2.1. Carter’s Anthropic Principle

To provide explanations for certain large-number coincidences in cosmology, in 1974, Brandon Carter introduced the ‘anthropic principle’. Applications of this principle allowed Carter to avoid explanations of these coincidences that used certain “exotic theories … involving departures from normally accepted physical conservation laws” (Carter, Reference Carter and Malcolm1974, p. 291). Carter’s original statement of the principle (which we’ll denote by AP) is

Carter goes on to say parenthetically (on p. 291) that “our situation … is inevitably privileged to some extent”. And this highlights a key aspect of its content as understood by Carter: that is, our (observational) situation must be somewhat special, to the extent that it must allow for observers. It isn’t necessarily clear what is meant by an observer—and this is one source of imprecision in AP (and other versions of it), which motivates EOP. But before we press this issue (in the following subsection), we’ll set the stage by briefly analyzing relevant aspects of AP.

AP can be illustrated by considering a scenario in which we will be performing an experiment (of a certain type)—where we are interested in what we can expect to observe for the outcome of that experiment. The conditions that are necessary for observers, according to AP, will delimit the possible observed outcomes of the experiment. Carter distinguishes two different types of anthropic principle. His ‘weak anthropic principle’ (WAP) is primarily about our spatiotemporal location within our universe, where the fundamental parameters and relevant initial conditions of our physical theories are held fixed. We will formulate it as follows.

So, if the observation of interest is the distance of our planet from our sun, that distance cannot be so small so as to preclude, on account of how hot it would be, the possibility of life (insofar as we understand it). And that distance cannot be so large so as to preclude, on account of how cold it would be, the possibility of life. His ‘strong anthropic principle’ (SAP) is primarily about the fundamental parameters and/or initial conditions of our physical theories. We will formulate it as follows.

So, if the observation of interest is the value of the cosmological constant, that value cannot be so large and positive so as to preclude the possibility of life, on account of it effecting so rapid an expansion of the universe that stars cannot form. And that value cannot be so large and negative so as to preclude the possibility of life, on account of it leading to a recollapse of the universe after an initial big bang such that stars do not have time to form. In this way, the nature of the ‘experiment’ to which AP is being applied may be thought of as involving either WAP or SAP or perhaps even a combination of the two.Footnote ² What is common between these principles is that they both warn that only a proper subset of a set of (physical) possibilities is to be expected to be observed: they both warn of selection effects. This common message will allow us to proceed by treating them in parallel.

Note also the following points about any formulation of the anthropic principle (such as AP, WAP, or SAP). (1) The principle ought to be understood in the context of a physical theory. In what follows, such theory-dependence will be assumed (and will be further analyzed later in this section). (2) The principle is typically stated in terms of an ‘expectation’ regarding observations (via “what we can expect to observe”). How to understand such an expectation (does it involve probabilities, for example) is not crucial for understanding the intended content of the principle and so we will set this issue aside.

It will help, in what follows, to write AP in slightly more formal terms. (We will switch between various ways of referring to the principle as is appropriate.) So, denote ‘observers’ by ${\cal O}$ and let $\left\{ p \right\}$ denote a set of possible values for the outcome of an experiment of interest. Then AP may be written as

(1) $${\rm{A}}{{\rm{P}}_1}:\left( {{\cal O}{\rm{\;can\;expect\;to\;observe\;}}{p_0} \in \left\{ p \right\}} \right) \Rightarrow \left( {\left\{ p \right\}{\rm{\;allows\;for\;}}{\cal O}{\rm{\;to\;arise}}} \right).$$

Here, ${p_0}$ is a possible value for the outcome of the experiment and the implication sign is to be understood within the context of an underlying physical theory. This principle certainly sounds reasonable: if the consequent is denied (so that no outcome in $\left\{ p \right\}$ allows for ${\cal O}$ to arise), then, given the theory, there will simply be no observers to observe an outcome in $\left\{ p \right\}$ .

In the more formal version [in Eq. (1)], note that the consequent contains the phrase “allows for”. And one might wonder whether a stronger phrase is permissible, one that indicates that, say, each element of $\left\{ p \right\}$ provides sufficient conditions for the existence of ${\cal O}$ . But this is to ignore important theoretical content that is built-in to ${\rm{A}}{{\rm{P}}_1}$ . Any such anthropic principle ought to be understood in the context of our current best physical theories. And in such theories, the notion of an ‘observer’ cannot be made precise. What one can plausibly describe are broadly interpreted conditions that would (likely) preclude observers—as we might qualitatively understand them. So, as described earlier, a planet that is too close to its sun or a cosmological constant that is too large and positive would each most likely preclude observers as we qualitatively understand them. Thus, we can (only) plausibly specify outcomes $\left\{ p \right\}$ that would not preclude (that is, would “allow for”) observers—in a way that is not sensitive to precisely how one defines an observer.

2.2. An effective interpretation

Assuming, then, that our current best physical theories cannot unambiguously describe observers, we will argue that one ought to think of the ‘observer’ referred to in the consequent of ${\rm{A}}{{\rm{P}}_1}$ in a very different way to the ‘observer’ in its antecedent. Developing this argument will be the task of this subsection and will lead us to EOP, which better captures what is meaningful in ${\rm{A}}{{\rm{P}}_1}$ .

Insofar as anthropic-style principles refer to what ‘we’—as beings carrying out and analyzing the results of experiments—should expect to observe, the notion of the observer in the antecedent of ${\rm{A}}{{\rm{P}}_1}$ should be clear. This notion refers to ‘us’ understood as such beings and no further elaboration (theoretical or otherwise) is needed. (After all, this is a principle that we will be using in our theorizing about the world in which we find ourselves.)

