of methodological naturalism? These would be relevant only if the appeal to God were considered as a scientifi c explanation, something that I am not assuming. Rather, God should be considered a philosophical or metaphysical explanation of LPU. So where does this leave us with regard to the burden of proof? The advocate of the fi ne-tuning argument will only need to argue that it is unlikely that all the cases can be given a natural explanation that removes their epistemic improbability without transferring that improbability up one level. And as argued in Sections 6.3.1 and 7.2, even if the fi ne-tuning of the constants of physics can be explained in terms of some set of deeper physical laws, as hypothesized by the so-called “theory of everything” or by an infl ationary multiverse, this would simply transfer the improbability up one level to the existence of these deeper laws.
2.6. Conclusion
There are many other cases of fi ne-tuning that I have not discussed, such as those extensively discussed by biochemist Michael Denton (1998). These latter consist of various higher-level features of the natural world, such as the many unique properties of carbon, oxygen, water, and the electromagnetic spectrum, that appear optimally adjusted for the existence of complex biochemical systems (Denton 1988, chaps 3–6, pp. 19–140). Presumably, these higher-level features of the universe are ultimately grounded in the laws, constants, and initial conditions of the universe. Nonetheless, they provide additional evidence that the fundamental structure of the universe is fi ne-tuned for life.
As illustrated by the case of Victor Stenger discussed earlier (Section 2.5), it should be pointed out that some physicists and scientists have been skeptical of some of the prominent cases of fi ne-tuning in the literature. As I have shown in detail elsewhere, in some cases this skepticism is warranted, but in other cases the physical arguments offered for fi ne-tuning are solid (see Collins 2003). Nonetheless, even if there are no cases of fi ne-tuning that are suffi ciently established to be beyond doubt, the argument would still have signifi -
cant force. As philosopher John Leslie has pointed out, “clues heaped upon clues can constitute weighty evidence despite doubts about each element in the pile” (1988, p. 300). This is especially true given that the clues in this case fall into a variety of distinct types – there are not only three distinct fundamental types of fi ne-tuning, but there are many distinct cases under each type. The only plausible response that a skeptic could give to the multitude of different cases of fi ne-tuning is to fi nd one or two overarching reasons that would undercut almost all the cases of fi ne-tuning in a single stroke. Given the diversity of the cases of fi ne-tuning, it is very unlikely that this will happen. In any case, in Section 7.2, I will address one such attempt, an attempt I call the “more fundamental law” objection, according to
which there might be some fundamental law (or principle) that entails all the cases of fi ne-tuning.
3. Epistemic Probability
3.1. The need for epistemic probability
According to atheist Keith Parsons:
If atheism is correct, if the universe and its laws are all that is or ever has been, how can it be said that the universe, with all of its ‘fi nely tuned’ features, is in any relevant sense probable or improbable? Ex Hypothesi there are no antecedent conditions that could determine such a probability. Hence, if the universe is the ultimate brute fact, it is neither likely nor unlikely, probable nor improbable; it simply is.
Further, even if the universe were somehow improbable, it is hard to see on the hypothesis of atheism how we could ever know this. If we were in the position to witness the birth of many worlds – some designed, some undesigned – then we might be in a position to say of any particular world that it had such-and-such a probability of existing undesigned. But we simply are not in such a position. We have absolutely no empirical basis for assigning probabilities to ultimate facts. (1990, p. 182)
Although commonly raised, Parson’s objection is deeply mistaken. It fails to recognize a common, nonstatistical kind of probability that some philosophers have called epistemic probability and others have called inductive probability (e.g. Swinburne 2001, p. 62).19 As Ian Hacking notes in his excellent study of the history of probability theory, the idea of probability was Janus-faced from its emergence in seventeenth-century Europe, with one side being the notion of statistical probability and the other side being the notion of epistemic probability:
On the one side it [the conception of probability] is statistical, concerning itself with stochastic laws or chance processes. On the other side it is epistemological, dedicated to assessing reasonable degrees of belief in propositions quite devoid of statistical background. (Hacking 1975, p. 12) So, for instance, when people say that the Thesis of Common Ancestry is probably true given the fossil and genetic evidence we currently have, they are clearly not talking about statistical probability, since this thesis is about a unique event in Earth’s history. The same holds for any claim about the probable truth (or “empirical adequacy”) of a scientifi c theory. In his treatise, A Treatise on Probability (1921), John Maynard Keynes further developed this conception, and there have been several recent attempts to provide a more precise account of this sort of probability (e.g. Swinburne 2001, chaps. 3 and 4; Plantinga 1993, chap. 9).
In conjunction with the Likelihood Principle, this sort of probability is extensively used in scientifi c confi rmation. Consider, for example, the arguments typically offered in favor 19. Swinburne (2001, p. 68), for instance, reserves the term “epistemic probability” for inductive probability that takes into account human cognitive limitations. I will use it more broadly to refer to what Swinburne calls “inductive probability.”
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of the Thesis of Common Ancestry, continental drift theory, and the atomic hypothesis.
The Thesis of Common Ancestry is commonly supported by claiming that a variety of features of the world – such as the structure of the tree of life – would not be improbable if this thesis is true, but would be very improbable under other contending, nonevolutionary hypotheses, such as special creation. Consider, for instance, the following quotation from evolutionary biologist and geneticist Edward Dodson, in which he summarizes the case for evolution, understood as the Thesis of Common Ancestry:
All [pieces of evidence] concur in suggesting evolution with varying degrees of cogency, but most can be explained on other bases, albeit with some damage to the law of parsimony. The strongest evidence for evolution is the concurrence of so many independent probabilities. That such different disciplines as biochemistry and comparative anatomy, genetics and biogeogra-phy should all point toward the same conclusion is very diffi cult to attribute to coincidence.
(1984, p. 68; italics added)
Similar lines of reasoning are given for accepting continental drift theory. For example, the similarity between the animal and plant life on Africa and South America millions of years ago was considered to provide signifi cant support for continental drift theory. Why?
Because it was judged very unlikely that this similarity would exist if continental drift theory were false, but not if it were true.
Finally, consider the use of epistemic probability in the confi rmation of atomic theory.
According to Wesley Salmon (1984, pp. 219–20), what fi nally convinced virtually all physical scientists by 1912 of the atomic hypothesis was the agreement of at least 13
independent determinations of Avogadro’s number based on the assumption that atomic theory was correct.20 For example, one method of determining Avogadro’s number is through observations of Brownian motion, that is, the random motion of very small particles suspended in a liquid, a motion that was postulated to be caused by the unequal, random impact of the molecules in the liquid. From this motion and the kinetic
theory of heat, one can calculate what the mass of each molecule must be in order to account for the observed motion, and then using that value one can obtain Avogadro’s number.
