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## The Traps of WAP and SAP

Let’s begin by quoting from Radford Neal:

There is a large literature on the Anthropic Principle, much of it too confused to address.

I’ve previously quoted John Leslie:

The ways in which ‘anthropic’ reasoning can be misunderstood form a long and dreary list.

My goal in this post is to go back to the original sources to try to understand the anthropic principle.

### Carter’s WAP

Let’s start with the definitions given by Brandon Carter in the original anthropic principle paper:

Weak Anthropic Principle (WAP): We must be prepared to take account of the fact that our location in the universe is necessarily privileged to the extent of being compatible with our existence as observers.

Carter’s illustration of WAP is the key to understanding what he means. Carter considers the following coincidence:

$H = m_p^3$ (1)

where $H$ is the Hubble constant and $m_p$ is the proton mass, both in Planck units. (Putting in the numerical values, $10^{-61} \approx 5 \times 10^{-58}$). Carter presents Dicke’s solution: if we turn equation (1) upside-down, then the left side is approximately the age of the universe, and the right hand side is the average lifetime of a star. The key is to note that the age of the universe (and thus H) changes with time. So the coincidence (1) is necessarily tied up with the question: why are we observing the universe now?

The answer to this question must depend on what an observer is. Observers in this universe (seem to) need to be made of atoms and take advantage of the complex chemistry of carbon and larger elements. These elements are not made in the Big Bang, and thus observers must wait for stars to form, synthesise the elements, disperse them in supernova and collect into planets. Thus, the time the universe takes to make observers is at least the lifetime of stars.

Let’s flesh out Carter’s WAP. We are considering our “location” in the universe, by which Carter means both our position in space and time. We first fix the laws of nature (L), the parameters of those laws (P), and cosmological initial conditions (I). We observe that the age of the universe is $t_0$. We now do the normal Bayesian thing of calculating the likelihood of the data given our hypothesis:

$p(t_0 | L \& P \& I)$ (2)

WAP then tells us what we must not do. We must not treat the universe as an experiment. The universe is not an apparatus we have set up in our lab and can observe whenever we like. The observation is part of the experiment. Equation (2) asks: when is this apparatus most likely to observe itself? The answer to this question depends on how the apparatus creates observers.

We have focused on time, but the same argument applies to space. The environment in which we find ourselves is highly atypical of the universe as a whole. For example, the density of the air in this room is $10^{27}$ times higher than the average density of the universe. We are way out in the tail of the distribution of overdensities. But once again, WAP tells us to ask the question: which parts of the universe will observe themselves? Observers require the chemistry of solid-phase matter, which requires much higher densities than the cosmic average.

A great example of WAP-in-action is this paper by Lineweaver and Egan, which calculates something analogous to (2) by considering the terrestrial planet formation rate, and then applying a time delay – observers will form on planets a certain time after the planet has formed. They consider a different coincidence to (1), namely why we are observing the universe at a time when $\Omega_{matter}\approx \Omega_{\Lambda}$. Without WAP, this seems to be an extraordinary coincidence – see their Figure 2. They show that there is “a large (68%) probability that observations made from terrestrial planets will result in finding $\Omega_{matter}$ at least as close to $\Omega_{\Lambda}$ as we observe today”.

### Carter’s SAP

We turn now to Carter’s formulation of the strong anthropic principle.

Strong Anthropic Principle (SAP):  “The Universe (and hence the fundamental parameters on which it depends) must be such as to admit the creation of observers within it at some stage.”

We can immediately see the difference between WAP and SAP. In WAP, we kept the laws, constants and initial conditions (the “physics”) of the universe constant and considered the probability of observing that we are at a particular place and time. In SAP, we no longer hold the physics of the universe constant and ask whether observers will form at any time and place. WAP asks “why here and now?”; SAP asks “why this physics?”.

Once again, Carter’s example of SAP is very helpful. Suppose we lived in a closed, matter-only universe. This universe has a finite lifetime – gravity causes it to re-collapse in a big crunch. Now, for observers to observe such a universe, stars must be able to live and die before the crunch. This implies the following relationship between cosmological and physical constants:

$N_p \gtrsim \alpha_G^{-2}$      (3)

where $N_p$ is the number of protons inside the radius of the universe $R(t)$, and $\alpha_G = m_p^2$ is the so-called “gravitational coupling constant” (defined using the proton mass). Numerically, $10^{81} \gtrsim 3 \times 10^{76}$.