But the notion of the observer in the consequent of ${\rm{A}}{{\rm{P}}_1}$ is ill-defined. Our current best physical theories are only able to account for background physical conditions out of which observers (as we qualitatively understand them) might arise. In such a theoretical context, observers as we qualitatively understand them are contingencies (or, more prosaically, ‘accidents’). And were such observers to arise, a sensible principle specifying what these observers can expect to observe ought to point to these background physical conditions and not, specifically, to conditions that relate to some ill-defined notion of an observer. Thus the ‘observer’ referred to in the consequent of ${\rm{A}}{{\rm{P}}_1}$ ought not to be an observer at all, but ought to be some physical entity with respect to which observers (like us) might arise. Such a principle would be consistent with the theoretical content and scope of our physical theories.

To further motivate such a principle, it might help to consider a misinterpretation of AP in which “our presence as observers” is assumed to (mistakenly) refer to a very specific observer: say, Joe, a 24-year old Cantabrigian. Then AP would read:

This, of course, is absurd (by design). For “what we can expect to observe” in performing, say, some cosmological experiment, ought to have little to do with Joe’s presence. But consider the conditions that are necessary for Joe’s presence. Joe cannot arise if Joe’s parents never meet. Nor can Joe arise if Joe’s grandparents never meet. Indeed, there is a very long list of conditions—such that were these conditions to not obtain, Joe would not arise. Some of these conditions (such as the meeting of Joe’s parents) ought not to have a hand in a sensible (theory-relative) anthropic-style principle—one that describes “what we can expect to observe”. If we exclude such irrelevant conditions, what conditions remain and precisely what do they pick out?

The remaining conditions won’t ostensibly refer to Joe or, more to the point, to anything like an observer. Take, for instance, the condition that energy-density perturbations (about the mean density) on spatial slices are small, very early on in the history of the universe. (This condition needs to hold for observers to arise but it ostensibly has very little to do with observers.) Such conditions, unlike “our presence as observers”, are indeed expressible in terms of our current best physical theories and are those upon which we believe we need to focus to extract a sensible (theory-relative) anthropic-style principle. The conditions are constrained in some way (for example, by physical considerations even earlier on in the history of the universe) but they aren’t constrained, in an essential manner, by considerations about ‘observers’—the ones referred to in the consequent of ${\rm{A}}{{\rm{P}}_1}$ .

Thus we are led to a reformulation of AP. This reformulation crucially makes use of our effective physical theories (which are our current best physical theories) and includes a term that we denote by $\delta $ , which (inter alia) serves to replace the notion of an ‘observer’ in the consequent of ${\rm{A}}{{\rm{P}}_1}$ . Here, $\delta $ is an entity that can be expressed within the context of an effective theory (or possibly multiple such theories): that is, there are rules that secure a correspondence between that entity and physical states and/or physical conditions of the theory. Given that $\delta $ is instantiated, observers (as we qualitatively understand them) may arise (as a cosmic accident). We dub this principle the effective observation principle (EOP).

The underlying effective theory might be an effective cosmological theory such as the ${\rm{\Lambda }}$ CDM model, coupled to the standard model of particle physics; but in more general applications of EOP, it might also involve, for example, nuclear theories or biophysical theories. As for generating $\delta $ in a given observational situation, we doubt that this can be achieved algorithmically. The issue is not that of finding a $\delta $ that constrains an observable, but that of finding more relevant $\delta $ s that yield suitably precise constraints. At the end of Sec. 3.1.1, we’ll exemplify considerations that may arise in selecting more relevant $\delta $ s. Nonetheless, we can characterize general features of $\delta $ —as detailed in the following subsection. For now, an example of a $\delta $ relevant to a cosmological setting could be energy-density perturbations on spatial slices, very early on in the history of the universe, which are drawn from a specific statistical distribution (such as perturbations that satisfy a gaussian distribution). Note ‘gaussian energy-density perturbations’ can indeed be interpreted (unambiguously) in the context of an effective theory such as that of ${\rm{\Lambda }}$ CDM—and that out of such perturbations, observers as we qualitatively understand them might arise (as we believe they may have).Footnote ³

It will be helpful to rewrite this principle along the lines of a more general version of Eq. (1). There we introduced $\left\{ p \right\}$ —a set of possible outcomes for some experiment (for a single cosmological parameter, for example). But we can think of this set more generally as a set of possible values for a vector $\vec p$ where each element of this vector denotes a parameter and/or a condition (such as an initial condition) required for the theory to yield predictions. Then we may consider a more formal version of EOP, denoted by ${\rm{EO}}{{\rm{P}}_\delta }$ .

(2) $${\rm{EO}}{{\rm{P}}_\delta }:\left( {{\cal O}{\rm{\;can\;expect\;to\;observe\;}}{{\vec p}_0} \in \left\{ {\vec p} \right\}} \right) \Rightarrow \left( {\left\{ {\vec p} \right\}\ {\rm{allows\;for\;}}\delta \;{\rm{to\;arise}}} \right).$$

So, although the set $\left\{ {\vec p} \right\}$ encodes necessary conditions for $\delta $ , and this set can also be thought to encode, via a consideration of $\delta $ , necessary conditions for ‘observers’ (as we qualitatively understand them), no further refinement of the conditions for $\delta $ as they relate to observers is stipulated. That is, observers as we might qualitatively understand them may or may not arise given (conditions for) $\delta $ . Indeed, our effective theories are such that we cannot further specify what we may expect to see.