The scientists reasoned that if atomic theory were false, then such an agreement between thirteen different determinations of Avogadro’s number would be exceedingly epistemically improbable – in Salmon’s words, an “utterly astonishing coincidence” (1984, p. 220). Indeed, if scientists had not judged the agreement to be exceedingly improbable if atomic theory were false, it is diffi cult to see why they would take it to count as strong evidence in its favor. On the other hand, the scientists reasoned, if atomic theory were true, such an agreement would be expected. Thus, by implicitly using the Likelihood Principle, they reasoned that these independent determinations of Avogadro’s number strongly confi rmed atomic theory.
It should be noted that one could not avoid this sort of reasoning simply by rejecting scientifi c realism, since even though antirealists reject the truth or approximate truth of certain types of well-confi rmed hypotheses, they still accept them as being reliable bases for future explanations and predictions – that is, in Bas van Fraassen’s (1980) terminology, 20. Avogadro’s number = 6.02252 × 1023. Avogadro’s number is defi ned as the number of atoms in 12 grams of carbon 12 and by defi nition is equal to the number of elementary entities in one mole of any substance.
they accept them as being “empirically adequate.” Consequently, instead of interpreting the confi rming evidence as evidence for a hypothesis’ truth, they accept it as evidence for the hypothesis’ empirical adequacy. This means that insofar as realists need to appeal to epistemic probabilities to support the approximate truth of a theory, antirealists will need to appeal to those same probabilities in support of a theory’s empirical adequacy – for example, antirealists would need to claim that it is highly improbable for the determinations of Avogadro’s number to agree if atomic theory were not empirically adequate.
Since some of the probabilities in the aforementioned examples involve singular, non-repeatable states of affairs, they are not based on statistical pro babilities, nor arguably other non-epistemic probabilities. This is especially evident for the probabilities involved in the confi rmation of atomic theory since some of them involve claims about probabilities conditioned on the underlying structure and laws of the universe being different – e.g. atoms not existing. Hence, they are not based on actual physical propensities, relative frequencies, or theoretical models of the universe’s operation. They therefore cannot be grounded in theoretical, statistical, or physical probabilities. Similar things can be said about many other related types of confi rmation in science, such as the confi rmation of quantum electrody-namics (QED) by its extraordinarily precise prediction of the gyromagnetic moment of the electron, which we shall discuss later in this chapter. Such cases, I contend, establish the widespread use of purely epistemic pro babilities in scientifi c confi rmation that are neither grounded in other types of probability nor in experience – e.g. the probabilities invoked in atomic theory clearly are not grounded in experience, since nothing like such an agreement had ever occurred before. We shall return to this in Sections 3.3.2 and 3.3.3
when I discuss the Principle of Indifference.
3.2. An account of epistemic probability
Having established the need for epistemic probability, we now turn to developing an account of it. Accounts of epistemic probability range from the so-called subjective theory to the logical theory. According to the subjective theory, epistemic probability amounts to nothing more than our purely subjective degree of belief in a claim with the only restriction on the rationality of our degrees of belief is that they obey the probability calculus and conform to our own personal inductive standards. In contrast, according to the logical theory, epistemic probability refers to some human-mind independent relation analogous to that of logical entailment. Despite its popularity, I shall say nothing more here about subjective theory, other than that it seems to lead to an extreme form of epistemic relativism. The reason is that it does not distinguish between rational and irrational inductive criteria. Given the right inductive criteria almost any set of beliefs can be made to cohere with the probability calculus – for example, the belief that the Sun and stars revolve around the Earth can be made to cohere with all the current evidence we have to the contrary. (For a further critique, see Swinburne (2001, pp. 73f) and Plantinga (1993, pp. 143f)).
On the other hand, at least two major problems confront the purely logical theory of epistemic probability. First, it is doubtful that we need to hypothesize a metaphysics of human-mind independent relations of logical probability between propositions to ground the rationality of all of our statements involving epistemic probability. As Keynes (1921, pp. 4, 32) pointed out, all we need is the existence of relations of rational support or expectation that are independent of merely personal or cultural beliefs and standards. Consequently, allowing for the relations of epistemic probability to be dependent on the contingent
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construction of human cognitive faculties fi ts much better with my overall approach of trying to stay as close as possible to the principles of reasoning that are part of everyday and scientifi c practice (see Section 1.1).
Second, purely logical probability would guide the expectations only of logically omniscient agents – that is, agents that could see all logical relations between propositions, including the relations of logical probability. Humans, however, are limited to a partial grasp of such relations, which is refl ected in the relevant usage of epistemic probability in science. For example, as Swinburne acknowledges, based on the current mathematical evidence, Goldbach’s conjecture (that every even number greater than two is the sum of two prime numbers) is probably true, but not certainly true. That is, the current evidence
– such as that Goldbach’s conjecture has been shown to be true for the fi rst trillion numbers and was claimed to be proven by some otherwise truthful mathematician – supports this conjecture, but not enough to justify our being certain that it is true. Since it is a mathematical claim, however, Goldbach’s conjecture is either necessarily true, or it is necessarily false, and thus its logical probability is either one or zero. The epistemic probability being invoked here, therefore, is not purely logical probability. Even if one does not agree that this sort of epistemic probability arises in mathematics, nonetheless it is clear that when judging the evidential support for a scientifi c theory, we are not aware of all the relevant logical relations between the evidence and the theory. Keynes, who made the degree of logical/epistemological probability of one proposition on another relative to human powers, recognized this issue. According to him:
If we do not take this view of probability, if we do not limit it in this way and make it, to this extent, relative to human powers, we are altogether adrift in the unknown; for we cannot ever know what degree of probability would be justifi ed by the perception of logical relations which we are, and must always be, incapable of comprehending. (1921, p. 33)
Following Swinburne, one could still attempt to take logical probability as the primary kind of epistemic (or what he calls inductive) probability, and then attempt to accommodate human limitations. The problem with this approach is that in order to use logical probability to make statements about the rational degree of credence one ought to place in a theory, or the amount by which we should take a body of evidence to increase our confi dence in a theory, one would need some account of how to translate degrees of logical probability to rational degrees of credence for beings subject to our limitations. Consequently, one would still need an account of another more human-centered type of epistemic probability that is relative to human cognitive powers to grasp these human perceptions of logical probability; in itself, logical probability only stipulates what the rational degrees of belief of a logically omniscient agent ought to be, not that of a mere human being. As far as I know, Swinburne does not provide any account that links the two together.
