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A common argument among theists is known as the design argument, and it asks: if something as complex as a watch implies a designer, would not human beings (whose bodies are certainly more complex than a watch) also imply a designer (namely, the theists' god of choice)? For a long time, this argument seemed rather compelling to a great many people, but then in 1859 Charles Darwin published his famous work "On The Origin of Species" in which he laid out a great deal of evidence for a theory of evolution based around natural selection. Evolution - since refined and generally accepted by the scientific community - has provided an alternate explanation for the complexity of living systems. Given enough time (and geological and astronomical data suggests the Earth has been around for billions of years), it seems perfectly possible for the most complex of creatures to emerge from the simplest of pseudo-life. Thus, the design argument - and its god - are no longer required to explain the complexity of living things.
Nowadays, the process of evolution is so well established as a scientific fact that a great many theists are forced to accept it (as a process designed by their god, of course). However, since Darwin's time, other areas of science (physics, astronomy and cosmology) have progressed and modern data has led to a new form of the design argument based around the apparent 'fine-tuning' of the universe to allow life.
Specifically, the Earth is located in what seems to be just the perfect place in the solar system for it to be life-supporting. If it were much closer to the sun or much further out, it would be too hot, too cold, or too irradiated to support life. Any number of other factors - from the makeup of the Earth's outer crust to the presence of water to the tilt of the Earth - seem to be in a rare range that allows life. If any of these factors were much different, humans could not have come about on Earth. Theists present these facts to suggest that our environment is fine-tuned for life. Our existence, according to the argument, is prohibitively improbable given the rarity of life-sustaining environments in a universe filled mainly with cold, empty space and the occasional fiery star and hot or frozen planet. Something needs to explain this occurrence of the vastly improbable, so the theists invoke their god.
What theists fail to take into account are observational selection effects and the weak anthropic principle. If a person wants to know the average size of fish in a pond, it's easiest to take a sample of reasonable size (say one hundred fish, for a pond of two thousand fish) and use the average size of that sample as the average size of the fish in the pond. Yet if the person sampling the fish is using a net which can only catch fish that are one or more feet long (because the holes in the net are too big for smaller fish), the sampling will be biased towards a larger result. Thus the choice of net is a selection effect, and it leads to problematic data. In the case of the fine-tuning argument, its proponents fail to realize that they are casting a faulty net. There are certain selection effects in our observations of the apparent fine-tuning which have to be taken into account before we can really say that our environment is fine-tuned in a way that requires an explanation like a god.
The weak anthropic principle gets at those observational selection effects. Basically, it states the trivially true - but still important to remember - fact that the environment of any observer (that is, life) must be such that the conditions allow life. Of course we don't observe ourselves in an environment that won't support life! So of course we do observe ourselves in an environment that will support life, no matter how rare that environment may be relative to the rest. So the fact that we are on Earth (or a place like it) and not in dark, frozen space is that we couldn't have been anywhere else. If life is going to arise, it's going to arise wherever it can (even if there is only one location in the entire universe!); if there were no places where it could arise, it obviously wouldn't and the question couldn't be asked by anyone.
Hence, the weak anthropic principle reminds us that we shouldn't be at all surprised - or require an explanation - when we find ourselves in an environment suitable to life. The theist is falling prey to observational selection effects because they are throwing a net designed to catch something that can only be caught in environments of type A, and then claiming the need to explain why their catches only come from environments of type A. In this case, it just so happens that the faulty net is a result of the weak anthropic principle, and this needs to be kept in mind before we assume that there is something that needs to be explained.