2.3. Characterizing the entity $\delta $

So, what are the key features of $\delta $ and how ought we to understand it?

$\boldsymbol \delta $ is expressible in terms of the theory.

The entity $\delta $ is to be understood as expressible in terms of the effective theory of interest. (This is indeed one of its virtues compared to the use of the notion of an ‘observer’ in the consequent of ${\rm{A}}{{\rm{P}}_1}$ .) To understand what we mean by this, note that any effective theory has limited resources to describe the physical world. For instance: the ${\rm{\Lambda }}$ CDM model of cosmology contains no terms and/or variables that directly encode specific facts about individual observers, such as your date of birth or the brand of car you may drive. Nor can one compute such facts from the theory. Compare this with the case of some specific statistical distribution of energy-density perturbations in the early universe. The ${\rm{\Lambda }}$ CDM model does contain variables that directly encode energy-density perturbations on spatial slices consistent with a given distribution. (The variables are simply the energy density as a function of space and time; the encoding occurs together with suitable conditions on those variables, which can indeed be expressed in terms of the model.) Candidate $\delta $ s for EOP have precisely this feature: there are terms and/or variables that appear in the effective theory of interest (together with suitable conditions that are expressible in terms of the theory) that directly encode $\delta $ .

$\boldsymbol \delta $ can be multiply realized.

Assumptions and results in cosmology (and astrophysics) are often expressed in terms of aggregate properties (such as statistical properties), which can be realized by multiple combinations of parameter values and/or values of initial conditions. So, for example, a $\delta $ that corresponds to energy-density perturbations that satisfy a specific statistical distribution at a time $t$ , such as a gaussian distribution, is consistent with various possible point-by-point specifications of conditions at a time earlier than $t$ . Or, for an example that is non-statistical in character, a $\delta $ that corresponds to energetic degrees of freedom in spacetime that satisfy certain ‘energy conditions’—where (inter alia) the energy density is non-negative—is consistent with various possible values of parameters and/or initial conditions. Thinking in terms of ${\rm{EO}}{{\rm{P}}_\delta }$ , this type of multiple realizability of $\delta $ amounts to the claim that there are multiple different vectors $\vec p$ that can all realize the same $\delta $ .

There are multiple candidate $\delta $ s for a given theory.

If we fix our theory of interest, there are multiple different entities $\delta $ that could reasonably complete our statement of EOP. Here is a sample:

1. (i) a low-entropy ‘initial’ state;

2. (ii) a gaussian distribution of energy-density perturbations in the early universe;

3. (iii) a precondition (that is, a necessary condition) for complexity—as understood in the context of a specific account of complexity that we will denote by $\mathbb{C}$ ;

4. (iv) the account of complexity $\mathbb{C}$ , itself.

How are we to select from amongst such candidates—taking into account that there may be relationships between them? The response we endorse is that one ought to select that entity that is as specific as possible about our observational situation (and is indeed expressible in terms of the theory). This is primarily for the descriptive and predictive power that one might expect from such a choice.Footnote ⁴

For example, compare option (iii) (a precondition for $\mathbb{C}$ ) with (iv) ( $\mathbb{C}$ itself). Our position favors the choice of (iv) over (iii) (assuming we have $\mathbb{C}$ in hand): option (iv) is more specific and therefore more useful as regards our observational situation. To see this, assume the following five preconditions for complexity to emerge, as described by Livio and Rees (Reference Livio, Rees, Sloan, Batista, Hicks and Davies2018), are preconditions for $\mathbb{C}$ . The preconditions relate to: (a) constraints on the strength of gravity; (b) charge-parity violation (so that the resulting universe contains more matter than antimatter); (c) energy-density perturbations of the right type; (d) non-trivial chemistry (effected by a particular balance between the strong nuclear force and the electromagnetic force); (e) constraints on the expansion rate of the universe. The choice of just one of these preconditions as a candidate for $\delta $ seems underspecified, compared to $\mathbb{C}$ itself. For instance, compare the following two ways of completing ${\rm{EO}}{{\rm{P}}_\delta }$ :

(3) $$\left( {{\cal O}{\rm{\;can\;expect\;to\;observe}}{{\overrightarrow {\;p} }_0} \in \left\{ {\vec p} \right\}} \right) \Rightarrow \left( {\left\{ {\vec p} \right\}{\rm{\;allows\;for\;complexity\;to\;arise}}} \right);$$

$$\left( {{\cal O}{\rm{\;can\;expect\;to\;observe\;}}{{\vec q}_0} \in \left\{ {\vec q} \right\}} \right) \Rightarrow $$

(4) $$\left( {\left\{ {\vec q} \right\}{\rm{\;allows\;for\;matter-antimatter\;asymmetry\;to\;arise}}} \right).$$

The first of these two options seems a lot more useful—at least insofar as we might expect to observe ${\vec p_0}$ rather than ${\vec q_0}$ . For $\left\{ {\vec p} \right\}$ is reasonably a proper subset of $\left\{ {\vec q} \right\}$ , as matter-antimatter asymmetry seems consistent with possibilities that aren’t as constrained as those by complexity. Even the conjunction of the five preconditions (a)–(e) seems underspecified compared to $\mathbb{C}$ itself. For the latter will include information beyond the preconditions, which could arguably be useful for understanding our observational situation.