I call the conception of epistemic probability that is grounded in our perceptions of logical relations between propositions, episto-logical probability. In contrast to the epistological account of epistemic probability, Alvin Plantinga (1993, chap. 9, pp. 159–75) has developed an account in which the relations of probability are grounded in the contingent construction of our cognitive faculties, which in many cases need not involve perceptions of logical relations. In his account, for instance, we think that the future will resemble the past, since those aspects of properly functioning human cognitive faculties that are aimed
at truth normally incline us to believe that the future will resemble the past. Similarly, we accept simpler theories over complex ones for the same reason. Because of their stress on the contingent construction of our cognitive faculties, I call views such as Plantinga’s noetic conceptions of epistemic probability.
The account of epistemic probability that I favor is one in which epistemic probabilities are grounded in some combination of both the contingent construction of our cognitive faculties and the perceived logical relations among propositions. For the purposes of this chapter, I will leave it as an open question which of these conceptions of epistemic probability – the logical, the epistological, the noetic, or some combination thereof – is ultimately correct. A word, however, needs to be said about a problem with Plantinga’s account of epistemic probability that is relevant for our purposes. Plantinga defi nes the conditional epistemic probability of a Proposition A on a Proposition B as follows:
Plantinga’s Defi nition of Conditional Epistemic Probability: P(A|B) = <x,y> iff <x,y> is the smallest interval which contains all the intervals which represent the degree to which a rational human being S (for whom the conditions necessary for warrant hold) could believe A if she believed B, had no undercutting defeater for A, had no other source of warrant either for A or for
-A, was aware that she believed B, and considered the evidential bearing of B on A. (1993, p. 169; italics added)
Plantinga’s account of conditional epistemic probability is a counterfactual account that defi nes epistemic probability in terms of the degree of warrant a rational human being would have in A if she believed B and had no other sources of warrant for A or –A. The italicized portion, which we shall call Plantinga’s condition of epistemic probability (CEP), primarily does the job of excluding contributions to our warrant which arise from our epistemic circumstances and other information besides B that is part of our background information k.
We cannot go into a detailed analysis of this account of conditional epistemic probability here. However, we shall consider one major objection that is relevant to the way in which we shall be using epistemic probability. As Bas van Fraassen has pointed out, Plantinga’s account does not account for those cases in which B could not be the sole source of warrant for A, an objection that Plantinga attempts to address (1993, p. 168–9). This problem arises in the case of the fi ne-tuning argument, since we claim that the epistemic probability of LPU is very small under NSU. Our own existence in a body, however, provides a source of warrant for LPU, and it is a source that we cannot eliminate without severely altering our cognitive faculties (or epistemic conditions) in a way that would undermine our rationality.
More recently, Richard Otte has given new teeth to the objection to Plantinga’s account raised by van Fraassen. Among other examples, Otte asks us to consider the following variation of one of Plantinga’s examples:
P (people are sometimes appeared to redly | I am appeared to redly)
According to Otte:
Intuitively this probability is 1; if I am appeared to redly, it must be the case that people are sometimes appeared to redly. But Plantinga claims that it is not possible for a rational person’s sole source of warrant for people are sometimes appeared to redly to be I am appeared to redly.
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Thus this probability is undefi ned according to CEP, even though it obviously has a value of 1. This example shows that CEP does not account for our intuitive notion of conditional epistemic probability. (2006, p. 87)
Otte locates the problem with Plantinga’s account in his use of counterfactuals, claiming that spelling out conditional epistemic probability in terms of counterfactuals is the wrong approach. Some sort of counterfactual element, however, is essential to any account of conditional epistemic probability if we are to connect degrees of conditional epistemic probability with actual rational degrees of belief, which we need to do if judgments of conditional probability are to serve as guides to life. This requirement, however, does not imply that we must provide a purely counterfactual analysis of conditional epistemic probability; all it demands is that counterfactuals will play some role in connecting conditional epistemic probability with rational degrees of belief.
Although I cannot develop this idea in detail here, I propose that conditional epistemic probability should be conceived as a relation between propositions that is in part determined by the contingent nature of our cognitive faculties. Through introspection, we have partial access to this relation. We typically determine the epistemic probability, P(A|B), of Proposition A on Proposition B – that is, the degree of rational inclination B should give us for A – by artifi cially creating in our own minds Plantinga’s conditions for CEP – that is, by “bracketing out” all other sources of warrant for A or –A, and any undercutting defeaters for A. Thus, for instance, suppose I see it raining outside but want to access the conditional epistemic probability of “it will rain today” (Proposition A) on Proposition B, where B is the conjunction of the claim that “www.weather.com has predicted a 25
percent chance of rain” and other background information, such as that www.weather.com is reliable weather predictor. In assessing this conditional probability, I block out all other sources of warrant for its raining today (such as seeing dark clouds on the horizon), except for Proposition B, and arrive at the correct conditional probability, P(A|B) = 0.25.
The fact that, for the cases when CEP applies, I come to know these counterfactual degrees of warrant by means of this “bracketing procedure” strongly suggests that epistemic probability should not be identifi ed with counterfactual degrees of belief. Rather, it should be considered a non-reducible relation of support or warrant existing between propositions that comes in degrees, is partially dependent on our cognitive faculties, and which we can know by introspection via the “bracketing procedure.” This relation in turn gives rise to the corresponding counterfactual degrees of warrant when CEP is met.
The fact that conditional epistemic probability should be considered such a relation existing between propositions that we determine by the “bracketing procedure” is supported by other examples. Consider, for instance, the conditional epistemic probability, P(A|B & k′), of the claim that “human beings exist today” (claim A) based on the claim that the asteroid that supposedly killed the dinosaurs missed planet Earth (claim B) and certain relevant scientifi c theories (k′) regarding the conditions necessary for the evolution of hominids. Given B and k′, I might judge it would be very unlikely that the dinosaurs would have become extinct, and hence very unlikely that humans would exist: that is, I would judge P(A|B & k′) << 1. The procedure I would go through is that of bracketing out all other sources of warrant for A except the relevant scientifi c theories, k′, and claim B, and then access the degree to which the remaining information warranted or supported A. For instance, I would bracket out all those everyday pieces of information that imply the existence of human beings. Since CEP cannot be met in this case, the existence of a conditional
epistemic probability in this case shows that identifying epistemic probability with counterfactual degrees of warrant (or belief or credence) cannot be right.
Part of the purpose of this section is to provide a theoretical underpinning for both the existence of conditional epistemic probability and the claim that P(A|B & k′) can exist for those cases in which the Proposition B & k′ cannot be the sole source of warrant for A.
Such a claim is crucial to the likelihood (and probabilistic tension) formulation of the fi ne-tuning argument, since LPU will be improbable only on background information k′
in which the information that embodied, conscious observers exist is subtracted out of our background information k (see Sections 4.3 and 4.4). Since all rational people believe that they are embodied, it is impossible for k′ & NSU to be the sole source of warrant for LPU.