One might wonder about the fact that environments of type A (that is, places that can sustain life) are so rare in our universe, and whether the chance of life popping up on any given one was so low that these environments' rarity makes life ever coming about prohibitively unlikely. However, as telescopes and other detection methods have improved recently, astronomers have discovered numerous other stars scattered throughout the galaxy, and it is commonly accepted now that planets are not unique to our solar system but exist by the billions in our galaxy alone (and there are estimated to be perhaps a hundred billion galaxies in the universe). With numbers like those, even the worst-case numbers for the percentage of planets that can support life leave an unimaginably huge pool of possible starting points for life. As for how it started here and could start elsewhere, that is a question for biochemistry and the growing field of abiogenesis, but there is good reason to believe it is not prohibitively unlikely. Given so many planets that can support life, it's not hard to imagine life arising somewhere, perhaps relatively often (granted, most all such planets would be lightyears from others, so one wouldn't exactly expect alien visitors even if life were as common as the most generous estimates).
One might think that this would put the nail in the design argument coffin once and for all (at least insofar as it is proposed as necessary to explain the existence of life), but there is a much more intricate twist on the idea which I have yet to cover. Aside from the supposed extreme unlikelihood of our coming into existence on one of the rare (but still amazingly numerous) planets that could sustain our existence, theists also cite data from modern physics which suggests that if certain fundamental constants of the universe had been significantly different, the universe would have been a place where life like ours could not come about. For example, if the ratio of the gravitational constant to electromagnetism were 10-33 instead of 10-39, stars would burn too fast to form heavy elements such as carbon, on which all life on Earth is based. If the strong nuclear force were slightly stronger, protons could not form; but if it were slightly weaker, the formation of stars would be impossible. These and a fair number of similar examples are presented by theists to suggest that the universe itself - at the fundamental level of physical laws - is fine-tuned to allow life. Looking at all the possible ways the universe could have worked (in which the laws of physics might have been drastically different), the number of possibilities in which the universe supports life is prohibitively small relative to the number of possible universes in which life is impossible. Thus, the fact that the universe is one of those very, very rare possibilities needs to be explained, and the theists once again assert god as the explanation.
Unfortunately, the weak anthropic principle may not be able to fully counter this in the way it does the previous argument. In the case of the previous argument, the theists mistakenly assumed that since a slight change in the "rare" setup would have made life on Earth impossible, something must have intervened to put Earth in the right place, or more generally to put life on the right planet - because these things were so statistically improbable they couldn't happen on their own. The mistake here is in ignoring the fact that there are tons and tons of planets out there on which life could have arisen, and it is a mistake to assume that ours is special (statistically improbable) because it is 'special' (a life-supporting planet and not dead space). There is no such statistical improbability of life showing up on one of the 'special' planets (the probably is 1, unavoidable, if life shows up at all), and so our being here isn't special.
In the case of the refined argument (dealing with fundamental physical laws), one might be tempted to assume that the same mistake is being made on a larger scale, and apply the same response. Life could only arise in a universe the laws of which supported life, of course. In this case though, the theist could counter that only one universe obtained out of an unimaginably huge (or infinite?) number of possible universes, and the type of universe that obtained (life-supporting) is prohibitively rare relative to that total number, so the existence of our universe rather than another is a statistical anomaly. The difference between this and the question of being on a "rare" planet is that we are no longer asking why life is showing up in the place it can show up (however rare that place might be, leading to statistical misunderstandings); instead we are asking why there is a place life can show up at all! It would be like revising the earlier question to ask why there are any life-supporting planets rather than none at all. That latter question is answerable by modern astronomy (we know how planets form, which places around stars would allow planets there to support life, and that the way planets are distributed will inevitably leave some in that place). Yet the more general universal question is not so easy to answer.
The simplest response would be that statistical anomalies simply happen, that the lotto may be an idiotic bet, but sometimes you really do win (that doesn't make the lotto any less of a bad bet, it just shows that statistical likelihood does not imply necessary obtaining). Our type of universe may be statistically less likely than the other type (because there are more possibilities of setups which would be inhospitable to life), but our individual universe is still statistically just as likely as any other individual possible universe. Rolling a number that is a multiple of 1,000 may be unlikely on a single role of a billion-sided die (because there are more numbers that aren't a multiple of 1,000 than are), but rolling 100,000,000 is just as likely as rolling 869,730,635. Thus, the easiest counter to the universal fine-tuning argument is that it makes a statistical fallacy in assuming that our individual universe is less likely than any other just because our type of universe is less likely than another type. Yes, it's possible that another universe could have obtained, and in that case the chances are it would not support life. But there was no probability-dodging in our universe obtaining (and thus no explanation necessary); it was just a lucky role of the dice for living beings. Therefore, there is no need for a designer god.