If we don’t have an account of complexity in hand, taking the conjunction of the five preconditions as a single candidate seems more preferable than taking $\delta $ to be just a single precondition. This selection, of the conjunction of the five preconditions over a single precondition, assumes that the relationship between the five preconditions and the $\delta $ of interest (in this case, complexity) is unambiguous. However, note that according to our current best physical theories, the relationship between such preconditions and the occurrence of complexity isn’t unambiguous. There are uncertainties about what may arise when we simultaneously vary multiple parameters of our current best physical theories. That (a)–(e) are all preconditions for complexity is made plausible by considering scenarios in which we vary the values of parameters one at a time, holding the others fixed. [Livio and Rees (Reference Livio, Rees, Sloan, Batista, Hicks and Davies2018) also consider scenarios in which two parameters are varied simultaneously.] If an unambiguous relationship can’t be ensured, then a $\delta $ that is less restrictive ought to be selected. Such considerations form part of the challenge, in applying EOP, of selecting appropriate theory-dependent $\delta $ s.

The candidates (i)–(iv) are ‘nested’, in that according to our current best theories of physics (perhaps supplemented by our understanding of biology), (i) can be thought of as necessary for (ii). Similarly, (ii) can be thought of as necessary for (iv). In this way, the response endorsed above—that we ought to choose the candidate for $\delta $ that is expressible in terms of the theory and is as specific as possible about our observational situation—recommends a choice amongst such options that is as far down the chain of implications as possible [namely, as far as possible to the right of the chain $\left( i \right)\Leftarrow \left( {ii} \right)\Leftarrow\left( {iv} \right)\Leftarrow \ldots $ ], but where one can unambiguously express the choice in terms of the theory. Thus (iv) would be preferable to (i), for it is unclear that the resulting version of EOP would appropriately constrain “what we can expect to observe”: it would be too permissive regarding such expectations.

Our remarks reveal that the entity that $\delta $ represents naturally arises in cosmic history before the relevant observers (those in the antecedent of ${\rm{EO}}{{\rm{P}}_\delta }$ ). It may persist during the act of observation (if $\delta $ was complexity, say) or it may not (if $\delta $ was, say, a low-entropy initial cosmological state). And the entity need not reach back to the beginning of the universe: it may emerge at some later point.

2.4. Addressing anthropocentrism in EOP

Our remarks in this section suggest a line on the question: to what extent should we consider ${\rm{EO}}{{\rm{P}}_\delta }$ to be ‘anthropic’? There is a sense in which ${\rm{EO}}{{\rm{P}}_\delta }$ is clearly not about humans as observers—it is not about humans at all: humans (and indeed observers more generally) aren’t unambiguously expressible in terms of our effective physical theories and so they can’t be central to the nature of the inference contained in ${\rm{EO}}{{\rm{P}}_\delta }$ .

Nonetheless, it is still via considerations of the conditions that are causally relevant for humans (and observers more generally) that we would go about choosing a suitable $\delta $ . Thus we take ${\rm{EO}}{{\rm{P}}_\delta }$ to be observer-class relative (in a mild sense)—a class that contains humans but also observers who are at least somewhat qualitatively like us. If we demand that EOP can account for totally different types of observers, different to anything we’ve experienced, then EOP may be rendered vacuous. If $\delta $ represented, say, slight inhomogeneities in the very early universe (conditions that are presumably amenable to very different types of observers) then the power of EOP as a principle is significantly compromised. Thus, while ${\rm{EO}}{{\rm{P}}_\delta }$ is not an anthropic principle as such, it rests on an inference from the observational situation (represented by $\delta $ ) in which we (as observers of a type with which we’re at least qualitatively familiar) find ourselves, to necessary conditions (represented by $\left\{ {\vec p} \right\}$ ) for such an observational situation.

Note that it has been remarked that the specific reference in AP to human observers is not needed to appropriately interpret the principle (McMullin, Reference McMullin1993; Wilson, Reference Wilson1991). Wilson (Reference Wilson1991, p. 169) says that the “problem is how to specify the required degree of anthropocentrism, that is, how to define the class of beings whose necessary conditions are singled out”. In our view the problem is instead about how to identify an entity (a $\delta $ ) that usefully relates to the observational situation of interest. We thus agree with Wilson that there’s a problem for anthropic-style reasoning, but that it shouldn’t be framed in the way he describes. Furthermore, it should be clear that some features upon which anthropic-style considerations might be based—and which, according to Wilson (Reference Wilson1991, p. 170), have been mentioned by others—such as “intelligence, morality, [and] consciousness”, are irrelevant. Such features are not plausibly within reach of our physical theories and so cannot practically be part of the story.

Indeed, as we will now show, human observers and observers more generally aren’t part of what is taken into consideration when AP is putatively applied in physical settings.

3.1. Cosmological settings

We address cosmological applications via two analyses. First, in Sec. 3.1.1, we present a non-probabilistic analysis of a putative application of AP as developed by Carter (Reference Carter and Malcolm1974). Secondly, in Sec. 3.1.2, we present a probabilistic analysis of an application as developed by Earman (Reference Earman1987).

3.1.1. A non-probabilistic analysis

Carter (Reference Carter and Malcolm1974) provides two examples of anthropic reasoning, one based on WAP and the other on SAP, that account for two separate large-number coincidences using standard physical theories. These accounts of the two coincidences get around the need to invoke non-standard physics. The inferential pattern underlying both examples is identical. (1) Identify proxies for the presence of observers. (2) Explicate the theoretical link between those proxies and our spatiotemporal location (in the case of WAP), or on fundamental parameters of the physical theory (in the case of SAP). (3) Derive a constraint on our spatiotemporal location (in the case of WAP), or on fundamental parameters of the physical theory (in the case of SAP). If the constraints—assuming further (standard) theoretical input—allow one to account for the coincidence, then the need for non-standard physics is avoided and the application succeeds. In what follows we focus on Carter’s example that relates to WAP (but a similar account can be given for the example that relates to SAP).