Hence, Plantinga’s CEP cannot be met for P(LPU|k′ & NSU). Despite these theoretical underpinnings, some still might question whether there can exist relations of epistemic probability in those cases in which the antecedent (B & k′) cannot be the sole source of warrant for the consequent (A).
To further support the existence of epistemic probabilities in these cases, consider the sorts of cases, such as those mentioned earlier, in which scientifi c confi rmation appears to depend on claims that some state of affairs S – such as the agreement of the various methods of determining Avogadro’s number – is expected under a certain hypothesis h, but very epistemically improbable under the negation of that hypothesis, ~h. Suppose we discovered for one of these cases that S was also necessary for our own existence. It seems clear that such a discovery would not itself undermine the confi rmation argument in favor of the approximate truth (or empirical adequacy) of the theory. If it did, then we could undermine the support based on the Likelihood Principle for many theories of physics simply by discovering that the state of affairs S predicted by the theory – for example, the degree of bending of light around the Sun predicted by General Relativity – was necessary for embodied conscious life. This seems clearly wrong. Thus, there must be a probability for P(S|~h & k′) in these cases, where k′ is some appropriately chosen background information that does not implicitly or explicitly include the fact that humans exist. (If k′ included that humans exist, then P(S|~h & k′) = 1, destroying any likelihood confi rmation; see Section 4.3 for more discussion on choosing k′ in cases like this.)
As a specifi c example, consider QED’s precise prediction of the deviation from 2 of the gyromagnetic moment of the electron to nine signifi cant digits, as discussed in Section 3.3.3. In terms of the Likelihood Principle, the reason this prediction has been thought to signifi cantly confi rm QED is that such a precise, correct prediction seems very epistemically unlikely if QED is not approximately true (or at least empirically adequate), but it is epistemically likely if QED is true.21 Suppose we discovered that this precise magnitude of deviation was necessary for the evolution of life in our universe. It seems clear that this would not undermine the confi rmation that this precise prediction gives to QED.
Finally, since no rational person could doubt LPU, it will often be useful to use the following conceptual device to intuitively grasp the relations of conditional epistemic probability for LPU conditioned on NSU & k′ and conditioned on T & k′. The device is to 21. The importance of this example, and others like it, is that insofar as the confi rmation in question admits a likelihood reconstruction, it clearly involves epistemic probabilities that cannot be reduced to statistical or theoretical probabilities (since the predicted value has to do with the basic structure of our universe), and it is one in which we can plausibly conceive of it as having some anthropic signifi cance.
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imagine an unembodied alien observer with cognitive faculties structurally similar to our own in the relevant ways, and then ask the degrees of credence that such a being would have in LPU given that he or she believes in NSU & k′ or in T & k′. This device of the unembodied alien observer should remove any lingering doubts about the existence of a conditional epistemic probability on background information k′ that we could not have – for example, when k′ does not implicitly or explicitly include the existence of embodied conscious beings.
Given that such an alien is possible, it could have k′ as its background information and thus would not have the aforementioned problem regarding the existence of an epistemic probability for LPU on k′; hence, the existence of LPU could signifi cantly confi rm T over NSU
for that being. It seems clear that if we met such a being and if we discovered that LPU
confi rmed T over NSU for that being, then it should do so for us too.22
I believe the various arguments I have offered establish both the crucial role of epistemic probabilities in scientifi c confi rmation and their existence between some Propositions A and B & k′ in those cases in which B & k′ could never be the sole source of warrant (or justifi cation) for A. Our next question is how to determine P(A|B & k′).
3.3. Determining epistemic probability
3.3.1. Introduction
Now that we know what we mean by epistemic probability, it is time to consider how it is justifi ed. In science, many times epistemic probability is determined by an appeal to intuition, such as many of the epistemic probabilities considered in the last section –
for example, those arising in conjunction with the Thesis of Common Ancestry, continental drift theory, and atomic theory. These probabilities clearly were not justifi ed by an appeal to statistical improbability – for example, we have no statistics regarding the relative frequency of life on a planet having those features cited in favor of evolution either under the evolutionary hypothesis or under some nonevolutionary hypothesis. Indeed, these judgments of epistemic probability were never rigorously justifi ed in any way. Rather, after (we hope) doing their best job of looking at the evidence, scientists and laypersons made judgments of what kind of world we should expect under each hypothesis, and then they simply trusted these judgments. This sort of trust in our judgments of epistemic probability is a pervasive and indispensable feature of our intellectual life. It is these sorts of intuitive judgments, I contend, that ultimately ground the claim that, given the evidence for the fi rst type of fi ne-tuning we discussed in Section 2.2 – that of the laws of nature – it is very epistemically unlikely that such a universe would exist under NSU
& k′.
Of course, the more widely shared these judgments are by those who are relevantly informed, the more seriously we take them. In this regard, it should be noted that, given the fi ne-tuning data, the judgment that LPU is surprising under naturalism is widely shared by intelligent, informed individuals, as evidenced by the various attempts to account for 22. For a denial of the kind of claim that I make here – namely, that the confi rmation must be the same for the unembodied being and us – see Sober (2005, pp. 137–40). For a critique of Sober, see footnote in Section 7.5.
Even if Sober is correct, my other arguments still hold for thinking there could exist an epistemic probability of LPU on k′ & NSU and k′ & T. Further, these arguments show that confi rmation can still occur even when there is an observer selection effect, such as the aforementioned thought experiments in which we discovered that the data in support of atomic theory or QED had anthropic signifi cance.
it, such as the multiverse hypothesis. Of course, the skeptic might object that scientifi c theories are testable, whereas the theistic explanation is not. But why should testability matter with regard to the acceptability of our judgments of epistemic probability? After all, testability is about being able to fi nd evidence for or against a theory in the future, not about the present likelihood of the theory or the likelihood of some body of data’s being the case if the theory is false or not empirically adequate. Thus, I contend, the merely intuitive judgments of epistemic probability in the case of fi ne-tuning are on as solid ground as many of those accepted in science that cannot be rigorously justifi ed. It is dishonest, therefore, to accept one sort of inference without rigorous justifi cation but reject the other merely because it purportedly lacks such justifi cation. At any rate, we shall present such justifi cation for judgments of epistemic probability in the case of the fi ne-tuning of the constants of physics, regarding which I shall argue that we can go beyond a mere appeal to intuition. Instead, we can provide a solid, principled justifi cation based on what I shall call the restricted Principle of Indifference, which we shall discuss in the next two subsections.23
3.3.2. Restricted Principle of Indifference
According to the restricted Principle of Indifference, when we have no reason to prefer any one value of a variable p over another in some range R, we should assign equal epistemic probabilities to equal ranges of p that are in R, given that p constitutes a “natural variable.”