One might inquire whether it is possible that some individual universes are more likely than others to be the one that obtains. Which is to say, the statistical response above assumes that the question of which universe obtains is analogous to rolling a gigantic, fair die: the probability of any given outcome is equal. Perhaps, though, the die is weighted in some strange, complex way so that it is more likely to land on a number that isn't a multiple of 1,000 not just because there are more numbers of one type than another but because there is an additional factor of weighting.
Now there is still a possibility of getting a multiple of 1,000 (a universe that could support life), and so one could argue that it is still analogous to rolling a fair die, just one with way more non-1,000-multiple numbers added in. This tactic seems to work, though it may appear ad hoc and thus counterintuitive. Ad hoc explanations are perfectly acceptable when seeking a counterexample to what is purported to be the only explanation - i.e. god - for given data. It also highlights a conundrum regarding the basic nature of probability: what does it really mean to say something has a probability of X if even the smallest of probabilities can still obtain? Or for that matter, one might simply argue that probability requires repetitions being possible, whereas the universe seems to be a single role and that's it.
Fortunately, there is a simpler answer that doesn't require this regress of probability. There is simply no reason to believe the universal dice are stacked in any way. There would need to be some sort of mechanism, and none come to mind or have been suggested. Thus, this final hypothetical move on the part of the theist needs some sort of elaboration and explanation as to what sort of thing would cause the universal die to be weighted. Until such a mechanism is provided, the objection is shaky, and the fine-tuning design argument seems to fail.
There is one other scenario which is worth mentioning. Throughout this paper I've been assuming that only one universe obtained from among a large number of possible universes. Some leading cosmologists, however, have began to suggest a multiverse theory in which our universe is merely one among a large (or infinite) number of universes, all of which obtained or will obtain (rather than being mere possibilities). The word universe has a somewhat ambiguous meaning, in that it is sometimes taken to mean all that exists (Absolutely Everything, in nice, big capital letters), and it is sometimes taken to mean all that is at least theoretically accessible to us, or able to somehow influence us or that which influences us (in other words, everything of the same ontological type as us and grass and fire hydrants and other physical objects in our spacetime). The idea of multiple universes may be incoherent under the former definition, but under the latter it at least makes sense.
Some multiverse theories posit an immense (or infinite) meta-space-time which encompasses a multitude (or infinity) of separate (inaccessible to each other), giant space-time regions, each its own universe. Others propose successive cycles of an oscillating universe (Big Bang, then Big Crunch, then Big Bang leading to a different universe, inaccessible to those in the first); or new, inaccessible universes forming on the other side of black holes in parent universes. Still others have suggested a many-worlds quantum theory that says reality continually branches into new universes for every possible outcome in every possible branch. Or it could be that universes "quantum-tunnel from nothing", as it has been put by some. Any number of explanations exist and are being considered by cosmologists, but the important thing is that if all possible worlds are actual worlds then there existing one of the rare life-supporting ones is not at all surprising (nor is it surprising that any life which arises will find itself in such a universe). So even if the universe is fine-tuned in some way that requires an explanation, there are alternatives to design, and often they fit better with the big picture science is coalescing these days.
I think it's safe to say that cosmology today is finally finishing the work Darwin started in tackling the argument from design. No longer is the god-of-the-gaps necessary to explain problematic observations: given time, it seems science can clear up the picture and remove the gaps itself. Science may not give us all the answers, and the answers it gives are by no means infallible, but it provides us with an alternate explanation that doesn't require invoking the supernatural.