Carter is primarily interested in a large-number coincidence that can be straightforwardly derived from considerations of the current age of the universe, ${t_0}$ . The coincidence is that (for a specific choice of units in which all physical quantities are dimensionless) the present (Hubble) expansion rate of the universe, denoted by ${H_0}$ , is of the order of magnitude of the cube of the mass of the proton, ${m_p}$ :

(5) $${H_0}\sim m_p^3.$$

(It is the inverse of each term in this relation that is large—of the order of ${10^{60}}$ .) It is well known that for certain simplified cosmological scenarios, one may estimate ${H_0}$ by the inverse of the current age of the universe, ${H_0}\sim t_0^{ - 1}$ , and so if one can show that ${t_0}\sim m_p^{ - 3}$ , one secures the coincidence. For our purposes, the coincidence is less important than how it is that Carter determines ${t_0}$ .

He estimates that ${t_0}$ is of the order of magnitude of the typical amount of time a star (somewhat more massive than our sun) spends on the ‘main sequence’: that is, undergoes hydrogen-burning to produce helium in its core (we’ll denote this time by ${t_{\rm{*}}}$ ). If, as Carter reasons, the current age of the universe was significantly shorter than this time ( ${t_0} \ll {t_{\rm{*}}}$ ), then the heavier elements would not have formed; if the current age of the universe were significantly longer than this time ( ${t_0} \gg {t_{\rm{*}}}$ ), then our galaxy would not have very many stars left in it and the stars it would contain wouldn’t have sufficient energy output to sustain observers. What is serving as a proxy for the presence of observers thus generally relates to processes that arise within stars of a certain type. A rough (conceptual) lower-bound on ${t_0}$ comes from considering heavy elements as necessary for observers while a rough upper-bound on ${t_0}$ comes from considering sufficiently energetic stars as necessary for observers. Carter indeed goes on to derive that ${t_{\rm{*}}}\sim m_p^{ - 3}$ : assuming that ${t_0}\sim {t_{\rm{*}}}$ one obtains ${t_0}\sim m_p^{ - 3}$ , thereby securing the coincidence. (This corresponds to an age that is on the order of ${10^{10}}$ years—indeed consistent with our observations.) Note that this is a single constraint on the current age of the universe—so that the lower and upper bounds are distinguished only conceptually—yielding a single order-of-magnitude estimate. This suits Carter’s purposes for he is interested in a large-number coincidence where order-of-magnitude precision is sufficient.

The key point for our analysis is that nothing like an observer has been uniquely singled out in Carter’s argument. Neither of the proxies that are assumed in Carter’s calculation are proxies for just an observer, let alone for (as stated in WAP) “our presence as observers”—and so it is not clear that WAP has been applied (even if one could uniquely interpret it). All sorts of physical states could be substituted for the notion of an observer and one would obtain the same estimate for ${t_0}$ via Carter’s reasoning. What has been identified in this example are two entities ${\delta _1} \equiv $ ‘heavy elements’ and ${\delta _2} \equiv $ ‘heavy elements and sufficiently energetic stars’, which are expressible in terms of our theories of cosmology and astrophysics. Given constraints associated with these entities, observers who arise as contingencies in this ‘observational situation’ would indeed obtain the estimate ${t_0}\sim m_p^{ - 3}$ . In this way, a version of EOP such as the one that follows has been identified and applied:

what we can expect to infer about the present age of the universe must be restricted by conditions, related to such a timescale, which are necessary for $\delta $ .

Can Carter’s estimate be improved upon by selecting more relevant $\delta $ s? Presumably, this involves accounting for more details of our observational situation. Although the presence of heavy elements and a sufficiently luminous star is important for our observational situation, so is the presence, on our planet, of certain chemical elements. Hogan (Reference Hogan2000, p. 1151) describes how radioactive nuclei and iron help to create the Earth’s magnetic field: one that is strong enough to protect the atmosphere against erosion by the solar wind—creating planetary conditions conducive to life. So, perhaps including (via new $\delta $ s) conditions for sufficient amounts of these nuclei to form would help to improve Carter’s estimate. However, one also needs to account for significant decreases in star-formation rates in galaxies—as estimated over the timescale of the present age of our solar system (Hogan, Reference Hogan2000, p. 1151)—which yield corresponding decreases in formation rates of salient radioactive nuclei. Some of these considerations may hinge on theoretical details that are less well established than for heavy-element production, so as to not improve Carter’s estimate. We’ll leave further analysis for future work. But it’s plausible that such considerations can address the same question (that of the present age of the universe) by potentially employing more relevant $\delta $ s (consistent with Carter’s ${\delta _1}$ and ${\delta _2}$ ).

3.1.2 A probabilistic analysis

We turn now to a probabilistic analysis of a putative application of AP, developed along Bayesian lines. We’ll focus on an anthropic account of the large-number coincidence in Eq. (5), as developed by Earman (Reference Earman1987). The core idea is that a low posterior probability for a standard cosmological theory given the coincidence is raised by virtue of (Bayesian) conditionalization on ‘life’. The new posterior probability for the standard theory given the coincidence and life is then approximately that for the non-standard (exotic) theory. In this way, one avoids the need for the non-standard theory.