A variable is defi ned as “natural” if it occurs within the simplest formulation of the relevant area of physics. When there is a range of viable natural variables, then one can only legitimately speak of the range of possible probabilities, with the range being determined by probabilities spanned by the lower and upper bound of the probabilities determined by the various choices of natural variables.
Since the constants of physics used in the fi ne-tuning argument typically occur within the simplest formulation of the relevant physics, the constants themselves are natural variables. Thus, the restricted Principle of Indifference entails that we should assign epistemic probability in proportion to the width of the range of the constant we are considering. We shall use this fact in Section 5.1 to derive the claim that P(Lpc|NSU & k′) << 1, where Lpc is the claim that the value for some fi ne-tuned constant C falls within the life-permitting range.
To see why the restriction to a natural variable is needed, consider the case in which we are told that a factory produces cubes between 0 and 10 meters in length, but in which we are given no information about what lengths it produces. Using our aforementioned principle, we shall now calculate the epistemic probability of the cube being between 9 and 10
meters in length. Such a cube could be characterized either by its length, L, or its volume, V. If we characterize it by its length, then since the range [9,10] is one-tenth of the possible range of lengths of the cube, the probability would be 1/10. If, however, we characterize it by its volume, the ratio of the range of volumes is: [1,000 − 93]/1,000 = [1,000 − 729]/1,000
= 0.271, which yields almost three times the probability as for the case of using 23. A rigorous justifi cation of the epistemic improbability of the initial conditions of the universe is a little trickier, since it presupposes that one can apply the standard measure of statistical mechanics to the initial state of the universe, something John Earman and others have questioned (see Section 2.4). We cannot pursue this issue further in this chapter, however.
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length. Thus, the probability we obtain depends on what mathematically equivalent variable we use to characterize the situation.
In the case of the constants of physics, one can always fi nd some mathematically equivalen t way of writing the laws of physics in which Wr/WR is any arbitrarily selected value between zero and one. For example, one could write Newton’s law of gravity as F = U 100 m 1 m 2/ r 2, where U is the corresponding gravitational constant such that U 100 = G.
If the comparison range for the standard gravitational constant G were from 0 to 10100G0, and the life-permitting range were from 0 to 109 G0, that would translate to a comparison range for U of 0 to 10U0 and a life-permitting range of 0 to 1.2U0, since 10U0 = 10100G0
and 1.2U0 = 109G0. (Here G0 is the present value of G and U0 would be the corresponding present value of U.) Thus, using G as the gravitational constant, the ratio, Wr/WR, would be 109G0/10100G0 = 1/1091, and using U as the “gravitational constant,” it would be 1.2U0/10U0, or 0.12, a dramatic difference! Of course, F = U 100 m 1 m 2/ r 2 is not nearly as simple as F = Gm 1 m 2/ r 2, and thus the restricted Principle of Indifference would only apply when using G as one’s variable, not U.
Examples such as that of the cube mentioned have come to be known as the Bertrand Paradoxes (see e.g. Weatherford 1982, p. 56). Historically, this has been thought of as the fatal blow to the general applicability of the Principle of Indifference, except in those cases in which a natural variable can be determined by, for instance, symmetry considerations such as in statistical mechanics. In the next section, however, we shall see that for purposes of theory confi rmation, scientists often take those variables that occur in the simplest formulation of a theory as the natural variables. Thus, when there is a simplest formulation, or nontrivial class of such formulations, of the laws of physics, the restricted Principle of Indifference circumvents the Bertrand Paradoxes.
Several powerful general reasons can be offered in defense of the Principle of Indifference if it is restricted in the ways explained earlier. First, it has an extraordinarily wide range of applicability. As Roy Weatherford notes in his book, Philosophical Foundations of Probability Theory, “an astonishing number of extremely complex problems in probability theory have been solved, and usefully so, by calculations based entirely on the assumption of equiprobable alternatives [that is, the Principle of Indifference]” (1982, p. 35). Second, in certain everyday cases, the Principle of Indifference seems the only justifi cation we have for assigning probability. To illustrate, suppose that in the last 10 minutes a factory produced the fi rst 20-sided die ever produced (which would be a regular icosahedron). Further suppose that every side of the die is (macroscopically) perfectly symmetrical with every other side, except for each side having different numbers printed on it. (The die we are imagining is like a fair six-sided die except that it has 20 sides instead of six.) Now, we all immediately know that upon being rolled the probability of the die coming up on any given side is one in 20. Yet we do not know this directly from experience with 20-sided dice, since by hypothesis no one has yet rolled such dice to determine the relative frequency with which they come up on each side. Rather, it seems our only justifi cation for assigning this probability is the Principle of Indifference: that is, given that every side of the die is macroscopically symmetrical with every other side, we have no reason to believe that it will land on one side versus any other. Accordingly, we assign all outcomes an equal probability of one in 20.24
24. A full-scale defense of the restricted Principle of Indifference is beyond the scope of this chapter. See Schlesinger (1985, chap. 5) for a lengthy defense of the standard Principle of Indifference.
In the next section, I shall fi rst offer a powerful reason for epistemically privileging those variables that occur in the simplest overall formulation of the relevant domain of physics.
I shall then show how this reason offers a further, strong support for the restricted Principle of Indifference based on scientifi c practice.
3.3.3. Natural variable assumption
Typically, in scientifi c practice, precise and correct novel predictions are taken to signifi -
cantly confi rm a theory, with the degree of confi rmation increasing with the precision of the prediction. We shall argue, however, that the notion of the “precision” of a prediction makes sense only if one privileges certain variables – the ones that I shall call the natural variables. These are the variables that occur in the simplest overall expression of the laws of physics. Thus, epistemically privileging the natural variables as required by the restricted Principle of Indifference corresponds to the epistemic practice in certain areas of scientifi c confi rmation; if scientists did not privilege certain variables, they could not claim that highly precise predictions confi rm a theory signifi cantly more than imprecise predictions.
We begin our argument by considering only cases in which the predictions of a theory are accurate to within experimental error. In such cases, the known predictive precision will be equal to the experimental precision of the measured quantity. Our fundamental premise will then be that everything else being equal, the confi rmation that a prediction offers a theory increases with the known precision of a prediction.
The experimental precision of a measurement of a quantity is dependent on the experimental error. In standard scientifi c notation, the experimental value of a quantity is often expressed as V ± e, where e indicates that one’s measuring apparatus cannot distinguish between values that differ by more than e. More precisely, to say that the experimentally determined value is V ± e indicates that we have a certain set degree of confi dence – usually chosen to be 95 percent – that the actual value is within the interval [V + e, V − e]. So, for example, one might measure the weight of a person as 145.3 lb ± 0.1: that is, the experiment gives us a 95 percent confi dence that the person’s weight is within the interval [145.4 lb, 145.2 lb].