Note the difference between (3) and (1). In (1), the left hand side is time dependent, so that the coincidence only holds “now”. In (3), there is no time dependence – the right side depends only on the proton mass in Planck units, and the left hand side remains constant as the universe expands. Thus, for the universe to permit observers at any time, (3) must hold.

Much of the confusion over SAP seems to have been generated by the word “must”. I agree with Leslie when he notes that:

… an ex-post-facto or consequential ‘must’ is meant, as in ‘Since the passport photo is labelled WIFE it must be of a woman’ or ‘The silverware has vanished so a burglar must have called’.

In other words, SAP is not saying that life is a metaphysical necessity, or that there must be some kind of biophilic principle in fundamental physics that makes the physics of the universe like friendly. SAP is not claiming that, somehow, we gave the universe its life-permitting properties.

Carter means for SAP to be like WAP, but applied to properties that are independent of time and space, like cosmological initial conditions and fundamental constants. Carter’s point is not that a universe which violates (3) is impossible, either physically or metaphysically. His point is that such a universe is unobservable. Given that we are observing, we must observe that (3) is true.

### Barrow and Tipler’s WAP and SAP

My real motivation for writing these posts is to show that the formulation of WAP and SAP in Barrow and Tipler’s The Anthropic Cosmological Principle is different to Carter’s. I’ve not seen anyone point this out before. To distinguish the two, I’ll call Barrow and Tipler’s versions btWAP and btSAP.

btWAP: “The observed values of all physical and cosmological quantities are not equally probable but they take on values restricted by the requirement that there exist sites where carbon-based life can evolve and by the requirement that the Universe be old enough for it to have already done so.”

Whereas Carter’s WAP only considered our position in space and time, btWAP considers “all physical and cosmological quantities”. Schematically, we can write:

Barrow and Tipler’s WAP = Carter’s WAP + Carter’s SAP

btWAP, by considering our position in space and time and all physical and cosmological quantities, applies anthropic reasoning to cases which Carter would consider separately under WAP or SAP. For example, Barrow and Tipler (pg 384) considers equation (3) to be a “striking example of how the Weak Anthropic Principle connects aspects of the Universe that appear, at first sight, totally unrelated.” Carter, as we saw, used equation (3) as an illustration of SAP.

What, then, is btSAP?

btSAP: The Universe must have those properties which allow life to develop within it at some stage in its history.

This statement could be interpreted as either (Barrow and Tipler, pg. 22):

(A) There exists one possible Universe ‘designed’ with the goal of generating and sustaining ‘observers’.

(B) Observers are necessary to bring the Universe into being.

(C) An ensemble of other different universes is necessary for the existence of our Universe.

As I hope you can see, none of these have anything to do with Carter’s SAP. These are not selection effects. In particular, note the difference between Carter’s must and Barrow and Tipler’s must. As we noted above, Carter’s must is consequential: given that we exist, the properties of this universe must permit life. Barrow and Tipler’s must, on the other hand, implies that there is some deeper reason why our universe must contain life.

A, B and C are possible explanations of how, against the odds, there exists a life-permitting universe. Carter’s WAP and SAP (and btWAP) are useful, succinct reminders that our existence is by no means a trivial piece of information when we come to calculate the probability of observing the universe we see around us. WAP and SAP are weaker and stronger (or perhaps, narrower and broader) applications of the same principle. Barrow and Tipler’s btSAP is something else entirely. It is not in any way a stronger version of WAP and/or SAP. To place them in the same category of ideas is very misleading.

You can see where people got the idea that the anthropic principle is quasi-religious. According to btSAP, point A, the (strong) anthropic principle is the idea that this universe was designed for us. No wonder the anthropic principle has such a bad name. It may hold the title of the most maligned tautology of all time.

### Conclusion

Stick with Carter. Call the anthropic principle AP = WAP + SAP = btWAP. Use Carter’s distinction between WAP and SAP if required. Forget btSAP – it is not a version of the anthropic principle. (Forget the final and participatory anthropic principle as well, for the same reason.) Or, at least, give it a different name.