We can explicate this core idea in the following way. That Dirac’s non-standard physical theory ${T_{\rm{D}}}$ is able to account for the large-number coincidence (which we’ll denote by ${\cal C}$ ) in Eq. (5) may be expressed probabilistically as $P({\cal C}|{T_{\rm{D}}})\sim 1$ . That the standard big-bang model of cosmology ${T_{\rm{B}}}$ cannot so account for the coincidence may be expressed as $P({\cal C}|{T_{\rm{B}}}) \ll 1$ . Thus, using Bayes’ theorem, we have [assuming equal priors, $P\left( {{T_{\rm{D}}}} \right) = P\left( {{T_{\rm{B}}}} \right)$ ] that

(6) $${{P({T_{\rm{D}}}|{\cal C})} \over {P({T_{\rm{B}}}|{\cal C})}} = {{P({\cal C}|{T_{\rm{D}}})} \over {P({\cal C}|{T_{\rm{B}}})}} \gg 1,$$

so that Dirac’s theory is overwhelmingly favored. If, however, we conditionalize (as Earman does) on the proposition ${\cal L} \equiv $ ‘Life such as ours now exists’, then this conclusion is modified. So, instead of using just the coincidence ${\cal C}$ as our total evidence, we now include ${\cal L}$ —so that our total evidence is ${\cal C} \wedge {\cal L}$ . In which case, using Bayes’ theorem [again assuming equal priors, $P\left( {{T_{\rm{D}}}} \right) = P\left( {{T_{\rm{B}}}} \right)$ ],

(7) $${{P({T_{\rm{D}}}|{\cal C} \wedge {\cal L})} \over {P({T_{\rm{B}}}|{\cal C} \wedge {\cal L})}} = {{P({\cal C} \wedge {\cal L}|{T_{\rm{D}}})} \over {P({\cal C} \wedge {\cal L}|{T_{\rm{B}}})}} = {{P({\cal C}|{\cal L} \wedge {T_{\rm{D}}})} \over {P({\cal C}|{\cal L} \wedge {T_{\rm{B}}})}}{{P({\cal L}|{T_{\rm{D}}})} \over {P({\cal L}|{T_{\rm{B}}})}}.$$

If we assume that the probability of life isn’t very different for the two theories, $P({\cal L}|{T_{\rm{D}}}) \simeq P({\cal L}|{T_{\rm{B}}})$ , then the ratio of the posterior distributions [on the far left-hand side of Eq. (7)]—also known as the posterior odds of ${T_{\rm{D}}}$ —depends just on the ratio $P({\cal C}|{\cal L} \wedge {T_{\rm{D}}})/P({\cal C}|{\cal L} \wedge {T_{\rm{B}}})$ . Given Carter’s argument (related to WAP) in the previous subsection, it is reasonable to assume that the two probabilities $P({\cal C}|{\cal L} \wedge {T_{\rm{D}}})$ and $P({\cal C}|{\cal L} \wedge {T_{\rm{B}}})$ are roughly the same (and each is also high) so that their ratio is roughly unity. This means that the posterior odds of ${T_{\rm{D}}}$ is roughly unity. Dirac’s theory is thus no longer favored and we have evaded the need for the non-standard physical theory.

But how well-defined is this line of reasoning? There are two key considerations. First, note that the key distributions that influence the conclusion (that we need not invoke non-standard physical theories), $P({\cal C}|{\cal L} \wedge T)$ and $P({\cal L}|T)$ (for particular theories $T$ ), are least subjective when the proposition ${\cal L}$ is expressible in terms of the theory $T$ . In which case it is appropriate to think of this proposition as a function (of sorts) of the theory ${\cal L}\left( T \right)$ . When ${\cal L}$ cannot be described in this way, an important question arises as to how one might actually compute such probabilities. That is, if ${\cal L}$ is not expressible in terms of the theory, what does it mean to assign a sharp number to the quantity $P({\cal L}|T)$ (for example)? Secondly, the comparison of such probabilities that is implicit in Eq. (7), is arguably well-defined if and only if the notion of ‘life’ is basically the same for each theory. That is, we require that either ${\cal L}\left( {{T_{\rm{D}}}} \right)$ should mean the same thing as ${\cal L}\left( {{T_{\rm{B}}}} \right)$ or, at the very least, any differences in how we interpret ${\cal L}\left( {{T_{\rm{D}}}} \right)$ and ${\cal L}\left( {{T_{\rm{B}}}} \right)$ should not be relevant. If there is a relevant difference, then it is not clear that it makes sense to compare a number such as $P({\cal L}\left( {{T_{\rm{D}}}} \right)|{T_{\rm{D}}})$ with $P({\cal L}\left( {{T_{\rm{B}}}} \right)|{T_{\rm{B}}})$ (a comparison indeed employed above).

Arguably, reasonable interpretations of ${\cal L}$ for each theory will be sufficiently similar. Life, as it is understood in the context of a theory in which the gravitational constant $G$ scales as ${t^{ - 1}}$ (as for Dirac’s theory ${T_{\rm{D}}}$ ) is not obviously different to life understood in the context of a theory in which $G$ is a constant (as for the standard big-bang model, ${T_{\rm{B}}}$ ). Thus, the second consideration above does not present an issue. The problem, however, is with the first consideration. For neither theory is it the case that the notion of life is expressible purely in terms of the cosmological theory. The dependence of the concept of life on each theory is thus unclear and this threatens how objective and thus well-defined may be probabilities such as $P({\cal L}|T)$ .