Scientists often speak of experimental accuracy/precision in terms of signifi cant digits.
Thus, in the above example, the precision is three signifi cant digits since one’s measuring apparatus cannot determine whether the person weighs 145.4 lb or 145.2 lb, and thus only three digits (i.e. 145) can be relied on to give one the value of the weight. Because sometimes zeroes can be “placeholder” digits that determine the order of magnitude of a quantity, these digits are not considered signifi cant. For example, the zeroes in 0.000841 are placeholder digits. To eliminate counting placeholder digits as signifi cant, one can simply express the measured value in terms of scientifi c notation, and then count the number of digits that are within the margin of error. Thus, a measurement of 0.000841 ± 0.000002
meters for the length of a piece of steel expressed in scientifi c notation becomes 8.41 ± 0.02 × 10−4 meters, which yields an accuracy of two signifi cant digits. This measure of precision will not work for those cases in which the measured value is zero, nor should it even be applied to those cases in which the measured value is less than the error.
A more precise way of thinking about this kind of precision is in terms of the ratio of the width, Wr, of the confi dence interval [V + e, V − e] to the value of V, with the restriction that V > e. Under this notion of precision, to say that the experimental value has a precision
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of δ means that Abs[Wr/V] < δ, where Abs denotes the absolute value of the quantity in brackets, and Wr denotes the width of the range [V + e, V − e] – that is, 2e. There is a rough correspondence between precision expressed in terms of this ratio and in terms of signifi -
cant digits: a precision of n signifi cant digits roughly corresponds to a ratio of one in 10n.
Thus, in our weight example, Wr = 2e = 0.2, and hence Wr/V = 0.2/145 ~ 1/1,000.
A more careful analysis would reveal that scientists only consider signifi cant digit (SD) precision as a guide to what I call WR precision, which is the ratio of the width, Wr, of the experimentally determined/predicted range for a variable to what is estimated to be the width of the expected range, WR, for the variable given the background information. The actual value, V, is then taken as a guide to the width of the theoretically possible range, and hence Wr/WR ~ Abs[e/V], where ~ means approximately. We shall return to this issue when we discuss QED, but for the purposes of this chapter, we are concerned only with showing that determining the degree of precision of a prediction – whether SD or WR precision –
depends on privileging the natural variable(s) as defi ned.
Finally, one might wonder why we cannot defi ne precision simply as the amount that, with a set degree of probability, the actual value could differ from the experimental value.
We could, but it would be a useless notion when it came to the question of experimental confi rmation. For example, how could we compare the confi rmatory value of a predictive precision of 1 kg with that of 1 μm? Or, is it really plausible to say that, for instance, a predictive precision of 20 signifi cant digits of the mass of the universe has less confi rmatory signifi cance than a prediction of one signifi cant digit of the mass of a hydrogen atom because the former is less accurate in terms of number of kilograms by which the actual value could differ from the predicted value?
We shall now argue that if either the degree of SD precision or the degree of WR precision is epistemically relevant, it follows that one must privilege variables that are closely tied with the natural variables. We shall start by showing that SD experimental precision depends on the variable one uses to represent the quantity being measured; consequently, in order to speak of precision in a nonrelative sense, one must specify the variable one is using to express the physical situation. To illustrate, consider the case of a cube discussed in the last section. The volume of a cube is equal to the third power of the length of its side: V = L3. Suppose we determine the length of the cube is 10 μm, to within a precision of 1 μm. Thus, expressed as a ratio, the SD precision is Abs[e/V] < 1/10, or one part in 10.
Roughly, this means that the length of the cube could be anywhere from 9 to 11 μm. In terms of volume, however, the cube can vary between 93 = 729 μm3 and 113 = 1,331 μm3.
This means that the experimental precision is (1,331–1,000)/1,000 ~ 1/3, or one part in three, if we take volume as our variable.
Now consider a theory that predicts that the length of the side of the cube is 10 μm.
This is equivalent to the theory predicting the volume of the cube to be 1,000 μm. In this case, the predicted value agrees with the experimental value to within experimental precision. If we ask what the known precision of the prediction is, however, we do not get a defi nite answer. If we consider the theory as predicting the length of the cube, we get one value for the known precision, whereas if we consider the theory as predicting the volume, we get another value for the precision. (Remember, since we are assuming that the theory predicts the correct value to within experimental error, the known predictive precision is equal to the experimental precision.) The moral here is that the known precision of a prediction depends on the mathematical variable – for example, L3 or L in the said example –
under which one is considering the prediction. Put differently, one can speak of precision of
an experimentally correct prediction only relative to the variable one uses to represent the predicted result. In analogy to Bertrand’s Cube Paradox for the Principle of Indifference, in the case of the aforementioned cube it seems that we have no a priori way of choosing between expressing the precision in terms of volume or in terms of length, since both seem equally natural. At best, all we can say is that the predicted precision is somewhere between that determined by using length to represent the experimental data and that determined by using volume to represent the experimental data.
For an illustration from actual physics, consider the case of QED’s astoundingly accurate prediction of the correction of the gyromagnetic ratio – called the g-factor – of the electron due to its self-interaction. QED predicted that, because of the self-interaction of the electron, the g-factor (gyromagnetic moment) of the electron differs from 2 by a small amount: g/2 = 1.001 159 652 38 ± 0.000 000 000 26. Very accurate experimental measurements yielded: g/2 = 1.001 159 652 ± 0.000 000 000 20. The precision of the said prediction of g/2 is one part in a billion.
Now determining the experimental value for g/2 is equivalent to determining the experimental value of some arbitrarily defi ned function U( g/2) of g/2, say U( g/2) = ( g/2)100.
Moreover, if QED predicted the value of g/2 to within experimental error, then it follows that it also predicted the correct value of U to within experimental error. The precision by which U is known, however, is one in 10 million instead of one in a billion, as in the case of g/2. Thus, in analogy to the case of probability, even to speak of the precision of QED’s prediction, we must already assume a certain natural variable. It seems that the only nonarbitrary choices are the natural variables defi ned earlier, which is what scientists actually use.
From examples like the one cited earlier, it is also clear that WR precision also depends on the choice of the natural variable, as we explained for the case of fi ne-tuning. So it seems that in order to speak of the predictive SD or WR precision for those cases in which a theory predicts the correct experimental value for some quantity, one must assume a natural variable for determining the known predictive precision. One could, of course, deny that there exists any nonrelative predictive precision, and instead claim that all we can say is that a prediction has a certain precision relative to the variable we use to express the prediction.