In a probabilistic setting, there are two ways forward. (1) Interpret probabilities such as $P({\cal L}|T)$ subjectively—perhaps as credences of agents who are assessing a cosmological theory $T$ . (2) Make the connection between such a theory and the notion of life more manifest. Assuming we are interested in objective comparisons, and thus focusing on (2), one would need to append theories to the cosmological theory $T$ so as to secure a characterization of life that accords with how it is (at least) qualitatively understood. For example, one may need to append theories of astrophysics, astrobiology, and biology. In which case the resulting characterization of life may more closely resemble our qualitative understanding, but only if we can ‘match’ these other theories in regimes where they may overlap. Such matching presents a hurdle because theories that relate more closely to life are relatively underdeveloped.

To summarize the above discussion: issues faced by probabilistic applications of AP are very closely related to concerns raised in our non-probabilistic analysis. AP makes reference to an entity (as in the consequent of ${\rm{A}}{{\rm{P}}_1}$ —namely, ‘observers’ or ‘life’) that cannot be uniquely interpreted in the context of the theories within which AP is meant to be understood and applied. In probabilistic cases, this ambiguity in what could be meant by ‘observers’ undermines the objectivity and thus the well-definedness of probabilities that are assigned—thus undermining arguments that rest on comparisons of such probabilities.

These worries are obviated if we reconceptualize the reasoning involved in terms of EOP. Then we are compelled to restrict attention to entities (namely, $\delta $ s) that are expressible in terms of the theory, $T$ . Indeed, the recommendation (as discussed in Sec. 2.3) is to choose that candidate that is expressible in terms of the theory and is also as specific as possible about our observational situation. This choice renders the underlying reasoning well-defined—whether it be non-probabilistic or probabilistic. Of course, in probabilistic situations, the issue remains as to how we ought to compute something like $P(\delta |T)$ for some cosmological theory $T$ —but, crucially, the question itself is well-defined. And, just as we discussed in the non-probabilistic situation, when estimates are made for probabilities of life—putatively based on AP (such as in Earman’s analysis)—what is actually being employed is some proxy for life (or observers). What has been identified (along with theoretical input related to cosmological modeling) is an entity (and conditions for such an entity) such that were it to obtain, observers might arise as an accident, but where that entity doesn’t refer to observers in an essential way. That is, a version of EOP has been identified and applied.

Note that we are not suggesting that EOP evades the hurdle (faced by AP), wherein the matching of theories is required to more closely connect $\delta $ —understood at the level of a cosmological theory—to the concept of an ‘observer’. For such a concept indeed seems better ‘schematized’ by the inclusion of (say) biological and chemical considerations. But this hurdle is made manifest in applying EOP—as a consequence of our general recommendation for selecting $\delta $ . This recommendation emphasizes two tasks: that of (i) unambiguously expressing $\delta $ in terms of the theory; and (ii) selecting $\delta $ such that it is as specific as possible about our observational situation. Accomplishing (ii) more closely connects $\delta $ to ‘observers’—so that EOP can presumably be more usefully applied, via considerations of a richer account of our observational situation.

In this way, we believe EOP provides conceptual clarity—over and above that provided by AP—that helps to highlight a further benefit of our reformulation of AP. That is, EOP describes explicit, specific (though non-algorithmic) tasks [(i) and (ii)] that may help to facilitate appropriate applications of anthropic-style reasoning in scientific practice. In particular, such tasks may help to yield better estimates for “what we can expect to observe”, whilst revealing (significant) limits to the precision of such estimates. (Towards the end of Secs. 3.1.1 and 3.2, we provide some thoughts on this front; we leave a more thorough analysis for future work.) Furthermore, considerations of EOP can make failures of anthropic-style reasoning more conceptually transparent—as for when one of the two tasks, (i) and/or (ii), is not adequately completed. In which case, for instance, uninformative constraints (such as very broad constraints) on observables may arise.

3.2 Beyond cosmology

Thus far we have analyzed EOP and its precursors, AP, WAP, and SAP, in expressly cosmological (and astrophysical) contexts. This is natural given that the first papers to use anthropic-style reasoning were focused on cosmological considerations (Dicke, Reference Dicke1961; Carter, Reference Carter and Malcolm1974). However, it has since become clear that anthropic-style reasoning can be applied in very different contexts, such as in particle physics and astrobiology.Footnote ⁵ We shall now apply EOP to an example that goes ‘beyond cosmology’. In particular, we will see how an example—presented by Barrow and Tipler (Reference Barrow and Tipler1986, Sec. 5.5) as an application of WAP to nuclear physics—is best reconstructed as an application of EOP, indeed in a local physical setting.

Prior to developing the example, note that EOP can be applied in a more ‘local’ sense whenever observers are part of a subsystem that is effectively isolated from its ‘environment’. What “we can expect to observe” corresponds to observables in that subsystem. The ‘effective isolation’ should be understood, theoretically, as a ‘decoupling’ between the system and its environment: so that one can capture any effects the environment may have on the subsystem in terms of degrees of freedom of the subsystem. Regarding the role of theory in such a setting, the particular variables or parameters we are looking to constrain when we apply EOP largely determine the relevant theory. If we are interested in a cosmological parameter, then $T$ will correspond to a cosmological theory, possibly coupled to an astrophysical theory. But any theories further ‘downstream’—such as astrobiological or biological theories will be irrelevant—for there is an effective decoupling of such theories from cosmological theories (and possibly also from classes of astrophysical theories). But if the parameter or observation of interest is more ‘local’ in nature—such as one that relates to a terrestrial phenomenon, then the relevant theory will be more geophysical in character; in which case cosmological theories (and perhaps astrophysical theories) will be irrelevant. In this way, the observation by Carter (Reference Carter2006, p. 2), that AP is “not intrinsically cosmological, but just as relevant on small local scales as at a global level” applies to EOP as well.

Turning to the example, Barrow and Tipler (Reference Barrow and Tipler1986, Sec. 5.5) ultimately derive a constraint on two parameters of physics, employing anthropic-style reasoning (putatively applying WAP). These two parameters are: the fine-structure constant of electromagnetism, ${\alpha _0} \approx 1/137$ , which mediates the strength of the electromagnetic force between charged particles; and an analogous constant for the strong nuclear force, ${\alpha _{s,0}} \approx 1/10$ , which mediates the strength of the strong force between nucleons (protons and neutrons). (In this subsection, a subscript ‘0’ will denote a value as measured in our universe.) There are two steps to this derivation. We’ll only touch upon the first (for it is standard) and will focus on the second (for this is where ‘anthropic’ considerations enter).

Thus, first, Barrow and Tipler (Reference Barrow and Tipler1986) rehearse a standard calculation wherein the nucleus, with a total of $Z$ protons and $N$ neutrons (and thus $A = N + Z$ nucleons), is described via a simplified phenomenological model (known as the ‘liquid drop model’). Their calculation yields the result that a nucleus is stable (that is, it won’t undergo fission) if the fissionability parameter ${Z^2}/A$ satisfies

(8) $${{{Z^2}} \over A} \lt {{2{a_s}} \over {{a_c}}} \approx 49.$$

Here, ${a_s}$ and ${a_c}$ are constants associated with the model, proportional to $\alpha _s^2$ and $\alpha $ respectively. The approximation in Eq. (8) uses measured values: ${a_{s,0}} \approx 17.23$ and ${a_{c,0}} \approx 0.7$ [as in Segrè (Reference Segrè1977)]. Carbon-12, for instance, with $Z = 6$ protons and $N = 6$ neutrons (so that $A = 12)$ has a fissionability parameter of ${Z^2}/A = 3$ and so is expected, according to this model, to be stable.

In the second step, where considerations putatively related to observers enter, Barrow and Tipler consider a scenario in which $\alpha $ and ${\alpha _s}$ can deviate from their measured values (in our universe). Such considerations yield an approximate upper bound for the fissionability parameter, wherein a nucleus is stable ifFootnote ⁶

(9) $${{{Z^2}} \over A} \lessapprox 49{\left( {{{{\alpha _s}} \over {{\alpha _{s,0}}}}} \right)^2}{{{\alpha _0}} \over \alpha }.$$

Interestingly, one can turn this relationship around and think of it as providing constraints on possible values of $\alpha $ and ${\alpha _s}$ , assuming that nuclei with a given value of the fissionability parameter are stable. (In this case, $A$ and $Z$ effectively correspond to two physical degrees of freedom.) Equation (9) immediately yields

(10) $${\alpha \over {{\alpha _0}}} \lessapprox {{49A} \over {{Z^2}}}{\left( {{{{\alpha _s}} \over {{\alpha _{s,0}}}}} \right)^2}.$$

If we now assume that Carbon-12, which is presumably necessary for life, is stable, we obtain (where, recall, ${Z^2}/A = 3$ for Carbon-12)

(11) $${\alpha \over {{\alpha _0}}} \lessapprox 16{\left( {{{{\alpha _s}} \over {{\alpha _{s,0}}}}} \right)^2}.$$

So goes the argument, according to Barrow and Tipler (Reference Barrow and Tipler1986), which putatively applies WAP and leads to a constraint [in Eq. (11)] on possible values of $\alpha $ and ${\alpha _s}$ . But note that this constraint has not been derived via a consideration of observers. Rather, the application identifies an entity—here, stable Carbon-12 atoms (which might at best be thought of as a proxy for observers)—such that were this entity to arise, “we” might arise. What has in fact been applied, therefore, in obtaining Eq. (11), is an instance of EOP—where $\delta $ corresponds to stable carbon atoms. This instance of EOP might read as follows:

The theory with respect to which this principle ought to be understood is, of course, the liquid-drop model. What it means for $\delta $ (as stable carbon atoms) to be ‘expressible in terms of the theory’, is that Carbon-12 atoms can be represented in the model by specifying values for numbers of nucleons ( $A$ ) and protons ( $Z$ ); and where the balance of the forces that appear in the model secures stability.

Now, the chances that we might arise presumably increase if one assumes conditions for an observational situation that better accords with our own. One might achieve this by demanding the stability of nuclei with larger fissionability parameters (smaller values of the factor $49A/{Z^2}$ )—which generally involves nuclei with more protons than Carbon-12. Indeed, one can straightforwardly show [from Eq. (10)] that sequentially demanding the stability of nuclei such as Nitrogen-14, Oxygen-16, and Sulfur-32—nuclei that are also generally thought to be important for life—has the effect of driving the least upper bound on the ratio $\alpha /\alpha _s^2$ closer to the observed ratio ${\alpha _0}/\alpha _{s,0}^2$ . This suggests that it is possible to obtain better estimates for “what we can expect to observe” by including more detailed theoretical considerations (albeit with presumably diminishing utility).