Such a claim, however, would amount to a denial that highly accurate predictions, such as those of QED, have any special epistemic merit over predictions of much less precision.
This, however, is contrary to the practice of most scientists. In the case of QED, for instance, scientists did take the astounding, known precision of QED’s prediction of the g-factor of the electron, along with its astoundingly accurate predictions of other quantities, such as the Lamb shift, as strong evidence in favor of the theory. Further, denying the special merit of very accurate predictions seems highly implausible in and of itself. Such a denial would amount to saying, for example, that the fact that a theory correctly predicts a quantity to an SD precision of, say, 20 signifi cant digits does not, in general, count signifi cantly more in favor of the theory than if it had correctly predicted another quantity with a precision of two signifi cant digits. This seems highly implausible.
Of course, strictly speaking, to make these sorts of comparisons of relative degrees of confi rmation, one need not privilege one particular variable for expressing the experimental result and calculating its precision, since one need not be committed to a specifi c degree of precision. Nonetheless, one must put some signifi cant restrictions on the functions of a given variable on which one bases one’s calculation of precision, for otherwise one cannot make any signifi cant comparisons. For example, in some cases, there might be
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several different equally simple ways of writing the laws of physics, giving rise to several different candidates for natural variables. In that case, one would simply say that the degree of precision fell into a certain range covering the different choices of natural variables.
Finally, consider a Likelihood Principle reconstruction of the confi rmation that QED
received from its correct, precise prediction of the correction to the g-factor of the electron.
Let QED represent the claim that QED is approximately true or at least empirically adequate; let ~QED represent the negation of this claim; fi nally, let e represent the fact that the correction to the g-factor of the electron falls within the experimentally determined range. Now, P(e|QED & k′) = 1, since QED entails that it will fall into the experimentally determined range. (Since e was old evidence at the time of QED’s prediction, k′ is our background minus this old evidence). The value of P(e|~QED & k′) will depend on the comparison range one chooses – that is, the range of plausible values for the correction to the g-factor given ~QED & k′. There is no precise way of determining this range, but given that without any correction, the g-factor is 2, it is reasonable to suppose that most physicists would have expected it to be no larger than 2. Suppose that it were reasonable to expect the correction to be no greater than ± 0.01, with no preference for any value between 0
and ± 0.01. This would yield a width, WR, for the comparison range of 0.02. If we let Wr be the range of the experimentally determined value of correction, and we used the restricted Principle of Indifference, we would arrive at P(e|~QED & k′) = Wr/WR ~10−7, yielding a large likelihood confi rmation of QED over ~QED.
The lesson here is that any support that correct, precise predictions provide for QED
over ~QED via the Likelihood Principle will involve using something similar to the restricted Principle of Indifference, with the epistemically privileged natural variables being those in the simplest formulation of the area of physics in question. The same can be said for the likelihood reconstruction of other cases of confi rmation based on precise predictions. Such likelihood reconstructions, if plausible, strongly support the epistemic role of the restricted version of the Principle of Indifference in scientifi c practice.
4. Determining k¢ and the Comparison Range
4.1. Introduction
To complete the philosophical groundwork for our argument, we shall need to provide some way of determining k′. Determining k′ will automatically determine the “possible universes” to which we are comparing the life-permitting ones – that is, what we called the
“comparison range.” We shall focus on the constants of physics, but everything we say applies to the laws and initial conditions of the universe with minor modifi cations. First, however, we need to get clear on what it means to vary a constant of physics.
4.2. What it means to vary a constant of physics
Intuitively, there is a distinction between laws and constants, and physicists usually suppose such a distinction. In current physics, most laws can be thought of as mathematical descriptions of the relations between certain physical quantities. Each of these descriptions has a mathematical form, along with a set of numbers that are determined by experiment. So,
for example, Newton’s law of gravity ( F = G m 1 m 2/ r 2) has a mathematical form, along with a number (G) determined by experiment. We can then think of a world in which the relation of force to mass and distance has the same mathematical form (the form of being proportional to the product of the masses divided by the distance between them squared), but in which G is different. We could then say that such worlds have the same law of gravity, but a different value for G. So when we conceive of worlds in which a constant of physics is different but in which the laws are the same, we are conceiving of worlds in which the mathematical form of the laws remains the same, but in which the experimentally determined numbers are different. It should be noted that the distinction between laws and constants need not be a metaphysical distinction, but only a conceptual distinction.
Now these constants of physics are relative to our current physical models, since these constants only occur within a model. Thus, any probabilities we calculate will only be relative to a certain model. Ian Hacking (1987, pp. 119–227) and Bas van Fraassen (1980, pp.
178–95), among others, have emphasized this model-relativism with regard to the relative frequency interpretation of probabilities. Under this interpretation, probabilities are understood as the limit of the ratio of the favorable outcomes to the total number of outcomes as the number of trials goes to infi nity. Since for most, if not all, cases these infi nite long-run frequencies do not exist in the real world, they ultimately must make reference to frequencies in idealized models, as van Fraassen has worked out in detail (1980, pp. 190–3).
Similarly, I shall assume, epistemic probabilities exist only relative to our models of the world and our other background information.
At least in the case of epistemic probabilities, this should come as no surprise, since it has to do with rational degrees of belief, which, of course, are relative to human cognition.
If one denies the model dependence of epistemic probabilities, then it is hard to see how any statements of epistemic probabilities will ever be justifi ed. One reason is that they are almost always grounded in conceptualizing alternative possibilities under some measure, as illustrated in Section 3.1 by the sort of epistemic probabilities used to justify the Thesis of Common Ancestry, continental drift theory, or the atomic hypothesis. But such conceptualizations typically involve implicit reference to some (often vague) model of how those possibilities are spread out. In fact, this was illustrated by use of natural variables in science discussed in Section 3.3.3.
The relevant models for the fi ne-tuning hypothesis are just the models given to us by our best theories in physics, just as if we calculated relative frequencies we should do so using the best models that we had in the relevant domain. At present, the best model we have is the Standard Model of particle physics. Sometimes, however, we can calculate the life-permitting range only for a constant that is less than fundamental, either because we do not have a fundamental theory or because of limitations on our ability to calculate. In that case, the most sensible thing to do is to go with the best model for which we can do calculations, as long as we consider only variations in the constant that fall within the limits of applicability of the model established by the deeper theory –for example, we could sensibly consider the consequences of varying mass using the model of Newton’s theory of gravity as long as the variation were within the range of validity of Newtonian mechanics dictated by Einstein’s theory of General Relativity.
Because we are considering only one reference class of possible law structures (that given by variations of the constants within our best theories and/or the ones we can perform calculations for), it is unclear how much weight to attach to the values of epistemic prob-
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abilities one obtains using this reference class. Hence, one 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 in Section 3.3.2), we obtain a certain epistemic probability, or range of probabilities. Then, as a second-order concern, we must assess the confi dence we have in this probability based on the various components that went into the calculation, such as the representative nature of the reference class. Given that there is no completely objective procedure for addressing this secondary concern, I suggest that the probability calculations should be thought of as providing supporting confi rmation, based on a plausible, nonarbitrary procedure, of the common intuitive sense that given the fi ne-tuning evidence, LPU
is very epistemically improbable under NSU. This evidence will be strengthened by the fact that there are many different fi ne-tuned constants of physics, and many different kinds of fi ne-tuning, so that the fi ne-tuning argument does not depend on one highly specifi c reference class. In light of this, we must keep in mind that our goal is not to provide some exact degree by which the fi ne-tuning evidence supports T over NSU. Rather, it is to show that the intuitive sense that LPU supports T over NSU 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 justifi ed, nonarbitrary procedure.25
4.3. Determining k′ : old evidence problem
In Premises (1) and (2) of our main argument in Section 1.3, the probability of LPU is conditioned on background information k′. As we mentioned in Section 1.3, we cannot simply take k′ to be our entire background information k, since k includes the fact that we exist, and hence entails LPU. To determine what to include in k′, therefore, we must confront what is called the “problem of old evidence.” The much-discussed problem is that if we include known evidence e in our background information k, then even if an hypothesis h entails e, it cannot confi rm h under the Likelihood Principle, or any 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 confi rm General Relativity even though it was known for over 50 years prior to the development of General Relativity and thus entailed by k.
An attractive solution to this problem is to subtract our knowledge of old evidence e from the background information k and then relativize confi rmation 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 we 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 25. Given the multiplicity of possible references classes, one could simply decide not to assign any epistemic probability to LPU. As with skepticism in general, such a practice would also undercut any sort of rational commitment to the approximate truth or empirical adequacy of a theory, since the epistemic probabilities used in justifying these theories also lack a complete fundamental justifi cation. Given that we are not skeptics, the best we can do is use the least arbitrary procedure available to assign epistemic probabilities.
is no unambiguous way of subtracting e from k. Consider the case of the fi ne-tuning of the strength of gravity. The fact, Lpg, that the strength of gravity falls into the 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 Lpg. Thus, if we were to obtain k′ by subtracting Lpg from our total background information k without also subtracting our knowledge of the existence of long-lived stable stars from k, then P(Lpg|k′) = 1.
To solve such problems, Howson says that 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 confi rms h becomes relative to our axiomatization scheme (1991, p. 550). Howson argues that in practice 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 he 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 defi ned by how we represent our background knowledge. Of course, in those cases in which there are alternative ways of axiomatizing 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 confi rms h. I agree with Howson that this is not necessarily a problem unless one thinks that the degree of confi rmation e provides h must be independent of the way we represent our background knowledge. Like Howson, I see no reason to make this assumption: confi rmation 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 fi ne-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 scientifi c 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 scientifi c realists who want scientifi c theories to “cut reality at its seams.”26 Furthermore, we have a particularly powerful reason for adopting this axiomatization when considering a constant of physics. The very meaning of a constant of physics is only defi ned 26. One might object that this procedure is only justifi ed under the assumption that we live in a deterministic universe, since, otherwise, the k we have chosen is not a true axiomatization of our knowledge. This is true, but it is diffi cult to see how the thesis that the world is indeterministic could be relevant to the legitimacy of the fi ne-tuning argument.
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A RG U M E N T 243
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.
It should be noted that although Howson’s method was developed in the context of subjective Bayesian conception of probability, his argument for this method does not depend on this conception. All it depends on is the claim that “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,” and that this is “the same thing as asking how much e boosts h relative to what else we know” (1991, p. 548). Anyone who subscribes to a probabilistic confi rmation account of evidence, according to which e counts as evidence for h if and only if knowledge of e increases our degree of confi dence in h, should at least be sympathetic to the underlying premises of his argument.
Finally, it is worth considering how the old evidence problem plays out in the method of probabilistic tension. As mentioned earlier, the major problem with Howson’s method is that the background information k′ depends on the subtraction procedure one uses. If we cast the fi ne-tuning argument in terms of probabilistic tension, as elaborated in Section 1.4, this problem can be avoided; we do not need to privilege any particular subtraction procedure. According to that method, both NSU and T should be elaborated in such a way that each of them entails LPU. (We called these elaborated hypotheses NSUe and Te, respectively.) Thus, LPU does not directly confi rm one of these elaborated hypotheses over the other. Nonetheless, the fi ne-tuning evidence creates severe probabilistic tension for NSUe but not for Te. Thus, it gives us a signifi cant reason to prefer Te over NSUe: if with respect to some domain, one hypothesis h1 has much more probabilistic tension than another, h2, then the probabilistic tension gives us strong reason to prefer h2 over h1, everything else being equal.
To determine the degree of probabilistic tension generated for NSUe by the fi ne-tuning evidence, we need to probe NSUe & k for hidden probabilistic tension related to the fi ne-tuning evidence. Now, when considering only a single constant C, the fi ne-tuning evidence only generates probabilistic tension because of the data Lpc. To bring this probabilistic tension out, we fi rst note that NSUe & k = NSU & k = NSU & k′ & Lpc, since k′ = k – {Lpc}
and hence k = Lpc & k′, where no particular subtraction procedure is specifi ed for –{Lpc}.
Then, we must consider all k′ such that k = k′ & Lpc, and take the true probabilistic tension of NSUe & k to be given by the lower bound of P(Lpc|NSU & k′), for all possible k′. We then follow a similar procedure for the probabilistic tension of Te & k. Using the lower bound guarantees that the information that a constant fell into the life-permitting range is not implicitly left in the k′ one uses to assess probabilistic tension, as we saw in our given
example for the fi ne-tuning of gravity where the existence of long lived, stable stars was left in k′. This is a determinate procedure that does not depend on any choice of subtraction procedure and demonstrates the power of the idea of probabilistic tension. (As an alternative to Howson’s method, one might also use this approach to determine k′ for the Likelihood Principle method mentioned, although we will not pursue this further here.) 4.4. Determining k′ : the EI region
Next, for any given fi ne-tuned constant C, we must determine the comparison range of values for C. My proposal is that the primary comparison range is the set of values for which we can make determinations of whether the values are life-permitting or not. I will call this range the epistemically illuminated (EI) range.27 Thus, given that the EI range is taken as our comparison range, we will say that a constant C is fi ne-tuned if the width, Wr, of the range of life- permitting values for the constant is very small compared with the width, WR, of the EI range.